some families of local systems over smooth projective varieties

Annals of Mathematics

Some Families of Local Systems Over Smooth Projective VarietiesAuthor(s): Carlos SimpsonSource: Annals of Mathematics, Second Series, Vol. 138, No. 2 (Sep., 1993), pp. 337-425Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/2946614 .

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Annals of Mathematics, 138 (1993), 337-425

Some families of local systems over smooth projective varieties

By CARLOS SIMPSON*

Introduction

One can try to study the fundamental group of a smooth, complex projective variety S by looking at spaces of representations of irn(S). If the space of representations is a union of isolated points, then this structure reduces to the discrete structure of the set of representations. So a natural problem is to try to find examples of varieties S where there exist nontrivial continuous families of representations (or, equivalently, local systems on S).

It is not too hard to see that if S is a projective algebraic curve of genus g > 2, then the moduli space of representations of rank r > 1 has dimension (2g - 2)r2 + 2; or, for example, if S is a variety with dim H1 (S, C) = a, then the space of representations of rank 1 has dimension a. Taking tensor products of pullbacks of families of local systems arising in these ways, we obtain some more families. To pursue the problem we ask: Do families other than these exist?

The idea in this article is to use the next natural construction, taking higher direct images of local systems. Suppose that f: X -* S is a smooth projective morphism and {Wt} is a family of local systems on X, chosen in a simple way. Let Vt = 13f*(Wt). This is a collection of local systems, and if the ranks are constant, then it is a continuous family. We can hope that {Vt} will be an interesting family of local systems on S.

The principal question that needs to be addressed is whether, if the family {Wt} varies nontrivially, the family of direct images {Vt} varies nontrivially. We are also interested in constructing examples where we can show that the family {Vt} does not, by some miracle, arise from a simpler construction such as the one described before. This article consists essentially of two parts. The

*Partially supported by the C.N.R.S., N.S.F. Grant DMS 91-57700 and the Alfred P. Sloan Foun- dation. The author would like to thank L. Katzarkov and D. Toledo for discussions posing the questions concerned here; K. Zuo for communicating his results and discussing his conjecture (cf. ?9); N. Katz for his course about monodromy groups; and specially J. Carlson for lengthy discussions, during which it became evident that one could find the examples presented here, and for explaining many of the required techniques.



338 C. SIMPSON

first (Sections 1-5) is devoted to answering our principal question in a fairly general situation. For this we develop the technique of taking the direct image of a harmonic bundle and its associated Higgs bundle. We give a way to calculate the spectral varieties of the Higgs bundles associated to Vt, as a way of verifying that the Vt vary nontrivially. The second part (Sections 6-8) is concerned with the construction of a particular class of examples and the verification of some additional properties about monodromy groups and possible factorization through morphisms to algebraic curves; these properties serve to show that our examples do not come from tensor products of pullbacks. The methods used in this part are all fairly well known, not original they were explained to me by J. Carlson but I have included detailed proofs, because it seems to me that it might be useful to have them collected in one place. The relationship between the two parts of the paper is that the results of the first part can be applied directly in the second part, although some examples constructed in the second part may be dealt with more easily by a topological method discussed in Section 2.

Here are some more details on the first part about direct images, the second part about some examples, and the origins of the questions addressed here.

Direct images. Suppose that f: X - S is a smooth projective morphism of algebraic varieties and {Wt} is a family of local systems on X. By taking the ith right-derived functor of the direct image, we obtain (if the ranks are constant) a family of local systems Vt = Rff*Wt on S. We would like to be able to show that if the Wt vary nontrivially, then so must the Vt. In any given example, the direct image can be calculated by topology-but this is likely to be difficult, and it is also difficult to check whether a given family {Vt} really varies nontrivially. We will give two different methods for proving this type of result. The first method is topological, involving the universal coefficients theorem. This works well in the case of the direct image by a Lefschetz pencil of an abelian variety. The second method is technically more complicated, but stronger; it involves calculating the spectral varieties of the Higgs bundles associated to the direct images Vt. It has the advantage of applying to any sufficiently ample Lefschetz pencil.

We state the result of the stronger method, with the following notation: Suppose that Z is a smooth, complex projective variety of dimension n + 1. Suppose that 1 is a very ample line bundle. Let U C IpN be the open subset of points in the full linear system of 1 corresponding to smooth hyperplane sections. Let Xu C U x Z be the incidence variety, so that the first projection f: Xu -* U is the universal family of hyperplane sections, mapping to Z by the second projection a: Xu- Z.



FAMILIES OF LOCAL SYSTEMS 339

THEOREM I. If 1 is sufficiently ample, then the following holds: For any local system V on U, there are only finitely many semisimple local systems Wt on Z such that Rnf*(a*Wt) V.

In particular, if the Wt vary in a nontrivial family, then the direct images Vt = Rnf*(a*Wt) must vary nontrivially.

Sections 3-5 are dedicated to the proof of Theorem I and related results. We use the theory of harmonic bundles and Higgs bundles developed in [4], [9], [15] and [25]. The main idea is to construct the direct image of a harmonic bundle by a smooth projective morphism such that the underlying local system is the direct image of the underlying local system on the domain. We give an algebraic construction of the corresponding direct-image Higgs bundle. If f: X -* S is a morphism of smooth projective varieties, then f is smooth over an open set in S, and so we can take the direct image of a harmonic bundle. We go further, analyzing what happens near the singular fibers of f. We have not done this in its utmost generality; for example, we usually restrict ourselves to the case where S is a curve. The direct-image harmonic bundle on an open set of S is a tame harmonic bundle these were discussed in [23]. We identify the filtered local system and filtered regular Higgs bundle that correspond to this harmonic bundle. This generalizes things, which have been known for some time in Hodge theory, for the direct image of the constant local system. The estimates in that case were due to Clemens [3] and, in our case, the proof is essentially the same.

We will use Theorem I in an application below. As the proof is rather technical, we have included in Section 2 a simpler topological argument, involving the universal coefficients theorem, which allows one in certain examples to verify that the family of direct-image local systems Vt varies nontrivially.

Construction of examples. Our aim is to construct a family of representations of the fundamental group of a smooth projective surface. This grew out of a result of [26], saying that any nontrivial family of Zariski-dense representations into SL(2, C) is essentially pulled back from a map to an algebraic curve with an orbifold structure. A natural question is whether this is true for representations into bigger groups. The families we construct will show that the answer is no.

We will construct some smooth projective surfaces S together with families of representations Pt of r1r(S). We obtain the Pt by taking higher direct images of rank-1 local systems and extending these from an open set to all of S. The main step is to be able to show that the representations Pt vary nontrivially. This is where we use Theorem I, or the shortcut of Section 2 mentioned above. In order to obtain an example contradicting the natural generalization of [26] we must also verify some other properties of our family.



340 C. SIMPSON

One of these is, of course, to show that the family is not a pullback via a map from S to an algebraic curve not even after going to a variety generically surjective over S. It is easy to construct families satisfying these properties by taking tensor products of representations pulled back from different curves. To rule out this sort of thing we seek an additional property, namely that the representations are Zariski dense in a simple group.

Here is an outline of the construction: Suppose that D1 C Ip12 is an irreducible curve, with only nodes and cusps, and let U1 = J'2 -D1. Let T be the quotient of iri(Ui) by the relation that -y2 = 1 for a path -y going once around D1 at a smooth point of D1. Let T' be the quotient of T by the intersection of all subgroups of finite index. A completed covering of Ui is a smooth projective variety S together with an open set Us C S and a finite etale morphism Us -* U1. An T'-completed covering of U1 is a completed covering such that the composition irl(Us) -* irl(Ul) - T' factors through a map irn(S) - T'. We show that an T'-completed covering exists.

Let Z be a smooth projective variety of odd dimension n + 1. Let 1 be a sufficiently ample line bundle on Z (cf. [10]). Let pN be the projective space of lines in HO(Z, 1). For each s E pN let f8 be a representative of the corresponding line and let X, be the zero set of f5. Let D C pN be the locus of s such that X. is singular. Choose a generic plane P2 C pN and let D1 = D n J2. We will see that D1 is an irreducible curve with nodes and cusps. Let U = p'N -D and U1 = I'2 -D1 and let

U XU a> Z

be the incidence variety. Suppose that W is a local system on Z. There is a local system V = Rnf*(a*W) on U whose fiber over a point s is Hn(XS, W). If -y is a path around D1 at a smooth point, then the monodromy of the local system V around -y has order 2. Thus the monodromy representation factors through the quotient T of 7r,(U). The kernel of any representation of T contains the intersection of all of the subgroups of finite index; so the monodromy of V factors through a representation p: T' -* GL(r, C). If S is an T'-completed covering of U1, then we obtain a representation inl(S) GL(r, C). In other words, the pullback of the local system V to Us extends to a local system on S. In this way we obtain a local system on a smooth projective surface S.

We introduce some notation for properties of representations. Suppose that G is a reductive algebraic group and F is a finitely generated discrete group. A representation p: IF - G is Zariski dense if the Zariski closure of p(F) is equal to G. This is equivalent to the condition that there is no proper algebraic subgroup of G that contains p(F). A representation p: IF - G is nonrigid if it fits into a family of representations that are not conjugate to each




other; that is, if there are a topological space T and a continuous morphism T -* Hom(F, G) (denoted by t | 4 Pt) such that p = Pt0 for a point to E T and, for any neighborhood JK of to, there exists t E JK such that Pt is not conjugate to p by any inner automorphism of G.

Suppose that F = irl(S) for some smooth projective variety S. We say that p: IF - G has the factorization property F1 if there exist a smooth quasiprojective variety Y and a smooth, quasiprojective, algebraic curve, with maps g: Y S and p: Y -* C, such that g*(p) factors through a composition irl(Y) -ri(C) -* G. We say that p has the nonfactorization property NF1 if it does not have the factorization property F1.

Suppose that H1 (Z, C) $& 0. Then the space of 1-dimensional representations of mri(Z) has positive dimension. Choose W to be a generic rank-1 local system on Z and let V = Rnf*(a*W). Let S be an T'-completed covering of U1 and let p(Vs): i1(S) -* GL(n, C) be the monodromy of the local system pulled back to Us and extended to S. Let G be the Zariski closure of the image of irn(S) in PGL(n, C). We may choose S so that G is connected, and we show that G is a simple group (Theorem 7.5 and Corollary 8.4). Let p: i1 (S) -? G denote the representation in G. It is Zariski dense (tautologically). We show that p has the nonfactorization property NF1 (Lemmas 8.1 and 8.5). The result stated in Theorem I above implies that p is nonrigid. The argument of Section 2 allows us to verify this more easily when, for example, Z is an abelian 3-fold.

We can sum up the results as follows:

THEOREM II. There exist a discriminant curve D1 C IpD2 with nodes and cusps, an T'-completed covering S of P2 - D1, a simple group G and a representation p: irn(S) -? G such that p is Zariski dense, nonrigid and has the nonfactorization property NF1.

This theorem is proved in Sections 6-8.

Origins. One of the questions that motivated this article was originally posed to me by D. Toledo (January, 1990) he asked whether nontrivial harmonic maps exist that do not come from variations of Hodge structure or factor through Riemann surfaces. This is equivalent to the question of whether there exist nontrivial families of representations. That question led to the work in [26], where the answer was (from this point of view) negative. The methods used there were special to the case of the group SL(2, C), and the analogy with the work of Culler and Shalen suggested that this might be an important distinction. However it was also natural to hope for a general statement saying that any nonrigid representation should essentially factor through a curve. L. Katzarkov has been working on this question-cf. [17]-and discussions



342 C. SIMPSON

about this with him in January, 1991 brought back the issue. It was obvious from the beginning that one had to do something to rule out constructions such as tensor product, but it was hard to find a good formulation. The cru- cial idea that one should demand that the representations be Zariski dense in a simple group is due to Katzarkov. In Spring, 1991, K. Zuo circulated his proof that nonrigid representations into SL(3, C) more or less factor through curves or surfaces (see [29]). This suggested that the natural generalization of [26] might be a statement about factorization through higher-dimensional varieties, rather than the strong statement about factorization through curves (cf. the discussion in Section 9 below).

The possibility of giving the examples presented here is due in large part to J. Carlson. In connection with his work with D. Toledo (also concerning questions of harmonic maps not factoring through Riemann surfaces), he had constructed variations of Hodge structure over smooth projective surfaces by taking the direct image of the constant local system and extending it across the discriminant locus (see [2]). He described how to do this in lengthy discussions in May and June of 1991. His methods (when applied to the case of direct images of other rank-1 local systems) allow the construction of families of local systems on smooth projective surfaces and the verification (via the Torelli theorem of M. Green) that they do not factor through maps to curves. Carlson also pointed out a result of P. Deligne, which shows that the monodromy group of a special point is the orthogonal group. By an analysis of what happens to the monodromy group under specialization, this leads to the property that the generic monodromy group is quasisimple.* After all this, the only remaining question was to show that the family of representations is not constant.

To finish the story, the families of local systems constructed here provide a positive answer to Toledo's question: There are harmonic maps that are not projections of variations of Hodge structure and do not factor through Riemann surfaces in any conceivable way. This corroborates the feeling one gets from the work of Carlson and Toledo [1] that it is difficult to prove that a harmonic map comes from a variation of Hodge structure and that one can only show this in certain cases.

Advice to the reader. This paper contains two parts. The first is dedicated to proving Theorem I; this takes up Sections 1-5. The second part is dedicated to proving Theorem II. This takes up Sections 6-8. The results are stated again, along with some applications, in Section 9. The second part uses the result of the first part. However one can get by with a much weaker statement,

*I have since replaced this part with a slightly more general discussion along lines suggested by N. Katz's course at Princeton University; see Section 7.




which is given in Section 2. Thus the reader who is interested primarily in Theorem II (i.e., Theorem 9.2) may read Sections 1, 2 and 6-8.

While the discussion in Sections 3-5 is somewhat brief, the attempt in Sections 1, 2, and 6-8 is to collect completely in one place the techniques (usually well known) needed for constructing our examples.

Notation. All algebraic varieties are defined over C. When discussing the topology of algebraic varieties, we refer (without further mention) to the usual topology of the associated complex analytic spaces.

1. Families of representations

Let T = Spec(A) be an affine scheme of finite type over C. Unless otherwise specified, we will assume that T is irreducible. Let F be a discrete finitely generated group. A family of representations of F of rank r, parametrized by T, consists of a projective A-module U of rank r with an action PA of F (compatible with the A-module structure). A framed family of representations is a family of representations together with an isomorphism U ' Ar.

Suppose that X is a topological space and F = 7ri(X). Then a family of representations of F is the same thing as a local system WA of projective A-modules of rank r on X. We will also refer to WA as a family of local systems. To define the fundamental group, a base point x E X must be fixed; the base point may also be a fixed universal cover. The projective A-module corresponding to a family of local systems WA is the fiber at the base point, (WA)x. A framed family of representations is a family of representations WA together with an isomorphism (WA)x - Ar'. We usually drop references to the base point in order to simplify notation.

Suppose that PA is a family of representations, corresponding to a family of local systems WA. For any C-valued point t E T, we obtain a representation Pt of F on the fiber Ut, corresponding to a local system Wt = WA 0 Ct. Here Ct denotes the field C, thought of as an A-algebra by the morphism dual to t -* T. The notion of a family of representations encodes the idea of a collection {Pt}teT of representations varying algebraically with t.

We will encounter a weaker notion: that of a local system of A-modules, WA. This is a locally constant sheaf of A-modules on X. For any A-algebra, A', we can form WA' = WA OA A', a local system of A'-modules. We say that WA is coherent if the stalks are coherent A-modules. In this case, there is an open set Spec(A') C Spec(A) such that WA is locally free of some rank r. Then WA' is a family of local systems. However the ranks of the fibers WC, can jump for values of t not in this open set. We do not even obtain a collection



344 C. SIMPSON

of local systems of the same rank. This problem will be dealt with further in Section 2 when A is a Dedekind domain.

Suppose that f: X -* S is a fibration of topological spaces and WA is a local system of coherent A-modules on X. Then the direct images VA = Wff*WA are local systems of coherent A-modules on S. However, even if WA were locally free (giving a family of local systems), the direct images need not be locally free. If the fiber of f is homotopic to a finite complex and WA is coherent, then the direct images are coherent. Thus, for an open set Spec(A') c Spec(A), the direct images give families of local systems VI on S.

These definitions can be globalized to replace Spec(A) by any scheme T. We use the notation {Wt}tET to refer to a family of local systems parametrized by T, even though the information contained therein is more than simply that of the collection of Wt.

Homomorphisms into algebraic groups. Suppose that G is an algebraic group. A family of homomorphisms from F into G indexed by T is a homomorphism PT: IF - G(T) into the group of T-valued points of G. For each geometric point t E T we obtain a homomorphism Pt: IF - G. We will also use the notation {Pt: IF - G}tET for a family like this.

Suppose that V is a representation of G. If PT is a family of homomorphisms of F into G, then by composing with the action of G on V, we obtain a family of representations VT of F. By its nature, this construction gives rise to families of representations that can be given frames: the locally free sheaf VT of (9T-modules is free, VT = V0COT. This indicates that we have not given the most general possible definition of a family of homomorphisms into G. One could require only that the homomorphisms be defined locally in the etale topology of T and that, on overlaps of open sets, the representations be conjugate (by functions into G, which are again only defined locally). We will avoid referring to this more general notion by introducing some ad hoc considerations as follows:

LEMMA 1.1. Suppose that VT is a family of representations of rank r of F. Then there are a covering T = UT,> by Zariski-open sets and, on each Tga family of homomorphisms pc, from F to GL(r, C) such that the families of representations VO, which result from the composition of p, with the natural representation of GL(r, C), are isomorphic to the restrictions VT IT,.

Proof. It suffices to take a covering such that VT ITC are free sheaves on T. F1

Suppose that H is an algebraic group. For any representation p: IF - H define the monodromy group to be the Zariski closure of the image p(F).




LEMMA 1.2. Let G c H be algebraic groups. Suppose that {Pt: IF -

H}tET is a family of homomorphisms from F to H parametrized by a reduced scheme T such that, for each t E T, the monodromy group Gt is conjugate (by an element of H) to G. Then there are a covering p: T' - T in the stale topology, a family of homomorphisms {(pr: IF - GC}ETi and a function h: T' - H such that pp(r)(Py) = h(r)Wo(y)h(r)-1 for all r E T' and -y E F.

Proof. Let N denote the normalizer of G in H. Then H/N is a space parametrizing all of the conjugates of G. For each t, the group Gt corresponds to a point f (t) E H/N. We claim that this is an algebraic morphism f: T -* H/N. There is a representation V of H such that G is determined by the set of lines, which it fixes in V. More precisely, if A c P(V) is the set of fixed points of the action of G, then

G = {h E H: for all A E A, g(A) = A}.

Let AT C T x P(V) be the set of fixed points of the action of F. It is a closed subset. There is a finite subset i C F such that AT is the set of points fixed by the elements of E (for example, a set of generators). The intersection with any fiber is the fixed-point set of Gt

AT n {t} x P(V) = A(Gt),

where A(Gt) is the set of fixed points of Gt. In particular the A(Gt) are all translates of A, so they have the same Hilbert polynomials. Hence the family AT is flat over T, and we obtain a morphism from T into the corresponding Hilbert scheme Hilb of subsets of IP1(V). On the other hand, the stabilizer of the subset A is equal to the normalizer N of G, so there is a morphism h 4 h(A) from H/N to Hilb. This induces an isomorphism of H/N with its image. The image of the morphism T -* Hilb is contained in the image of H/N -*

Hilb (scheme theoretically, since T is reduced). Hence we obtain a morphism T -* H/N sending t to the point f (t) corresponding to Gt. The quotient morphism H -* H/N is smooth, so there are an etale covering p: T' - T and a lifting to a function h: T' - H, with Gt = hGh-1. Now set p,(-y) =

h-1pp(,)(-y)h. This is the required family of representations in G. [1

Moduli spaces. Let F be a finitely generated group. The Betti representation space

def ZB(F, r) = Hom(F, GL(r, C))

is an affine variety parametrizing framed representations of F. It represents the functor that associates to T the set of families VT of representations of F with framings VT - O'. The group GL(r, C) acts on 1ZB(F, r) by conjugation.



346 C. SIMPSON

There is a universal categorical quotient

MB(F, r) = 1ZB(F, r)/ GL(rC).

This is an affine variety, constructed by setting its coordinate ring equal to the ring of invariant functions on 1ZB(F, r). The semisimplification of a representation is the direct sum of the irreducible subquotients in a Jordan- Holder series. Two representations of F are said to be Jordan equivalent if their semisimplifications are isomorphic. The points of MB (F, r) parametrize Jordan equivalence classes of representations of F.

Any family VT of r-dimensional representations of F gives rise to a morphism T - MB(IF, r). Locally in the Zariski topology of T we can choose a frame for the family and get a morphism to 1ZB(F, r). The projection to MB(F, r) is independent of the choice of framing, so the locally defined morphisms glue together to give T - MB (F, r).

Suppose that G is a reductive algebraic group. Then there are an affine scheme 1ZB(F, G), parametrizing homomorphisms from F to G, and its affine categorical quotient MB(F, G) = JZB(F, G)/G by the action of conjugation.

Suppose that F = irl(X) for a smooth projective variety X. If G C GL(r, C) is a faithful representation, then the resulting morphism MB(F, G) -* MB(F, r) is finite (by [27]). It follows that if G -* H is an injective homomorphism of algebraic groups, then the resulting morphism MB(F, G) MB(F, H) is finite.

Nonstationary families. Suppose that T is a reduced and irreducible scheme. A family VT of representations of F is nonstationary if the corresponding morphism T - MBg(F, r) is not constant. We have an alternative characterization in terms of traces of elements. For each -y E F, the function tr(-y)(t) = tr(pt(-y)) is a regular function on T (here Pt denotes the action of F on Vt).

LEMMA 1.3. A family VT is nonstationary if and only if the functions tr(-y) are not all constant.

Proof. Procesi's theorem in [20] implies that the functions tr(-y) generate the coordinate ring of MB(F, r). Hence the map T -* MB(F, r) is nonconstant if and only if these functions are not all constant. [1

Suppose that G is a reductive algebraic group. A family PT of homomorphisms from F to G is nonstationary if the corresponding morphism T MB(F, G) is nonconstant.

Suppose from now on that F = 7r1(X) for a smooth projective variety X.




LEMMA 1.4. Suppose that f: G -* H is an injective homomorphism of reductive groups. Then a family of homomorphisms PT from F to G is nonstationary if and only if the family of compositions fOPT is nonstationary.

Proof. Composition with f induces a morphism MB (F, G) -* MB (F, H). Hence, if the family of compositions is nonstationary, the original family must be. This morphism of moduli spaces is finite; so if PT is nonstationary, then f 0 PT is nonstationary. [1

LEMMA 1.5. Suppose that F' C F is a subgroup of finite index. Then a family of homomorphisms PT from F to G is nonstationary if and only if it is nonstationary when restricted to F'.

Proof. Since, by Lemma 1.4, we may compose with a faithful representation of G, we may assume that G = GL(r, C). There is a morphism MB(F, r) -* MB(F', r) expressing restriction; so if VT is stationary, then the restriction to F' is stationary. Suppose that the restriction to F' is stationary. If -y E F, then there is a positive integer k such that yk E F'. The eigenvalues of pt(yk) are constant as t varies. But the eigenvalues of ak determine those of -y up to finitely many choices. As T is irreducible, this implies that the traces of Pt(-y) are constant; hence VT is stationary as a family of representations of F. E

Recall that if G is any algebraic group, each element has a Jordan decomposition g = g9gs into commuting semisimple and unipotent parts. This decomposition is functorial with respect to morphisms of algebraic groups.

LEMMA 1.6. Suppose that PT is a family of representations of F in a connected reductive group G. Then PT is nonstationary if and only if there exists -y E F such that the semisimple parts Pt(y)s are not all conjugate.

Proof. Choose a faithful representation V of G. By Lemma 1.4, the family PT is nonstationary if and only if the corresponding family of representations VT is nonstationary. This occurs if and only if, for some -y E F, the eigenvalues of -y acting on VT are not constant. This occurs if and only if the images of Pt(y)s are not all conjugate in GL(r, C). Let 0 C G be a maximal torus. The set of conjugacy classes of semisimple elements is the quotient of 0 by the Weyl group. But 0 is included in a maximal torus in GL(r, C) and, again, the set of conjugacy classes of semisimple elements is the quotient by the (finite) Weyl group. This implies that the map from the set of semisimple conjugacy classes in G to the set of semisimple conjugacy classes in GL(r, C) is finite. Hence the Pt(Q)s form a constant family of conjugacy classes in G if and only if they form a constant family in GL(r, C). [1



348 C. SIMPSON

LEMMA 1.7. Suppose that f: G -+ H is a homomorphism of reductive groups with a finite kernel. Then a family of homomorphisms PT from F to G is nonstationary if and only if the family of compositions f 0 PT is nonstationary.

Proof. As in Lemma 1.4, the morphism MB(F, G) - MB (F, H) implies that if the family of compositions is nonstationary, then the original family must be. Suppose that the original family PT is nonstationary. Let Go denote the connected component of G. The family of homomorphisms r -* GIGO is constant, so there is a subgroup r' C r of finite index with pt(r') C GC. By Lemma 1.5, the family PT lp is nonstationary. Hence there is an element a E r' such that the conjugacy classes of Pt(y)s are not constant. The map from the set of conjugacy classes of GC to the set of conjugacy classes of HO is finite; so the conjugacy classes of fpt (y), in HO are not constant. Hence the family f 0 PT IF' is nonstationary as a family of representations in HO. Since HO injects into H, the family is nonstationary as a family of representations in H (by Lemma 1.4). By Lemma 1.5, f 0 PT is nonstationary. O

COROLLARY 1.8. Suppose that G is a reductive algebraic group whose connected component has a finite center. Let Ad denote the adjoint representation of G on Lie(G). Then a family of homomorphisms PT from r to G is nonstationary if and only if the functions t | 4 tr(Ad(pt(-y))) are not all constant.

Proof. The hypotheses imply that the adjoint representation Ad has a finite kernel. By Lemma 1.7, PT is nonstationary if and only if the family Ad OPT is nonstationary. By Lemma 1.3, this can be tested by looking at the trace functions. a

Families of rank-1 local systems. For representations of rank 1, the group GL(1, C) = C* is commutative, and so its conjugation action is trivial. Thus

MB = Hom(7ri (X), C*).

This is an affine variety. If 91,... , gA are generators for 7r,(X) with relations ri, ... , ri, then MB = Spec(A) with

A [ti I t11 1, . . .,k It0-1]/I

where I is an ideal that comes from the relations ri. In this case, the space MB is a fine moduli space. There is a universal family of representations

iri (X) - GL(1, A) = A*

gi H-> ti*




and hence a universal family of local systems {Wt}tESpec(A). In particular, if T is a reduced irreducible affine variety and T -* MB is a nonconstant morphism, then we can pull back to get a nonstationary family of rank-1 local systems, {Wt}teT.

In the rank-1 case, MB forms an algebraic group. The operation is the tensor product of rank-1 local systems. Since C* is abelian, the homomorphisms from 7rl are the same as the homomorphisms from H1 (X, 2). We can write H1(X, 2) = Za ? Hi,tor, where Hitor is a finite abelian group. Then a = dim H1 (X, C), and if H'tor denotes the dual finite abelian group, we have

MB = (C*)a x Hvltore

In particular dim(MB) = dimH1(X,C). A nonstationary family of rank-1 local systems exists if and only if this dimension is positive.

There is a real-analytic subset Muni parametrizing the unitary local systems. In terms of the above description we have

Muni = U(1)a X Hv

If we take, for T -* MB, any one of the subgroups of the form C* (C*)a, then T n MWi will be equal to U(1) C C* and, in particular, Zariski dense in T. The union of the images of all possible such group homomorphisms is Zariski dense in (C*)a.

2. The universal coefficients theorem

In this section we come to our first method for showing that the direct image of a nonstationary family is nonstationary. It has the advantage of being simple and topological, but it only applies in a limited range of cases.

Suppose that f: X -* S is a smooth projective morphism to a smooth base S. Suppose that WA is a sheaf of locally free A-modules on X, giving a

def family of representations Wt = WA 0 Ct for t E T = Spec(A). Set

M= Rf*(WA).

These are locally constant sheaves of A-modules on S. The stalks might not be locally free over A. Set

Vt' = RWf*(Wt).

LEMMA 2.1. The ranks of Vti vary upper semicontinuously in t. Suppose that A is an integral domain. If the ranks are constant, then M' are locally free and Vt' = M' 09 Ct. Conversely, if M' are locally free, then the ranks are constant.



350 C. SIMPSON

Proof. Since the fibration f: X -* S is topologically locally trivial, the direct images are locally constant sheaves on S whose stalks are equal to the cohomology groups M2(s) = H2(XS, WA) of the fibers X,. It suffices to prove the lemma for the stalks at s. Let UA denote a typical stalk of WA at a point in X. Choosing a covering of X, with all multiple intersections contractible, we can calculate the cohomology H2(Xs, WA) by a Cech complex, which has the form

C' = ... -+ k

Uki+1 ..

The cohomology is a coherent module over A. On the other hand, Vti(s) is the cohomology of the complex C OA Ct. The elements of C are projective A-modules. Let P be a projective resolution of Ct. Then Vt(s) is also the cohomology of the double complex 0 0 P'. There is a spectral sequence for calculating this cohomology, with

EP q = Tor-q (MP (s ),C Ct) =X Vp+q (S)

For the semicontinuity, it suffices to consider the case where A is a discrete valuation ring. Then the spectral sequence degenerates at E2 (because EP q = 0 for q $& 0, -1). By decomposing each MP(s) into a direct sum of a free module and a torsion module over A, one easily sees that the dimensions of each term can only increase when t moves from the generic point to the closed point of Spec(A). Hence the total dimension of the E2 term is semicontinuous, implying that the dimension of Vt (s) is semicontinuous in t. Suppose that the dimensions of the Vt7(s) are all the same. Reasoning from the highest-degree terms first, we find successively that the ranks of MP(s) OA Ct are the same for all t; hence the MP(s) are locally free, and the higher-Tor terms vanish. If the MP(s) are locally free, then Vti (s) = Mi (s) 0 Ct Ct.

The spectral sequence appearing in this proof gives rise to what is known in topology as the universal coefficients theorem. Suppose that A is a Dedekind domain. Let rT denote the A-torsion subsheaves of M2 and let N2 = M2/Tr. The N2 are locally constant sheaves of locally free A-modules; hence they give families of representations of 7r,(S). We will think of these as the "true" families of direct images of the Wt. For generic t E Spec(A), the modules ri are not supported near t, and the lemma above implies that the Nt' = Mt' are also equal to the usual direct images. We investigate what happens when the dimensions of the usual direct images jump; this occurs at the support of the torsion parts Td.

LEMMA 2.2 (Universal coefficients theorem). There are exact sequences of local systems of A-modules on S:

0 -* M2 OA Ct -* Riw*(Wt) -* Tor' (MZ+l, C) -* 0.




Proof. This follows from the spectral sequence that appeared in the previous proof. The fact that A is a Dedekind domain implies that the Tort vanish for i > 2. Thus the spectral sequence degenerates at E2 and gives the required exact sequences. Note that the morphisms of the exact sequence are defined canonically, so they are morphisms of locally constant sheaves on S hence the exact sequence is one of locally constant sheaves on S. El

COROLLARY 2.3. There is a filtration 0 C Vlit C V2it C V3it = Vt' by local subsystems, with

= T2 OA Ct,

V22t /1= Nz OA Ct;

the semisimplification of V3t/V2"t is isomorphic to the semisimplification of Ti+l OA Ct.

Proof. The first two steps in the filtration come from the fact that the exact sequence 0 -ri - M- N2 -* 0 splits locally (A is regular of dimension 1, so the local rings are principal ideal domains). To identify the third quotient note that if z is a uniformizing parameter at t, then (locally at t) we have an exact sequence

OV-+A z ,A -+ t -- ,

and hence

0 -* Torl(Ti+lCt) I - Ti+l Z -i+T- i+l 0 t C? .

Consider the rT (localized at t) as locally constant sheaves of C-vector spaces on S. Recall that the semisimplification of a locally constant sheaf is the direct sum of the simple quotients of a Jordan-Holder series and this is, up to isomorphisms, independent of the choice of the Jordan-Holder series. Taking two Jordan-Holder series for Td, one compatible with the image of the multiplication by z and the other compatible with the coimage, we see that the uniqueness of the semisimplification implies that the semisimplifications of TZ+1 0 Ct and Tor1 (ri+1, Ct) are isomorphic. [1

Remark. Suppose that we know that the local system Vi = R'f*Wt is semisimple. Then all of the subquotients occurring in the above corollary are semisimple, even (from the next case of Ri+lf*Wt) the TZ+l 0 Ct. Hence in this case we may conclude that

Vi /vi2 i+1 A

V3,tV 2,t T OACt

although the isomorphism is not canonical. This remark applies to the direct images of rank-1 local systems by Lefschetz pencils (Corollary 2.8 below) and, in fact, any time Wt is semisimple (by Corollary 4.5 below).



352 C. SIMPSON

Lefschetz pencils. We will give a preliminary description of Lefschetz pencils and apply the previous results in some cases. This discussion will be taken up again several times later in the article. Everything presented below is well known. We give the details for convenience. The presentation is influenced by lectures of C. Voisin (Sophia-Antipolis, 1991).

Suppose that Z is a smooth projective variety of dimension n + 1. Let L be a very ample invertible sheaf on Z, equal to 0z(1) for some projective embedding of Z. Let jpN denote the projective space of lines in HO(Z, L) (or, equivalently, of hyperplanes in the projective space containing Z). For each S E pN let fs denote an element in the line corresponding to s and let Xs C Z denote the hypersurface defined by the equation fs(z) = 0. Let X c jpN X Z denote the incidence variety, consisting of the set of (s, z) such that z E Xs. Let 7r: X __ jpN and a: X -* Z denote the projections on the two factors. If y __ jpN is any variety mapping to jpN, let Xy = X X1PN Y. Keep the notation 7r: Xy -* Y and ay: Xy -* Z. Let D c pN denote the closed set of values of s for which Xt is singular and let U denote the open set that is the complement of D.

Suppose that W is a semisimple local system on Z. Let

= R'wr*(a* W)

be the direct-image local systems on U. Let Hi(Z, W) denote the cohomology of W on Z, considered as a constant sheaf on U. There is a natural morphism H2(Z, W) -* V2; let V1ew denote the cokernel. We recall some basic results about the cohomology of Lefschetz pencils.

LEMMA 2.4. Vniew = 0 for i 7& n.

Proof. First note that the hard Lefschetz theorem works with coefficients in any semisimple locally constant sheaf such as W. In other words, the morphisms

Acl(L)2: Hn+1-i(Z, W) -* Hn+l+i(z, W)

are isomorphisms for all 0 < i < n + 1. This may have been known for some time, but in any case it follows from the Lefschetz decomposition theorem in [23]. The Morse-theoretic version of Lefschetz theory implies that Z is obtained from Xs by attaching cells of real dimension > n + 1. This implies that the restriction map

Hz(Z, W) -* Hz(XsI W x8)

is an isomorphism for i < n and is injective for i = n. For classes of degree i > n, the restriction map factors as

Hz(Z, W) -* HZ+2(Z, W) -* H2(Xs, W IKx)J




where the first map is a cup product with the hyperplane class and the second is the inverse of the Poincare dual of the restriction isomorphism mentioned for i < n above (for the dual local system). In particular the hard Lefschetz theorem implies that the restriction is surjective for i > n. This gives the lemma. [1

Remark. The referee pointed out that the hard Lefschetz theorem is not true for non-semisimple local systems; similarly the statement of the above lemma may fail if W is not assumed to be semisimple.

LEMMA 2.5. There are no fixed vectors in the monodromy representation of Vn.

Proof. For this we restrict to a one-parameter Lefschetz pencil. Let 1P1 C pN be a generic line. Let Uo = U n JP1 and Do = Dn JP1. Since the morphism 7rw(Uo) - r(U) is surjective, the invariants of Vn on Uo are the same as those on U.

The variety X = X1pi is obtained by blowing up Z along a smooth subvariety B (which is the intersection with a generic pN-2). Let E denote the exceptional locus. Then E r B x P1, the map wr JE giving the projection on the second factor. Here wr denotes the morphism X -* JP1. The singular fibers of wr each contain one double point and are otherwise smooth. Let ax denote the map X -* Z. Use the notation W for pullbacks or restrictions of W to various varieties, introducing a subscript if necessary such as Wx = a* (W).

We treat the structure near the singularities. Let so be a point in jPl such that the fiber Xo = 1r- (so) has an ordinary double point. The inverse image of a disk around so retracts to Xo. Let s be a point near so so that X, is smooth. There is an embedded n-sphere in X, called the vanishing cycle 6; one obtains Xo by adding to X, an (n + 1)-ball with boundary 6. The local system W is trivial on 6, so we can contract cohomology classes with coefficients in W, with 6 (the answer is an element of the stalk of W at the double point). Let 6* denote the cochain that is Poincare dual to 6 on X, The monodromy transformation obtained by moving s around so may be represented by a diffeomorphism of X, that has the geometric effect of replacing a cochain u by u + c(u, 6)6* plus a coboundary, where c is a nonzero constant. Since W is trivial in a neighborhood of the singularity, we may assume that W is trivial on the subset, where the monodromy diffeomorphism differs from the identity. The monodromy then has the same effect on cochains with coefficients in W, where the contraction (u, 6) yields an element of the stalk of W (at the double point), which can be multiplied by 6* to give a cochain with coefficients in W. The occurring coboundary, supported where the diffeomorphism is different from the identity, becomes the coboundary of



354 C. SIMPSON

a chain with coefficients in W. We get the same formula u 4 u + c(u, 6)6* for the monodromy transformation on H2(Xe, W). The subspace of classes in H2(X,, W), which are invariant under the monodromy transformation, is the same as the subspace of classes that yield 0 when contracted with 6. By the Mayer-Vietoris principle, this is the same as the subspace of classes that have extensions over the ball bounded by 6 in other words, classes that extend to Xo; and the extension is unique for classes of degree i < n. Thus, for i < n, the retraction from X, to Xo gives an injection H2(Xo, W) c H2(Xs, W) with the image equal to the subspace of invariant classes.

The space of sections of R'7r*Wx over a disk neighborhood of so is equal to H2(Xo, W). The previous paragraph concludes that, for i < n, the restriction morphism from this space of sections to H2(X8, W) is injective with its image equal to the subspace of classes invariant under the monodromy. This subspace of invariant classes is equal to the space of sections of R'7r*Wx over the punctured disk. Recall that Vi is the restriction of R'7r*Wx to the complement U of the set of singularities and let j: U JDPl denote the inclusion. For i < n, the natural morphism RWr*Wx -* j*(V2) is an isomorphism. In particular the sheaves R'7r*Wx are constant sheaves equal to Hi(Z, W) for i < n, and Rnwr*(WX) = j*(Vn). The subspace of constant vectors in Vn is equal to the subspace of constant vectors in j* (Vn); so to prove the lemma it suffices to prove that Hn(Z, W) -* HO(1P1, Rn~r*Wx) is surjective.

We will use the Kiinneth formula for E - B x P1 and also for the cohomology of JP1 with coefficients in a constant sheaf. From the Leray spectral sequence for 7r: X -* JP1 we obtain a five-term exact sequence:

H2(1PD1) 0 Hn(X., W) -n H-(X, W) -nXW) H0(1P1 RnwrW )

- H2(P1) 0 Hn-1(XsI W) -* Hn+1(X, W).

We claim that the last map is injective. The composition of this map with the restriction map Hn+1 (X, W) -* Hn+1 (E, W) is equal to the composition of

H2(P1) o Hn-1(X.I W) -* H2(P1) o Hn-1(B W)

with the inclusion given by the Kiinneth formula. But B is a smooth hyperplane section of X, of dimension n - 1; so the Lefschetz theory applied there shows that Hn-1(X,, W) - Hn-1(B, W) is injective. This shows the claim.

Now we obtain a three-term right exact sequence:

H2(1P1) H (X W) -* H--(XW)- Ho(PlR WX) 0, R

On the other hand, the Leray spectral sequence for a: X -* Z gives a three- term sequence exact in the middle:

Hn (Z W) -- Hn(X, W) -- H2(1P1) ( Hn-2(B, W).




Again, Lefschetz theory for B C X, implies that HI-2(Xs, W)- Hn-2(B, W). Considered together with the previous exact sequence, this gives the desired surjection Hn(Z, W) -* H0(oP(1, Rn~r*WX) to prove the lemma. El

COROLLARY 2.6. There are no fixed vectors in the dual (Vnew)*.

Proof. Poincare duality gives an isomorphism between (Vn)* and Rnwr*(a*W*). The fixed vectors of (Vn)* are Hn(Z, W*) (by Lemma 2.5 applied to W*). But (Vnnw))* is the kernel of (Vn)* Hn(Z, W)*. The composed morphism

Hn(Z, W*) -- Hn(Z, W)*

is the composition of the Lefschetz isomorphism

Hn (Z, W*) -_Hn+2 (Z, W*)

with the isomorphism of Poincare duality. This implies that there are no fixed vectors in (Vnew) *

LEMMA 2.7. Suppose that the local system W has rank 1. Then Vnnew is irreducible.

Proof. Keep the notation established in the proof of Lemma 2.5. The fundamental group 7rw(Uo) is generated by loops -yi going around the points si of Do. The monodromy of the cohomology Hn(Xs, W) around each of these loops is (if nontrivial): semisimple with one eigenvalue -1 and the rest 1, if n is even; and unipotent with the rank of (-y - 1) equal to 1, if n is odd. This is well known for the case of a constant sheaf, but the reason is purely local at the double point, and W is isomorphic to the constant sheaf in a neighborhood of any double point (see Lemma 6.5 below). The fact that the discriminant locus D is irreducible implies that the images of -yi in 7rw(U) are all conjugate. Hence the monodromy representation of Vnnew is generated by conjugates of one transformation -y of the above form. Suppose that V1 C Vnnew is a proper invariant subspace, and let V2 = Vnnew/Vi be the quotient. By the nature of -y, it must act trivially on either V1 or V2 (or both). But the subspace Vi is preserved by all elements g of the monodromy group; so any conjugate gyg-1 also acts trivially (on V1 or V2, respectively). Thus either V1 is a subspace of fixed vectors or V2* is a subspace of fixed vectors in (Vnnew)*. This contradicts Lemma 2.5 or Corollary 2.6. Hence Vnnew is irreducible. [1

COROLLARY 2.8. If W has rank 1, the local system Vn is semisimple, the direct sum of the subspace of fixed vectors with the irreducible part VnneW.

Proof. From the proof of Corollary 2.6 we have (Vn)* = Hn(Z, W*) ED (Vnnew)*. Taking the dual gives a decomposition of Vn as a direct sum of a trivial local system and an irreducible one. [1



356 C. SIMPSON

Families. We turn to the question of direct images of families of representations by Lefschetz pencils. Keep the same geometric notation as in the previous subsection. Suppose that A is an integral domain and WA is a family of local systems on Z parametrized by Spec(A). Again replace the subscript A by t to denote the tensor product with Ct for t E Spec(A). We say that WA is a family of semisimple local systems if each Wt is semisimple.

Let H2(Z, Wt) be considered as a constant sheaf on the base U. Let M' = R'7r*(a*WA) be the direct-image local system of A-modules, as defined before. For each t E Spec(A) let Vt7 = R'wr*(a*Wt) and Vtnwew = Vtn/H2(Z, Wt).

LEMMA 2.9. Suppose that WA is a family of semisimple local systems and Spec(A') C Spec(A) is an open-affine subset such that the dimensions of H'(Z, Wt) and the ranks of Vt/ are constant for t E Spec(A'). Then the Vtnnew fit together into a family of local systems indexed by Spec(A').

Proof. We may set A = A'. The assumptions imply that Mn and Hn(Z, WA) are locally free. We have a morphism Hn(Z, WA) , Mn. The Lefschetz theorem implies that, for each t, the morphism between fibers is injective. This implies that the quotient

Qnf = Mn/H (Z, WA)

is locally free over A. The fiber Qn is isomorphic to Vtnew. This provides the required structure of a family of local systems. [1

Suppose now that Spec(A) is an irreducible smooth algebraic curve (i.e., A is a Dedekind domain). Let N2 denote the quotients of M' by their torsion submodules Ti, as before. We have a family of local systems Nt' = NiOACt indexed by t E Spec(A). Fix a point t E Spec(A) and let r1 denote a generic point in Spec(A). Denote the jumps in dimensions by

62(Xs) = h2(Xs, Wt) - h2(Xs, Wn)

and 62(Z) = h2(Z. Wt) - h2(Z. Wq)

(where the lowercase h denotes the dimension of the cohomology group). We have 62(X,) = 62(Z) for i < n and 6'(Xs) - 6i+2(Z) for i > n (by Poincare duality). Note that

h2(Xs, Wq) = rk(N2)

and, by Lemma 2.2,

62(Xs) = dimT2 0 Ct + dim Tr+l 0 Ct.




PROPOSITION 2.10. Suppose that WA is a family of rank-1 local systems on Z. Suppose further that 6'(Z) > 6'(X,). Then the family {NtT}teT is nonstationary.

Proof. By Corollary 2.8 the local system N7' is a direct sum of an irreducible local system V1nnnew (which is not constant if it is nonzero) and the constant local system Hn (Z, Wi,). A constant local system Hn (Z, Wt) is contained in the local system Vtn. Hence the dimension of the space of fixed vectors in the semisimplification of Vtn is at least hn (Z, Wt). If we choose a Jordan-Holder filtration compatible with the filtration given by Corollary 2.3, we find that the dimension of the space of fixed vectors in the semisimplification of Nn 0 Ct is at least

hn(Z, W) - dim(Trn 0 Ct) - dim(Tn+1 0 Ct) = hn(Z, Wt)-6 (Xs)

= hn(Z, Wn) + 6n(Z) - 6n(X.).

Under our hypotheses, this is strictly bigger than hn(Z, W.). Hence the semisimplifications of Nn 0 Ct and Nn 0 C ', are different (this includes the statement that rk(Nn) > 0). Thus the family is nonstationary. [1

COROLLARY 2.11. Suppose that 6n+1(Z) > 6n+2(Z). Then the family of the local systems Ntn is nonstationary.

Proof. Recall that

6'(X,) = dim(T2 (0 Ct) + dim(Tr+l 0 Ct)

Therefore the 62(X8) for i < n - 1 determine dim(r2 0 Ct) for i < n. If we denote the torsion in H2(Z, WZ,A) by 4z, then the same statement holds, with the same formulas. From the Lefschetz theorem, dim(ri 0 Ct) = dim(Tz 0 Ct) for i < n. Similarly, using the Lefschetz theorem for degrees above the middle degree, we find that dim(Tr 0 Ct) = dim(Tz+2 X0 t) for i > n. Hence from the universal coefficients theorem we get

6n(X,) = dim(,Tzn (0 Ct) + dim(Tzn3 0 Ct) = 6n(Z) + 6n+2(Z) - 6n+ (z)

Thus the difference occurring in the present statement is the same as in the previous proposition: 6n (Z) - 6n (X,) = 6n+1 (Z) - 6n+2 (Z). D

COROLLARY 2.12. Suppose that WA is a family of rank-1 local systems on Z parametrized by a smooth irreducible curve Spec(A). Suppose that there is a point t E Spec(A) such that the corresponding jumps satisfy 6n+l(Z) > 6n+2(Z). Let Spec(A') C Spec(A) be an open set such that the family {Vt~new}teSpec(A/) is defined. Then this family is a nonstationary family of irreducible local systems indexed by Spec(A').



358 C. SIMPSON

Proof. Note that since Nt/ is the direct sum of Vt'new with some constant local systems, the condition that Nt/ be nonstationary (deduced in the previous lemma) implies that the family of Vntnew is nonstationary.

Example. Suppose that Z is an abelian variety of dimension g = n ? 1. Then the moduli space for local systems of rank 1 is a fine moduli space, given by

MB(Z, 1) (?*)29.

We can choose a curve Spec(A) -* MB(Z, 1), which passes through t corresponding to the trivial system and through a general point r1 of MB(Z, 1). This gives a family of rank-1 local systems WA. The cohomology with coefficients in a nontrivial local system vanishes,

Hz(Z, Wn) = 0,

for all i. On the other hand, the dimensions of the cohomology groups with constant coefficients are given by binomial coefficients

hi (ZI Wt) = hi (Z) ( )

Hence we have the formula

62 (Z) = (29)

In particular, since 2g is even, the middle binomial coefficient 6n+1 (Z) is strictly bigger than the next one 6n+2(Z). The conditions of Corollary 2.12 are satisfied by taking the direct image of the family WA by a Lefschetz pencil, we obtain a nonstationary family of local systems {Vtnnew} on U c pN.

3. Direct images of Higgs bundles

In this section we will discuss direct images of Higgs bundles. We refer to [15] and [25] for general definitions and properties of Higgs bundles. There is also an expository discussion in [26].

Suppose that X and S are smooth quasiprojective varieties and f: X S is a smooth projective morphism. Suppose that E = (E, 0) is a Higgs bundle on X. We define the higher direct images HbDOI(X/S, E) as follows: Let Qx/s(E) denote the complex of sheaves

*QZ/ $ ) E AO Qi+1l X E ...

(the relative Dolbeault complex of E on X/S). Now define

HbDOI(X/SI E) = R'f* (Qx/s(E)) I




where Rif* denotes the right-hyperderived functor. These are coherent sheaves on S. We give them structures of Higgs sheaves in the following way: Let Q (E) denote the Dolbeault complex of E on X, with differentials given by multiplying by 0. Let I1 = I1Q' (E) denote the subcomplex that is the image of f*Q' Q (E) and let 12 = I2Qx(E) denote the image of f*Q 0 IlQ (E). Note that the relative Dolbeault complex is the quotient

Qx/S(E) = Q (E)11

and we have an isomorphism

f *Qs X Q. s(E) IlQ' (E)/12.

Hence we get an exact sequence of complexes

0 - f*Q1 0 QX/S(E) - Q(E)/I2 Qxs(E) 0.

The hyperdirect image of the complex on the left is

Rif* (f*Q' 0 QX/s(E)) = Q' ( Rif* (Q'/s(E)).

So the coboundary map for the hyperdirect images gives a morphism

A: H1DOI (X/S, E) - Hb01 (X/S, E) 0 Q.

One can verify that p A po = 0 (this will also follow from the description given below); so (HiDOI(X/S, E), (p) is a Higgs sheaf on S.

If the HiDOI(Xs, E) have the same dimensions for all s E S, then the direct-image sheaves HiDOI(X/S,E) are locally free, with HbDOI(X/SE)s =

HbDo0(X E). In this case, the direct images are Higgs bundles.

Algebraic calculation. We give a slightly different way of calculating the direct image. This is based on an idea that Hitchin pointed out to me some time ago, which is that the Dolbeault cohomology with coefficients in a Higgs bundle is in some sense concentrated at the zeroes of the Higgs field. In the relative case, this gives rise to a picture where the spectral curve of the direct image comes from the family of zeroes of the Higgs fields on the fibers.

Recall that the operation 0 gives E a structure of a coherent sheaf on the cotangent bundle T*(X) (see [26]). We denote this sheaf by 8. Let p: T*(X) -* X denote the projection. The direct image p*E is equal to the coherent sheaf E. The sheaf 8 is finite over X (this follows from the previous statement, since p is affine).

On T*(X) there is a tautological section a of p*Q1 This projects to a section of P*Q1/s, and we can make a complex A by setting

A' = p*Q



360 C. SIMPSON

with the differential given by the wedge product with a. Then we have

Q (X/S, E) = p* ( (0 A).

The complex 8 0 A is supported on a scheme, finite over X, so this direct image is equal to the derived direct image. We have

H2(X/S, E) = R2(fp)* (8 (0 A).

On the other hand, let D = f*T*(S) C T*(X). Let WD/T*S denote the restriction of p*Qx to D. Then A is a resolution of WD/T*S[-n]. (This notation means the complex formed by placing the sheaf WD/T*S in degree n.) Furthermore the components of A are locally free. Hence the complex, which appears above, is a left-derived tensor product

L 8KA =0(9WD/T*S [-n]

The left-derived tensor product can be calculated in two ways by taking projective resolutions of either factor. Choose a finite resolution Q -* 8 by locally free sheaves over T*(X). Then there is a quasiisomorphism in the derived category between 8 0 A and Q 0 WD/T*S[-n]. In particular,

Hi(X/S, E) = R2(fp)* (Q 0 WD/T*S [-n])

We can now give the Higgs structure of H2(X/S, E) by describing the corresponding sheaf F on T*(S). Note that there is a natural map

g: D -T*(S)

and that we have the factorization fp ID = psg, where Ps denotes the projection from T*(S) to S.

Let E C T*(X) denote the support of 8 (the spectral variety). Let R = E n D be the intersection of the spectral variety with the subset D, which is the pullback of the zero section in T*(X/S). The cohomology sheaves of the complex Q 0 WD/T*SE-n] are set theoretically supported on R. (Certainly the second factor is supported on D; and the complex Q is an exact complex of locally free sheaves outside A, hence the tensor product is exact outside E.)

Note that E is proper over X and X is proper over S. Therefore the closed subset R C E is proper over S and, in particular, the image g(R) is closed and proper over S.

We claim that the hyperderived direct images Rj9*(Q0 WD/T*S[-n]) are supported set theoretically on g(R). To see this, note that there is a spectral

k j-k k . sequence with the E2' term Rkg *(7-k (Q 0 WD/T*S[-n])), converging to the hyperderived direct image. As the cohomology sheaves appearing here are supported on R, their higher direct images are supported on g(R). This




implies that the hyper-derived direct images are suppported on g(R). These images are coherent sheaves (since g is proper). Hence there is an infinitesimal neighborhood g(R)k of g(R) such that R'g* (Q (0 WD/T*S [-n]) are supported scheme theoretically on g(R)k. The map Ps: T*(S) -4 S is affine and g(R)k C

T*(S) is a closed subset, proper over S; so the morphism g(R)k -4 S is finite. Hence the projection Ps,* is exact on coherent sheaves supported on g(R)k. Consequently

Rj(psg)*(Q 0WD/T*S[-n]) =Ps,*(Rjg*(Q 0 WD/T*S[-n])).

Set F = Rig*(Q 0 WD/T*S[-rn]). From the previous equation and the earlier discussion we have

H2(X/S, E) = Ps,

The reader may verify that F is the sheaf on T* (S) that corresponds to the Higgs sheaf (HW(X/S, E), (p) on S.

We describe the spectral sequence for hyperderived direct images more precisely. First of all, note that the cohomology sheaves of the left-derived tensor product are by definition the Tor of the factors:

Hi (Q 0 WD/T*S[-n]) = Torn3 (8wD/T*S)-

Hence the spectral sequence for the hyperderived direct images has

E'2- = Rig* (Torn+J i(,i W/DT*S)) = Rig* (Q 0' D/T*S[-f]) = z.

The terms Torf+ i(8,WD/T*S) are supported on R. We will show below that the spectral sequence is compatible with the decomposition of R into connected components. Note also that the spectral sequence is compatible with an etale base change over T*(S). So it will follow that the spectral sequence is compatible with the decomposition of R into connected components locally over T*(S) in the etale topology.

Note. The notation used in the following lemma and its proof is to be considered independently from the notation of the remainder of the article.

LEMMA 3.1. Suppose that C is a finite complex of coherent sheaves on a scheme X1 such that the cohomology sheaves are supported on a closed subset R1 C X1. Then there is a quotient C -* B inducing an isomorphism on the cohomology such that B is supported scheme theoretically on an infinitesimal neighborhood (R1)k of R1.

Proof. Let I denote the sheaf of ideals defining R1. We will proceed by induction on the length of the complex. We can assume that the first term in the complex is at degree 0.



362 C. SIMPSON

We claim that there exists m such that ImC0 * C' is injective. For this, note first that the kernel of ImCo ,- C' is contained in HO = ker(C0 -* C') and this HO is supported on R1. Let F be the subsheaf of Co of sections supported on R1. It suffices to show that there is an m such that ImCo nF = O. By Noetherian induction and localization we reduce the case to the following: C' is a finitely generated module over a Noetherian local ring with J as the maximal ideal; F' C C' is the submodule of sections supported at the closed point; and we have to show that JnC' n F' = 0 for some n. By Krull's theorem, nn JnC' = 0. Thus nn(JnC' 0 F') = 0; but F' is a module of finite length, so there is an n with JnC' n F' = 0. This completes the proof of the claim.

Define a subcomplex D c C with D2 = C', for i > 1, and Do = ImCO. The quotient C /D is supported set theoretically on R1. There is a quotient D1 - E with E = C' for i > 2, E = 0 and E1 = Cl/ImCO. Here the kernel is the complex ImC0 --+ ImC7, with a trivial cohomology. The complex E has its cohomology supported set theoretically on R1 and its length is less than the length of C. By the inductive hypothesis we can find a quotient E -* A, which induces an isomorphism on the cohomology and such that A is supported set theoretically on R1. Composing, we get a quotient D1 -* A, again inducing an isomorphism on the cohomology. Let K be the kernel of D1 -* A; it is an exact complex. It is also a subcomplex of C, and we can let B = C /K . Note that we have an exact sequence

and so B is supported set theoretically on R1. [1

Apply this lemma to the complex Q 0 WD/T*S [-n], considered as a complex of sheaves on D. We obtain a quotient complex B of coherent sheaves on D, supported on an infinitesimal neighborhood Rk of R. The morphism Q ( WD/T*S [-n] -* B induces an isomorphism of cohomology sheaves. Hence it induces an isomorphism of hyperderived direct images and also an isomorphism of spectral sequences. Thus the hyperderived direct images and the spectral sequence have a natural action of 9g,(ORk) In particular the idempotents corresponding to the decomposition of Rk into connected components give decompositions of the hyperderived direct images and the spectral sequence. Note that the components of Rk are the same as those of R. Write

R = R(l) U ... U R(m)

as a disjoint union of connected components. Then we have a decomposition of the hyperderived direct image

-Fi = (il) ( " .. d m)C(i )




with PI supported set theoretically on g(R(,)). The Torl decompose as

Tor (8,wD/T*S) = G Tor (8,wjDpT*s)(i)

with Tory (8, WD/T*S)l supported set theoretically on R(i). Finally the spectral sequence decomposes into spectral sequences for each R(i) with

E'12-j (R(i)) = Rig* (Torn+j (8, WD/T*S)(l)) X)

As noted above, the same discussion holds after localization in the etale topology of T*(S).

The advantage of these last remarks is that we can make use of the degeneration of some component of the spectral sequence without worrying about the degeneration of other components.

COROLLARY 3.2. Let n be the relative dimension of X/S. For s E S let Es and D, denote the fibers over s in T*(X) Ix, Suppose that y E Rs= 0, n D, is an isolated point. Then g(y) E T*(S) lies in the support of IF1.

Proof. Let R1 denote the union of irreducible components of R containing y. Note that R1 is finite over S, hence also over T* (S). We can localize everything in the etale topology of S such that R1 becomes a connected component of R. Then, in the above decomposition of the spectral sequence, consider the part corresponding to R1:

E"_j j(R) = Rjg*(Torn+jii(8,wD/T*S)D ) X E.

Since the sheaves Torn+f-i(8, WD/T*S)1 are supported on R1, they are finite over T*(S). Hence the higher direct images vanish, and there is only one nonzero row in the E2 term of the spectral sequence. The spectral sequence degenerates, leaving

E= g* (Tornfi (8,w/T*S)1).

In the case where i = nr, the point y is contained in the support of 8 0 wD/T*S; so g(y) is contained in the direct image. [1

The subset R, is the part of the spectral variety corresponding to the zeroes of the Higgs field restricted to the fiber X, The corollary above says that an isolated point here contributes to the spectral variety of the direct image.

Analytic calculation. Here is an analytic calculation of the direct image. Suppose that we are in the case where the HDOI (XeE) all have the same dimension. Each of these fibers can be calculated as the cohomology of the



364 C. SIMPSON

complex A (XI E) of C?? forms on X. with coefficients in E, and with the differential given by the operator D" = a + 0. These cohomology groups fit together into a C? vector bundle (a section is C' if it can be represented by a family of forms a, on X5, which vary smoothly with s). To give this vector bundle the structure of a Higgs bundle we need to describe its operator D". Suppose that a, is a smooth family of forms representing a smooth section [as] of the bundle of cohomology groups. The choice of a C' metric on X gives a splitting of the space of forms on X; so we get a form a on X with coefficients in E such that a, = a Ix. Suppose that a is a vector field on S. Lift it to a vector field (which we also call a) on X. Put

Ds[a](a)= [(a L1Dxa)JxJ.

Here the symbol I denotes the contraction of a vector field and a form, re- ducing the degree of the form by one. Note that D" (a) IxK = D" (a) = 0. Thus (a I D" a) IxK does not depend on the choice of the lifting of a (two different liftings would differ by vertical vector fields). One can check that if 3 is a form with all 3 JxK = 0, then

D/X (a Ko) = (a I1D//3) K

(using the fact that the projection of a to f*T(S) is constant along the fibers X,). Since D" a Kx. = 0, this applies to give

D" (a 1 D" a) IKx = 0.

Hence [(a I D"a) Ix] defines a section of the bundle of cohomology groups. Using the above formula, one sees that it is well defined, independent of the choice of a restricting to representatives of [a,]. Since this defines the value of D" [as] on any vector field, it defines a 1 form on S. The bundle of cohomology groups is naturally isomorphic to the C' bundle underlying HzDOI(X/S, E). We leave it to the reader to check that the operator D" defined by the above formula is the same as the D" operator from the Higgs-bundle structure of HDO1 (X/S, E), as defined previously.

Nonsmooth morphisms. We finally discuss the direct images of Higgs bundles by some nonsmooth morphisms. Our treatment is inspired by the methods of Steenbrink [28] in Hodge theory.

Suppose that X and S are smooth projective varieties and f: X S is a morphism that is smooth over the complement of a divisor B C S. Let D = f ̀ (B). Assume that D is a divisor with normal crossings and (for simplicity) the reduced components Di are smooth. Write D = >iaiDi with ai positive integers. Recall that Q (log D) and Q' (log B) are the sheaves of differential forms, logarithmic along D or B. For example, if D is defined in a




neighborhood of a point by the vanishing of a product of coordinate functions zal .zk, then Q' (log D) has a local frame consisting of wedge products of dzj/zj (1 < j < k) and dzj (1 > k). Define

~X/S~logD) =Q' (log D) Qi Xs/ g D) =f*(Ql (log B)). Q'-1(log D)

Suppose that (E, 0) is a Higgs bundle on X. Then we can define the logarithmic relative Dolbeault complex Q S(E, log D) to be the complex with the terms Q' /S(log D) E and the differential given by the wedge product with 0. Then put

HDO1(X/S, F, log D) = Rif* (Qjs(E, log D)).

As in the smooth case, the connecting morphism for an appropriate exact sequence gives a Higgs field

p: HbDO(X/S, E, log D) -* H 01(X/S, E, log D) 0 Q (log B).

We get what might be called a logarithmic Higgs sheaf (Fi, (p) on (S, B) (see [19] for the origin of this name).

Remark. This construction can also be made if (E, 0) is a logarithmic Higgs bundle on (X, D)-that is, if 0: E -*E Q0 (log D).

This construction may or may not represent the "good" direct image. If we suppose that S is a curve, it will turn out to represent the good direct image if all of the multiplicities ai of components of D are 1. Otherwise the logarithmic direct image will be an approximation; the true direct image is a filtered regular Higgs bundle. We will give an algebraic characterization of the good direct image in case S is a curve, but the proof must be postponed until the next section.

Suppose that X, S, D, B and f are as above, with S a smooth curve. Suppose further that (E, 0) is a Higgs bundle on X, which is a direct sum of stable Higgs bundles with vanishing Chern classes. Note that any coherent sheaf on S gives a locally free sheaf after dividing by the Os-torsion. Put

HDO1(X/S, E, log D, If) = HDO1(X/S, E, log D)/torsion.

We now proceed to define the good ith direct image, a filtered regular Higgs bundle F = {F a}. (Note: we combine the filtrations at all b E B together so that Fa means what was called F(a.. in the notation of [23].)

Let v be the least common multiple of the multiplicities ai of components of D. Choose a finite covering p: S' -* S ramified over B and let B' = p 1(B) (set-theoretic inverse image). For each b' E B' let ,u(b') denote the multiplicity of b' in the scheme-theoretic inverse image of B (thus ,u(b') - 1 is



366 C. SIMPSON

the ramification index at b'). Assume that all of the ,u(b') are divisible by v. Let X' be a desingularization of X x s S' and f': X' -* S' be the morphism. Let D' = (f')-1(B') and assume that D' is a divisor with normal crossings and smooth components. Let E' be the pullback of E to X' (via the natural morphism X' -* X). We obtain a morphism of Higgs sheaves

I: p*HboI(X/S, E, log D, lf) -* Hb01(X'/S', E', log D', If).

For a E lR define Fa to be the sheaf of meromorphic sections f of HzO1(X/S, F, log D, lf), regular on S - B, such that, near b E B, I(p*f) vanishes to order > al-u(b') at each b' E pl (b).

PROPOSITION 3.3. With the operator p induced by that of the Higgs bundle HbO1(X/S,FlogD,lf), F = {Fa} becomes a filtered regular Higgs bundle, which is a direct sum of stable, filtered, regular Higgs bundles of degree 0. Furthermore Fo = HDO1(X/S, E, log D, If).

The proof will be given in the next section. We call F the ith direct image and also employ the notation

F = HDO1(X/SI E).

In the next section we will show that if W is the local system on X corresponding to E, then the direct image of W corresponds to F, in the correspondence of [23].

Duality. Here are some algebraic statements about duality for direct images of Higgs bundles. They will be used in the next section to prove the above proposition. One can see, from the failure to have a completely nice duality theorem, that HDO1(X/S, E, log D, lf) is not necessarily the good direct image.

Keep the hypotheses that S is a curve, f: X -* S has relative dimension n, B C S is the reduced divisor over which f is not smooth and the divisor D = f 1 (B) has normal crossings. We assume that E is a Higgs bundle on X.

We refer the reader to [13] for an explanation of duality for complexes of sheaves with coherent cohomology. There is a morphism f! from a derived category of complexes on S to the derived category of complexes on X. The duality theorem ([13], Ch. III, Thm. 11.1) says that

Rf*R Hom(Y, f!OS) = R Hom(Rf*., Os)

We have to identify f!Os. Since X and S are smooth projective varieties, we can use the formula for composition. Let g: S -- Spec(C) be the projection. Then

(gf!C = wx[n ? 1],




where ws = Q1 and wx = Qn+1 are the dualizing sheaves and [k] denotes the operation of shifting by degree -k. The composition formula f!g!C = (gf)!C gives f!wS = wx [n]. Since ws is an invertible sheaf, we can divide by it to get f !IOs = Wx/S[n], where

WX/S = Wx 0 (f*WS)l

Suppose that A is a finite complex of projective Ox-modules. Then we do not need to derive the Hom on the left-hand side of the duality statement, and we get

Rf* Hom(A', Wx/s) = R Hom(Rf*A', Os) [-n].

There is a spectral sequence for computing the right-hand side of the equation. Since S is a curve, ExtZ(*, Os) vanishes for i > 2 (this refers to Ext sheaves on S). Therefore the spectral sequence degenerates and gives a collection of exact sequences. The duality statement can then be written as

0 -- Extl(RZ+lf*A IOS) - Rn-if* Hom(A, wx/s) -- Hom(Rf*AOS) -v 0.

The surjection is given by the cup-product pairing together with a map

T: Rnf*WX/S - S

Over S - B, where f is smooth, WX/S IX-D is isomorphic to Qn and r coincides with the usual isomorphism

Hn(X-D/S-B, QX-D/S-B ) OS-B-

We can describe Wx/S in terms of logarithmic differentials. Note that Q>/s (log D) is locally free, equal to the quotient of Q> (log D) by f*Q5 (log B), and

QX/S(log D) = AQ>/S(log D).

In particular Q/5(log D) is the determinant of QX/5(log D), so we get the formula

QX/S(logD) = Q+1(logD) 0 (f*Ql (logB))-l.

Let Dred denote the reduced divisor underlying D. Then we have Qn(log D) =

wx 0 OX(Dred) and, similarly, Q (logB) = ws 0 Os (B). As the pullback of B is D, not Dred, we get

QX/S(logD) = wx 0 OX(Dred) 0 f*(ws 0( Os(B))-

= wx 0 f*(ws)-l ? OX(Dred - D)

= WX/S ? OX(Dred - D).



368 C. SIMPSON

This isomorphism, composed with the usual isomorphism (9x(D-Dred) IX-D OX-D, coincides with the identification

0(X-D)/(S-B) = W(XD)/(S B)

We will apply the duality statement to the logarithmic Dolbeault complex A = Q,/s(E, log D). The components are locally free sheaves on X. The formula

Qx/s (log D) = Hom (Qn-7 (log D), Qn/S (log D))

gives

QX/s(E, log D) = Hom(Qj,/s(E*, log D), Qx/S(log D)[-n]).

Putting these things together, we get the following statements:

LEMMA 3.4. There is an exact sequence

0 ,~ Ext (H'+O(X/S, ElogD), IOs) H2n-i (XIS, E* ? Ox (D - Dred), log D) Hom (HR01 (X/S, E, log D), IOS) -? 0.

For the locally free direct images there is an isomorphism of logarithmic Higgs bundles on S

H2n yi (XIs, E* 0 Ox(D - Dred), log D, lf) - H 01(X/S, E, log D, lf)

This isomorphism is fixed by the property that it is equal to the usual duality over S - B.

Proof. The first statement follows from what has been said so far. For the second statement, note that taking Hom into Os kills the torsion and takes the dual of the locally free part, and that the Ext1 term is torsion. We leave to the reader the horrible job of checking that all of these isomorphisms are compatible with the Higgs fields on the direct images. O

COROLLARY 3.5. If all the components of D occur with multiplicity 1, then

H7z (X/S, E*, log D, lf) - HDO1(X/IS E, log D, lf)

Proof. In this case, D = Dred.

This corollary indicates that if all of the components of D have multiplicity 1, then H'DO(X/S, E, log D) might be the good direct image. We will see in the next section that this is the case and, more generally, that the filtered regular Higgs bundle provided by Proposition 3.3 is the good direct image for any multiplicities of components of D.




4. Direct images of harmonic bundles

In this section we will discuss direct images of harmonic bundles. For basic existence theorems, refer to [4], [9], [15], [22]. For notation and properties, we refer again to [25] and, in the case of noncompact curves discussed later on, we refer to [23].

Suppose that X and S are smooth quasiprojective varieties and f: X -- S is a smooth projective morphism. Suppose that W is a harmonic bundle on X. This comprises an underlying C' vector bundle, which we denote by W, a local system given by a flat connection D acting as an operator on W, and a Higgs bundle given by an operator D" on W, such that there exists a harmonic metric relating these two operators (see [25]). We will suppose that a harmonic metric K is fixed. The hermitian metric K may be thought of as a bilinear pairing

K: WxW-*C?(X),

where W denotes the complex conjugate C' bundle. The complex conjugate bundle has a section w whenever w is a section of W, and the correspondence w w-4 UP is C-antilinear. Let D' = D - D" and define an operator D" by multiplying the (1, 0) part of D" by -1. Thus, if D" = a + 0, we have D"= O-0. We can now write the formula stating that the metric K relates the operators D" and D:

K(Dcv, w) + K(v, D'w) = OK(v, w) for sections v and w of W. We can rewrite this formula in another useful way by contracting with a vector field a. Let Co, be the vector field obtained by multiplying the (1, 0) part of of by -1. Denote by -f the complex conjugate vector field. Then the above formula is the same as

K(D"v(Co),) I) + K(v, D'w(ff)) = OK(v, Uw)(f).

The pairing K can be extended to a pairing between W-valued and W-valued forms, with the value of the pairing being a scalar-valued form. This is done by wedging together the forms and applying the pairing K to the coefficients. Note that we can also use the notation A for a form with coefficients in W, when f is a form with coefficients in W. This is antilinear in the sense that if f = wr for a section w and a scalar-valued form 77, then

W = qIt, where r1 is the complex conjugate form. If a and f are W-valued forms of degree i and j, respectively, then we get

K(a, t),

a scalar-valued i + j form. We now turn to the question of direct images. The direct image of the

local system W is a local system V on S whose fiber at s E S is HZ(X8, W, D).



370 C. SIMPSON

Since f is topologically a fibration, these groups all have the same dimensions. The restriction of W to X, is a harmonic bundle, so we have an isomorphism of cohomology groups

RDO1 (XS I WI D) HDO (XS I WI D")

(see [25]). In particular the Dolbeault cohomology groups all have the same dimension, and they fit together into a vector bundle HDO1(X/S, W, D"). This has a structure of a Higgs bundle on S, as described in the previous section. The fiber-by-fiber isomorphism between cohomology groups is a C' isomorphism (because the C' structures can be defined by the condition that harmonic representatives vary smoothly, and the spaces of harmonic forms are the same on both sides). Thus we get a C' isomorphism between the local system (V, D) and the Higgs bundle HDO1 (X/S, W, D"). To simplify notation we will say that we have an operator D" on V giving a structure of a Higgs bundle- this operator may be described analytically, as was done in the previous section, and the operator D may be described similarly.

One can also construct a natural metric on V. This depends on choosing a Kdhler metric w on X, which induces Kihler metrics on the fibers X, An element of a fiber V, is a cohomology class [a,] in either de Rham or Dolbeault cohomology of the harmonic bundle W on X, The de Rham and Dolbeault cohomology groups are represented by the same space of harmonic forms (see [25]) (which gives the isomorphism between them). We may assume that a, is a harmonic form. Then let I[a8]12 [ las IL2 be the L2 norm of the form a, on X, (made using the Kihler metric chosen above).

PROPOSITION 4.1. The L2 metric on V relates the operators D and D" in the required way so that (V, D, D") is a harmonic bundle and the L2 metric is a harmonic metric. This is called the ith higher direct image of the harmonic bundle W.

Proof. This proof depends on an expression for the L2 metric by means of the Riemann bilinear relations. We first need to discuss the Lefschetz decomposition. In the de Rham or Dolbeault cohomology H'(XS, W) of the harmonic bundle W, there is a subspace of primitive cohomology P'(XS, W), represented by primitive harmonic forms. The spaces of primitive harmonic forms are the same in both cases, so the primitive subspaces correspond under the isomorphism between de Rham and Dolbeault cohomology. We have decompositions

Hi (X, W) = Gw&k A pi-2k(X, W). k

Again the decompositions occur at the level of harmonic forms and are the same in both cases. Furthermore the decompositions are orthogonal for the




L2 norm on harmonic forms, and the morphism a O-* wk A a from pi-2k to Hi is an isometric inclusion.

The union of subspaces of primitive classes forms a C' subbundle of the bundle of cohomology groups, as s varies in S. Denote it by P (X/S, W).

We claim that P (X/S, W) is preserved by the operators D and D" and that the morphism pi-2k(XIS, W) V = HI(X/S, W) is compatible with the operators D and D". This is because the space of primitive classes Pi (X/S, W) can be defined as the kernel of the operation of wedging with w +l; the morphism is given by wedging with a power of w. We show that the operation of wedging with w is compatible with D' (the proof for Ds being similar). Note that 0(w) = 0, so D"(a A w) = D"(a) A w. From the formula in the analytic calculation of D' we have

Ds[a, A Aw,](u) = [( I D"a) Ix. A ws + (D" a) A (u L w) I1x]. But since D" a I1x, = 0, the second term vanishes and we get

D" [a, A w8] (u) = D" [a,] (u) A [w8],

as required. It follows that the Lefschetz decomposition is a decomposition of (V, D, D") into bundles with operators, isomorphic to the bundles of primitive cohomology.

The L2 metric is the orthogonal direct sum of the metrics induced from the primitive cohomology. So it suffices to prove that the operators D and D" are related by the L2 metric on the bundle of primitive cohomology classes PI(X/S, W).

Use the Riemann bilinear relations. These say that for primitive i forms, a and A, we have

(a,3 )L2(Xs) = (constant)J K(Ca, A wnZ-,

where Ca denotes the form obtained by multiplying the parts of a of type (p, i - p) by (-1)P. The inner product of two cohomology classes is given via their harmonic representatives in the above formula. The constant depends only on the dimension of X, and the degree of the forms involved.

In fact, to get the inner product of two cohomology classes, it suffices to note that one of a or 0 is a harmonic representative and the other is any representative of its Dolbeault or de Rham cohomology class. This is because of the formula

K(D'Cv, T) + K(v, Dw) = DK(v, I),

which holds also for forms and implies that

K(C(D"a) I/) + (-1)deg(a)K(CaD'0) = OK(Ca 3).



372 C. SIMPSON

By the hermitian symmetry of K we also have

-K(C(D'a),/) + (-1)deg(a)K(Ca, D110) == OK(Ca,/ )

(the extra minus sign comes from moving the operation C). If a is harmonic, then D"a = 0 and D'a = 0; changing 0 by D"(v) or D(v) = D'(v) + D"(v) then changes the integrand in the Riemann bilinear relation by 0(u1) + O(U2),

where ui and U2 are forms of degree 2n - 1. The integral is left unchanged, as claimed.

A similar point is that the formula of the Riemann bilinear relations can be used to take the inner product of two cohomology classes if only one of the forms is primitive. This follows from the orthogonality of the Lefschetz decomposition.

We can now proceed with the proof of the proposition. Suppose that a and 0 are forms on X whose restrictions to the fibers X, are primitive harmonic forms. These correspond to sections [a] and [0] of the bundle of primitive cohomology P (X/S, W). Suppose that as is a vector field on the base, with any lifting to a vector field denoted by of on X. The operator giving the Higgs-bundle structure on V contracts with the vector field as, according to the formula

D" [a] (as) = [l I D"a],

where the brackets on the right indicate the class in Dolbeault cohomology of the enclosed; similarly the flat connection on V is given by the formula

D[a]](as) = [flIDa],

with the brackets indicating the class in de Rham cohomology of the enclosed. The operator D' is by definition the difference of these. By the linearity of the above formulas, D'[a](as) Jx8 is represented by a harmonic form that differs from of I D'a by addition of D(ul) + D"(u2) for forms u1 and U2. In particular, when we take the inner product of D'[a] (as) Ix. with a cohomology class represented by a harmonic form, we may use the formula of the Riemann bilinear relations. In order to prove the proposition we must verify that

(D" [a] (Cos ), []) L2 + ([a], D'[0] (0)) L2 - ([a], [/]D)L2(0).

Evaluate this at a point s E S. Using the above formulas and the Riemann bilinear relations turns the left-hand side into

(constant) J [K(C((Co,) I D"a) ),f ) + K(Ca,-i1 I D'3)] A Wn-i

= (constant) J [K(o I DC(Ca), +) ? K(Ca, a I D) A An-i v~~~




The general formula uL(aAb) = (a a) Ab+ (-1)deg(a)aA (al b) shows that the previous formula is equal to

(constant) J [i I (K (Dc (Ca)vf) + K (Ca, D) )] ni

plus two terms that vanish, because D" (Ca) Jx8 = 0 and D' Jx8 = 0. The harmonic property of K gives

K(D'c(Ca), 3) + K(Ca, D'/) = oK(Ca, /). Our formula is now equal to

(constant) J c1L(K(CaI) A wn-) + (OK(CaR3)) A (a w, a)

The restriction of OK(Ca, t3) to X, is equal to 0, as it is a sum of two terms- one involving D"a and the other involving D'o which restrict to 0 on X, because of the assumption that a Ix8 and 0 Ix8 are harmonic. Therefore

(constant) J (0K(Ca, )) A (o I 1nwi) = 0.

Now our formula is equal to

(constant) | I O(K(Ca,) Awn i),

which in turn is equal to

((constant) | K(Ca,3) A n-i) (us).

This is equal to the right-hand side of the equation that was to be verified. This completes the proof of the proposition. O

COROLLARY 4.2. Suppose that X and S are smooth projective varieties and f: X -> S is a smooth morphism. Suppose that W is a semisimple local system on X corresponding to a Higgs bundle E. Let V = R'f.W denote the higher direct image of the local system. Then V is semisimple, and the Higgs bundle on S that corresponds to V is the higher direct image HDO1 (X/SI E).

Proof. The higher direct image of the harmonic bundle corresponding to the local system W is a harmonic bundle with an underlying local system V. The Higgs bundle underlying this harmonic bundle is by definition the higher direct image of E. The fact that V underlies a harmonic bundle implies that V is semisimple. O

Remark. In the same situation, the argument shows that if E is a polystable Higgs bundle with vanishing Chern classes on X, then the higher direct



374 C. SIMPSON

image HDO1(X/S, E) is a polystable Higgs bundle on S with vanishing Chern classes.

Poincare duality. For the arguments below we need to discuss the relationship between the L2 metric and Poincare duality. Let W* denote the dual harmonic bundle on X (with the induced dual of the chosen metric). We have a nondegenerate pairing of local systems

Rzf*W 0 R2n-ifW* C

by taking a wedge product of forms and integrating along the fiber. This pairing induces an isomorphism

R2n-if*W* (Rif*W).

LEMMA 4.3. This isomorphism induced by Poincare duality is an isometry with respect to L2 metrics on the higher direct images possibly modified by multiplying by universal constants on the different pieces of the Lefschetz decomposition.

Proof. This is a pointwise statement, so we may restrict our attention to one fiber X, From the Lefschetz decomposition, the L2 metric is a direct sum of isometric copies of the restrictions to primitive cohomology. Similarly the Poincare-duality pairing is isomorphic to a direct sum of copies of pairings of the form

4: Pi(X8, W) 0 Pi(X8, W*) C

obtained by wedging the forms together with wAn- and integrating. It suffices to prove that this pairing gives an isometry between P (X8, W) and Pz(X,, W*)*; in other words, for a primitive harmonic i form, a, that

a -

(constant) sup I' (a,f ) I .

We can consider the harmonic metric on W as a morphism A: W -> W*. It is an isometry. The harmonic condition for the metric on W implies that D"W(C0) = W(CD'0) and D'W(Co) = -W(CD"10) (these are equivalent to the formulas given previously in terms of the pairing K). We also have Ayp(C)3) = -W(CAO3). By the Kdhler identities, this implies that if 3 is harmonic, then W(Co) is harmonic. The Riemann bilinear relations give

(ax, ) = (constant)4(a,so(C~)).

The signs of the constants alternate with i, and the absolute values might, as far as I know, be different for different parts of the Lefschetz decomposition (if




not, one could improve the statement of the lemma). The s(C/) run through all harmonic forms when the f do so. Now

(constant) I sup I4 (a, ( (C ))I = sup I(a(fl)I

= inf A1

= l.

The first line is from the Riemann bilinear relations; the second results because 0 p(C$) is an isometry; and the third is a property of inner products. This completes the verification of the lemma. I

Nonsmooth morphisms. Corollary 4.2 is not directly applicable for our purposes, since we must deal with families that are not smooth or, equivalently, with a base S that is quasiprojective, but not projective. Some further work is needed. For this we must refer to [23] for notation and properties of tame harmonic bundles. Because the analysis there is restricted to the case of quasiprojective curves, we must make the same restriction on our base.

PROPOSITION 4.4. Suppose that X is a smooth projective variety and W is a harmonic bundle on X (with chosen harmonic metric). Suppose that S is a smooth projective algebraic curve and f: X -- S is a morphism. Let So C S be the open set over which f is smooth and let Xo C X be the inverse image. Let V be the ith higher direct image of the harmonic bundle W Ixo, with the L2 harmonic metric described above. Then V is a tame harmonic bundle on So, and the corresponding filtered local system has a trivial filtration (with its jump at 0).

Proof. This proof is essentially taken from Clemens's article [3]. The question is local near a singularity s E S - So. We will show that if {v(y)} is a flat section of V defined on a sector of the punctured disk centered at s, then for any E > 0 there is a constant such that

Iv(y) IL2 < (constant) IyI.

The same will then hold for the higher direct images of W*. So by Lemma 4.3 we get the same estimate for the norms of flat sections of V* with respect to the dual metric. By [23] this implies that the harmonic bundle is tame (flat sections have polynomial growth) and that the corresponding filtration on the local system (by order of polynomial growth) is trivial, centered at 0.

The statement of the estimate seems to depend on the choice of the L2 metric, which depends on the choice of the metric on X. However any two harmonic metrics on a harmonic bundle are mutually bounded. This is



376 C. SIMPSON

because the connection 0 + 0, which can be constructed from the operators D and D", is unitary with respect to any harmonic metric. Two harmonic metrics are mutually bounded at one point and, by translation via the unitary connection, they are mutually bounded everywhere. Note that this argument makes no reference to the compactness of the base.

In view of this we may resolve the singularities of the family f: X -- S and use a Kihler metric on the resolution. We obtain a new variety X mapping to X such that the map is an isomorphism over Xo and the singular fibers of X -) S are divisors with normal crossings. The harmonic bundle W pulls back to a harmonic bundle on X with the same restriction to X0 (and hence the same higher direct image V on So). The L2 metric on V induced by a metric on X is harmonic, and it suffices to prove our estimate for this metric. Thus we may replace X by X; so we may assume that the singular fibers of f are divisors with normal crossings.

Choose a point yo in the sector. Suppose that we have constructed C' diffeomorphisms Oy: Xyo -- Xy, equal to the identity for y = yo and varying smoothly with y in the sector. Suppose that these are lifted to isomorphisms from W Jx,0 to W Ix,. Choose a C' form a on Xy, with coefficients in W, representing the cohomology class v(yo). Then the form Oy* (a) represents the class v(y) on Xy. The L2 norm of the cohomology class is less than or equal to the L2 norm of any representative (since the harmonic representative is the one with the smallest norm). Therefore, to obtain our bound, it suffices to give a subpolynomial bound for

I I0Y'* (C) 1

We will choose the diffeomorphisms y and give bounds for the L2 norms in the neighborhood of a point x in the singular fiber Xs. These diffeomorphisms can then be glued together to a family of diffeomorphisms defined globally, by deforming them slightly at the boundary of the neighborhood and gluing them with a partition-of-unity argument. The local estimates are unchanged by the deformation and add up to give the required global estimate. This is discussed in [3] and we will not go into further details.

We can choose a coordinate t on S with t(s) = 0 and t(yo) = 1/2. We can also choose coordinates zo, ... , Zn for X in the neighborhood K of x, with zi(x) = 0, such that the function f: X -- S is given by

z ao... zant 0 n

The ai are positive integers, at least one of which is strictly positive. The picture is a product of the picture, involving the coordinates with ai > 0, and a trivial picture (with no change in norms of the differential forms under




the deformations) in the directions of coordinates with ai = 0. The desired estimate is conserved under the taking of a product with a trivial situation; so we may (and will) assume from now on that all ai > 0.

Write t = peir and zi = rieki in polar coordinates. The equation of the family splits into two equations,

rao ... ran = p 0 nl

and aofo + * = 71.

Let ENt denote the intersection of the fiber Xy (for t = t(y)) with the neighborhood. We may assume that our neighborhood is a polycylinder, given by ri < 1. Let M denote the region ri < 1 in (IR+)n+l and let Mt denote the hypersurface given by the first of the previous two equations. Choose a map OMt: M1/2 -A Mt such that fMt preserves the boundaries of M and, on the boundaries, coincides with the corresponding choice in a smaller dimension (this will allow the local maps to be approximately patched together). Then define

'Ot : A'/2 -E t

by ,ot (ro I .. *, rni 60 , ()=(,mt (rol .. * * rn), I 0 + 7q/ao,**,

The metric on Xt is comparable to the restriction of the standard metric involving the coordinates zi. We may also choose a flat trivialization of W over jK and note that the harmonic metric is bounded with respect to a constant metric. Hence we may work with the standard metric on AT/ and the constant metric on a trivial flat bundle of coefficients W.

We can assume that the maps fMt have uniformly bounded distortion as t approaches the origin. Hence we may ignore the distortion in the Mt direction. The remaining distortion is in the angular direction of the (i. The norm of a differential form goes as the inverse of the product of the distortions in the directions involved. The maximum distortion of a differential form at a point (r, () in the image of ot is therefore bounded by the inverse of the volume of the torus given by fixing the value of ri and letting (i vary according to the second of the above equations. We can think of Et as fibered over Mt with the fibers being these tori. Let Vol(r) denote the volume of the torus at the given value of r = (ro, .. ., rn). The square of the maximum distortion of the differential form, times the volume of the torus, is equal to Vol(r)-1. The volume of the tori are approximated by

Vol(r) sup rjiro rn



378 C. SIMPSON

(this approximation becomes a bound when appropriate constants are in- serted). From these facts we get

J kbt,*(a)I2 < (constant) Jt(infri)(ro rn)-<dVol(Mt).

For i = 0, ... , an let Mti denote the part of Mi where ri is the smallest of the ro,..., rn. On Mti, the form dro... dri ... drn approximates (to within a bounded multiple) the volume form. Hence our integral is bounded by a constant times

n ?ZJ d log ro.. d log ri ..dlogrn. i=O ,i

For simplicity we may consider just the i = 0 term. The region Mt,o is given in the coordinate system ri, ... , rn by

rj < 1, j= 0, . .., n, rj? > Pr -a, .r-an j = 0O,... ,n.

Introduce coordinates sj = - log rj and a = - log p. Then the region Mt,o is given by

Si >O., a - als -* *-ansn>Si

ao

The integral in question is the volume of this region in the euclidean metric of the coordinates si. The second equation implies in particular that si < a/aj, so the volume is bounded by a constant times an. Hence the integral in question is bounded by a constant times I log pen. This provides the estimate we are looking for, hence concludes the proof of the proposition. O

Remark. In the situation of the proposition, let E denote the Higgs bundle associated to W and let F denote the filtered regular Higgs bundle on S associated to the local system of V with trivial filtrations. The restriction F Is0 is the Higgs bundle underlying the higher direct-image harmonic bundle V so that F Iso = HbiD0(Xo/So, E). Note that we have not obtained a description of F on S and, in fact, we have not even obtained a description of the algebraic structure; so the isomorphism F IsO - HbiD(Xo/So, E) is at this point only analytic.




COROLLARY 4.5. Suppose that X is a smooth projective variety and W is a harmonic bundle on X (with a chosen harmonic metric). Suppose that S is a smooth projective variety of any dimension and f: X -> S is a morphism. Let So C S be the open set over which f is smooth and let Xo C X be the inverse image. Let V be the ith higher direct image of the harmonic bundle W Ixo. Then the local system underlying V is semisimple.

Proof. Suppose that S' C S is a curve that is a general complete intersection of hyperplane sections. Let X' = X xS S' (it will be smooth already, so there is no need to resolve singularities). Since W Ix' is a harmonic bundle, the proposition above applies, showing that the local system V Is/, when given trivial filtrations at the points of S' - S6, underlies a tame harmonic bundle. In particular the filtered local system is a direct sum of stable filtered local systems of degree 0. When the filtrations are trivial, the stability condition is equivalent to irreducibility of the local system. This implies that V Is/ is semisimple. Since 7rn(So) -+ 7ri(So) is surjective, the local system V is semisimple. C]

COROLLARY 4.6. Suppose, in the situation of the previous corollary, that the local system of V extends to a local system Vs on S. Then the restriction to So of the harmonic bundle corresponding to Vs is the same as the higher direct-image harmonic bundle V. If E denotes the Higgs bundle corresponding to W and F denotes the Higgs bundle corresponding to Vs, then there is an analytic isomorphism F IsO HbDOI(Xo/SoI E).

Proof. Suppose first that S is a curve so that we are in the situation of the proposition. The restriction of the harmonic bundle corresponding to Vs to So is also a harmonic bundle corresponding to the local system of V with a trivial filtration. By the uniqueness result of [23], this restriction coincides with the higher direct-image harmonic bundle V. In particular the restriction of F to So is the Higgs bundle underlying V, namely HDOI(Xo/SoI E).

Now consider the general case where S has any dimension. The previous corollary implies that Vs is semisimple, so it corresponds to a harmonic bundle, which we also denote by Vs. Suppose that S' c S is a smooth curve that is a complete intersection of hyperplane sections, sufficiently general that X' = X x s S' is smooth. By the statement for the 1-dimensional base of the previous paragraph, the harmonic bundle Vs Is/ is the higher direct image of the harmonic bundle W Ixo. But formation of the higher direct image of a harmonic bundle is compatible with restriction (the higher direct image of the local system certainly is, and the L2 metric is defined pointwise these



380 C. SIMPSON

facts suffice, though one can also see directly that formation of the operator D" on the higher direct image is compatible with restriction). Therefore Vs IS, = V I s. Note that the isomorphism on the level of local systems is the tautological one; so this statement says that D" IsO/ and D" Is/ are the same. But for any point s E So and tangent vector a E T(S)8 there is a sufficiently general curve S', as above, which passes through s and whose tangent space contains the vector a. If v is a section of V defined near s, we get D" (v)(a) = D" (v)(a). Hence D" = D" , which shows that the restriction of the harmonic bundle Vs is the harmonic bundle V. D

Filtered regular Higgs bundles. We will show that the algebraic constructions given at the end of the previous section serve to identify the filtered regular Higgs bundle associated to the higher direct image of a harmonic bundle. In particular we will prove Proposition 3.3 and show that the filtered regular Higgs bundle F constructed there is the one that corresponds to the higher direct-image harmonic bundle. Again this generalizes some things in Hodge theory (work of Schmid and Steenbrink) for the case of constant coefficients.

Suppose that f: X -- S is a morphism, with X smooth projective and S a curve. Define divisors B C S and D = f-1(B) C X, as in Section 3, and suppose that D is a divisor with normal crossings. Suppose that W is a harmonic bundle on X, with a corresponding Higgs bundle (E, 0). Let V be the ith higher direct-image harmonic bundle on S - B. As noted above, it is a tame harmonic bundle. The associated Higgs bundle is, by definition, HDO1((X-D)/(S-B), E). Fix a harmonic metric K on E and a Kdhler metric on X. Then the Higgs bundle H 1((X -D)/(S - B), E) has an L2 metric, which can be given by taking the minimum of the L2 norm of representatives for the analytic Dolbeault cohomology classes in the fibers X,

Transport the L2 metric by the isomorphism

HDO1 ((X - D) / (S - B), E) - H01 (X/S, E, log D, lf) I S-B

to obtain a metric L. It may be considered as a singular metric on HDO1(X/S, E, log D, lf) with singularities over B.

PROPOSITION 4.7. Suppose that f is a holomorphic section of Hz 1(X/S, F, log D, lf), defined near b E B. Let t be a local coordinate vanishing at b. Then for any ? > 0,

If IL < ItI_-

for small values of ItI. Proof. This is similar to Proposition 4.4. Let X be a neighborhood of b in

S, and let M = f-1(K). Then the holomorphic section f can be represented




as a class in the cohomology of Qj /S(E, log D) on M. This cohomology can be calculated by the Dolbeault resolution, namely by a complex of the form

@ Aoq(M, QX/S(E, log D)) p,q

with the differential given by a+ 0. In particular f can be represented by a form

71 E @ A, Q(MI QP S (E, log D)) . p+q=i

The restriction of f to each fiber X, (s E X-b) is a representative for the analytic Dolbeault-cohomology class corresponding to f(s) E HbDOI(X8, E). In particular the norm of the cohomology class is less than or equal to the L2 norm of 71 Ixx. This norm is calculated using the harmonic metric K on E and the Kdhler metric chosen for X (restricted to X8). However we can use a partition-of-unity argument and a collection of local trivializations of E, as well as standard metrics on coordinate charts. The resulting estimate will be comparable with the true norm. Choose a coordinate patch P C X near z E D, with coordinates xo, . .. , xn and a coordinate t on S, so that the map is given by t = x0 ...xk as usual. Then we can write

/= YiJ dxi A d-J

with dj = dxjj A/\. A dxjq and

dxj = Ai, / A /\ Tdxzi+1 A /\ dxip

where iR < k for u < r and iu > k for u > r. The coefficients YIJ are C? sections of E. We obtain an estimate of the form

/17712 < C J dxj12.

On the other hand, the same type of argument as given in the proof of Propo- sition 4.4 shows that, for any ? > 0,

I idXI12 < et(S)t-e

for s near b. This provides the required estimate. C



382 C. SIMPSON

COROLLARY 4.8. Let G = {G.>} denote the filtered regular Higgs bundle corresponding to the tame harmonic bundle V. Then the analytic isomorphism

HzbOG(X/SFlogD,if) IS-B - C IS-B extends to an injection of coherent sheaves

HDO1(X/S, E, log D, If) C Go.

In particular the isomorphism HbO1(X/S, F, log D, if) IS-B -G IS-B is meromorphic at B.

Proof. This follows from the characterization of Go as the sheaf of sections g with growth II < ItJ-" for any ?> 0 (see [23]). The morphism is injective, because it is an isomorphism on an open set, and the coherent sheaves have no torsion. For the last statement, note that a morphism of coherent analytic sheaves is algebraic. C]

The next step is to use duality to get an estimate for the dual metric L* on HbDOI(X/S, E, log D, If)*.

LEMMA 4.9. Suppose that v is bigger than any of the multiplicities of components of D and A is a holomorphic section of HDOI(X/S, E, log D, lf)* defined near b E B. Let t be a coordinate on S at b. Then, for any ? > 0, the estimate

AIlL < JtJ("-v)1v-

holds for small values of ItI.

Proof. Recall from the end of the previous section that

Hz n(X/S E,logD,lf) H~jli(X/S, E0 8 Ox(D - Dred), log D, If).

Over S - B this is the isomorphism given by Poincare duality. In particular, if we give E* the dual harmonic metric and Ox(D - Dred) the metric (singular along D) that corresponds to the constant metric on OX-D, then the isomorphism above takes the metric L* to the induced L2 metric on the right (up to modification by different constants on different parts of the Lefschetz decomposition-cf. Lemma 4.3). Employ the same technique as in Proposition 4.7 for estimating the norm of a holomorphic section. The only difference is that we have to take into account the norm of a C?? section of Ox(D - Dred). This is bounded (in a coordinate chart, by the same notation as usual) by a constant times

def S1-ak /3 = sup I XO Jlao ... JXk~k X00.. Xak = t, lxi |< 1 a0 ak

As ai < v, we have IXIai > Jxilv. Hence, on the region indicated, we have

IXoIv... IXkI <? It|,




and hence 3< ltI(l'-)/lv. We have to multiply the bound obtained in Proposi- tion 4.7 by /3. This gives the stated estimate. C]

COROLLARY 4.10. The injection of Corollary 4.8 is an isomorphism

HDO1(X/S, E, log D, If) = Go.

Proof. We show slightly more, namely that the left side contains GO for any ? > 0 and 3 > -1/v + 3? (in particular /3 = 0). Fix such ? and 3 and suppose that g E Gi and

A E HDO1(X/IS E, log D, If)*.

By the previous estimate we have

JAl < ltl|(' v1l-6

and also 191 < ItI,3-,F

Since (1-v)/v- + d -F > e-1, we have

JA(g)J < Itl.

But A(g) is a meromorphic function, and this implies that it is holomorphic. The fact that A(g) is holomorphic for all sections A of the dual bundle implies that g E HzDO(X/S, E, log D, lf). This proves the corollary. C]

COROLLARY 4.11. Suppose that the local monodromy transformations of the local system V at points b E B are unipotent. Then the filtered regular Higgs bundle obtained by setting FQ = HbDO(X/S,E,logD,lf) for -1 < a < 0 and extending by FQ+1 = tF, is naturally isomorphic (via the given isomorphism over S - B) to the filtered regular Higgs bundle G associated to the tame harmonic bundle V.

Proof. The jumps in the filtration of G are the arguments of the eigenvalues of the monodromy transformations of V. If the monodromy is unipotent, this means that the jumps occur only at integers. The previous corollary shows that F = G. C]

PROPOSITION 4.12. The filtered regular Higgs bundle F, defined before Proposition 3.3 in the previous section, is naturally isomorphic (via the given isomorphism over S - B) to the filtered regular Higgs bundle G corresponding to the tame harmonic bundle V. Also, Fo = HDO1(X/S, E, log D, If).

Proof. Let v be the least common multiple of the multiplicities of the components of D (as in Proposition 3.3). Suppose that b E B and let s be a



384 C. SIMPSON

nearby point. There is a map ir: X- Xb, compatible with the monodromy transformation y (of letting s go around b). The fibers of ir are disjoint unions of tori permuted by -y; but the number of components in each orbit under the action of -y divides v. Consideration of the Leray spectral sequence for ir shows that the monodromy transformation acts quasi-unipotently on the cohomology of Xs, with eigenvalues that are vth roots of unity. This classical argument works for cohomology with coefficients in W, since W Ix. ir*(W Ixb).

From this it follows that if p: S' -> S is a ramified covering with ,t(b') dividing v, as in the definition of F, then the monodromy transformations of p* V around b' E B' are unipotent. By Corollary 4.11 the filtered regular Higgs bundle F', defined by giving HDO1(XI/S', E', log D', lf) the trivial filtration with a jump at a = 0, is isomorphic to the filtered regular Higgs bundle G' corresponding to the tame harmonic bundle p*V. The definition of F says exactly that p*(F) = F'. Formation of the filtered regular Higgs bundle associated to a harmonic bundle is also compatible with pullback (cf. [23]), SO G' = p* (G). Since knowledge of the pullback of a filtered regular Higgs bundle serves to characterize the filtration, we get F = G. C]

Remark. We can now prove Proposition 3.3 by noting that all of the required properties hold for G (see [23]). C]

If we call F = Hb01(X/S, F) the ith higher direct image of E, then we can paraphrase Proposition 4.12 as saying that the filtered regular Higgs bundle corresponding to the higher direct image of a harmonic bundle is the same as the higher direct image of the Higgs bundle that corresponds upstairs.

We have restricted our attention to the case where S is a curve for lack of a good theory of filtered regular Higgs bundles in higher dimensions. However we can obtain a statement in the case where this is not needed.

THEOREM 4.13. Suppose that f: X -- S is a morphism between smooth projective varieties and suppose that there are normal crossing divisors D C X and B C S such that D = f-1(B) and f is smooth outside D. Let W be a harmonic bundle on X and V be the ith higher direct-image harmonic bundle on S. Suppose that V extends to a harmonic bundle Vs on S. Let E be the Higgs bundle corresponding to W and Fs be the Higgs bundle corresponding to Vs. Then

Fs = HDO1 (X/S, E, log D)**

Proof. The double dual of a coherent sheaf is the universal reflexive sheaf into which it maps. Note that we have an analytic isomorphism between F and HDO1 (X/S, E, log D) over S - B. For any smooth curve S, C S set B1 = B n S1, X1 = X x s S1 and D1 equal to the inverse image of B1 in X1 . Choose a general




family of smooth complete intersection curves S1. For an open subset of curves in the family, X1 will be smooth and Di will have normal crossings, and

HDO1(Xi/S1, E, log Di, lf) = HDO1(X/S, E, log D) Isl

In any case, Fs Is, will be the Higgs bundle associated to the extension of the higher direct image V Is,. In other words, if we provide Fs Is, with a trivial filtration at the jump a = 0, then it becomes the filtered regular Higgs bundle associated to V Is,. Corollary 4.11 implies that

Fs s, = HD01 (Xi /S,, E, log D1, lf) = HDO1(X/IS E, logD) Isle

Let U C S be the subset where HDol(X/S,E,logD)** is locally free. Our isomorphism between this locally free sheaf and Fs, over U - B, may be considered locally as given by a matrix of analytic functions with singularities along B. We know that this matrix (and its inverse) are holomorphic when restricted to a family of smooth curves (which we may assume are transverse to B). A version of Hartogs's theorem implies that these matrices and their inverses are holomorphic on U. Hence Fs and Hb01(X/SFlogD)** are analytically isomorphic over U. The complement of U has codimension at least 2, so sections of HDOI(X/S, E, log D)** (which become analytic sections of Fs defined on U) extend to sections of Fs. Thus HDO1(X/S, E, log D)** is a reflexive coherent subsheaf of Fs, and they are equal outside codimension 2. This implies that they are equal, which proves the theorem. C1

Remark. Suppose that the hypotheses of the theorem hold, except that B and D are not necessarily divisors with normal crossings. Choose a resolution of singularities X' -+ 5' so that the corresponding divisors have normal crossings. Let p: S' -> S denote the (birational) projection. If V extends to a local system Vs on S, let Fs be the corresponding Higgs bundle. Let E' be the pullback of E to X'. Then p* (Fs) is the Higgs bundle on S' corresponding to the extended higher direct image. The theorem gives

p*(Fs) = H 01(X'/S', E', log D)**.

Since p is projective with connected fibers, we get

Fs = p* (Hb01 (X'/S', E', log DU)**).

Thus Fs can be constructed algebraically.



386 C. SIMPSON

5. Higher direct images of nonstationary families by Lefschetz pencils

Suppose that Z is a smooth projective variety of dimension n + 1 with a very ample invertible sheaf L. Let pN denote the projective space of lines in HO(Z, 1). For each s E pN there corresponds an n-dimensional hyperplane section X, C Z. Let U c pN denote the Zariski-open subset of s such that X, is smooth.

Let Xu C U x Z denote the incidence variety of (s, z) such that z E X, Let a: Xu- Z and f: Xu -- U be the two projections. The morphism f is smooth. For any local system W on Z let V denote the nth higher direct image on U,

V = Rnf*(a*W).

If {Wt}tET is a family of local systems on Z such that the Vt have the same ranks, then the Vt fit together into a family {Vt}tET of local systems on U.

We say that L satisfies the second-order condition if, for any z E Z and any u E mZ/m3 (where mz denotes the maximal ideal in Ozz), there exists a section v E HO(Z, L) such that v is equal to u times an invertible section in L 0 OzZ/m3 and such that the zero set of v is smooth outside z.

THEOREM 5.1. Suppose that L is very ample, and ample enough to satisfy the second-order condition. Suppose that {Wt}teT is a nonstationary family of semisimple local systems on Z, indexed by an irreducible variety T. Sup- pose that the nth higher direct images Vt form a family of local systems on U (in other words, they have the same ranks for all t). Then {Vt}tET is a nonstationary family.

Proof. Let r denote the rank of the Wt. Since MB(Z, r) is affine, the image of the map T -> MB(Z, r) is not contained in any compact set. There is a sequence ti E T, which eventually goes outside any compact set in MB(Z, r). Let { (Ei, 6i) } be the family of Higgs bundles corresponding to the Wti. Let Pi(x) be the monic characteristic polynomials of Oi (the coefficient of xr-j is a section of Symi Q1). Let P be a compact set in the affine space of such polynomials, not containing the trivial polynomial xr. Write

Pi(x) = A Qi(A71x)

for polynomials Qi E P. The set of points in MB(Z, r) corresponding to Higgs bundles, with a fixed bound for the characteristic polynomial, is compact (cf. [27] or [26]). Therefore Ail -+ oc for our sequence of representations.

Since P is compact, we may go to a subsequence so that Qi -- Q,,. For each i let Pi C T*(Z) be the subvariety defined by the equation Qi. Then




Ei = Ai Ti is the spectral variety of Ei-in other words, the support of the corresponding coherent sheaf Si on T*(Z).

Let X = Xu. We have the natural morphisms

j: a*T*(Z) T*(X)

and b: a*T*(Z) T*(Z),

the first being a morphism of bundles over X and the second covering the morphism a: X -> Z. Let <>i = j(b-'(Fi)) c T*(X). Note that j*(b*Si) are the sheaves on T* (X) corresponding to the Higgs bundles a* (Ei) on X. Hence the spectral variety of a*(Ei) is j(b-1(>i)) = Ai(i. Let D = f*T*(U) C T*(X) with the morphism g: D -- T*(U). We state the next step in the proof as a lemma.

LEMMA 5.2. There exist s E U and y E T*(X) Ix. such that y is an isolated point in the intersection (VD,,) nD, and g(y) is not in the zero section of T* (U).

Proof. Choose a general point x in TI', not in the zero section of T* (Z). Let z denote the image in Z. Then there is a usual open neighborhood X of x such that TI' nOX is a section of T* (Z) corresponding to a 1 form ,B defined on the usual open neighborhood of z.

We claim that there exists s E U such that z E X, and such that the restriction ,B Ix. has an isolated zero at z. First note that ,B is closed. To see this choose a smooth projective variety, surjective over T'1, and pull back the canonical section of T*(Z). This gives a smooth holomorphic 1 form, which on a smooth projective variety must be closed. This form is locally the pullback of so, so /3 is closed. Now, since /3(z) $, 0, on the appropriate usual neighborhood

we can choose coordinates uo, ... , un for Z with zi (z) = 0 and /3 = duo. By the second-order condition we can choose a section v E HO(Z, 1) such that the zero set of v is smooth, and near z we have

V = (uo + u2 + 0+ u 2+ )V1'

where vi (z) $& 0. Let s be the point in U corresponding to the section v. Note that u,... ,un form coordinates for X, at z, and we have

/3 Ixx = uldul + .

+ undun. Thus /3 has an isolated zero at z, which proves the claim.

Let (A = j(bT1 ('I n K)). It is an open subset in (DOO. We have a morphism of projection

q: T*(X) -T*(X/U),



388 C. SIMPSON

and D is the inverse image of the zero section. Since tI' n A( corresponds to the section /3, (DI corresponds to the pullback a*(3). In particular DI is one to one over its image in X and, hence, the restriction of q to (IV is one to one over its image. Hence (<Dv), n D8 consists of one point if and only if the intersection of q((J?1f),) with the zero section of T*(X/U) Ix8 consists of one point. But T*(X/U) x8 = T*(X8), and q((IDX),) is the section corresponding to the restriction (a*/) Ix8. By assumption this has an isolated zero at z (and by making the neighborhood JV small enough, we can assume that it has no other zeroes). Hence the intersection of q((IDX),) with the zero section of T*(X8) consists of one point, and so ((IV), n D8 consists of one point y.

Note that y is the value in T*(X) of a*(3) at the point (s, z). Since L is very ample, the linear system has no base points and the map a: X -> Z is smooth. The 1 form /3 does not have a zero at z (since, by assumption, x was not in the zero section of T*(Z)). Hence a*(3) does not have a zero at (s, z), and so y is not in the zero section of T* (X). This implies that g(y) is not in the zero section of T*(U), completing the proof of the lemma. C]

We continue with the proof of the theorem. Since the subschemes (Di approach (Doc, Lemma 5.2 implies the following result: For i > 0 there exist points yi that are isolated points of the intersections ((i), n D8, such that yi -- y. Since D is invariant by scaling, the Aiyi are isolated points of the intersections (Ai(i), n D8. As g(y) 7& 0 in T*(U)8, the norms of g(Aiyi) in T*(U)8 go to infinity.

Let F= HDO(X/U, a*Ei)

and let Fi denote the corresponding sheaf on T*(U). The spectral variety of a*Ei is Ai2Ji. We have Aiyi, which is an isolated point in the intersection (AiJ?i), n D8. Hence, by Corollary 3.2, the image g(Aiyi) is contained in the spectral variety of Fi. From the conclusion at the end of the previous paragraph, it follows that, as i -+ oc, the spectral varieties of Fi are not all the same.

Choose a generic IP1 C jpN and let Uo = U n IV'. By making the choice generic, we may assume that the spectral varieties of Fi Ju0 are not all the same. Restrict the family of hyperplane sections to a family fo: Xo -+ Uo. Note that

HnDO1(Xo/Uo, (a*Ei) luo) = Fi luo

Recall that Vi is the higher direct-image local system on U obtained from Wi so that

Vi jU0 = Rn(fo)*(a*Wi xo).




By Corollary 4.5, Vi luo is semisimple. Let viluar denote the tame harmonic bundle corresponding to the filtered local system given by Vi Iuo with trivial filtrations. Then, by Proposition 4.4, vilar is the higher direct-image harmonic bundle; it has Fi Ju0 for underlying Higgs bundle. Since the spectral varieties of Fj Ju0 are not all the same, this implies that the harmonic bundles ViHar iIU0 are not all the same, so the local systems Vi Iuo are not all the same. Since these local systems are semisimple, the corresponding points in the moduli space MA(lrl(Uo),r') are not all the same (here r' is the rank of the Vi). Since the moduli-space construction is functorial, the points in MB ( 1 (U), r') corresponding to the local systems Vi are not all the same. Hence the morphism T -+ MB(7r1(U), r') is not constant, and the family of local systems {Vt}tET on U is nonstationary. The proof of Theorem 5.1 is now finished. O

COROLLARY 5.3. Keeping the same notation as in the theorem, suppose that IPm C IN is a generic linear subspace (m > 1). Suppose that h: B u n IPm is a finite etale covering. Then the family of local systems {Vt lB = H*(Vt)}tET on B is nonstationary.

Proof. For a generic linear subspace, the Lefschetz theorems say that rni l(u m) - ri(U) is surjective. Hence the family of local systems restricted to U n IPm is nonstationary. Now Lemma 1.5 implies that the family of pullbacks to B is nonstationary. C]

COROLLARY 5.4. With the notation of the previous corollary, there are only finitely many semisimple local systems W of rank r on Z with a given nth higher direct image V IB on B.

Proof. The space of representations RZI (rn (Z), r) decomposes as a finite union of locally closed subvarieties Tj such that there are ri with rk(Vt) = ri for any higher direct image Vt of a local system Wt parametrized by a point t E Ti. We may (by refining) assume that this decomposition is compatible with the constructible subset of t such that Wt is semisimple. Consider one of the Ti with all Wt semisimple. The family of higher direct images {Vt IB}tET2 gives a morphism Ti -+ MB(lrl(B), ri). Suppose that T' C T is a connected component of a fiber of this map. The family of higher direct images for t E T' is stationary. By Corollary 5.3 the family {Wt}teT' must be stationary in other words, the morphism T' -> MBI(irj(Z),r) is constant. Since the Wt parametrized by t E T' are all semisimple, this implies that they are all isomorphic. There are only finitely many connected components in any fiber of Tj -+ MBI(7rn(B),ri), so there are only finitely many isomorphism classes parametrized by t E Tj that give the same higher direct image. As there are only finitely many Ti, this proves the corollary. C1



390 C. SIMPSON

Remark. In the theorem and subsequent corollaries we could weaken the hypothesis that L satisfies the second-order condition by putting in any hypothesis that makes the proof of Lemma 5.2 work. We have made the statements as they are above for simplicity.

In order to complete the proof of Theorem I of the Introduction we have to bound the rank of W in terms of the rank of V.

LEMMA 5.5. Given K. > 0 choose L to be sufficiently ample to get the following estimate: If W is a local system of rank r on Z, then the higher direct-image local system V on U has rank > ir.

Proof. Let X, denote a typical smooth hyperplane section in the linear system corresponding to C. Then

Z (- 1)'hz(XsI W Ixj) = r Z(-1)zhz(Xs, C).

On the other hand, the Lefschetz theorems give HZ(X8, W Ix) - HZ(Z, W) for i < n, and these determine HZ(Xs, W Ix) for i > n by Poincare duality. We can triangulate Z and calculate Hi (Z, W) by a complex of finite-dimensional vector spaces whose dimensions are proportional to r, but otherwise independent of W. Hence there is a constant 61, independent of L, such that

hn(X., W IX.) > r(hn(X8, C) - 61).

We estimate hn(X,, C) from below by the dimension of the top component in the Hodge decomposition, h0(X,, Q ). Taking the residue of differential forms with poles along X, gives an exact sequence

0 Qz X,

so that

ho(X8, Qn ) > ho(Z, Qnz+1 ?L) - ho(Z, Qn+l) - hl(Z, Qn+').

In particular, by choosing L to be sufficiently ample, we can assume that h0(X,, Qn ) > r, + 61. Then we have hn(X,, W Ix) ? > r, as desired.

This allows us to complete the proof of Theorem I. Fix i > 0 and choose a sufficiently ample L, as in the last lemma. Then, for a fixed V, we have the bound rk(W) < <-1 rk(V) for any local system W on Z with a higher direct image equal to V. Now Corollary 5.4 applies to show that there are only finitely many such W. O

Example. When Z is an abelian variety. We will return to the example discussed in Section 2, the case when Z is an abelian variety. We will describe more geometrically the picture of what happens with the higher direct image




of a rank-1 Higgs bundle with a nonzero Higgs field. It was considering this example that led to the Higgs-bundle methods developed in Sections 3-5.

A rank-1 Higgs bundle (E, 0) consists of a line bundle E and a holomorphic 1 form 0. Since End(E) = (Oz, it follows that 0 is a scalar-valued 1 form. We change notation and denote this form by a. It is a constant section of Q1 (92+1. We assume that a #A 0, which implies that a(z) 7# 0 at all z E Z. Suppose that f: Z - P1 is a general Lefschetz pencil from a sufficiently ample linear system. Here Z is the blowup of Z along the base locus. The fibers of f are hyperplane sections Xs, smooth for s in a Zariski-open U C JP1. We can assume that a Jx8 has only isolated simple zeroes (i(s),...,(k(s). These might be interchanged when s moves around inside U; but we can make U smaller so that the sections (i (s) are not ramified. Let R C Z denote the set of points (i(s) as s ranges in U. Then a IR is equal to a section of f*T*(U). We obtain a map g: R -- T*(U). The coherent sheaf F on T*(U) corresponding to the higher direct-image Higgs bundle F is

F = g*(E IR).

In particular the spectral variety of F is g(R). Over a point s E U there is one point in the spectrum for each zero (i(s) of a IR. These points correspond to eigenforms that can be calculated from a. (The (i (s) are points where the level sets of the functions f a and f are tangent and the eigenforms are essentially the Lagrange multipliers relating these two functions.)

6. Even-dimensional hyperplane sections

The results in this section were explained to me by J. Carlson. They are treated in [2] (and apparently they have been known for some time), but we give the details here for completeness.

Let Z be a smooth projective variety of odd dimension. Put dim(Z) = n + 1 with n even. Let L be a very ample invertible sheaf on Z. We will assume that L is sufficiently ample (cf. [10]). This means that a very ample Lo is fixed, and we assume that 1 ? fJ-1 is very ample. The Lo will be fixed according to our requirements as we proceed.

Let pN denote the projective space of lines in H0(Z, L). For each s E pN let f8 denote an element in the line corresponding to s and let X, denote the hypersurface defined by the equation f8(z) = 0. Let X C JpN X Z denote the incidence variety, consisting of the set of (s, z) such that z E X, Let ir: X __ pN and a: X -- Z denote the projections on the two factors. If y _pN is any variety mapping to JpN let Xy = X XpN Y. Keep the notation ir: Xy -- Y and a: Xy -- Z. Let D C pN denote the closed set of values



392 C. SIMPSON

of s, for which Xt is singular, and let U denote the open set, which is the complement of D.

Let 1p2 c jpN be a generic plane. Let Ui = U f 1P2 and D1 = Dn 1p2.

PROPOSITION 6.1. Assume that L is sufficiently ample. Then the following general position statements hold: The curve D1 C p2 is irreducible and has only nodes and cusps. A point s is a smooth point of D1 if and only if X, has a single ordinary double point. It is a node of D1 if and only if X, has exactly two double points. It is a cusp of D1 if and only if X, has exactly one singularity x; the defining equation f8 is given by

MsZ0,** Zn) = ZO 3 Z2 + ***+ Z 2

where Zo,.. ., Zn form an analytic system of local coordinates vanishing at x.

Proof. First we establish some notation. For s E jpN we have chosen a section f8 of L. For any point z E Z we can evaluate f8(z), the value being in the line 4Z. If f8(z) = 0, we can evaluate the derivative f5(z) E 4Z 0 T*(Z)z. Similarly, if f8(z) = 0 and f5(z) = 0, we can evaluate the second derivative,

f,'(z) E 4Z 0 Sym2 T*(Z)z.

We can think of the second derivative as a symmetric bilinear form on T(Z)z with coefficients in 4Z.

We will consider three projective-space bundles

1' Q 7Z

Z (Z x Z)0 T(Z)0 where (Z x Z)0 denotes the complement of the diagonal and T(Z)0 denotes the complement of the zero section. They come with maps

p pN,

Q ~*IpN,

such that the fibers over points of Z, (Z x Z)? or T(Z)0, respectively, are projective spaces in JpN. They are defined as follows: For z E Z the fiber Pz is the projective space of s E JpN such that f8(z) = 0 and f5(z) = 0. For (y, z) E (Z x Z)0 the fiber Q(yz) is the projective space of s E JpN such that fs(y) = 0, f5(y) = 0, f8(z) = 0 and f5(z) = 0. For z E Z and v a nonzero vector in T(Z)z the fiber lZ(zv) is the projective space of s E JpN such that f8(z) = 0, f5(z) = 0 and f5'(z)(v,w) = 0 for all vectors w E T(Z)Z. If LC is sufficiently ample, then these conditions define smooth projective-space bundles, as required.




The image of the map P __ pN is exactly the discriminant locus D. This proves that D is irreducible and, hence, a generic section D1 = D n p2 is irreducible.

If z E Z and s E Pz, then the zero set X, of f, is singular at z. This singularity is an ordinary double point if and only if f"'(z) is a nondegenerate bilinear form. The bilinear form is degenerate if and only if s E 7Z(zv) for some vector v. Hence the image of the map 7Z -- D C pN is the set of s such that X, has a singularity worse than a double point. The image of Q -- D C pN is the set of s such that X, has at least two singularities. The complement of the union of the images of Q and 7? consists of the set of s E D such that X, has exactly one ordinary double point.

If L is sufficiently ample, then the generic element of the image of P1, Q or 7 has no worse than the required singularities. This is based on the fact that the generic element of a linear system is smooth away from the base points. We may assume that, for any y, z E Z, the restriction morphism

H0(Z, L) - /M4 (t/M4

is surjective. Then for z E Z and a generic choice of s E Pz, the second derivative f,'(z) is a nondegenerate form. Hence X, has an ordinary double point at s. Since z is the only base point of the linear system Pz (true if L is sufficiently ample), this implies that X, has exactly one ordinary double point. Similarly, for a generic s E Q(yz), the forms f,'(y) and f,'(z) are nondegenerate. Thus X, has ordinary double points at y and z and, as these are the only base points of the linear system Q(yz), the variety X, has exactly two ordinary double points. Finally, for a generic s E Z(z~v), the variety X, has z as its only singularity, and at z the Taylor expansion of f5 begins with a quadratic form with v as its only nondegenerate direction, plus a generic cubic term, plus higher-order terms.

The images of Q and R are irreducible, of dimension N-2 in pN, So a generic p2 meets them in a finite set of generic points. Hence we may decompose the discriminant locus in J2 as a disjoint union D1 = A1 U A2 U A3, where A1 is an open set in the irreducible curve D1 and A2 and A3 are finite sets: A2 is the intersection of J2 with the image of Q and A3 is the intersection with the image of 7R. We have the characterization that s E A1 if and only if X, has exactly one ordinary double point; s E A2 if and only if X, has exactly two ordinary double points; and s E A3 if and only if X, has exactly one singularity whose defining equation has a Taylor series as described at the end of the last paragraph.

We investigate the third case more precisely. Suppose that s E A3 and let z denote the singularity of X, Choose a trivialization of L near z (we will make use of this freedom of choice below) so that the defining equation f5



394 C. SIMPSON

may be considered as a holomorphic function. We may choose local analytic coordinates wo, . .. , wn such that the Taylor series of f, at z is

f5 = Wo3 + W 2+ *+ W 2 + * * C(W) + **

where c(w) is a cubic term not involving w3. Now make a change of coordinates so that wo = xo and wi = x +q(x... X) = 1, .. , n). The cubic term becomes xA + c(xo, .. ., xn) + 2xlql + + 2xnqn. Since c involves only terms that include one of x1,... ,xn, the quadratic terms qi can be chosen so that the cubic term reduces to xA. Then

as= X3 + X12 + * * * 2 O +**

with the remaining terms of degree 4 or more. Let I denote the ideal generated by x?, i = 1,..., n. In the local analytic coordinate ring, modulo I, we can divide f8 by xA and take the cube root to get a unit u (which can be extended to an analytic function). In other words, f8 = (xOu)3 modulo I. Set zo = xou. Now write

= z6 + X1 + + Xn + ai4x + + anon

with ai analytic functions. They must vanish at the origin, since we know the quadratic term of f8 already. Therefore 1 + ai are units, and we can take their square roots. Let zi = xi(1 + ai)1/2 for i = 1, ... , n. Then ZO,... , Zn form a system of local analytic coordinates at z, and we have

f= z3 + Z 2+ + Z 2

To complete the proof we must analyze the family of hypersurfaces X, near a point s on D1. To conserve subscripts let f denote f8. The generic J2 is given by a 3-dimensional space of sections of L; one basis vector is f, and we may choose two others denoted by g and h. Note that the space spanned by g and h is generic with respect to f.

Suppose that s E A1 and z is the double point. Then we may assume that g(z) =h 0 and h(z) = 0, but h'(z) = 0. Then we may choose a trivialization of L such that g corresponds to the function 1. We may choose local coordinates so that f = zA + + Z2. Then the family of hypersurfaces near s can be given as a family indexed by two parameters E and 6 with the equation

f^=Zo2 + ***+ Z2 + E + 6ih.

Let &i denote the partial derivatives with respect to zi. Then we have

&ifeb = 2zi + 6&ih.

The system of equations &if,,6(y) = 0 has a unique solution y = y(6) near y(O) = z (the equations, and hence the solution, are independent of F). To




solve the equations concretely note that h'(z) is generic; so we may assume that the &ih are units at z. Then ui = 2zi/&ih is again a system of coordinates near z, and the equations become ui(y(6)) = 6. The y(6) is an analytic map from a neighborhood of 0, in the 6 line, to Z. Now put E(6) = -f0,(y(6)). It is the only value of E such that f?,,(y(6)) = 0, hence it is the only value of E such that X?,, has a singularity near z. Since X, is smooth away from z, so is X?,, for small values of E and 6. The curve E = F(6) is a smooth curve near the origin in the F-6 plane. It is the discriminant locus near s in p2, which shows that the open set A1 of the discriminant locus is smooth.

We note, for use in the next paragraph, that the derivative of F(6) with respect to 6 is 0 at 6 = 0. To see this recall that F(6) = -f(y(6)) - 6h(y(6)). Since f(z) = 0 and f'(z) = 0, and y(O) = z, the term f(y(6)) has derivative equal to 0 at 6 = 0; and since h(z) = 0 and h(y(6)) = 0 at 6 = 0, so the derivative of 6h(y(6)) is 0 at 6 = 0.

Suppose that s is a point in A2. Then there are two ordinary double points y and z on X, We may proceed as above, with sections g and h, such that g(y) = 0, h(y) = 0 and h'(y) = 0, whereas h(z) = 0, g(z) = 0 and g'(z) = 0. Again let f,,6 = f + Fg + 6h. By the same analysis as above, we find a curve F = Fy(6) such that fy (6),6 has a singularity near y and a curve 6 = 6z(F) such that f6,6z(6) has a singularity near z. The discriminant locus near s is the union of these two curves. The two curves meet at the origin; by the note in the preceding paragraph, they are tangent respectively to the 6-axis and the i-axis. Hence they are transverse, and so their union has a node at s. Thus the discriminant locus D1 C PE1 has a node at each point of A2.

Suppose that s E A3 and let z denote the singularity of X, As before, we may assume that the family of hypersurfaces is given by the two-parameter family of sections f,,6 = f +Fg+6h. We can trivialize ? so that g corresponds to the unit section and then choose coordinates (as explained several paragraphs ago) so that the equation becomes

f?, = zO3 + Z2 + .. +z3 + E+ 6h.

To complete the proof of the proposition we just have to show that the discriminant locus has a cusp at s. Solve the system of equations aif,,6(y) = 0 for a function y(6) (as before, it does not depend on e). These equations are now

3z2 + 6&oh = 0,

2zin+i60ch =c0 (it= 1t ..h , n)a

Since h'(z) is generic, we can assume that the Oih are all units. Introduce a



396 C. SIMPSON

new system of coordinates with ui = -2zi/&ih, for i = 1,...,n, and uo = (-&oh/3)-l/2zo. Now the equations become

Uo =f,

ui = T2 (il..n),

and 6 = T2.

Finally set 6(T) =- (y( )) We have curves y(r) in Z and (E(r), 6(T)) in a neighborhood of s in J2 such that the X?,, have a singularity at y near z if and only if y = y(T) and (F, 6) = (E(T), 6(r)) for some r near the origin. Since X, has no singularity other than z, the X?,,, for small values of (s, 6), have no singularities except those near z. Hence the discriminant locus D1 near s is the image of the curve (E(r), 6(r)). In terms of the new coordinates ui, the function f has the Taylor expansion

f = Couo+ clUl+ * + cnun+ unulq+ * ** + Unqn+ *

where ci are constants, qi are quadratic expressions and the remaining terms have order 4 or more. Also the Taylor expansion for h is

h = &oh(O)zo + ,

and zo = (-Ooh(O)/3)1/2uo +...

with the remaining terms quadratic in r in both cases. Hence

h = -1/3X(oh(O))3/2T +....

Note also that the constant co is fixed at

co = (-aoh(O)/3)3/2.

Putting this all together, we have

6(T ) =-f (y(T) - T2h(y(T) )

= -((_1/3)3/2 + (-1/3)1/2)(Ooh(O))3/2T3 +

This first term is nonzero, so we can find a new coordinate El, which is a function of e with a nonzero derivative at the origin, such that El (r) = r3.

Thus, in terms of the analytic coordinate system (,E, 6) near s E p2, the discriminant locus D1 is the curve l= 6; this has a cusp at the origin s. This shows that the points of A3 are cusps of D1. We now have the trichotomy that D1 has smooth points, nodes and cusps, and these are exactly the points of A1, A2 and A3, respectively. We have already seen that, for s E Ai, the hyper-




surface X, has exactly the required singularities. The proof of the proposition is complete. R

Suppose that s E D1. Let B denote a small ball in p2 centered at s. Let FU denote the fundamental group of B1- (Di n 1). Let Ts denote the quotient of Fs by the relations -y2 = 1 for all elements -a E Fs represented by loops around the smooth points of irreducible components of D1 n 1.

LEMMA 6.2. For any s E D1, the group T, is finite.

Proof. If s is a smooth point of D1, then F '- Z and T, Z/2. If s is a node, then FU s Z D Z and T, ' Z/2 D Z/2. Suppose that s is a cusp. Let U be a neighborhood of the origin in C and let p: B -- U be a projection in a generic direction. Let the set L = p-'(O) C 1, K = B - (Di n 1) - L and M be a neighborhood, in 1, of the complement of a small disk around s in L. Then -rl (M) = Z and -7ri (M n) = Z x 7Z. On the other hand, the projection p: K -- U-0 is a fibration, with the fiber F equal to a disk with two punctures. By appropriately choosing the shape of the ball 1, we can assume that there is a section a: U -- B1- (Di n 1) of the projection. Choose a base point on this section and let r denote the image by a of the loop going around the origin in U -0. There is a monodromy action a: ir1 (F) --,ri (F) corresponding to the operation of moving paths along as the fiber is moved once around the origin, with the base point kept in the section a(U - 0). The fundamental group of K is a semidirect product generated by r and r1r(F) with the relation rur_1 - a(-y). Now apply the Seifert-Van Kampen theorem: iri(1 - (Di n 1)) is the amalgamation of ir1 (K) and iri (MA) = Z over iri (K n.M) = Z x Z. The element r generates the kernel of 7r1(KM n M) -- 7r1(M), and this map is surjective. Hence the amalgamation is equal to the group obtained from 7r1 (K) by adding the relation r = 1. In view of the semidirect product described above, this means that Irl (1 - (Di n B)) is equal to the group obtained from i1 (F) by adding the relations -y = a(a) for all -y E xi (F). When the fiber F is turned once around the origin, the punctures wind around each other 3/2 times. Let b: xi (F) -- xi (F) represent the effect of the punctures winding around 1/2 of a time; then a = b3. There are two generators, which we call a and /3, representing paths that go around the two punctures (in the same direction, and in the same direction as the winding). The operation b has the effect

b(a) = /3, b(i3) = /3a/3.

Iterating this, we get

b2(o) = 3aIP-1i3-1iPi = papa-1p-1

b3p(/) = (/3a/3i)/(/3a/3i)/3i(/3&i/3i)

= /3a3aTB-1ce-,i-



398 C. SIMPSON

(whereas b3(a) = b2(/)). Thus 7ri (1 - (D1 n 1)) has generators a and 3 with relations

a. = papa-1p-1

13 = 3cece,-1e-1)3-1.

These are both equivalent to the relation

ao3a = p3a:3

(the relation of a braid group). Hence FU is generated by a and 3 with the above relation. The elements a and / are paths that go around the discriminant locus D1 (they are conjugate, as Df n 1 is connected-this can be seen from the above relations). Therefore we obtain Ts by adding the relations a2 = 1 and 32 = 1. Now multiplying the braid relation by 3 on the right gives (a/3)2 = 3a, whereas multiplying by 3 on the left gives (13a)2 = a/3. Hence (a/3)4 = a/3, and so (a/3)3 = 1. The conjugate of a/3 by either a or /3 is equal to /3a = (a/3)2. Hence the cyclic subgroup of order dividing 3, generated by a/3, is a normal subgroup. In the quotient by this normal subgroup we have a = 3; thus the quotient is generated by one element of order 2. So it is either a group of order 2 or trivial. Thus Ts is finite, of order dividing 6. (One can see for example, by looking at our local systems on the complement of the cusp-that ao3 is not trivial; hence Ts is the symmetric group S3.) This proves the lemma. O

Let F = 7ri(Ui) and let T denote the quotient of F obtained by adding the relation -y2 = 1, where -a is a loop going once around D1. Note that all such loops are conjugate to each other, as D1 is irreducible. Let T' be the quotient of T by the intersection of all subgroups of finite index.

LEMMA 6.3. There exist a Galois covering Us -- U1 and a completion to a smooth projective variety S D Us such that the map irl(Us) -Y T' factors through rxi(Us) -- i(S) -- T'.

Proof. For each point s E D, we have a homomorphism j8: Ts -- T. For every normal subgroup Ti C T of finite index, let Ti,, = j-'(Ti) C T, As the subgroups Ti decrease, the Ti,, form a decreasing sequence of subgroups of T, By Lemma 6.2, T, is finite. Hence this sequence of subgroups is stationary, eventually equal to T,,,. There are only finitely many groups T, that occur (one for the smooth points s and one for each singular point). Hence there is a normal subgroup of finite index Ti C T such that Ti,, = T,,,, for all s. Let Fi C F be the pullback to a subgroup of finite index in F and let Us be the covering of U1 with the fundamental group Fi. Let Us C S be a smooth projective completion.




We claim that there is a factorization 7rl(Us) -- ri(S) -- T'. The quotient 7ri(S) is obtained from irl(Us) by adding some relations of the form ( = 1, where ( is a path going around some component of the divisor S - Us. Note that, aside from the path going from the base point and back again, ( is localized near a point on S - Us. Hence there are a point s E D1 and a small neighborhood t3 of s such that ( is conjugate to a path in the inverse image of t3 - D1. This inverse image decomposes as a disjoint union A1 U ... U Ah. The fundamental groups irl(Aj) are, up to conjugacy in Fi = iri(Us), all of the subgroups of the form ri n grFg-g for g E I7. Since Fi is normal, these are the F-conjugates of Fi', = ri n IF,. Note that Fim, is the pullback to F of Ti,,. But also Tis = Toos is contained in the intersection of all subgroups of finite index in T. Hence ]i,, maps to 1 in T'. The same is then true for all of the iri(Aj) (the F-conjugates of FiX,). The path ( is in one of these, so it maps to 1 in T'. This shows that the map iri(Us) T T' factors through the quotient 7rl (S). O

LEMMA 6.4. Suppose that p: IF -- GL(n, C) is a representation such that p(-y) has order 2. Then p factors through T'.

Proof. Since p(Q-2) = 1, p factors through T. It is well known that if u E T is contained in the intersection of all subgroups of finite index, then p(u) = 1.

(Here is the argument, which works for any linear representation p: There exists a subring A C C of finite type over Z such that the representation p has coefficients in A. Note that A is an integral domain. Let I be a maximal ideal of A. Then A/Ik are finite, but the I-adic completion A = lima A/Ik contains A as a subring (by Krull's theorem). Let PA: T -Y GL(r, A) denote the representation that yields p upon extending scalars to C and let K C T denote the kernel of PA. It is the same as the kernel of p and also the same as the kernel of the representation of PA. Let Kk denote the kernel of the representation PA/Ik obtained by reduction modulo 1k. Then Kk are subgroups of finite index, but K = nk Kk. Hence K contains the kernel of T -- T'.) D

Suppose that W is a local system of C vector spaces of rank r on Z. Let

V = Rnin*(a*W)

denote the nth higher direct-image sheaf on U1. It is a local system of rank xr on U1, where X is the topological Euler characteristic of X,. This may be computed from ci (L) and ci (TZ) by the Riemann-Roch theorem. Let p: xin(Ul) -- GL(Xr, C) denote the monodromy representation of the local system V.



400 C. SIMPSON

LEMMA 6.5. Suppose that s is a smooth point of D1. Then the monodromy of the local system V around the discriminant locus D1 (at s) has order 2 with exactly r eigenvalues equal to -1.

Proof. This is the same as in the case of constant coefficients, discussed in [5]. We will outline the argument here. We can restrict the family to a JIEJ, which intersects the discriminant locus transversely at s, and consider the monodromy of the resulting Lefschetz pencil around s. Let z be the double point of X, and choose a ball B around z. The monodromy transformation when t goes around s can be made to act on the Mayer-Vietoris sequence

*.. -~ HI-1(Xtn w, W) -- H'(Xt, W) -- H'(Xtfnl1, W)DHI(Xt- B, W)

Furthermore the spaces Xt n 01 and Xt -1B retract to the fiber X, so that the monodromy acts trivially on those terms. As B is simply connected, W 1B- Cr. Hence HW(Xt n B, W) 2 H(Xtn B, C)r. One can see that the space Hn(Xtfn 3, C) is 1 dimensional, spanned by the "vanishing cycle" (and H (Xt n 1, C) = 0 for i 7k 0, n). One can see, in the case where the relative dimension n is even, that the monodromy transformation acts by multiplication by -1 on the vanishing cycle. (This also follows from the Mayer-Vietoris sequence and the fact that the monodromy transformation on HW(Xt, C) is a reflection; cf. [5].) Now the r-dimensional space in which the monodromy acts by -1 must be split from the rest of the Mayer-Vietoris sequence, where the monodromy acts trivially. Hence the monodromy transformation on Hn(Xt, W) has order 2 with r eigenvalues equal to -1.

COROLLARY 6.6. Let h: Us -- U1 be the finite etale covering given in Lemma 6.3, with the smooth projective completion i: Us >- S. Then the local system h*(V) extends to a smooth local system Vs = i*h*(V) on S. In other words, the representation p o h* factors through 7rl (S).

Proof. By Lemmas 6.4 and 6.5 the monodromy of the representation p factors through T'. By Lemma 6.3 the representation restricted to 7rl(Us) must factor through 7ri(S). C

We close with a well-known remark about F = iri(Ul), which will be used in the next section.

PROPOSITION 6.7. Let -y denote any path going from the base point to a point near the smooth part of the discriminant curve D1, once around D1, and back to the base point by the same path. Then F is generated by all of the elements of the form u-yu1 (u E F).

Proof. The Lefschetz theorem says that if U2 = Ui n PJ for a generic I c JR'2, then 7rl (U2) --+ i (Ui) is surjective. But U2 is a punctured projective




line; its fundamental group is generated by loops around the punctures. In Ui these loops are conjugate to -y (because D1 is irreducible). O

Combined with Lemma 6.5, this says that if W is a rank-1 local system on Z, then the monodromy of the higher direct image V is generated by conjugate reflections.

7. Monodromy groups

All algebraic groups will be over C. Suppose that F is a finitely generated discrete group, H an algebraic group and p: F --+ H a representation. The monodromy group is the Zariski closure G of the image p(r) or, equivalently, the smallest algebraic subgroup of H containing the image. The representation p is reductive if its monodromy group G is reductive. In the case H = GL(n, C), a representation p is reductive if and only if the representation on the vector space Cn is a direct sum of irreducible representations.

An algebraic group G is simple if it has no normal algebraic subgroups. This implies, in particular, that it is connected and reductive and that the center is trivial. A group is quasisimple if the center ((G) is finite and the quotient G/M(G) is simple.

We study how the monodromy groups vary in a family of representations. If G C H is an algebraic subgroup, we call any subgroup of the form hGh-1 an H-conjugate of G.

LEMMA 7.1. Suppose that {Pt: F -* H} is a framed family of representations and G C H is an algebraic subgroup. Then the subset of t E T, such that pt(F) is contained in an H-conjugate of G, is a constructible subset.

Proof. The set of (t, h) E T x H, such that pt(F) c hGh-1, is closed. Its image in T is therefore constructible. C1

LEMMA 7.2. Suppose that H is a reductive group and {Pt: F -+ H} is a family of homomorphisms. Suppose that for each t E T the monodromy group Gt of Pt is reductive, the connected component has finite center and there is a uniform bound on the number of connected components. Then the H-conjugacy classes of the Gt vary in a constructible way: there is a partition T = U Ti into constructible sets and, for each set, there is a subgroup Gi C H such that Gt is H-conjugate to Gi for t E Ti.

Proof. There are, up to H-conjugacy, only finitely many possible subgroups Gt that can occur. On the level of connected components, Go, one can see this by choosing maximal tori and looking at the root systems. Afterward there are, up to conjugacy, only finitely many subgroups Gt/G? c N(GC)IG?



402 C. SIMPSON

with a given bound on the number of elements. Now Lemma 7.1 implies that the sets of t corresponding to the conjugacy classes are constructible. C1

Caution. Monodromy groups do not always vary in a constructible way. For example, the family of 1-dimensional representations of F = Z is parametrized by A = p(l) E C*. The monodromy group changes whenever A is a root of unity; the set of roots of unity is not constructible (its complement does not even contain a Zariski-open set of C*).

Monodromy generated by conjugate reflections. In this subsection we will discuss the special properties of monodromy groups that are generated by conjugate reflections. Deligne's Theorem 4.4.1 of [6] is a prototypical example. It is the one brought to my attention by Carlson in connection with his examples, and one can use it to obtain the information necessary to construct our examples. We will present some results that are slightly more general, learned in Katz's course at Princeton cf. [16]. The proofs are sketched for the reader's convenience.

Suppose that T is a group with an element -y such that T is generated by the set of all the conjugates u-yu-1, u E T. Suppose that p: T -Y GL(r, C) is a representation such that p(-y) is a reflection (i.e., it acts semisimply with an eigenvalue -1 of multiplicity 1, and an eigenvalue 1 with multiplicity r - 1). Let V denote the representation space Cr with the action of T.

LEMMA 7.3. On the determinant of the representation det(V), T acts through the group {?1}.

Proof. The element -y acts by -1. The image of T in C* is generated by conjugates of the image of 'y, hence the image is just {+1}. C1

LEMMA 7.4. Suppose that V is semisimple. Then it decomposes into V = Vo e V1, where T acts trivially on Vo and V1 is irreducible. If the representation V1 can be expressed as a tensor product, one of the factors must have dimension 1.

Proof. Decompose this space into V = Vo E V1, where Vo is trivial and V1 has no fixed vectors. Note that the representation V1 has the same property, namely that oy acts by a reflection. If VI = V2 ? V3, then the 1-dimensional eigenspace for -1 must lie in one of the factors, say, V2. Thus r acts trivially on V3. But any g E r preserves the decomposition, so that grg-1 acts trivially on V3. Thus F acts trivially on V3, contradicting the fact that V1 has no fixed vectors.

Suppose that V1 = Wi 0 W2 with dim(Wi) > 2. Note that r cannot act as a scalar in both factors. Suppose that r acts with eigenvalues al 5# a2 5 ...




on W1. Then for each eigenvalue bi of the action of r on W2 we obtain at least two distinct eigenvalues, a1bi and a2bi, for r. Thus one of the two is -1. This occurs for values of bi accounting for multiplicity at least 2; hence the -1 eigenspace of r would have dimension at least 2. This is not the case, however, so the representation cannot be a tensor product. L

THEOREM 7.5. Suppose that V is irreducible and dim(V) > 5. Then there are three possibilities:

(a) the image of p is finite, (b) the image of p is not finite, but is contained in the normalizer of a

maximal torus in GL(r, C), or (c) the monodromy group G of p has a connected component Go, which

is quasisimple of positive dimension. In case (c), the representation remains irreducible when restricted to GO.

Proof. Let Go denote the connected component of G. If h E G and W C V is a Go-invariant subspace, then so is h(W) (from the formula gh(w) = h(h-1gh(w))). Decompose V into G0-isotypic subspaces

V= ?Ui Ail

where Ui are irreducible representations of Go and Ai are vector spaces. Then, for any h E G, h(Uj 0 Ai) = Uh(i) 0 Ah(i) for a permutation of the indices i 4 h(i) in this way, the finite group G/Go acts on the set of indices. This action must be transitive, since V is irreducible as a representation of G. The morphism h: Ui 0 Ai -* Uh(i) 0 Ah(i) intertwines the representation of GO and the representation twisted by the automorphism Ad(h) of Go. Hence there are isomorphisms rihi: Ui -* Uh(i) intertwining the representation and the twisted representation. In particular the dim(Uj) and dim(Ai) are all the same.

Suppose that the reflection r permutes some indices nontrivially. Fix i such that r(i) 54 i. Let Vr denote the subspace of codimension 1 of vectors fixed by r. Then

(Uz 0 Ai) n Vr c (Ui0 Ai) n (Ur(i) 0 Ar(i)) = 0.

Hence dim(Ui 0 Ai) = 1. Thus all of the different isotypic subspaces are 1 dimensional. The group of elements of GL(n, C) that preserve the isotypic decomposition is a maximal torus, and the group of elements that preserve the set of isotypic subspaces (possibly permuting them) is the normalizer of the torus. Thus, in this case, G is contained in the normalizer of a maximal torus, which is conclusion (b).

On the other hand, suppose that the element r of r C G acts trivially on the set of indices i; so then all of G/G0 acts trivially. Since V is irreducible



404 C. SIMPSON

as a representation of G, this implies that there is only one index i (which we may temporarily drop). There is an isomorphism j: U -* U intertwining the action of Go and the action twisted by r. Hence

r(rj 1)-1: U0A - U0A

is a morphism of Go-modules. Thus it equals 1 0 m for m E End(A). We have r = rj 0 m, which implies that one of A or U is 1 dimensional. If U is 1 dimensional, then Go acts on V by scalars. However the action of G on det(V) is contained in the subgroup {?1} C ?*; so the group of scalars that can act on V is finite. This implies that Go is trivial, which is case (a).

Suppose instead that A is 1 dimensional. Then V = U is irreducible as a representation of Go. We have to show that Go is quasisimple in order to complete our verification of case (c). Since V is irreducible, the center acts by scalars. Hence the morphism from the connected component of the center of Go to C*, corresponding to det(V), is finite. As the image is contained in {?1} (by Lemma 7.3), this implies that the center is finite. In particular g - Lie(G0) is a semisimple Lie algebra.

Write g = 01 (... ?k. The irreducible representation V decomposes into a tensor product

V = V1 X *Vk,

where Vi is an irreducible representation of gi. This factorization is unique (as follows from the highest-weight parametrization of representations). In other words, the representations Vi are uniquely specified, and the isomorphism of the tensor product with V is unique up to a scalar. Since V is a faithful representation of g, the Vi are faithful representations of gi. Since the 0i are semisimple, this implies that dim(Vi) > 2.

Our element r acts on g by the adjoint action. It permutes the factors of the direct sum decomposition by a permutation (of order 2), which we denote by p. Let V' (resp. Vi') be the representation with the same underlying vector space as V (resp. Vi'), but with the action composed with Ad(r). We have

V = pl p(k)

(with the order still corresponding to the decomposition of 0). The automorphism r gives an isomorphism r: V _ V'. By the uniqueness statement, this factors into isomorphisms r = ri with ri: Vi -VI(i). By the uniqueness of the isomorphisms, R2 = 1 implies that r is a scalar.

If there is only one index i, then g is simple, as desired. If there are two indices i = 1, 2, which are permuted by p, then r =

a(aO b), where a: Vi -* V2 and b: V2 -* V1, and a: V2? V1 -* V1 0 V2 is the isomorphism expressing the commutative property of the tensor product.




Further we have ab = A and ba = A-1, where A is some scalar. Since aba = Aa = A-1a, we have A = ?1. Let ej be a basis for V1 and let fj = aej. Then

r(ej 9 fk) = a(fj 0 Aek) = Aek 9 fj.

Thus ?r is a permutation of the standard basis elements of the tensor product. Since R2 = 1, this permutation is a product of transpositions that commute. However, since dim(V) > 5, we have dim(V1) > 3, and there must be at least three transpositions:

el 0 f2 ' e2 (0 fi,

el 0 ff3 e3 ( fi,

e2 0 ff3 e3 0 f2.

In particular this contradicts the fact that r is a reflection. If there are more than two indices, or two indices that are not permuted,

then p is not transitive. We can divide the set of indices into two subsets, preserved by p, and express r as a tensor product of automorphisms. As before, this implies that one of the factors has dimension 1. This contradicts the fact that dim(V) > 2, noted above. Thus the only possibility is that there is only one index i, and g is simple (hence Go is simple modulo its center). This gives case (c) of the theorem. El

Behavior in families. We will consider how the possibilities in Theorem 7.5 vary in families. Suppose that {Pt: T -* GL(r,C)}tET is a family of representations of T such that PtQy) are reflections. Suppose that each Pt is irreducible. In this case we have T = T(a) U T(b) U T(,), where T(p) are disjoint subsets so that property (p) of Theorem 7.5 holds for Pt when t E T(o (p = a, b or c).

LEMMA 7.6. The subset T(,) is constructible.

Proof. Jordan's theorem says that there is a bound b such that, for any finite subgroup H C GL(r, C), there is a maximal torus e C GL(r, C) such that H n e has an index < b in H (cf. [16]). There is a subgroup T1 C T such that any subgroup of an index < b in T contains T1. Then, for any t E T such that property (a) or (b) holds, the image pt(Ti) is contained in a maximal torus e. On the other hand, if property (c) holds, then pt(Ti) cannot be contained in a maximal torus (since the Zariski closure of pt(Ti) contains GC, which is quasisimple). The subset T(,) is thus the complement of the set of t such that pt(Ti) is contained in a maximal torus. All the maximal tori are conjugate, so this set is constructible by Lemma 7.1. L



406 C. SIMPSON

COROLLARY 7.7. Suppose that case (c) of the theorem holds for some generic geometric point ti E T. Then there are a subgroup G C GL(r, C), such that the connected component is quasisimple of positive dimension, and a nonempty Zariski-open set T' C T, such that for each t E T' the monodromy group Gt of Pt is GL(r, C)-conjugate to G.

Proof. We may assume that T(p) are defined over Q. If case (c) holds for a generic geometric point t1, then ti E T(,), which implies that T(,) is dense. In particular there is a nonempty Zariski-open set T1 C T(,). Now apply Lemma 7.2 to the family {Pt}tET1. Note that there are only finitely many conjugacy classes of Go C GL(r, C) that can occur, since the representation of Go is irreducible; and, for each one, N(G?)/G? is finite. Thus there is a uniform bound for the number of connected components of Gt, and so Lemma 7.2 applies. One of the components in the constructible decomposition must contain a nonempty Zariski-open set T'. If G is the corresponding monodromy group, then the statement of Lemma 7.2 gives the desired conclusion here. LI

Remark. In the situation of the previous corollary, recall that Lemma 1.2 implies there are a nonempty T", etale over T', and a family of Zariski-dense representations {VT: T -? G}TET" such that the pullback of the family {Pt} is conjugate to the family of compositions {T - G c GL(r, C)}.

Applications to families of hyperplane sections. The results described above apply directly to the situation described in Section 6. We use the same notation as in that section. Let {Wt}tET be a family of rank-1 local systems on Z and assume that the corresponding nth higher direct-image local systems Vtnew on Ui = Ip2 - Di fit together into a family {Vt,new}tET of local systems of some rank r. By Lemma 6.5, the monodromy transformations of Vtnew around smooth points of D1 are reflections and, by Proposition 6.7, the monodromy group of the local system Vt new is generated by conjugates of any one of these reflections. Furthermore, Lemma 2.7 says that the Vt,new are irreducible. By choosing L sufficiently ample, we may assume that r > 5 (cf. Lemma 5.5, which is independent of the rest of Section 5). Let Gt denote the monodromy group of Vt. Theorem 7.5 provides a trichotomy of possibilities (a), (b) and (c) for Gt.

LEMMA 7.8. Suppose that case (c) holds for some generic geometric point t, E T. Then there are a connected quasisimple subgroup G C GL(r, C) acting irreducibly; an T'-completed covering S D Us -) U1 of p2; and a nonempty Zariski-open set T' C T such that the following hold for each t E T': The pullback of Vt,new to Us extends to a local system VtSnew on S and the monodromy group of Vt,S,new is GL(r, C)-conjugate to G. Furthermore there are an




stale morphism p: T" -* T' and a family of Zariski-dense representations { Pt: iri(Ul) -+ G}teT" such that the local systems Vp(t),S,new are obtained by the composition of Pt with the inclusion G C GL(r, C) for t E T".

Proof. Corollary 7.7 provides a Zariski-open subset T' C T and a subgroup G' C GL(r, C) whose connected component is quasisimple of positive dimension such that Gt is conjugate to G' for all t E T'. Let G be the connected component of G'. It acts irreducibly on Cr by the additional statement in Theorem 7.5. By Lemma 6.3 there exists an T'-completed covering S. By Corollary 6.6 the monodromy representation of Vtnew factors through a representation T' -- Gt, and the pullback of Vtnew extends to a local system Vtsnew on S. Let b be the number of connected components of G'; we may assume that irl(S) maps into the intersection of all subgroups of T' of index < b. Since the index of Go C Gt is < b, the representation 7rl(S) -* Gt has its image in Go. On the other hand, the image of 7ri(S) is a subgroup of finite index in VT; so the Zariski closure of the image in Gt is a subgroup of finite index. Hence this Zariski closure, the monodromy group of Vtsnew, is Go. Note that this is conjugate to G. The last statement of the lemma follows from Lemma 1.2. 0

8. Torelli theorems

In this section we will recall a general Torelli theorem of M. Green. We use it to show that our families of higher direct images do not factor through curves and also to follow up Corollary 7.7 of the previous section.

We begin by defining a weak notion of factorization through a curve. Suppose that F = nl(S) for some smooth projective variety S. We say that p: F -- G has the factorization property F1 if there exist a smooth quasiprojective variety Y and a smooth quasiprojective algebraic curve C with maps g: Y - S and p: Y -- C such that g*(p) factors through a composition 7r1(Y) 7ri1(C) -* G. We say that p has the nonfactorization property NF1 if it does not have the factorization property F1.

LEMMA 8.1. Suppose that p: 7ri(S) -* G is a Zariski-dense representation into a quasisimple group. Let ((G) denote the center and Iv: G -? G/((G) the projection. Suppose that G c GL(r, C) is a representation of G and let V denote the corresponding local system on S. Suppose that V satisfies the nonfactorization property NF1. Then v o p satisfies the nonfactorization property NF1.

Proof. Suppose that v o p satisfies F1. Let

S4 Y --C



408 C. SIMPSON

be the maps giving the factorization. We may assume that they are smooth. There is a representation 4: 7ri(C) -* G/((G) such that g*(v op) = p*Q(/). We may replace C by a finite etale cover and Y by the corresponding fiber product; so we may assume that 4 lifts to a representation p: 7r (C) -* G. There is a representation rj: 7rl(Y) --4 ((G) such that g*(p) = Tlp*(W). We may replace Y by a finite etale cover so that r1 is trivial. Thus p satisfies F1. It follows immediately that the local system V does too. This is a contradiction. LI

We would like to show that the representations p: 7ri (S) -* G, constructed in the previous sections, satisfy NF1. By the above lemma it suffices to show that the local systems Vt,s,new satisfy NF1. For this purpose we need to consider some Torelli theorems for variations of Hodge structure.

Recall that a complex variation of Hodge structure on a smooth variety S is a local system V, together with an indefinite hermitian form, and an orthogonal C' decomposition V = ( VP, satisfying the axioms that the form is (-1)P-definite on VP and the VP vary within the restrictions of the Griffiths transversality condition (i.e., they vary in the direction of VP-1 in holomorphic directions, VP+1 in antiholomorphic directions). Let rp dim(VP) and let

D = U(rl + r3 + . * *, r2 + r4 + ..

U(rl) x U(r2) x ...

To give a complex variation of Hodge structure with given rp is the same as giving a representation of 7ri(S) into U(rl + r3 + , r2 + r4 + ) and an equivariant map

4): S-)TD

such that 4 is holomorphic and horizontal with respect to a natural complex structure and distribution in the tangent bundle of D.

We record some facts about variations of Hodge structure: (1) Suppose that f: X -* S is a smooth projective morphism of smooth

quasiprojective varieties. Suppose that W is a complex variation of Hodge structure on X. Then the higher direct images Vi = Rzf*(W) are complex variations of Hodge structure on S in a natural way.

(2) Suppose that U C S is an open subset of a smooth projective variety. If Vu is a variation of Hodge structure on U, and if the local system Vu extends to a variation of Hodge structure Vs on S, then Vs is a complex variation of Hodge structure (restricting to the given Vu).

(3) If V is a complex variation of Hodge structure on a quasiprojective variety S, then V decomposes as a direct sum of complex variations of Hodge structure whose underlying local systems are irreducible.

(4) Suppose that g: Y -* S is a surjective morphism of smooth projective varieties with connected fibers. Suppose that V is a local system on S such




that g*V has the structure of a complex variation of Hodge structure on Y. Then V has the structure of a complex variation of Hodge structure pulling back to the given one on g*V.

Item (1) is due to Deligne's Khhler identities for variations of Hodge structure (see [7]). Item (2) is an extension theorem of Griffiths (see [11]). Item (4) is proved in [25] (note that, under the hypotheses, 7ri(Y) - 7ri(S) is surjective).

For item (3) note first that the category of complex variations of Hodge structure is semisimple. Suppose that V is an irreducible object. Then A = H0(End(V)) is a semisimple algebra with a Hodge structure (due to results of Schmid [21]), and the part of type (0,0) consists only of scalars. If V is not irreducible as a local system, then there is an element a E AP-P for p 7? 0. Hence a is nilpotent; so ker(a) =A 0, but ker(a) is a subvariation of Hodge structure, contradicting the hypothesis. Thus the irreducible complex variations have irreducible local systems.

Our examples. We return to the situation of Section 2, keeping the notation established above Lemma 2.4. Suppose that W is a rank-1 local system on Z. Then W is a complex variation of Hodge structure if and only if it is unitary, that is, if the monodromy is contained in U(1) C C*. In this case, the pullback a*W is a variation of Hodge structure on Xu; hence the higher direct image V = Rn7r*(a*W) is a variation of Hodge structure on U C pN.

PROPOSITION 8.2. There is a number 6 such that, for any sufficiently ample choice of 1, the following holds: If W is a unitary rank-1 local system, the differential d1 of the classifying map for the variation of Hodge structure V has rank > N - 6 at generic points of U.

Proof. This is a simple variant of Green's Torelli theorem in [10]. His proof concerned the case of constant coefficients W ? C, but it extends immediately to the case of rank-1 twisted coefficients. Green shows that the map from the deformation space of X, to the tangent space of D has maximal rank. One can see that, for sufficiently ample , the tangent space of the linear system is equal to the infinitesimal deformation space of X, plus the space of tangent vector fields on Z that preserve ?. Let 6 denote the dimension of this space of tangent vector fields. The dimension of the infinitesimal deformation space of X, is at least N - 6. So by Green's theorem the generic rank of d1 is at least N -6. Note that by Noetherian induction we can choose L to work for all choices of W. OI

Choose a generic plane P2 C pN and let U1 = U n p2. Let V1 = V Iu1U



410 C. SIMPSON

COROLLARY 8.3. If L is chosen to be sufficiently ample and the plane p2 sufficiently general, then the differential of the classifying map for V1 has maximal rank at generic points of U1.

Proof. The differential of the classifying map on U1 is the restriction of d4 to U1. If we choose ? ample enough that N > 6 + 2 and choose a general P2, then this restriction will have maximal rank at generic points. L

COROLLARY 8.4. Suppose that ? is sufficiently ample and W is a rank-1 local system with a corresponding higher direct image V1 on U1. Let Vlnew be the irreducible component of V1 obtained by dividing out the trivial components (cf. Lemma 2.7). Then, in the trichotomy given by Theorem 7.5, case (c) holds: the monodromy group G has a connected component Go that is quasisimple of positive dimension.

Proof. Note that if ? is sufficiently ample, then dim(Vt,new) > 5, so The- orem 7.5 applies. In cases (a) or (b), there would be a finite covering U2 -+ U1 such that the pullback V2 of Vinew has a monodromy contained in a maximal torus. Then, by item (3) above, V2 decomposes as a direct sum of rank-1 variations of Hodge structure. But the classifying map for a rank-1 variation of Hodge structure is constant (as the classifying space D is a single point). Thus the classifying map 4v2 is constant, so the classifying map of V1 is constant. This contradicts the previous corollary. L

Suppose that S is an T'-completed covering of U1. Then the pullback of V1,new to the open set Us is again a complex variation of Hodge structure and, by item (2) above, the extended local system VSnew (given by Corollary 6.6) is a complex variation of Hodge structure on S. The conclusion of Corollary 8.3 still holds: the differential of the classifying map 4)S,new is of maximal rank at generic points of S.

LEMMA 8.5. The local system Vs,new on S satisfies the nonfactorization property NF1.

Proof. Suppose to the contrary that VSnew satisfies the factorization property F1. Then, since Vs is the direct sum of VSnew with a trivial representation, Vs also satisfies F1. Thus there are an irreducible quasiprojective variety Y and a quasiprojective curve C with maps g: Y -* S and p: Y -* C such that g*(Vs) = p*(Nc) for a local system N on C. We may assume (by going to the Stein factorization if necessary) that p has connected fibers.

Let C, be a smooth projective completion of the curve C and let Y1 be a projective completion of Y. Let Y2 be the closure of (1, g,p)(Y) in Y1 x S x C1. Let Y3 be a resolution of singularities of Y. There are maps, which we still




denote by g and p, from Y3 to S and C1, respectively. On the open set p 1(C) C Y3, the local systems p*(Nc) and g*(Vs) are isomorphic. This is because they are isomorphic on the open subset Y and the fundamental group of Y surjects onto the fundamental group of p-1(C). But the local system g*(Vs) is defined on all of Y3. It follows that the monodromy transformations of Vc around the points z E C0 - C have finite order (the order around z divides the greatest common divisor of the multiplicities of components of P 1(Z))

There exists a finite covering p: C2 -- C1, ramified at the points of C -C, such that so*(Nc) extends to a local system N2 on C2. The argument for this is similar to the argument of Lemma 6.3.

Let Y4 be a resolution of singularities of an irreducible component of Y3 xC0 C2. It again has morphisms g: Y4 -* S and p: Y4 -* C2, and we have g* (Vs) p* (N2). Furthermore p: Y4 -- C2 has connected fibers. The local system g*(Vs) is a variation of Hodge structure on Y4. We may apply remark (4) from the beginning of this section to conclude that N2 is a variation of Hodge structure on C2 such that the isomorphism p*(N2) _ g*(Vs) is an isomorphism of variations of Hodge structure. This implies that the classifying map for g*(Vs) factors through the map of universal coverings p: Y4 -* C2. In particular the real rank of the differential of the classifying map for g* (Vs) is at most 2. But g is a submersion, so the generic rank of the differential of the classifying map for g*(Vs) is the same as the generic rank of the differential of the classifying map for Vs, which in terms of real dimensions is 4. This is a contradiction. Thus Vs cannot satisfy F1; and neither can VS,new. Thus VS,new satisfies NF1, as claimed. O

Remark. The statement obtained in Corollary 8.4, that Go is quasisimple of positive dimension, is preserved by pulling back to finite covers (i.e., by going to subgroups of finite index). Hence this conclusion holds for the monodromy of the local system VS,new on S.

Generic families. Suppose that {Wt}tET is a family of rank-1 local systems on Z. Assume the following condition:

Condition. The subset of t E T such that Wt is unitary is Zariski-dense.

It is possible to choose T C MB (Z, 1) so that this condition is satisfied. The irreducible parts of the nth higher direct images form a collection

of local systems Vtnew. We may restrict everything to a Zariski-open subset T' C T so that the Vt,new form a family. Since the set of t E T corresponding to unitary local systems Wt is Zariski dense, the set of such t E T' is also Zariski dense. For each such t, the local system Vtnew underlies a complex variation of Hodge structure, and Proposition 8.2-Lemma 8.5 apply. The subset of t E T',



412 C. SIMPSON

such that case (c) in Theorem 7.5 holds, is thus constructible by Lemma 7.6 and Zariski dense by Corollary 8.4, so it must contain a Zariski-open subset. By making T' smaller, we may assume that case (c) always holds.

Choose an T'-completed covering S of P2 (Lemma 6.3). By Corollary 6.6 we obtain a collection of local systems Vt,s,new on S. By choosing S and T' appropriately (as in Lemma 7.8), we may assume that the monodromy groups Gt are connected and all conjugate to a single connected quasisimple group G. The group G acts irreducibly; in other words, the Vtsnew are irreducible. Making T' smaller and taking an etale covering T", we obtain a family of representations Pt: 7rI (S) -* G indexed by t E T" (Lemma 1.2). These representations are Zariski dense. Lemma 8.5 applies to a Zariski-dense subset of t E T", the subset of values corresponding to unitary local systems Wt: for these values of t, Vtsnew satisfies the nonfactorization property NF1. But if NF1 holds for Vtsnew, then it holds for the corresponding representation Pt into the monodromy group G. Hence there is a Zariski-dense subset of values t E T" such that Pt satisfies NF1.

By Lemma 8.1 the projections of these Pt into G/((G) also satisfy NF1.

Remark. We have not shown that F1 and NF1 are constructible properties. Clearly F1 is preserved under specialization. Hence, if {Pt} is a family of representations, then the subset of values of t, such that F1 holds, is a countable union of closed subvarieties of the parameter space; and the subset, where NF1 holds, is the complement of such a set.

Growth of the dimension of the monodromy group. Choose a rank-1 unitary local system W on Z such that the monodromy group of the corresponding extended higher direct image Vs on S is the generic G of the previous subsection.

LEMMA 8.6. Let 6 be as in Proposition 8.2. Then there is a bound dim(G) > 4(N - 6).

Proof. Let G' be the monodromy of the original higher direct image V on U; it has the same connected component as G. Note that V is a complex variation of Hodge structure. Let G' be the real Zariski closure of the image of 7rl(U) in G'. One can reduce the structure group of the variation of Hodge structure V to obtain a G'-variation of Hodge structure. This implies, in particular, that G' is a real form of G'. Furthermore we have a Hodge decomposition of Lie(G') ?R C = Lie(G'):

Lie (G') = @0

P




There is a compact subgroup K C G' with Lie algebra g0o?, giving a classifying space

D(G'R= Ga/K.

The classifying map for V factors through a classifying map

A : CU-+D (G'R

and T is holomorphic and satisfies Griffiths's horizontality conditions. In particular we have a bound

dimc(4D((U)) < dim?(xF(()) < i?0 )

On the other hand, the sum @02P-2P is the complexified Lie algebra of the maximal compact subgroup of G' , and there is a reflective symmetry dim(gPP) = dim(g-AP). Hence

dimc(g-1,1) < 1 dimc(G'). -4 Putting these together with the estimate of Proposition 8.2, we obtain the estimate dimc(G) > 4(N - 6). L

COROLLARY 8.7. Appropriately choosing the line bundle L allows the dimension of the generic monodromy group G to be made arbitrarily large.

Proof. We can choose ? to make N = dim Ho (Z, L) arbitrarily large. OL

Lemma 8.6 implies that, as L becomes more and more ample, the rank of the higher direct image V increases. However this argument is only valid for higher direct images of rank-1 local systems. In Lemma 5.5 we gave a different calculation that is valid for any W.

9. Applications

A counterexample. We first deal with the question our examples are designed to answer, namely whether the result of [26] generalizes to bigger groups. We present a natural hypothesis that would constitute such a generalization.*

Question 9.1. Let G be a simple group. Suppose that S is a smooth projective variety and {pt: 7ri (S) -- G}tET is a nonstationary family of Zariski- dense representations. Must the Pt then satisfy the factorization property F1 (cf. ?8)?

*The formulation (to me) of the question in this form is due to L. Katzarkov.



414 C. SIMPSON

The result of [26] says that this is true for G = PSL(2, C) (note that we may choose Y so that the pullback of the representation lifts into SL(2, C), so the statement of [26] applies).

If one takes a pullback of a family of local systems from a curve, then the resulting local system will satisfy F1. If one takes a tensor product of local systems, the resulting local system will have either a nonsimple monodromy group or the same monodromy representation as one of the factors (when projected modulo the center). Hence the easy examples of families of local systems mentioned at the beginning of the Introduction do not contradict this hypothesis.

The examples we have constructed show that the answer to Question 9.1 is no, so that the result of [26] does not generalize in this way. More precisely we can sum up some results of the previous sections as follows:

THEOREM 9.2. There exist a surface S and a family of representations {Pt: 7ri(S) -* G}teT into a simple group G such that all the Pt are Zariski dense, the family is nonstationary and the Pt satisfy NF1 for a Zariski-dense subset of t E T. The group G can be taken to have arbitrarily high dimension.

Proof. Fix an odd-dimensional smooth projective variety Z such that H1 (Z, C) / 0 and choose a line bundle ? sufficiently ample to satisfy the requirements of the rest of the argument. Take a generic 2-dimensional Lef- schetz pencil of hyperplane sections in the linear system determined by 4, as described in Section 6. Let S be a smooth, projective, T'-completed covering of ]P2, ramified over the discriminant locus of the Lefschetz pencil (Lemma 6.3).

Choose a family of rank-1 local systems {Wt}tET on Z and look at the corresponding collection of local systems Vt,s,new on S (formed from the higher direct images Vtnew under the Lefschetz pencil by pulling back to Us and extending to S cf. Corollary 6.6). There is an open subset T' C T such that { VtSnew}tET/ forms a family of local systems on S.

We may assume that the family of Wt was chosen so that the set of t, with Wt unitary, is Zariski dense. Then, as discussed in the subsection "generic families" in Section 8, we obtain an etale morphism T" -- T, a quasisimple group G and a family of representations {Pt: 7rl(S) -) G}teT". There is a representation v of G such that v o Pt are isomorphic to the monodromy representations of Vt,S,new. The images of the representations Pt are Zariski dense in G, and there is a Zariski-dense subset of t such that Pt satisfy NF1.

We must verify that the family of representations Pt is nonstationary. By Lemma 1.4 it suffices to verify that the family of Vt,S,new is nonstationary. There are two ways to do this. The first is to note that we could choose the family {Wt} to be nonstationary and apply Corollary 5.3 to conclude that the family of Vt,S,new is nonstationary. As this uses the analysis of Sections 3-5,




we indicate another way. We may choose the family {Wt}wtj such that the parameter space T is a smooth curve, with a point to E T, where the jumps in dimension defined in Section 2 satisfy 6nn+l(Z) > 6nn+2(Z). For example, this holds if Z is an abelian variety, Wto is the trivial local system and Wt is nontrivial for nearby values of t. Note that we can choose a family such as this that also satisfies the previous requirement that the set of unitary Wt be Zariski dense. Now Corollary 2.12 and Lemma 1.5 imply that the family of local systems Vt,S,new (for generic t E T) is nonstationary.

In order to obtain a family of representations into a simple group, project from G to the quotient by its (finite) center. All of the required properties are preserved (cf. Lemmas 1.7 and 8.1).

This completes the verification of the required properties. Note finally that by choosing ? sufficiently ample we can insure that the dimension of the group G is arbitrarily high. This was shown at the end of Section 8. 0I

Remark. Theorem II, stated in the Introduction, follows from this construction. Note that if p is a Zariski-dense representation that fits into a nonstationary family, then p is nonrigid.

Harmonic maps. The examples referred to in Theorem 9.2 provide examples of harmonic maps that do not come from variations of Hodge structure or maps to Riemann surfaces. This provides a positive answer to the question of Toledo referred to in the Introduction.

THEOREM 9.3. There exist a smooth projective surface S, a simple group G and an equivariant harmonic map

0: S - G/K,

where S is the universal covering of S and K C G is a maximal compact subgroup; then the following properties are satisfied: The map X does not factor locally through a holomorphic map from S to a Riemann surface; the image of X is not contained in any translate of a proper subspace of the form G1/K1 (for any real group G1 C G and maximal compact K1 C Gl); and X does not come from a variation of Hodge structure (a notion explained in the proof below).

Proof. By Theorem 9.2 there exist a surface S, a simple group G and a nonstationary family of Zariski-dense representations p: ir (S) -4 G satisfying NF1. Associated to each representation Pt there is an equivariant harmonic map qt: S -4 G/K (see [4]).

Let f: G c-+ GL(r, C) be a faithful representation of G. Then for each t we obtain a local system Vt. The harmonic bundle associated to Vt corresponds to an equivariant harmonic map

At: S -* GL(r, C)/U(r),



416 C. SIMPSON

related to the previous maps by At: f o Ot. One says that Ot comes from a variation of Hodge structure if the harmonic bundle Vt underlies a complex variation of Hodge structure. In this case, O/t is a projection of the classifying map of the variation of Hodge structure. We say that Ot comes from a variation of Hodge structure if the composition O/t does. From the discussion of [25], it follows that this condition is independent of the choice of the representation f.

The classifying map for a variation of Hodge structure satisfies the Grif- fiths transversality condition, which implies that the image is transverse to the fibers of the map D -* GL(r, C)/U(r). Hence the dimension of the image of the classifying map is the same as that of the associated harmonic map. Recall from the proof of Theorem 9.2 (and Proposition 8.2) that, for some values of t, the representation Pt comes from a variation of Hodge structure where the differential of the classifying map is generically injective. Thus the image of the classifying map has real dimension 4-and the same is true for the image of the harmonic map Qt. The harmonic maps vary continuously with t, so there is a usual open set To C T such that the dimension of the image of Ot is 4 for t E To. In particular Ot cannot factor locally through a map to a Riemann surface.

The functions tr(Adpt(Qy)) are algebraic, hence holomorphic, functions of t. They are not all constant, since the family is nonstationary (Corol- lary 1.8). Hence there is a nonempty open set of values T1 C To such that tr(Adpt(y)) are not all real numbers.

Suppose from now on that t E T1. This implies that Pt (ri (S)) is not contained in any real form of G (otherwise the real form of the Lie algebra would be preserved by the adjoint representation, so the tr(Ad Pt(y)) would be real). The same argument works for the composition with the representation f, so f o pt(7ri (S)) is not contained in any real form of GL(r, C). In particular f ? Pt cannot be the monodromy of a complex variation of Hodge structure, for then it would be contained in some U(p, r - p) C GL(r, C). Thus the qt do not come from variations of Hodge structure.

Finally suppose that the image of kt is contained in a translate of a proper subspace G1/K1. After conjugating by an element of G, we may assume that the image is contained in G1/K1 with K1 = GC n K. We may also assume that the image contains the identity coset. Then the equivariance of Ot implies that pt(7ri (S)) C Gi . K. In particular, the real algebraic Zariski closure of Pt(7rli (S)) is a proper subset of G. The real algebraic closure is a real algebraic subgroup H c G, which is complex-Zariski dense. The real Lie algebra Lie(H) spans Lie(G) over C. Suppose that u E Lie(G). Then we can write u = al +ia2 with a, and a2 in Lie(H). Suppose that v E Lie(H) n i Lie(H). Then

[U, V] = [al, V] + i[a2, v],




and both terms are again in Lie(H) n i Lie(H). Hence Lie(H) n i Lie(H) is a normal (complex) subalgebra. Since G is simple, Lie(H) n i Lie(H) = 0. This implies that H is a real form of G. This contradicts the fact, established above, that the image of Pt is not contained in any real form of G. Hence the image of qt is not contained in any proper subspace G1/K1. a

Examples of the Riemann-Hilbert correspondence in elementary form. Suppose that R is a smooth variety and (N, V) is a vector bundle of rank k with an integrable relative connection on S x R over R. This means that

V: N--N0QXxR/R

satisfies Leibniz's rule and the integrability condition V2 = 0. This may be considered as an algebraic family of vector bundles with integrable connections {(Nr, Vr)} indexed by r E R. The sheaf of sections u of Nr, such that Vr (u) = 0, forms a local system 14 on S. These local systems vary analytically with r. The map from the set of vector bundles with integrable connections to the set of local systems is known as the Riemann-Hilbert correspondence. In the present situation, it is described by an analytic map R -* MB(S, k) sending r to the point corresponding to the local system 14 (note, however, that this map completely describes the isomorphism classes of the Vr only in case they are semisimple). In general this analytic map will be given by coordinate functions that are highly transcendental.

Suppose that k = 1 and the underlying vector bundle N is trivial, N =

OSxR. Then the connection can be described by V = d + A, where A is a section of Q If1,.. , Wk is a basis for Ho(S, Q ), then we can write A = ailw + . + akWk with ai(r) algebraic functions on R. If ̂ y E 7r,(S), the monodromy around -y of the connection Vr is given by

Pr(Y) = exp ( ai(r) ji Thus the coordinate functions for the map R -+ MB(S, 1) are exponentials of algebraic functions on R.

Our methods allow us to obtain some examples in higher rank.

THEOREM 9.4. There exists an algebraic variety S with an algebraic family of rank-k vector bundles with integrable connections {(Nr, Vr)}rER such that the analytic map R -* MB(S, k), given by the corresponding family of local systems {Wr}rR, is nonconstant and is given by coordinate functions that are polynomials in exponentials of algebraic functions on R. The local systems Wr are irreducible with quasisimple monodromy groups that can be chosen to have arbitrarily high dimension.



418 C. SIMPSON

Proof. We obtain the (N, V) by taking higher direct images of connections on trivial bundles of rank 1, as in the previous examples.

The higher direct images of a vector bundle with integrable connections can be defined in algebraic terms (similar to the definition for Higgs bundles given in Section 6; the method for constructing the connection on the hyperderived direct image of the relative de Rham complex is that of [12]). This definition is compatible with the Riemann-Hilbert correspondence and the process of taking higher direct images of the corresponding local systems. We can also make sense of the family of higher direct images of a family of vector bundles with connections if the higher direct images all have the same rank.

Suppose that {(Oz, Vr)} is an algebraic family of connections on the trivial bundle over Z. corresponding to the analytic family of local systems {Wr}. Choose a family of hyperplane sections, as described in Section 2, and let S be an T'-completed covering of p2, branched over the discriminant locus. Let (Nro, Vr) denote the higher direct images on Us of the pullbacks from Z to Xus of (Oz, Vr). Each Nr has a unique extension to a vector bundle with an integrable connection on S. By an argument involving constructible sets, there is a Zariski-open subset R' C R such that the {(Nr, Vr)}rER1 form an algebraic family. The corresponding local systems are the extended higher direct images Vr. The coordinates of the points in MB(S, k') corresponding to Vr are expressed algebraically in terms of the local systems Wr. Hence the map R' MB(S, k') given by the Riemann-Hilbert correspondence applied to {(Nr, Vr)}rERt has coordinates that can be expressed as algebraic functions of exponentials of algebraic functions on R'.

Finally note that the image of the set of connections of the form (Os, V) in MB (S, 1) is Zariski dense, so the monodromy group of the generic example of the form (Nr, Vr) is equal to the generic monodromy group discussed in Section 7. Thus Corollary 8.7 tells us that by choosing L appropriately, we can make dim(G) arbitrarily large.

Remark. We have ignored the question of whether the original rank-i objects (Oz, V) can be recovered uniquely from their higher direct images. Corollary 5.4 shows that, in our examples given by sufficiently ample hyperplane sections, the map from the set of rank-i local systems to the set of higher direct images is quasifinite. Hence the original (Oz, V) can be recovered from the higher direct image (N, V) by solving some algebraic equations. Thus the correspondence between the isomorphism classes of Wr and (Nr, Vr) can be expressed in elementary terms if one allows the solution of algebraic equations.

Remark. The correspondence between general rank-i bundles with connections and rank-i local systems can be expressed in terms of functions that




are slightly less elementary than exponential functions, namely exponentials of integrals of meromorphic forms on Z x R/R. Call these quasielementary functions. They are the coordinate functions of the isomorphism of complex analytic manifolds MB(Z, 1) A MDR(Z, 1). By taking higher direct images, we again get examples of the Riemann-Hilbert correspondence expressed in terms of these functions.

Higgs bundles corresponding to local systems in elementary form. A similar situation occurs for the correspondence between Higgs bundles and local systems. For the case of rank 1, recall that (assuming H1 (X, Z) is torsion free)

MB (X, 1) = ((C*)2g U(1)2g X (R+*)2g

and MDol(Xi 1) OVA X C9,

where A = Pico(X) and C9 = HO(X, Q1) The correspondence between Higgs bundles and local systems is the product of real-analytic group isomorphisms A V U(1)29 and C9 r (JR+*)2g. The first is given by some exponentials of integrals of differential forms of the second kind on X; the second is given by the exponential composed with a linear isomorphism C9 rv R2g whose matrix is composed of period integrals. In particular the morphism C9 -) MB is given in an elementary way in terms of complex conjugation and exponentials. The full isomorphism MDo0 l MB is given in terms of some types of quasielementary functions (here one may have to expand slightly the class of quasielementary functions).

Again we obtain examples, where the correspondence between Higgs bundles and local systems can be expressed in elementary (or quasielementary) terms, by taking higher direct images of families of rank-1 local systems.

THEOREM 9.5. There exists an algebraic variety S with an algebraic family of rank-k Higgs bundles {(Fr, c0r)}rER (stable Higgs bundles with vanishing Chern classes) such that the continuous map R -4 MB(S, k), given by the corresponding family of local systems {Vr}rER, is nonconstant and is given by coordinate functions which are polynomials in exponentials of polynomials of algebraic functions and their complex conjugates, on R. The local systems Vr are irreducible, with quasisimple monodromy groups that can be chosen to have arbitrarily high dimension.

Proof. Apply the same procedure as used for constructing the example of Theorem 9.2, starting with a family of rank-1 Higgs bundles of the form (Er, Or) such that Er rv( Oz and Or are a family of holomorphic 1 forms. The corresponding family of local systems Wr is expressed in terms of exponentials



420 C. SIMPSON

of real parts of linear functions of 0r E HO(Z, QD). Note that the space of local systems on Z that can arise is a real form of MB, hence it is Zariski dense.

We obtain a variety S, with an open set Us, and a family of harmonic bundles VUsr on Us, which are higher direct images of pullbacks of Wr. (Note that there is a Zariski-open set of r, where the ranks of the higher direct images are constant, and this maps into a Zariski-open set of MB(Z, 1), where the ranks of the higher direct images of the local systems are constant; thus the collection of higher direct images can be made to fit together into a family.) In our example, the VUsr extend to harmonic bundles Vr on S. By Theorem 4.13 and the subsequent remark, the associated Higgs bundles Fr on S may be expressed algebraically in terms of the Er. Note that the higher direct images we construct are irreducible, so the corresponding Higgs bundles Fr are stable with vanishing Chern classes. Furthermore, by Corollary 8.7, the monodromy groups can be chosen to have arbitrarily high dimension.

Remark. If we allow the line bundles Er to vary, then we obtain some more families where the corresponding local systems can be expressed in quasielementary terms. a

We might hazard an outlandish conjecture (as usual, contained in the class of "infinitely hard" conjectures that posit the existence of some geometric objects):

Conjecture 9.6. Any nonstationary family, where the correspondence between Higgs bundles, local systems and vector bundles with an integrable connection is expressed in quasielementary terms, is obtained from the class of families of rank 1 by repeated higher direct images, extensions, inverse images, tensor products and direct sums, and by taking subquotients.

Remark. In all of the cases discussed above, there are analytic or C' isomorphisms between the vector bundles underlying the local systems, integrable connections or Higgs bundles. These isomorphisms are probably not expressed in elementary terms only the isomorphism classes of the objects are.

In order to obtain examples where the Riemann-Hilbert correspondence could be expressed in elementary terms, we had to start with vector bundles with a connection of the form (N, V), where N 2v Oz. Similarly, in order to obtain examples where the correspondence between Higgs bundles and local systems could be expressed in elementary terms, we had to start with Higgs bundles (E, 9) with E rv Oz. However there are not any nontrivial families of rank-1 local systems such that the Higgs bundles and the vector bundles with a connection both have this form. (This can be seen infinitesimally at




the identity local system it amounts to the statement that the subspace of the Hodge filtration does not meet the subspace of real cohomology classes in H1(X, C).) Admittedly this is rather weak justification, but we can make the following conjecture anyway:

Conjecture 9.7. There are not any nonstationary families such that the correspondence between local systems, families of vector bundles with integrable connections and Higgs bundles is expressed in elementary terms.

Coutnterexamples to the inverse scattering problem for the Riemann- Hilbert correspondence. Suppose that X is a compact Riemann surface and {(Nr, Vr)}rER is an algebraic family of vector bundles with an integrable connection on X. Then the Riemann-Hilbert correspondence gives an analytic map R -? MB(X). One can view this as the result of a scattering process: the differential equation V(u) = 0 "scatters" the solution when it is continued around a loop in the fundamental group; the "settings" of the machine are the parameters r E R and the scattering data are the monodromy representations.

This leads to an inverse-scattering question (cf. [24] for another more special version). Let I' be a Riemann surface group. Given an algebraic variety R and an analytic map R -* MB(f), suppose that this comes from a family of vector bundles with an integrable connection on a Riemann surface X. Does the analytic map determine the Riemann surface and the family of connections?

It seems possible that the answer could be a qualified yes, in some cases involving representations of rank 2 (this feeling comes from some special results for the rank-2 case in [24]). However the examples we have constructed (stated in Theorem 9.2) show that the answer is no in the higher-rank case, even in spite of various obvious qualifications that could be made. For there is a nonstationary family of vector bundles with the integrable connection (Nr, Vr) on a surface S. Then, for any curve X C S, the monodromy of the restriction (Nr, Vr) Ix is given by the composition of the monodromy on S and the morphism 7ri(X) -* 7rI(S). The morphism of fundamental groups remains unchanged if X is moved in a continuous family; so we can obtain several different curves such that the analytic maps R -4 MB(F) = MB(X) are the same.

One might still hope to recover, from the scattering data R -4 MB1(F), a universal variety S with a family {(Nr, Vr)} such that any family of equations on a Riemann surface X, which gives the same scattering data, comes by pulling back via a map X -4 S.

This inverse-scattering question is closely related to Question 9.1. The apparent hope of solving the inverse-scattering problem in the rank-2 case originally led me to the result of [26].



422 C. SIMPSON

Factorization through higher-dimensional varieties. The examples we have constructed in this article are complementary to work (independently) of Zuo and of Katzarkov. Their aim is to narrow the possibilities for having families of representations of 7ri(X) as a function of geometric properties of X. One can hope to have, in the future, a fairly complete picture of what can or cannot happen.

Zuo informed me in June, 1991 of his conjecture that something like the following, a weaker statement similar to Question 9.1, should be true. It is apparently the correct generalization of the result of [26].

Conjecture 9.8 (Zuo). Suppose that G is a simple group. There is a number ((G) such that the following holds: Suppose that S is a smooth projective variety and {Pt: 7r,(S) -4 G}tET is a continuous family of Zariski- dense representations that are not conjugate to each other. Then there exist a variety Y with a generically surjective map g: Y -* S, a quasiprojective variety C of dim < ((G) and a map p: Y -4 C such that g*(pt) factors through p*: 7i1(Y) -4 71r(C).

Zuo has proved this (or, rather, the corresponding statement for the case of a group that is quasisimple) for G = SL(3) and ((G) = 2 (see [29]). He gives a more precise statement about which coverings Y are needed. He is working on the general case.*

Katzarkov has obtained some results in this direction, including a factorization statement for any group, but one that assumes some other conditions (see [17]). Katzarkov gives some geometric properties of Higgs bundles that have simple monodromy groups, but do not factor through curves.

Here are a few remarks which may serve as a bridge between our formulation and that of Zuo. In keeping with the notation established in Section 8, we could call the conclusion of the conjecture a factorization property F((G). We could include without loss of generality the hypotheses that the map g is smooth and topologically a fibration, with connected fibers. This is because Y and C can always be replaced by Zariski-open subsets. The condition that the representation g*(Pt) factor through p*: 7r,(Y) -4 7ri(C) is then equivalent to the apparently weaker condition that, when restricted to the fibers Yc = p-1(c) c Y, the representation g*(Pt) have a monodromy group smaller than G. To see this, note that since g is topologically a fibration, the fundamental group of the fiber is a normal subgroup of 7ri(Y) with quotient 7rl (C). The Zariski closure of g*(pt)(7rq(Yc)) is therefore a normal subgroup of G.

*Added in revision. Zuo has now proved this for G = SL(n, C) with n = n-1; it looks as if ((G) should be the rank of G in general.




Since G is simple, this implies that g* (Pt) (-l (Ye)) is trivial and hence that the representation factors through 7ri(C). In his treatment of the SL(3) case, Zuo has concluded that there exist a covering Y and map p to a curve or surface such that the representation becomes reducible when restricted to the fibers. This implies a reduction of the monodromy group.

Directions for further research. Finally we will enumerate various prob- lems for further research that are suggested by the things done above.

First of all, it should be possible to analyze a wide variety of examples by the methods presented in Sections 3, 4 and 5. This includes examples where the map f: X -* S is obtained not from a Lefschetz pencil, but in some other way. For example, one can construct smooth morphisms of projective varieties where the fibers are curves, and it might be interesting to see what happens in those examples.

Second, the examples we have constructed show that there are some fundamental groups of smooth projective varieties which are interesting from the point of view of their representations. It might be worthwhile to try to calculate 7ri(S) precisely in some of the examples we have constructed or, for example, to calculate rl (PN - D).

Conjecture 9.8 should be proved, if possible, with a more precise description of the intermediate space Y in the factorization (in the manner of [29]). Once this is done, it will be interesting to try to find sharp bounds for the dimension ((G) of factorization. The first thing would be to try to construct examples of families of local systems on higher-dimensional smooth projective varieties, not factoring through varieties of smaller dimension. One would hope to show that the ((G) cannot be bounded uniformly in G. It would also be nice to know what is the smallest group G occurring in our (or other) examples.

More ambitiously, one would hope to have a complete picture of what kinds of representations can exist. The best situation would be to have a reasonably concrete list of varieties C with representations p of Inl(C), such that any representation of 7ri(X) is a pullback of one of the p by a morphism X -* C (if necessary, admitting an intermediate space X +- Y -* C, which is under control). One part of this problem is to do the same for nonstationary families of representations. To complete the statement, something like the conjecture of [25] (that rigid representations are motivic) would be needed.

The most technical direction for further work is in the subjects treated in Sections 3 and 4. The subjects of those sections are generalizations to harmonic bundles of some things that have been known for some time for variations of Hodge structure. It would be good to continue to generalize what



424 C. SIMPSON

is known from Hodge theory, for example, to find some kind of generalization of the Clemens-Schmid exact sequence for higher direct images of harmonic bundles or to analyze the degenerations in more detail.

Once the theory of tame harmonic bundles is satisfactorily extended to higher-dimensional quasiprojective varieties, it will be necessary to give a general treatment of the higher direct images of a tame harmonic bundle by any map of quasiprojective varieties f: X -4 S. Even further in this direction would be a Dolbeault theory to parallel the theory of D-modules and perverse sheaves. This is probably worth pursuing, because it is sometimes easier to compute in the Dolbeault theory than in the de Rham or topological Betti theories (our proof of Theorem 1 provides an example of this principle).

One might make the following conjecture: that the decomposition result of Beilinson, Bernstein, Deligne and Gabber holds for the higher direct image of any irreducible perverse sheaf by a projective morphism. Note that Corollary 4.5 provides such a result in the case of the higher direct image of an irreducible locally constant sheaf W on X by a smooth projective morphism, from an open set in X to an open set in a variety S.

UNIVERSIT9 PAUL SABATIER (C.N.R.S.), TOULOUSE, FRANCE

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(Received November 8, 1991)



some families of local systems over smooth projective varieties

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