hirzebruch's proportionality theorem in the non-compact case · inequalities needed for the...

Inventiones math. 42, 239- 272 (1977) /F/ve//t/onr mathematicae �9 by Springer-Verlag 1977

Hirzebruch's Proportionality Theorem in the Non-Compact Case

D. Mumford

Harvard University, Department of Mathematics Cambridge Massachusetts 02138, USA

Dedicated to Friedrich Hirzebruch

In the conference on Algebraic Topology [-7] in 1956, F. Hirzebruch described a remarkable theorem relating the topology of a compact locally symmetric variety:

x =o/r, D = bounded symmetric domain, F=discrete torsion-free co-compact group of automorphisms of D

with the topology of the extremely simply rational variety /), the "compact dual" of D. (See w for full definitions.) His main result is that the Chern numbers of X are proportional to the Chern numbers of/5, the constant of proportionality being the volume of X (in a natural metric). This is a very useful tool for analyzing the structure of X. Many of the most interesting locally symmetric varieties that arise however are not compact: they have "cusps". It seems a priori very plausible that Hirzebruch's line of reasoning should give some relation even in the non-compact case between the chern numbers of X and of/5, with some correction terms for the cusps. The purpose of this paper is to show that this is indeed the case. We hope that the generalization that we find will have applications.

The paper is organized as follows. In w 1, we make a few general definitions and observations concerning Hermitian metrics on bundles with poles and describe an instance where such metrics still enable one to calculate the Chern classes of the bundle. This section is parallel to work of Cornalba and Griffiths [6]. In w 2, which is the most technical, we prove a series of estimates for a class of functions on a convex self-adjoint cone. In w 3, the results of w 1 and w 2 are brought together, and the Proportionality Theorem is proven. One consequence is that D/F has the property, defined by Iitaka [8], of being of logarithmic general type. Finally, in w 4, we analyze the step from logarithmic general type to general type and reprove a Theorem of Tai that DIE is of general type if F is sufficiently small.

240 D. Mumford

w 1. Singular Hermitian Metrics on Bundles

In this section we will not be concerned specifically with the locally symmetric algebraic varieties D/F, but with general smooth quasi-projective algebraic varieties X. When X is not compact, we want to study the order of poles of differential forms on X at infinity, and when E is moreover a vector bundle on X, we want to study Hermitian metrics on E which also "have poles at infinity". This situation has been studied by Cornalba-Griffiths [6]. The following idea of bounding various forms by local Poincar6 metrics on punctured polycylinders at infinite is due to them. More precisely, we choose a smooth projective compactification .~:

X c X

where ) ~ - X is a divisor on X with normal crossings.

Then we look at polycylinders:

[A =uni t disc] A ~ ) f \ r = d i m X !

where A r n ( X - X ) = ~ uni~ of coordinate hyperplanes~ {Z 1 = 0 , Z 2 = 0 , . . . , Zk=O .}

hence:

Ar n X = ( A * ) k x A r-k.

In A* we have the Poincar6 metric:

ds 2 = Idzl 2 [zl2(log Izl) 2

and in A we have the simple metric [dzl 2, giving us a product metric o n (A*) k x d ' -k which we call co tv).

Definition. A complex-valued C ~ p-form I/on X is said to have Poincar~ growth on X"- X if there is a set of polycylinders U, c ,~ covering A"- X such that in

Hirzebruch's Proportionality Theorem in the Non-Compac t Case 241

each U,, an estimate of the following type holds:

[q(tl, . . . , tp)l 2< C, ''(p)t'wv.~ol, tO . . . . . c~(mtv., p, tp)

(a l l t 1 . . . . , tp tangent vectors to X at some point of U~ n X). it is not hard to see that this property is independent of the covering U, of

- X (but unfortunately it does depend on the compactification .~). Moreover, if ~1, q2 both have Poincar6 growth on ) ~ - X , then so does ~/1 A q2. This leads to the basic property:

Proposition 1.1. A p-form 11 with Poincard growth on X - X has the property that for every C ~ (r-p)- form ~ on X,

j" I~A~I< +oo X - X

hence t l defines a p-current [t/] on f(.

Proof Since ( has Poincar6 growth, we are reduced to checking that if t/is an r- form with Poincar6 growth, then

In a polycylinder U,, this amounts to the well-known fact that for all relatively compact V c c U , , the Poincar6 metric volume of Vc~(A*kxA "-k) is finite. QED

Definition. A complex-valued Coo p-form ~/ on X is good on f f if both q and dq have Poincar6 growth.

The set of all good forms q is differential graded algebra for which we have the next basic property:

Proposition 1.2. I f q is a good p-form, then

d([~]) = [d , ] .

Proof. 1 By definition of d([q]), this means that for all Coo(r -p -1 ) - forms ( on 2,

J d ~ ^ ~ = - S q^d~. X - X X - X

This comes down to asserting that if U, is a tube of radius e around _~ - X. then

lira ~ (q A r = O. ~ 0 0 0 ~

If we take, for instance, r = 2 and set up this integral in local coordinates x, y on -~ near a point where ) ( - X has 2 branches x = 0 and y = 0 , then this comes

i Compare Cornalba-Griffi ths [6], p. 25

242 D. Mumford

down to the assertion

[dx] 2 ]dyl =0 lim x 5>_~ Ixla(log Lxl) 2 lYI" I(log Lyl)I

which is easy to check. The general case is similar. QED

Next, let/~ be an analytic rank n vector bundle on )?, let E be the restriction of/~ to X and let h: E ~ I E be a Hermitian metric on E. For such h we define "good" as follows:

Definition. A Hermitian metric h on E is good on X if for all x e X - X and all bases e~ . . . . , e, of/~ in a neighborhood A ' of x in which J ? - X is given as above

k

by Iq zi=O, if hij=h(ei, ej), then i = 1

i) [hJ,(deth)-l<=C log]zll , f o r s o m e C>0, n>__l, i =

ii) the 1-forms (Oh. h-1)ij are good on 2c~ U.

The first point about good Hermitian metrics is that given (E, h), there is at most one extension/~ of E to 2 for which h is good. This follows from:

Proposition 1.3. I f h is good, then for all polycylinders A~c2 in which X - X is k

given by l~ zi=0, i = I

F(A', ft.) = {s~F(A" c~ X, E)] h(s, s) <= C . (~. log Izzl) 2n, for some C, n}.

Proof The inclusion " c " is immediate. As for " ~ " , if s= ~a i ( z ) e ~ is a i = 1

holomorphic section of E on A r n X , for which h(s, s) is bounded as above, then it follows that

la~(z)l _< C'(~ log Izil) 2m, for suitable C', m.

Therefore z~ �9 ai(z ) is bounded on A r, hence is analytic, hence at(z ) is

meromorphic with simple poles on J ? - X . But as no inequality

1 iz12 ~ C(log Izl) zn

holds, at(z) is in fact analytic. QED

The main result of this section is the following:

Theorem 1.4. I f ft. is a vector bundle on X and h is a good Hermitian metric on E =/~lx, then the Chern forms Ck(E, h) are good on $2 and the current [Ck(E, h)] represents the cohomology class Ck(ff~)~H2R(x, tI~).

Hirzebruch's Proportionality Theorem in the Non-Compact Case 243

Proof. Let h* be a C ~ Hermitian metric on/~. Define

O=dh . h -1, O* =dh* . h *-1,

K=~O, K* = 00".

Intrinsically, K and K* are Hom(E,E)-valued (1,1)-forms and 0 - 0 " is a Horn (E, E)-valued (1, 0)-form. According to results in Bott-Chern [5], for each k there is a universal polynomial Pk with rational coefficients in the forms K, K* and 0 - 0 * such that on X:

Ck(E, h) - Ck(E, h*) = d(Tr Pk(K, K*, 0 - 0")).

Now K, K* and 0 - 0* are forms good on 9(, hence the (k, k)-form Tr Pk(K, K*, 0 -0") is good on 3(. It follows that Ck(E, h) is good on )~ and that

[Ck(E , h)] = d [ t r Pk] + [ek(E, h*)]. QED

represents c~(g)

w 2. Estimates on Cones

The results of this section are purely preliminary. We have isolated all the inequalities needed for the general proportionality theorem which involve only the cone variables (cf. w 3, definition of Siegel Domain), and worked these out in this section.

The object of study then is a real vector space V and

CcV,

C an open, convex, non-degenerate ((7~(pos. dim. subspace of V), or equivalently, 31eV, />0 on C) cone. Most of our results relate only to those C which are homogeneous and self-adjoint; for any C, we let G c G L ( V ) be the group of linear maps which preserve C, and say C is homogeneous if G acts transitively. If, moreover, there is a positive-definite inner product ( , ) on V for which

(~= {x~V] (x, y ) > 0 , all y~C}

we say C is self-adjoint. The classification of these is well known (see [1], p. 63), as in the fact that all such arise by considering formally real Jordan algebras V and setting

C = {x 2 [ x~ V, x invertible}.

All convex non-degenerate cones C carry several canonical metrics on them. First of all, there is a canonical Finsler metric on C, which is analogous to the Caratheodory metrics on complex manifolds ([10], p. 49):

V x ~ C , t~Tx, c~- V, let:

]/(t)] p~(t)=sup .

l~C l(x)

244 D. Mumford

(Another canonical Finsler metric, analogous to Kobayashi's metric in the complex case, can also be defined�9 First introduce on the positive quadrant IR+

�9 dx 2 dy 2 x IR+ the metric ~ + ~ - ; then we have on any cone the definition:

, fcanonical length in ) Px(t)=~the cone C ~ ( R x + R t ) ~

= -~ a~

if a 1 > 0 and a2<O are determined by x + a 1 t, x + a 2 teC, q~C. We won't need this second metric however.)

The advantage of the Finsler metric is that (as in the complex case) it behaves in a monotone way when you replace C by a smaller (or bigger) cone:

P r o p o s i t i o n 2.1. i) I f C is" an open convex non-degenerate cone in V, and a t C, x6C, t6V, then

px+,(t)c <= px(t)c.

ii) I f CI c C z are 2 open convex non-degeherate cones in V, then for all x6C~, t~V,

px(t)c~ > px(t)c2.

(The proofs are easy.) Now suppose C is homogeneous and self-adjoint. Then one can introduce a

Riemannian metric on C as follows. Chose a base point e~V which we take as the identity for the Jordan algebra and let ( , ) be an inner product on V in terms of which G=tG. Then ([1], p. 62), C is self-adjoint with respect to ( , ) . Moreover, K =Stab(e) is a maximal compact subgroup of G and if g =1 @ p is a Cartan decomposition with respect tO K and * : G ~ G the Cartan involution, then:

1) (gx, g * y ) = ( x , y ) , hence K=exp(~) acts by orthogonal maps while P = exp (p) acts by self-adjoint maps,

2) (gx)- 1 = g*(x- 1) (here x - 1 is the Jordan algebra inverse).

Now identifying Te, c with V, we use ( , ) to define a Riemannian metric on C at e: since it is K-invariant~ it globalizes to a unique G-invariant Riemannian metric on C, which we write ds~. For later use, we need the following formula:

Lemma. Take tl, t2 ~ Tx, c, and let f ( x ) = ( tz,X-1) where x-1 is the Jordan algebra inverse�9 Then

ds2(tl, t 2 ) = - D , ~ f

where Dr, is the derivative of functions on V in the direction t 1.

Proof Let g~exp(p) carry e to x. Then by G-invariance:

ds 2, x(tl ' t2 ) = ds 2, ~(g- 1 tl ' g- 1 t2 ) = ( g - i tl ' g - I t2 )


and

D , f ( x ) = Dg i t~(fo g)(e).

But

f o g(x) = (t2, (gx) -1 )

= (t2, g-1 . (X- 1)) = ( g - l t 2 , X-1 ).

If 6xe V is small, then (e + 6x) -~= e - 6 x +(terms of lower order), hence

Dg-ltl(fo g ) ( e ) = - (g-~ t 2, g-1 t~). QED

On a homogeneous cone C, any 2G-invariant metrics are necessarily comparable, so we can deduce, from the monotone behavior of the Finsler metric p, a weaker monotonicity for ds~:

Proposition 2.2. i) I f C is a self-adjoint homogeneous cone in V and a t (_., then there is a constant K > 0 such that

ds~,x+,(t,t)<=K.dS~,x(t,t ) all teV, xeC .

ii) I f C 1 c C2 are 2 self-adjoint homogeneous cones in V, then there is a constant K > 0 such that

dsZc2,x(t,t)<K.ds~,,x(t,t), all teV, x e C 1.

(The proofs are easy.)

The main estimates of this section deal with the following situation:

C a self-adjoint homogeneous cone, G = Aut~ C) (o means connected component), C, = cone of positive definite n x n Hermitian matrices, p: G --* G L(n, C) a representation, H: C ~ C, an equivariant, symmetric map with respect to p.

Here (p, H) equivariant means:

H(gx)=p(g)H(x)tp(g) , all xeC, geG, (1)

while (p, H) symmetric means:

p(g*)=H(e ) . ' p (g ) - l .H(e ) -1, all geG. (2)

Note that A~-.H(e).LA-1~ H(e)-1 is just the Cartan involution of GL(n, ~) with respect to the maximal compact subgroup given by the unitary group of H(e): calling this ~, we can rewrite the symmetry condition (2):

p(g*) = p(g)". (2')

246 D. Mumford

Condition (2) is actually independent of the choice of e: if e 'eC is any other point, and g~--~g*' is the corresponding Cartan involution then one can check that (1)+ (2) imply:

p(g*') = H(e')tp(g)- ' H(e')- '. (2")

We will also need for applications the slightly more general situation where p is fixed but H depends on some extra parameters t ~ T with T compact. In this case, we ask that (1) and (2) hold for each H t. Note that we can change coordinates in Cn, to get a new pair:

p'(g) =ap(g)a- 1,

n~(g)=aHt(g)'5,

satisfying the same identities. In this way, we can, for instance, normalize the situation so that

H,o(e)=l.

hence:

p(K) ~ U(n),

, , (self-adjoint matrices, commuting~ ptexp 0) ~ I.~with Hi(e), all t ;

This normalization will not affect our estimates. The first of these is:

Proposition 2.3. For all 2 > O, there is a constant K > 0 and an integer N such that

II~(x)ll and IdetH~(x)l-l<K.(x,x) ~, all x~ (C+2 .e ) .

Proof. Let A~exp(p ) be a maximal R-split torus. Then C = K .A . e and C + 2 e = K . (A e + 2 e). Write x = k(a(e)). Then

Ilnt(x)ll = IIn,(ae)ll

= lip(a)" Hi(e). p(a)ll

IIp(a)ll 2. Ilnt(e)ll.

We may change coordinates in Cn by a unitary matrix so that all the matrices p(a) are diagonalized. Write then p(a)=zi(a)-6ij where Zi: A ~ R* are characters. Now we may coordinatize A by

A ~" , A . e ~ - R ' + c V , e~--~(1,...,1).

Let a~,. . . , a r be coordinates in R'+. Then

zi(a)= I~I aj',, sijER. jffil


Note that if ae~C+2e , then ae~F,+ +(2 . . . . . 2), i.e., ai>2, all i. Hence

Up(a)[] 2 < max (zi(a)) z l < i < n

r

<= K 1 �9 K 2 ((ae, ae)) ~ . . . . ,j)r

hence

IlHt(kae)ll < KI " Kz" Ka ((kae, kae)) t . . . . . j)r.

The same proof works for [det[-1. QED

Next, let ~s V, and let D~ be the derivative on V in the direction ~. We wish to estimate the matrix-valued functions

(DeH,). H/~: C ~ Mn(~).

To do this, we prove first:

Proposition 2.4. For all 1 <_<_a, fl <=n, let (Dr t. H;-1)~p be the (~,fl)th entry in this matrix. There is a linear map

C~t~,t: V---~ V

depending continuously on t such that

(Dell,. H;- 1),~(x)= ( C,e,,(~), x - 1>.

Moreover C,g,, has the property:

For some reason, I can't prove this by a direct calculation, but must resort to the following trick:

Lemma. Let C c V be a convex open cone, let e~C, and let f: C--~r be a differentiable function. Suppose that for all W = V, dim W = 2, and e~ W, f is linear on C c~ W. Then f is linear.

Proof The hypothesis means that

f (ax+be)=af (x )+b f (e ) , all x~C, a,b~IR+.

So we may extend f to all of V by the formula

f ( x ) = f ( x + a e ) - a f ( e ) , provided x + a e ~ C .

248 D. Mumford

Note that

f ( t x ) = f ( t x + t a e ) - at f (e)

= t ( f ( x + a e ) - af(e))

= tf(x).

Thus for all x~V, n=>l:

f ( x ) :__f(x/n) - f (O) n

. ' . f (x)=Dxf(O)

and the right hand side is linear in x. QED

We prove (DcHt. H;-1),o(x -1) is bilinear in ~,x. Since it is linear in ~, it suffices to find a basis {~k} of V such that (DckH t �9 Hi- l) ,~(x- 1) is linear in x. In fact, we will show that for every ~eC, (DcHt. Hi-1)~(x -'1) is linear in x. To do this, by the lemma, it suffices to show that for every W c V with dim W = 2, ~ e W, (D~H t �9 H7 1),o(x-1) is linear on C n W. We will do this for all a, fl at once, so in verifying this we can change coordinates in C".

What we do is this: we let ~ be a new base point of C and we change coordinates so that Hto(~) = I .. This reduces us to verifying (Dell t �9 H;-1),~(x-1) is linear on C ~ W when dim W = 2 , eeW. Now any such W is part of a subspace of V of the form A. e, where A c exp (p) is a maximal IR-split torus (of course, this p corresponds to the new choice of e). Moreover, as above, we can diagonalize p on A:

p(a 1 . . . . , a,) = (zi(a) fig,i), Zi(a) = ~I a~ ~j" j = l

Then

Ht(a e) = p(a) z Ht(e), p(a)H t (e) = Ht(e ) p(a).

Since e=(1 . . . . . 1), it follows:

o, (n , (ae) ) = i~1 ff~ai H'(ae)"

Using this, one calculates:

r

(Dell t �9 H i- a),a(ae)= ( ~ 2s~---2 / 6~ ~j= l aj l

and since on Ae, x -1 is given by (a 1 . . . . . a,)~---~(a~ 1 . . . . , a ; 1), this proves that (D~Ht.Hi-1)~,tj is linear in x -1 on every W = A e .

Now say ~, q~C, (r q ) = 0 . For a suitable coice of maximal torus A, we have

~, r/cA. e. In the coordinates (a 1 . . . . . a,) on A . e , let

~=(~1 . . . . . ~,),

,7 = ( ,71 . . . . ,,7~).


Since ( , ) on Ae makes A self-adjoint, it is a quadratic form of the type (ae, b e) = ~ 2 i a / b i, so (r t / ) = 0 means that for every i, r or r/i=0. A calculation like that just made shows:

(D,H,. HZ1)~#(oe)= ( ~ 2s~j~ t .~# \j= 1 aj 1

i.e.,

(D~" H," Ht-1)~I3(x-1)'A.e= (j~12s~j~ jaj" 6~,).

This is clearly zero if x = ~. QED

For the next result, suppose 6 is any vector field in the manifold of values of t. Then

Proposition 2.5. For all vector fields 6 on T,

( 6 H I �9 H t 1)(X)

is independent of x, i.e., depends on t alone.

Proof. By equivariance:

3 Ht(g x ) = p(g). 6 Ht(x) . ' p(g)

hence

(6Ht "Ht 1)(g. e) = p(g). (6H t �9 H t- 1)(e). p(g)- 1.

By symmetry:

p(g*). 6H,(e) = 6Ht(e ). 'p(g)- I

hence

p(g*). (3Hr. H;- ')(e). p(g,)- i = (6 H,. n i- ')(e).

Together, these imply the Proposition. QED

Proposition 2.4 gives us estimates on iIDcH t �9 H;-lll. To work these out, we fix a maximal flag of boundary components of C. In the notation of [1], p. 109, choosing this flag and the base point eeC is equivalent to choosing in the Jordan algebra V, a maximal set of orthogonal idempotents:

e = / 3 1 + . . . + /3 r .

Let

C~ = boundary component containing ~+ 1 + " " + ~,.

Then

250 D. Mumford

is the flag. Also let

~ = C w C ~ w C 2 u . . . w C , w(O)

and let

A= i ~'%" i=1

Let P be the parabolic group which stabilizes the flag { C~}. Our estimates are based on:

Proposition 2.6. (1) Let ~ e C and let ~'leV satisfy

( ~ , ~ ) = O~ _ ~ < G , @ = o .

t/~C

Then for every compact set o~cP, there is a K > 0 such that:

[(~'l,x-1)l<Kl/ds2,~(~l,~l), all xeo~.a.e.

(2) Let ~leC, ~'leV be as above. L e t ~2EC. Then for every compact set a~cP, there is a K > 0 such that:

[ds2,x(~'~, ~2)] ~ KF/ds~,x(~, ~1)" ]/dscZ,~(r ~2)-

Proof. We will use the Peirce decomposition of V defined by the idempotents %:

v=| % i_<j

where

a i + aj XE Vij ~ E aie," x = - ~ - x.

This decomposition is orthogonal with respect to ( , ) and C k is an open cone in the subspace @ V w If ~IECk, then note that

k<i~_j

~1• | % i<__j<_k

and

@ V~j~ (the boundary component of C corresponding to e 1 +. . . + ek). i<-_j<=k

This boundary component is open in

i<j<k o r

~.)

(~ Vii. Thus i<j<k


To prove (1), let x = g a e , g~co, a e = ~ a i e i. Then

<~,1, x - 1) = <g-1 ~'l,(ae)- 1)

= - < ( g ~0.,~i> i = l ai

where (g- 1 ~'l)ii is the component of (g- 1 ~'1) in V/i. But co preserves the flag, so

g- 1 ~6(~) Vij i<--j k<j

tOO. Thus

< ~ ' I ' X - - I > = i = k + l ~ l ( (g - l~] ) " ' e i>"

As g varies in co, ((g-l~'l)ii, ei) is bounded, hence

i = k + l a i

Let e ~k~ = ek+ ~ +. . . + e r be the base point in C k and note that, again by compact- ness of co, there is a K 2 >0 such that

g--l~l~Ck+KEe~k), all g~co,

hence

ds2,x(KEetk) ,K2e~k))<ds~,x(g-l~l ,g-l~O, all x e C .

Thus

d s 2 , g a e ( ~ l , 4 1 ) = d s 2 , a e ( g -1 ~ y,, g-1 ~1)

> g 2 d s 2 a e (e(k), e(k))

= K2ds2,e(a- I e(k), a- i etk))

= K 2 i ~k+l~<ei'ei>

The same procedure proves (2). If ~ e C k, ~2 e Cz, the argument goes like this:

2 r - 1 - dsc,g.e(~l.~2)=< a (g l~i) ,a-1(g-l~2)>

1 = ~ --<(g-x~'l) i j , (g-1~2), j>

[ i ~ _ j \ a i a j i t < i | ~k < j /

<- g ~ ~ ~ " )_2 -~ i= 1 ai ~ j = k + l aj

<K~K2I,/ds2~(r162 �9 1,/ds2~(~2,~2). QED

252 D. M u m f o r d

We can put everything we have said together as follows: Suppose N = dim V and 41 . . . . . ~N~Cspan V. Define a simplicial cone a c C by

N

i=1

Let l~: V~IR be a dual basis: li(~j)=6~j. Then for all p,H t as above, we have the estimates:

Proposition 2.7. For all vector fields 6,6' to the T-space, and a ~ C there is a constant K > 0 such that for all x~Int(a + a):

K IIDr Ht- l(x)l[ <

= li(x ) - li(a ) '

II 6H,. n ~ 1 (x)II <= K,

K [[O~,(o~jn,. H? 1)(X)I [

<= (li(x ) -l i(a)). (lj(x) - lj(a)) '

IIOr �9 n,- 1)(x)ll = 0,

K [16(oe,n,. n 21)(x)ll < l , (x)- l~(a)'

116(6'H~. a , - 1)(x)l[ __< g .

Proof Combine (2.4), (2.5) and (2.6) and the formula ds2(tl, tz)= - D t l ( ( t 2, x - 1 ) ) to get estimates in terms of ds2(r ~i) on sets co. A. e. Then apply (2.2)(i) and (ii) to the inclusion Int(a+a)--~ C, plus Ash's theorem that I n t ( a + a ) c c o . A .e if co is large enough ([1], Ch. II, w plus the formula

ds z = ~ dl2i i=a lZi

for the canonical metric on the homOgeneous convex cone a. QED

There is one final estimate that we need. For this, we first make a definition:

Definition. A linear map T: C " ~ C" is called p-upper triangular if the following holds: For all maximal R-split tori A c G , let X ( A ) = H o m ( A , ffJm) be the character group of A. As is well known, there is a basis 71 . . . . . 7, of X(A) such that the weights of A acting on V are contained in 7i+Yj, i ~ j (and contain 2~, 1 < i < r). Partially order the' characters by defining

EnlYi2>Emi~i i f ni>mi, all i.

Diagonalize p(A):

r @ v~. 2EX(A)

Then T is p-upper triangular if for all A, all 2o~X(A),

T( | V3=- | V~. ,~--> 3,o A~)-o


The estimate we need is:

Proposition 2.8. I f T is p-upper triangular, then for all a t C(F), there is a constant K > 0 such that

I ln t (x ) . 'T .n , (x ) - l l l < K , all x E C ( F ) + a .

Proof. Take a as a base point of C(F) and pick any maximal torus A so that A e -IR+,- " a = ( 1 . . . . . 1), C ( F ) = K . a . e . Therefore

C(F)+ a = { k a e l k ~ K , a = ( a 1 ...,an), ai> 1, all i}.

Change coordinates in C n so that

\/21(a)0 0 ) p ( a ) = | "..

2,(a) Now

n t (x ) . t ~. Ht(x )- 1 = p(k)nt (e ) p(a) 2. 'p(k) .t ~. tp(k)- 1 p(a)- 2 H t (e)- ' p (k) - t

so it suffices to bound [Ip(a)2.t(p(k) - 1. T.p(k)) .p(a)-2[[. This means we wish to bound:

12i(a)2 2j(a)- 2(p(k )- 1. T. p(k))~i [

when k ranges over K, and a = ( a 1 . . . . . a,) satisfies ai> 1, all i. This is equivalent to:

(p(k)- 1. T. p(k))ji ~ 0 ~ 2j ~ 2 i ( ,)

(weights of A being partially ordered as in the definition). But for all i define

W,=p(k)( | r ~j > hi

(where eieC" is the ith unit vector). Note that k A k - 1 maps W~ into itself and that W~ is one of the sums of weight spaces referred to in the definition. Therefore T ( ~ ) ___ ~ , hence

( p ( k ) - ' rp(k) )e ,~ (~ ff?,.ej Zj > 2i

which is precisely (,). QED

w 3. The Proportionality Principle

Let D be an r-dimensional bounded symmetric domain and let F be a neat 2 arithmetic group acting on D. Then X = D / F is a smooth quasi-projective

2 Recall that following a definition of Borel, a "neat" arithmetic subgroup F of an algebraic group ~ c GL(n,r is one such that for every x~F, x~e, the group generated by the eigenvalues of x is torsion-free. Every arithmetic group F' has neat arithmetic subgroups of finite index

254 D. M u m f o r d

variety, called a locally symmetric variety, or an arithmetic variety. In [1], Ash, Rapoport, Tai and I have introduced a family of smooth compactifications ) f of X such that X - X has normal crossings. We must first recall how .g is described locally. At the same time, we will need various details from the whole cumbersome appartus used to manipulate D so we will rapidly sketch these too. All results stated without proof can be found in [1].

By definition, D ~ K \ G, where G is a semi-simple adjoint group and K is a maximal compact subgroup. Inside the complexification G c of G, there is a parabolic subgroup of the form P§ �9 K e (P§ its unipotent radical which is, in fact, abelian and Kr the complexification of K) such that K=Gc~(P+ .K~) and G. (P§ Kc) open in G c. This induces an open G-equivariant immersion

O ' , /5

K \ G P+.Kr162

Here I~ is a rational projective variety known as a flag space and G C is an algebraic group acting algebraically on/} . L e t / ) be the closure of I) in D. The maximal analytic submanifolds F c / 5 - D are called the boundary components of D. For each F, we set

N(F) = {geG I gF =F}, W(F) = unipotent radical of N(F), U(F)=center of W(F), a real vector space of dimension k, say,

V(F)= W(F)/U(F): known to be abelian, centralizing U(F). Via "exp", we get a section and write W(F) set-theoretically as V(F). U(F). Also dim V is e v e n - let it be 2 l.

Next splitting N(F) into a semi-direct product of a reductive part and its unipotent radical, we decompose N(F)further:

N(F)=!Gh(F ) �9 GI(F ) �9 M(F)). V(F). U(F),

direct product mod finite central group

where

a) G 1 �9 M . V- U acts trivially on F, G h mod a finite center being Aut~

b) G h �9 M. V. U commutes with U(F), G~ mod a finite central group acts faithfully on U(F) by inner automorphisms

c) M is compact.

Here F is said to be rational if F n N(F) is an arithmetic subgroup of N(F). Mod F there are only finitely many such F, and if F1,..., Fk are representatives:

k

X w U (Fi/F c~ N(Fi)), i=1

with suitable analytic structure is Satake-Baily-Borel's compactification of X.


Next, for each F, we define an open subset D r c / ) by

DF= U g-D. geU(F)c

The embedding of D in D r is Pjatetski-Shapiro's realization of D as "Siegel Domain of 3rd kind". In fact, there is an isomorphism:

D r ~ U (F ) r x F

such that not only N(F) but even the bigger group G h �9 ( M " Gl) C �9 V•" Ur acts by "semi-linear transformations":

(x, y, t)F--~(Ax +a(y, t), Bty+b(t) , g(t))

(A, B t matrices, a, b vectors) and

O = {(x, y, t) I Im x + l,(y, y)e C(F)}

where C ( F ) c U(F) is a self-adjoint convex cone homogeneous under the G,- action on U(F) and lt: ~ l x ~1~ . U(F) is a symmetric IR-bilinear form.

Moreover

U(F)~-group of automorphisms of D: (x, y, t)~--~(x +a, y, t), aeU(F),

U (F)c = group of automorphisms of D (F): (x, y, t ) ~ ( x + a, y, t), ae U(F)r

W(F)~-group of automorphisms of D: (x, y, t)~-~(x +a(u, t), y+b(t) , t) and the group V(F) acts, for each t, simply transitively on the space ~ of possible y-values.

There is a technical lemma which we will need about this action:

Lemma. Let toeF, eoeC(F ), and let uoeU(F)r be the map (x, y, O~--*(x +ieo, y, t). Let e=(ieo, O, to) be a base point of D, so that Stabs(e)= K, a maximal compact of G and Stab6r ) = K~. 1~ Moreover, Staba,(eo)= K z is a maximal compact in G I. Since G t ~ Stab (0, 0, to), u o (G t) Uo i ~ Stab (e) and we may look at

0~: G 1 -c~176 , Stab~(e) ------~m~ Kr

I f * is the Cartan involution of G l with respect to Kl, then:

ct(g*) = ct(g).

Proof. This is a straightforward calculation, for instance, using the fundamental decomposition of g = L i e G via sl(2)rcg (cf. [1], p. 182) and the description of L ie (M. G~) in this decomposition for the standard boundary components Fs given in [1], p. 226.

We now describe local coordinates on X. Recall that )f is not unique but depends on the choice of certain auxiliary simplicial decompositions. We need not recall these in detail. The chief thing is that each X is covered by a finite set of coordinate charts constructed as follows:

1) take a rational boundary component F of D,

2) take {~1, .-., ~k} a basis of Fc~ U(F) such that ~i~C(F)c U(F) and in fact,

~ i ~ C ( F ) L ) C l k . ) C 2 u . . . u C I = C , where C ( F ) ~ _ C I ~ C 2 ~ . . . ~ _ C t is a flag of

256 D. Mumford

boundary components (cf. w and at least one ~ is in C(F): say

~ , ..., ~m~C(F), ~ra+ 1,- . . , ~keC(F) - C(F),

3) let li: U ( F ) e ~ r be dual to {~i}, i.e., li(~j)=6i i, 4) consider the exponential:

D ~ ( U ( F ) r )

( e 2~ '~ '1~ . . . . . e2~'~%y, t)

D/F c~ U (F)c(IE *k x •' x F)

X

5) Define (D/Fc~U(F)) ~ to be the set of P e C k x C ~ x F which have a neighborhood U such that

U c~ (r k X tEz x F) c (D/F ~ U (F)).

Note that m

(o/rc~ U(F)) ~ ~i~=1 {(z, y, t) l z=(z , , ..., Zk), Zi=0} =S(F, {~i}).

6) The basic property of X is that for suitable F, {~}, the covering map p extends to a local homeomorphism

~: (D/F c~ U (F))~-o s

and that every point of X is equal to/~((z, y, t)), where z~ -- O, some 1 _ i_< m, for some such F, {~i}.

We now come to the main results of this paper. Let Eo be a G-equivariant analytic vector bundle of rank n on D. E o is defined by the representation

a: K--~GL(n, C)

of the stabilizer K of the base point e+ ~D in the fibre ~2" of E o over e+. We complexify a and extend it to P+ �9 K c by l~ting it kill P§ Then a defines a G~- equivariant analytic vector bundle E o on D also. In the other direction, we can divide E o by F obtaining a vector bundle E on X. Since K is compact, E o carries a G-invariant Hermitian metric ho, which induces a Hermitian metric h on E. We claim:

Main Theorem 3.1. E admits a unique extension E to X such that h is a singular Hermitian metric good on .Y.

H i r z e b r u c h ' s P r o p o r t i o n a l i t y T h e o r e m i n t h e N o n - C o m p a c t C a s e 2 5 7

These various bundles are all linked as in this diagram:

E o - r estfi_cti_on - ~ E 0 _ _ d e s c e n t _ + E - - -~x!"-~si-~ - ~ E

1 1 1 D , D * X , ) ( .

Proof We saw in w 1 that/~, if it existed, has as its sections the sections of E with

growth O log Izil along X - X . To see that the set of these sections \ \ i = 1 / t

defines an analytic vector bundle on )(, it suffices to check this locally, e.g., on (D/Fc~ U(F)) ~. But now the bundle /~0 restricts to a bundle E r on D e with N(F). U(F)r acting equivariantly. Now note that the subgroup U(F)~ acts simply transitively and holomorphically on the first factor U(F) of D v in its Siegel Domain presentation. Since 117 ~ x F is contractible and Stein (F is another bounded symmetric domain), it follows that E v has a set of n holomorphic sections e 1, . . . , e, such that

i) e i is U(F)e-invariant, ii) el(x), . . . , e,(x) are a basis of Ev(x ), all xeD~.

Dividing by Fc~ U(F), E F descends to a vector bundle E~ on tE *k x C l x F, which is also globally trivial via the same basic sections e l, . . . , e,. We can then extend E} to ~2Rx C tX F so as to be trivial with these basic sections. We must show that along S(F, {~}) the sheaf of sections of this extension is exactly the sheaf of

sections of E) on (D/Fn U(F)) with growth O log IzJ on the coordinate k i

hyperplanes U (zl =0). Equivalently, this means that h(ei, Q) and (det h(e i, e j))-1 i = 1

have this growth. To do this, it is convenient to use a 2nd basis of E), which is C ~ but not analytic. Note that V(F). U(F)e acts simply transitively on the 1st 2 factors of D r. So we can find e'l . . . . . e',eF(De, Ee) such that

i') e i is V(F). U(F)r ii') e'i(x) . . . . . e',(x) are a basis of Er(x), all xeDv.

iii') On (0, 0) x F, e' i = % hence are holomorphic sections.

{e~} and {el} are related by an invertible U(F)e-invariant matrix S; so that ISul and Idet SI-1 are uniformly bounded on subsets

compact subset) k

([~*)kx of l l ]ZxF c ( C ) x l I ]~xF.

Therefore it is enough to check that h(e~, ej), (det h(e~, e~))-1 have the required growth.

Now if geG~(F), note that because g normalizes V. U e, ge' i is another V. Ue- invariant section of E}. Therefore

g" (el)= i rig(t)" ej (here t is the coordinate on F). j = l

258 D. Mumford

Since g(U(F)r x 112 t x {t})= U(F)e x i~ l x {t}, it follows that for each t,

g~-~p,(g)=(rij(t))

is an n-dimensional representat ion of Gt(F ). In fact, as g(0, 0, t) =(0, 0, t), this is just the representat ion of the stabilizer of (0, 0, t) restricted to G~(F). This shows that Pt is a ho lomorph ic family of algebraic representat ions of G, (this is not trivial because G, has a positive dimensional center). Since Gt is reductive, we may change our basis {el} so that Pt is in fact independent of t.

Now consider the functions hit = h(e' i, e~) on D. Since h, el and e~ are U . V- invariant, so is hl j , hence in Siegel domain notat ion, it is a function of u = Im x + lt(y, y) and t, i.e., is a function on C(F)• F. For each fixed t~F, and variable u~C(F),

U,(u) = (hlj(u , t))

is a map

(cone of pos. def. n x n] . C(F)-* C, = \Hermi t ian matrices !

I claim that (p, Ht) satisfy the hypothesis of w In fact,

Ht(gu), J = h(ge I, gej)

= Y'. r,k(g) h(e'k, el) rjt(g ) k , l

=(p(g ) . Hi(u). 'p(g))ij.

Let e = ( i e 0, 0, to) be a base point of D. Since h is a K-invar iant metric on E0,

h(ke~, kej)(e) = h(e'i, ej)(e),

Complexifying, we get too:

h(ke'i, kej)(e) = h(e'i, ej)(e),

all k~K.

all k6Kq:.

Let uo~U c be given by (x, y, t)~--~(x + ie o, y, t). Then for all g~Gl(F), UogUol(e) =e, so U o g U o l = k . p , where k~Kc , peP+. By the lemma above, u0g*Uo 1 = k . p' for some o ther element p'EP+. Therefore:

o r

h(e'i, ej)(e)=h(pe'i, p' ej)(e) (since P+ acts trivially on Eo(e))

= h(kpe'i, kp' ej)(e)

= h(u o gUo l(e'i), u o g* u o 1 (ej))(e)

= ~ rik(g) h(e'k, el)(e) ~ l(g*) (since u o e' i = e' i, all i)

Hto(eo ) = p(g) H,o(eo ) tp(g.).

In fact, the same holds if t o is replaced by any t~F as follows easily using the Gh(F)-invariance of h and the fact that Gz(F ) and U(F)r commute with Gh(F ).


Thus we have the full situation of w In particular, we have available all the bounds of w 2.

As above, to describe local coordinates near boundary points of )(, choose a simplicial cone:

k

~ = ~ IR+-~i C C(F); ~isC(F).,~l <_i<_m i=1

and let I i be dual linear functionals on U(F)r Then if (x, y, t) are Siegel domain coordinates on D(F), and zi=e 2'~tax), then (z, y, t) at points where at least one z i is 0(1 <i<m) are local coordinates on )f. Moreover each point PsS(F, {~i})c)f has an open neighborhood whose intersection with X is contained in the image of

{(x, y, t) ] u = Im x + It(y, y)~a + a, y s Y', t6F'}

for some a as above, as C(F), Y' (resp. F') a relatively compact subset of Y (resp. F). Note that l og] z i [=-2n l i ( Imx ). At all points P~S(F,{(i}), we have to estimate h(s' i, sj) -+1 in terms of

choose C that ~ 1 in of P ) . This is the large enough SO < a neighborhood

same as estimating h(sl, sj) -+~ in terms of

(y, l,(u) + c ) 2"

(choose C large enough so that l~(u)+ C > 0 in a neighborhood of P). But if l~(u) + C > 0 , (~li(u)+ C) 2u is comparable with (~l,(u)2) u, hence with (u, u> u. This is exactly the estimate that Proposition 2.3 gives us. Next we have to estimate the connection and the curvature. Now in terms of a holomorphic trivialization of E o, the connection is given by 0h. h -1. What we have is good control of 61h.h -1 and 62(6 th .h -1) for all vector fields 61, 6 2 in terms of the real analytic trivialization given by {el}. Write

e i = ~ aij ej, j=l

A = matrix (aifl,

H ~ = matrix h(e i, e j).

Then

H a " = A . H . ' A .

From this you calculate:

Connexion in~ trivialization ~ = OH "". (H a")- 1

{ei} / t ~ A . A - I + A . O H . H - 1 . A - I + A �9 H . (OtA �9 t A - 1 ) H - 1 A - 1 ,

260 D. Mumford

connexion in\ trivialization ~ = d(O H a" . (H a") - 1)

{e~} / = d ( O a . A - X ) + d ( a �9 O H . H - x . A - x)

+ d ( A . H . (O'A. t ~ - a ) . H - 1 . A - l )

= d ( O A . A - a ) + d A �9 (OH. H - X ) �9 A - x

+ A �9 d (OH. H -x)- A -x + A . O H . H -x �9 A -x �9 dA �9 A -x

+ d A . [ H . ( O * A . ' A - 1 ) H - X ] . A-X

+ A ( d H . H - x ) [ U �9 (991. tA-x) . H -X ] . A -x

+ A . [ H . d(O ~ . t A - l ) . H - x ] �9 A -1

+ A . EH . (O'A . tTt- X) . H - X] . (dH . H - X) . A - x

+ A . [ H . (OtA �9 ~Zl-x) �9 H - x ] �9 A - x . d A . A -1.

Therefore, since A is a C a metric on ){, to show that the connexion and its differential have Poincar6 growth on )( it suffices to prove that the 4 forms:

0 H . H -1,

d ( O n . H - 1),

H . ( 0 ~ I . ' d - 1 ) . H - l , '

H . d ( 0 ~ . ' 4 - 1 ) . H-X

have Poincar6 growth on X. To check this for the first two, note that H is a function on C ( F ) x F , hence it suffices to bound 6 1 H . H -1 , 6 2 ( 6 x H . H -x) for all vector fields fix, 6z on C ( F ) x F. But the Poincar6 metric is given in Siegel coordinates by:

ds z = ~ [dzilZ z. z ~ - E [ d y ' I E + E Idt't2

( l o g - ' ) Izil= C

tdl~(x)[ 2 ~_~ Idyll 2 +~" Idt~l 2 =Y" (/,(Im x )+ C') 2

(choose C large enough so that l i ( I m x ) + C > O near the boundary point in question). Therefore the bounds of Proposition 2.7 imply that OH. H -x and d ( O H . H - 1 ) have Poincar6 growth.

To check the result for the last two, we need to know what sort of a function A is. Firstly, since e i and e' i are both Ue-invariant, A is Ue-invariant, i.e., is a function of y and t alone. Therefore it has derivatives only in the y and t directions, so to say these have Poincar6 growth is just to say they are bounded along X'. Next, for all t o , F , the action of Von vector space of points (y, to) puts a complex structure on V. Thus it defines a splitting Vr Vto + @ Vto, where V. + to are complex subspaces and V~o acts trivially on the points (y, to), while V~o + acts simply transitively. If we fix one t o, then Vt + still acts simply transitively and holomorphically on the vector spaces of points (y, t) for t near t o. Thus it is natural to choose our holomorphic basis e i to be in fact V~ + . Ue-invariant.


Moreover, it is easy to see that G~. V normalizes Vto + �9 U c for all t o, hence that the action of Gt " V in terms of the holomorphic basis is given by

g(el)= ~ ~ij(t).ej j = l

where

g~-~(g) = (matrix ~ij(t))

is a representation of G~. E Comparing p and ~, we get:

P(g) = g* A - l " Pr(g)" A.

Since ei=e' i on (0, 0 ) x F , P(g)=Pt(g) for all g~G, and A(0, t )=I , . Thus if v(0, t) = (y , t) ,

I. = p(v)= A(y, t)- 1. ~t(v) . I , ,

o r

A(y, t)=pAv).

Now we use the simple:

Lemma. Let a be an algebraic representation of G I �9 V, and let a o be the restriction of a to G I. Then for all veV, a(v) is ao-upper triangular.

Proof. Let A ~ GI be a maximal IR-split torus. As in w there is a basis 71 . . . . . 7, of the character group of A such that A acts on the vector space U(F) containing the cone C(F) through the weights 7i +7i- Then its action on V(F) is through the weights 7i (cf. [1], p. 224). Now if V/(F)~ V(F) is the root space corresponding to 7i, and if we diagonalize a(A):

r | w~, ~eX(A)

then

V,(F)(W~)= W~ + ~,

hence V(F) acts in a ao-upper triangular fashion. QED

Thus A(y, t) is p-upper triangular for all y, t, hence so are

A-I . t~A and d(A-I .OA) .

Applying Proposition 2.8, it follows that

H . t ( A - I . S A ) . H - X and H . ' ( d ( A - X . ~ A ) ) . H - X

are bounded in a neighborhood of every point of )~, as required. This completes the proof of the Main Theorem.

A natural question is whether these vector bundles/~ are in fact pull-backs of vector bundles on less blown up compactifications of D/F. Thus Baily and Borel

262 D. Mumford

defined in [2] a "minimal" but usually highly singular compactification (D/F)* of D/F. Unfortunately /~ is only rarely a vector bundle on (D/F)* (we will see below one case where it is however). However, in [1], Ash, Rapoport, Tai and I defined not only smooth compactifications of D/F but also a bigger class of compactifications with toroidal singularities (cf. [9]). These are important

because when you try to resolve (D/F)*, often there is a D/F with relatively simple structure on the boundary but still with some toroidal singularities. It is easy to see that the construction above of/~ goes through equally well on all of these compactifications: it gives vector bundles on all of them such that whenever compactification a dominates compactification b, then extension a is the pull-back of extension b.

The Main Theorem, plus Hirzebruch's original proof of his proportionality theorem for compact locally symmetric varieties X, gives us easily the proportionality theorem in the general case:

Proportionality Theorem 3.2. As above, f ix:

X = an arithmetic varieiy D/F, D = K \ G,

.~=a smooth compactification as in [1],

= compact dual of D.

Then there is a constant K, which in terms of a natural choice of metric on D is the volume of X, such that for all:

Eo = Gr analytic rank n vector bundle on

defined by a representation of Stabo~(e ) trivial on P+,

ff~= corresponding vector bundle on X,

the following formula holds:

C~(/~) =(__ 1)dim X. K . c~(ff~o), all ,=(c t I . . . . . ~,), ~, ~zi=dim X.

Proof As above, choose a G-invariant Hermitian metric h o on E o. By the Main Theorem, h o defined a "good" Hermitian metric h on E, hence its Chern forms Ck(E, h) represent the Chern classes of /~. But on D, Ck(E, h) are G-invariant forms, so:

c~(ff~) = ~ c~(E, h) X

= S c~(Eo ' ho) F' al Domain

F e D

= vol (F). c~(Eo, ho)(e ).

Now if G c is a compact form of G:

Lie G = t 0 0 ,

Lie G c = f ~ ip


then /~o has a unique GC-invariant Hermitian metric/~ equal to h 0 at e. So

/3

=vol (/)). c'(/~ o,/~)(e).

T h e n - a n d this is the essence of Hirzebruch's remarkable p r o o f - a simple local calculation shows (cf. [7]):

ek(E, h)(e)=(- 1) k Ck(Eo, h~).

This proves the result. To apply this result, it is important to describe the bundles E as closely as

possible. Firstly, we can characterize their sections, precisely as a special case of the general definition of automorphic forms given by Borel [3]. Let p : K-* GL(n, ~) be a representation of K, and let

E o = G x K ~ "

be the associated G-equivariant vector bundle over D =K ' - . G (i.e., E o--set of pairs (g,a), mod(g,a)~(kg, p(k)a)). E o has a complex structure as follows: complexify p and extend it to K c.P+ to be trivial on P+. Then E o is the restriction to D to the bundle

/~o = Gr x tKr247

on /), and in the definition of/~o, everything is analytic. Borel introduces a measure of size on G by:

Hglla = tr(Ad g*- 1. g)

* - C a r t a n involution on G w.r.t. K,

and defines holomorphic p-automorphic form f to be a function

f: G--,~"

such that

(1) f(kgT)=p(k)f(g), all keK, 7~F, (2) f induces a holomorphic section of Eo,

(3) [f(g)l<C'llgt]~, some n > l , C>0 .

Then one can show:

Proposition 3.3. In the above notation:

F(X, E) ~- {vector space of holomorphic p-automorphic forms}.

Sketch of Proof The problem is to check that the bound (3) is equivalent to requiring that the corresponding section of g over X has growth O((~ log [zil) 2")

264 D. Mumford

along Jr. But IlgllG defines a measure of size on D and on X by:

Vx~D: Ilxllo=llgllG i f x corresponds to the coset K.g ,

V~eX, image of x: II~lLx=min IIy(x)llo=min IIgTLIG. ~'EF ~er

Then holomorphic p-automorphic forms are clearly holomorphic sections s of/~ over X, such that

h(s,s)(x)<Clllxll~x, some n > l , C1>0.

But if d o is a G-invariant distance function on D, then it is easy to see (using G = K . A. K) that do(x, e) and log Ilxll are bounded with respect to each other. In another paper [4], Borel has proven that if x is restricted to a Siegel set = co. A t �9 e c D, then

min do(x T, e) ,~ do(x, e),~ d a( a(x), e) ~EF

(here x=co(x)a(x) .e and ~ means the differences are bounded). Applying this to a subset of a Siegel Domain of 3rd kind of the type {(x ,y , t ) lyeV' , teF' , Re xeU' , Im x-l~(y, y)ea + a} where U'c U, V' c V,, F' c F are compact subsets

and a c C(F) is a simplicial cone and aeC(F), we see that

min do((x, y, t) 7, e) "t~F

can be bounded above and below by expressions

C210g(( Imx- l , (y ,y) ,e ) ) , eeC(F), C 2 > 0

hence IlXEIx can be bounded above and below by expressions

( Im x - l,(y, y), e)", e~ C(F), n ~ 1.

Describing a as l i> 0 as above (ll linear functionals on U), this is of the same size as

(Z li(Im x) + C3)"

and as z i = e 2~ Z,(x), this is equal to

(-~log(bzil/C4))". Q E D ~

Next, there are 2 particular equivariant bundles where we can describe /~ more completely:

Proposition 3.4. a) I f E o = 0~, the cotangent bundle, with canonical G-action, then E = O ~ (log), the bundle on X whose sections in a polycylinder A"c X such that

A n n ( ~ ? - X ) = ~) (coordinate hyperplanes)

i= 1 \ z i = 0


are given by

f dz i a~(z)--+ f

i=1 zi i = k + l bi(z) dzv

b) I f E o is the canonical line bundle ~2"n, then E is the pull-back of an ample line bundle (_9(1) on the Baily-Borel compactification X* of X. The sections of (9(n) are the modular forms with respect to the nth power of the canonical automorphy factor given by the jacobian, hence (9(n), n~ O, is the very ample bundle used by Baily and Borel to embed in X* in IP N.

Proof. Using Siegel Domain coordinates (x, y, t) on D(F), a U(F)e-invariant basis of t2~tF) is given by {dx i, dyi, dtk}. Therefore these span the corresponding bundle on X near the boundary F. But here

{Z i = e2ni li(x), Yi' tk}

are coordinates and these differentials are ~ dzi, dy2, dtk~. This proves (a). ~ zi )

To prove (b), recall that X* is set-theoretically the union of X and of F/FnN(F) for all rational boundary components F. Moreover, if P e F / F n N ( F ) c X * , then there exists a neighborhood U ~ X * and an open set V c D such that V maps to Uc~X and V is a (Gt(F). V(F). U(F))nF-bundle over U n X. Now say {sl} is the U(F)r holomorphic basis of ~o on D(F) used to extend E over the F-boundary points of .(. If we verify that each sl is (G~(F). V(F). U(F))nF-invariant it follows that E is trivial on U n X and moreover, if n: X ~ X * is the canonical birational map, then {s~} are a basis of ~z,/~ over U. Thus ~./~ is a vector bundle which pulls-back to E on X. Now in the case in question, E is a line bundle, sl can be identified with the differential form

(A dx,) A (A dyj) A (A dtk) i j k

on D(F), and G~. V. U acts on it by multiplication by the Jacobian determinant in the Siegel Domain coordinates. But Baily-Borel ([2], Prop. 3.14) showed that the Jacobian on (Or" V. U)nF was a root of unity. Since F is neat, it is one and (Gt" V. U ) n F indeed fixes sl. The last assertion is just a restatement of Pro- position 3.3 for this special case. QED

The following consequence of the proportionality principle seems to be more or less well known to experts, but does not seem to be contained in any published articles:

Corollary 3.5. Let L = (~2~6) - 1 be the ample line bundle on D, and let

P(1) = z( L |

be the Hilbert polynomial of D. Let 7r: R-* X* map a smooth compactification of X onto Baily-Borers compactification. Let n 1 = d i m ( X * - X ) . Then there exists a

266 D. Mumford

polynomial Pl(l) of degree at most n 1 such that for all l > 2 :

[cusp forms on D ] dim ~ v o l ( X ) * P ( l - 1) +PI(I). I_w.r.t. F of weight lJ

Proof. The R i e m a n n - R o c h theo rem gives us a "universa l po lynomia l " Q such that if L is any line bundle on a smoo th project ive variety W,, then

)~(L) = O(c, (L) ; c1(f21) . . . . . c,,(f2~v)).

Therefore if n = d im D,

( - 1)" vol(X) �9 P( - l) = ( - 1) n vol(X) �9 Z((O~-) | ')

= ( - 1)"vol(X). Q(lc,(O~); c,(f2~) .... ,c.(O~)) = Q(lcl(Y2 ~ (log)); C 1 ( ~ c ~ l ( l o g ) ) . . . . . c,(f2~-(log)))

by Propor t iona l i ty T h e o r e m 3.2. Consider a typical t e rm

[lc x (f2~(log))] k. c~(f2~-(log)), ]c~] + k = n.

N o w by Propos i t ion 3.4b:

c 1 (f2~-(log)) = re* H,

H an ample divisor on X*. Let n 1 = d i m ( X * - X ) . If k > n 1, the cycle class H k on X* is represented by a cycle suppor ted on X alone, hence so is n*H k. Thus if k > n l

(1. n* H) k. c~(f2~-(log))=(l �9 x ' H ) k. c~(f2~-).

Therefore

Q(lct(O~(log)); c, (f2~(log)), ..., c,(O~- (log)))

= Q(lcl (f2~(log)); c I (12~-) . . . . . c,(f2~)) + (polyn. of degree < n:)

= X(f2~(log) | + (polyn. of degree < nl)

= ( _ 1), )~(f2~ (log)|174 f2~) + (polyn. of degree < nl)

by Serre duality. Thus for suitable P~ of degree at mos t n l :

vol(X) �9 P(t - 1) = Z(Q~ (log) | - 1) @ f2~) - P~ (l).

But since (f2~(log)) | is genera ted by its sections and maps X to X* of the same d imens ion for N~>0, K o d a i r a Vanishing (cf. [13]) applies if 1>2 and we have

h ~ ( log ) ' - ~ | f2]) = vol (X) . P ( I - 1) + P~ (l).

Hirzebruch's Proportionality Theorem in the Non-Compac t Case 267

The left-hand side is exactly the space of sections of ~2](log) t which vanish on the boundary. By Proposition 3.3, these are exactly the cusp forms of weight I. QED

w 4. Applications: General Type and Log General Type

The purpose of this section is to consider the application of the preceding theory to the question of when D/F is of general type, and to reprove as a consequence of our theory the following theorem of Y.-S. Tai ([1], Ch. IV, w 1).

Tai's Theorem 4.1. I f F is any arithmetic variety acting on a bounded symmetric domain D, then there is a subgroup F o c F of finite index such that for all F 1 ~ F o of finite index, the variety D/~ is of general type.

We recall that if X is any variety of dimension n, we say that X is of general type, if for one (and hence all) smooth complete varieties )( birational to X, the transcendence degree of the ring

(~ r(,Y,(n~) | N=O

is (n+ 1). More generally, the transcendence degree of this ring minus one is called the Kodaira dimension of X.

Recall that Iitaka [-8] has recently introduced a complementary theory of "logarithmic Kodaira dimension" for arbitrary varieties Y In fact, he first chooses a smooth blow-up Y' of Y and then a smooth compactification I 7 of Y' such that I 7 - y ' has normal crossings and defines t21(log) as the complex of 1- forms

dz i ai(z)---}- ~ ai(z)dzi

i=1 zi i = k + l

k

if, locally, 17-Y' is given by 1-[ zi=O. By definition Q~(log)=Ak(O~(log)). He i = l

then looks at the "logarithmic canonical ring":

R = (~) F(F,,t2~(log| N=0

He shows that this ring, as well as all other vector spaces of global forms with logarithmic poles (obtained from decomposing (2~(log)| ... | t2~(log) under the symmetric group and taking global sections) are independent of the choice of Y' and I7. He then defines the logarithmic Kodaira dimension of Y to be the transcendence degree of R minus 1. We may restate Proposition3.4(b) in this language as follows:

Proposition 4:2. I f F is a neat 3 arithmetic group, then D/F is a variety of logarithmic general type, i.e., its logarithmic Kodaira dimension equals its dimension.

3 Some hypothesis on elements of finite order is needed because H/SL 2 (Z)---A l which is not of log general type!

268 D. Mumford

Proof In fact, by Proposition 3.4, R is just the homogeneous coordinate ring of the Baily-Borel compactification of D/F.

Note that D/F of logarithmic general type is weaker than saying D/F is general type.

I would like to add one comment to his theory which, in some cases, makes it easier to apply: one does not need smooth compactifications, but merely a toroidal compactification 17 of Y' (cf. [9], p. 54). This means that locally Y' c I 7 is isomorphic to (~* ) "cX, , where X~ is an affine torus embedding (i.e., (r is Zariski-open in X,, X, is normal affine and translations by (E*" extend to an action of ((E*)" on X,). On X~, define f2~,(log) to be the sheaf generated by the ((E*)"-invariant 1-forms. Carrying these over, we define I2-~(log) to be the coherent sheaf of 1-forms on 17, regular on Y', isomorphic locally to f2Jc~(log). If 17'---~ 17 is an "allowable" modification of toroidal embedding of Y ([9], p. 87), then p*(f2~(log_))~12~,(log). In particular, there is always a smooth allowable modification Y' ([9], p. 94). So Iitaka's spaces of forms with log poles can be calculated equally well on a smooth 17 or a toroidal 17.

This extension is helpful in checking the analog of the ~tbove Proposition for the moduli space of curves:

Proposition 4.3. Let OYil(") be the moduli space of smooth curves of genus g with level n structure. I f n > 3, then OJl(') is of log general type. - - - - g

Sketch of Proof The proof follows the ideas of [12], w 5 very closely. Let Hg be the Hilbert scheme of e-canonically embedded smooth curves of genus g. Let H(g")---~ H be the covering defined by the set of level n structures on these curves. _ g -

Let H_, 9J/_s_be the compactified spaces allowing stable singular curves as well. Let //~), 9J/~"' be the normalization of H g, gJ/j in the coverings H~"),~ ("'_._g . The group G=PGL(v ) ( v = ( 2 e - 1 ) ( g - 1 ) ) acts on Hg and on H~"), freely on the latter,

- - "-~ - ~(") -~ iq(")/C, We have the diagram: so that 9J/g = Hg/G, ._.g ___g ,_.

r)(n)

where D and c~ are the universal curves. Recall from [12] the notation: whenever p: C ~ S is a flat family of stable curves,

2 = Agp,(O~c/s),

O9c/s = relative dualizing sheaf

and if p is smooth over all points of S of depth zero, then A c S is the divisor of singular curves and

6 = ~Os(Zt ).


Now on all 3 families above, we wish to show 2 1 3 | -1 and the sheaf of top logarithmic forms t2"(log) are isomorphic line bundles. Firstly, for p : / ) g ~ Hg, we proceed like this: a) a simple modification of the proof of Theorem 5.10 [12] shows:

213 | 6 - 2 _~ A 3 g- 3 (p, ((2~/a | o9~/a)).

b) Since //g represents the functor of e-canonical stable curves, Ta,,tcl is canonically isomorphic to the vector space of deformations of C. This has a subspace consisting of deformations of the e-canonical embedding where C doesn't change, and a quotient space of the deformations of C alone:

0 ~ Lie G --~ Tn~, to--+ Ext ~ (f21, (tic) --* 0

or dually:

0 -~ H~ | COc) -~ f2~. | IK(C)--,(Lie G)'---~ O.

Therefore globally, we get

0--~ p,(f2b/a | COO/a)--+ Q]~ --~ (Lie G)' | (9.% ~.0

hence if m = dim//g,

Hg

.'. Q ~ ( 1 o g ) ~ 213 @ 6 - 1

Secondly, for p: B~n)~I4~ n), 213 @6-1 pulls back to the analogous sheaf on //tgn). Moreover,_because /4~n)~Hg is ramified only along A which has normal crossings, H~ n) has toroidal singularities and O~(log) pulls back to f2~,)(log). Finally both bundles descend to ~tg,) and by Proposition l.4 [11], are still isomorphic. Finally, it is proven in 1-12] (Th. 5.18 and 5.20: cf. diagram in w that 213@6--1 is sample on ~" ) . QED

In certain cases, there is a way of deducing that coverings of a variety of log general type are actually of general type. To explain this, suppose we are given a smooth quasi-projective variety Y,, and a tower of connected 6tale Galois coverings:

n~: Y~-* Y,, group F~.

We assume that any 2 covers n, ,na are dominated by a third one nr:

y / "~

270 D. Mumford

Let 17 be a smooth compactification of Y with normal crossings at infinity. Extend the covering Y~ to a finite covering

7~t I ]7 ___~ 1,7

be defining 17, to be the normalizations of 17 in the function field of Y~. We now make the definition:

Definition. The tower {~,} is locally universally ramified over 17- Y if Jot all x~ 17 - Y, we take a nice neighborhood of x:

A n t Y,

A" c~(17-Y)= (union of coordinate hyperplanes)

\Z 1 = 0 , ...,Zk-~-O

then for all m, there is an ~ and a commutative diagram:

~ 1 (A") \ ~ 1 (A")\ U s it~

. ~ ( m . , . ,z.). A n / ~ z l ' ...~Zk~Zk+l~ ..

(Z 1 . . . . ,Zn~

In other words, lr~-~(A"n(17-Y)) is cofinal in the set of all unramified coverings of A" c~(17- r).

Then we assert:

Proposition 4.4. Let Y c Y be as above and let n,: Y,--~ 17 be a tower of coverings unramified over Y and locally universally ramified over 17-Y. Then if Y is logarithmically of general type, there is an % such that for all cq such that the covering n~ dominates n,o, Y~ is of general type.

Proof Let A = 17- Y, n = d i m Y and let O) =f2~z(log). Then we know that

a) for some N, there are differentials r/o . . . . , r / ,~F(s *N) such that ql/rlo . . . . . rl,/qo are a transcendence base of the function field C(Y),

b) h~ n+l i f N > N o.

From (b) it follows that

r(~, co| r(A, o~ | @ ~ ~)

has a non-zero kernel for some N: let ~ be in the kernel. Replacing ~/i by ql | ~, we may assume that all qi are zero on A. Now let's examine locally what happens to qi when lifted to a covering of I7: let A"c 17 be a polycylinder such

/n that A " n ( 1 7 - Y ) = V I I 1 z~ . Write out qi: \ i= 1 /

k rIz, (dz - az"tN, i= 1 \ Z1 Zk !


z TM l<_i<k , w~=z~, k+l<_i<_n and let ai ho lomorphic . Let w~= ~ , _ _

TOm: An-~ A"

be the cover ing of the z-polycyl inder by the W-polycylinder. Then

dw i 1 dz i - , l < _ i < k ,

w i m z i

dwi=dz l , k + l <_i<_n

hence

~*(rh)=mkUai ' i~ lw~" \ -w-- 1 w k / "

So if m > N , rc*(th) is a ho lomorph i c differential form on A'. N o w for each xe17 - Y, fix a ne ighborhood U~ c 17 of this type and a cover ing

which, over Ux, domina tes rc N. 17- Y is covered by a finite n u m b e r { U J of U~'s, so we can find one cover rc~o which domina tes all the covers rc,~,). I c la im that if re,1 domina tes re,o, hence rc,(x.~, then rc*~(qi ) has no poles on a desingular izat ion 17"1 over 17,,. This is clear because it has an open cover ing by open sets V~ sitting in a d i a g ra m

U ~

so g*(rh) regular rc, l(t/i ) regular on V i. But now rc, l (qi/tlo) are a t ranscendence base of the function field of 17~1 so I~, is of general type. Q E D

Let 's consider the case Y = D/F again. I f F is an ar i thmet ic group, then for every posit ive integer n, we have its level n subgroup F(n), i.e., if

F = ff(TZ.), ~ algebraic g roup over Spec Z

then

r(n) = Ker [ i f (Z) ~ Cg(Z/n Z)].

I t 's easy to see tha t

~.: n/r(n)-~ o /r

272 D. Mumford

is locally universally ramified over D/F-D/F. In fact, let F be a rational boundary component. Then near boundary points associated to F, the pair

D/F c D/F is isomorphic to

D(F)/%(F) c~ F c (D(F)/U(F) n 7Z)~

II C*" x C " • F ~" x ~ " • F.

Thus if U c D/F is a small neighborhood of a point corresponding in the above chart to (O,y,t), ~I(Un(D/F)) is isomorphic to U(F)nF. Thus we must check that for all n, there is an m such that

U(F) n F(m) c n. U(F) c~ F.

But if F is rational, U(F) is an algebraic subgroup of the full group (~ which is defined over ~ , so this is clear. So now Proposition 4.2, Proposit ion 4.4 and this remark altogether imply Tai's theorem.

It is now known that this same method will show that at least some high- non-abelian levels of 9J~g are varieties of general type too. It is not simple however to check that the Teichmuller tower is locally universally ramified at infinite. This has recently been proven by a ingenious use of dihedral level by T.-L. Brylinski.

References

1. Ash, A., Mumford, D., Rapoport, M., Tai, Y.: Smooth Compactification of locally symmetric varieties. Math. Sci. Press (53 Jordan Rd., Brookline, Mass. 02146), 1975

2. Baily, W.L., Borel, A.: Compactification of arithmetic quotients of bounded symmetric domains. Annals of Math. 84, 744 (1966)

3. Borel, A.: Introduction to Automorphic Forms. In: Symp. in Pure Math., Vol. IX (AMS, 1966), p. 199

4. Borel, A.: Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem. J. Diff. Geometry 6 (1972)

5. Bott, R., Chern, S.: Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta math. 114, 71 (1965)

6. Cornalba, M., Griffiths, P.: Analytic cycles and vector bundles on non-compact algebraic varieties. Inventiones math. 28, 1 (1975)

7. Hirzebruch, F.: Automorphe Formen und der Satz von Riemann-Roch. In: Symposium In- ternacional de Topologia Algebraica, Unesco, 1958

8. Iitaka, S.: On logarithmic Kodaira dimension of algebraic varieties. (To appear) 9. Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings. Springer Lecture

Notes 339 (1973) I0. Kobayashi, S.: Hyperbolic manifolds and holomorphic mappings. M. Dekker Inc., 1970 11. Mumford, D.: Geometric Invariant Theory. Berlin-Heidelberg-New York: Springer 1965 12. Mumford, D.: Stability of projective varieties. L'Enseignement Mathematique 1977 13. Mumford, D.: Pathogies III. Amer. J. Math. 89 (1967)

Received May 8, 1977

hirzebruch's proportionality theorem in the non-compact case · inequalities needed for the...

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