complex differential geometry · complex differential geometry roger bielawski july 27, 2009...
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![Page 1: Complex Differential Geometry · Complex Differential Geometry Roger Bielawski July 27, 2009 Complex manifolds A complex manifold of dimension m is a topological manifold (M,U), such](https://reader033.vdocuments.mx/reader033/viewer/2022052803/5f1facae6e5afd7a86303794/html5/thumbnails/1.jpg)
Complex Differential Geometry
Roger Bielawski
July 27, 2009
Complex manifolds
A complex manifold of dimension m is a topological manifold (M,U),such that the transition functions φU φ
−1V are holomorphic maps
between open subsets of Cm for every intersecting U,V ∈U.
We have a holomorphic atlas (or “we have local complexcoordinates on M.”)
Remark: Obviously, a complex manifold of dimension m is a smooth(real) manifold of dimension 2m. We will denote the underlying realmanifold by MR.
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Example
Complex projective space CPm - the set of (complex) lines in Cm+1,i.e. the set of equivalence classes of the relation
(z0, . . . ,zm)∼ (αz0, . . . ,αzm), ∀α ∈ C∗
on Cm+1−0.In other words CPm =
(Cm+1−0
)/∼.
The complex charts are defined as for RPm:
Ui = [z0, . . . ,zm];zi 6= 0 , i = 1, . . . ,m
φi : Ui → Cm, φi([z0, . . . ,zm]) =
(z0
zi, . . . ,
zi−1
zi,zi+1
zi, . . . ,
zm
zi
).
Example
1. Complex Grassmanian Grp(Cm) - the set of all p-dimensional vectorsubspaces of Cm.2. The torus T 2 ' S1×S1 is a complex manifold of dimension 1.3. As for smooth manifolds one gets plenty of examples as level sets ofsubmersions f : Cm+1 → C. If f is holomorphic and df (theholomorphic differential) does not vanish at any point of f−1(c), thenf−1(c) is a holomorphic manifold.For example Fermat hypersurfaces:(
(z0, . . . ,zm);m
∑i=0
zdii = 1
), d0, . . . ,dm ∈ N.
4. Similarly, homogeneous f give complex submanifolds of CPm.5. Complex Lie groups: GL(n,C), O(n,C), etc.
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Almost complex manifolds
A complex manifold of (complex) dimension m is also a smooth realmanifold of (real) dimension 2m. Obviously, the converse is not true,but it turns out that there is a characterisation of complex manifoldsamong real ones, which is much simpler than the existence of aholomorphic atlas.
Identifying R2n with Cn is equivalent to giving a linear mapj : R2n → R2n satisfying j2 =−Id.Let x ∈M and let (U,φU) be a holomorphic chart around x .Define an endomorphism JU of TxMR by JU(X) = φ
−1U j φU(X).
JU does not depend on U, and so we have an endomorphismJx : TxMR → TxMR satisfying J2
x =−Id.The collection of all Jx , x ∈M, defines a tensor J (of type (1,1)),which satisfies J2 =−Id and which is called an almost complexstructure on MR.
Definition
An almost complex manifold is a pair (M,J), where M is a smooth realmanifold and J : TM → TM is an almost complex structure.
Thus a complex manifold is an almost complex manifold. The converseis not true, but the existence of complex coordinates follows fromvanishing of another tensor.Remark: Obviously, an almost complex manifold has an evendimension, but no every even-dimensional smooth manifold admits analmost complex structure (e.g. S4 does not).Remark: S6 admits an almost complex structure, but it is still an openproblem, whether it can be made into a complex manifold.
Definition
A smooth map f : (M1,J1)→ (M2,J2) between two complex manifoldsis called holomorphic if ψV f φ
−1U is a holomorphic map between
open subsets in Cn, for any charts (U,φU) in M1 and (V ,ψV ) inM2. This is equivalent to the differential of f commuting with thecomplex structures, i.e. f∗ J1 = J2 f∗.
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The complexified tangent bundle
Let (M,J) be an almost complex manifold. Since J is linear, we candiagonalise it, but only after complexifying the tangent spaces. Wedefine the complexified tangent bundle:
T CM = TM⊗R C,
and we extend all linear endomorphisms and linear differentialoperators from TM to T CM by C-linearity.Let T 1,0M and T 0,1M denote the +i- and the −i-eigenbundle of J. It iseasy to verify the following:
T 1,0M = X − iJX ; X ∈ TM, T 0,1M = X + iJX ; X ∈ TM,
T CM = T 1,0M⊕T 0,1M.
Remark −J is also an almost complex structure.T 1,0(M,−J) = T 0,1(M,J).
Holomorphic tangent vectors
Let M be a complex manifold. Recall that we have three notions of atangent space of M at a point p: T R
p M = TM - the real tangent space,
T Cp M - the complexified tangent space, and T 1,0
p M (resp. T 0,1p M) - the
holomorphic tangent space (resp. antiholomorphic tangent space).Let us choose local complex coordinates z = (z1, . . . ,zn) near z. If wewrite zi = xi +
√−1yi , then:
TpM = R
∂
∂xi,
∂
∂yi
,
T Cp M = C
∂
∂xi,
∂
∂yi
= C
∂
∂zi,
∂
∂zi
,
where
∂
∂zi=
12
(∂
∂xi−√−1
∂
∂yi
),
∂
∂zi=
12
(∂
∂xi+√−1
∂
∂yi
).
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Consequently:
T 1,0p M = C
∂
∂zi
, T 0,1
p M = C
∂
∂zi
,
and T 1,0p M (resp. T 0,1
p M) is the space of derivations which vanish onanti-holomorphic functions (resp. holomorphic functions).Now observe that:
V ,W ∈ Γ(T 1,0M
)=⇒ [V ,W ] ∈ Γ
(T 1,0M
),
andV ,W ∈ Γ
(T 0,1M
)=⇒ [V ,W ] ∈ Γ
(T 0,1M
).
The Newlander-Nirenberg Theorem
Theorem (Newlander-Nirenberg)
Let (M,J) be an almost complex manifold. The almost complexstructure J comes from a holomorphic atlas if and only if
V ,W ∈ Γ(T 0,1M
)=⇒ [V ,W ] ∈ Γ
(T 0,1M
).
An almost complex structure, which comes from complex coordinatesis called a complex structure.Remark: An equivalent condition is the vanishing of the Nijenhuistensor:
N(X ,Y ) = [JX ,JY ]− [X ,Y ]− J[X ,JY ]− J[JX ,Y ].
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As for the proof: we just have seen the ”only if” part. The ”if” part isvery hard. See Kobayashi & Nomizu for a proof under an additionalassumption that M and J are real-analytic, and Hormander’s”Introduction to Complex Analysis in Several Variables” for a proof infull generality.
Definition
A section Z of T 1,0M is a holomorphic vector field if Z (f ) isholomorphic for every locally defined holomorphic function f .
In local coordinates Z = ∑ni=1 gi
∂
∂zi, where all gi are holomorphic.
Definition
A real vector field X ∈ Γ(TM) is called real holomorphic if its(1,0)-component X − iJX is a holomorphic vector field.
LemmaThe following conditions are equivalent:
(1) X is real holomorphic.
(2) X is an infinitesimal automorphism of the complex structure J, i.e.LX J = 0 (LX - the Lie derivative).
(3) The flow of X consists of holomorphic transformations of M.
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The dual picture:
We can decompose the complexified cotangent bundle Λ1M⊗C into
Λ1,0M =
ω ∈ Λ1M⊗C; ω(Z ) = 0 ∀Z ∈ T 0,1M
andΛ0,1M =
ω ∈ Λ1M⊗C; ω(Z ) = 0 ∀Z ∈ T 1,0M
.
We haveΛ1M⊗C = Λ1,0M⊕Λ0,1M.
and, consequently, we can decompose the k -th exterior power ofΛ1M⊗C as
Λk M⊕C = Λk (Λ1,0M⊕Λ0,1M)
=M
p+q=k
Λp (Λ1,0M)⊗Λq (Λ0,1M
).
We writeΛp,qM = Λp (Λ1,0M
)⊗Λq (Λ0,1M
),
so thatΛk M⊕C =
Mp+q=k
Λp,qM.
Sections of Λp,qM are called forms of type (p,q) and their space isdenoted by Ωp,qM. In local coordinates, forms of type (p,q) aregenerated by
dzi1 ∧·· ·∧dzip ∧dzj1 ∧dzjq .
Remark: The above decomposition of Λk M⊗C is valid on any almostcomplex manifold.The difference between ”complex” and ”almost complex” lies in thebehaviour of the exterior derivative d .
Theorem
Let (M,J) be an almost complex manifold of dimension 2n. Thefollowing conditions are equivalent:
(i) J is a complex structure.
(ii) dΩ1,0M ⊂ Ω2,0M⊕Ω1,1M.
(iii) dΩp,qM ⊂ Ωp+1,qM⊕Ωp,q+1M for all 0≤ p,q ≤m.
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Proof.The only non-trivial bit is (ii) =⇒ (i). Use the following elementaryformula for the exterior derivative of a 1-form:
2dω(Z ,W ) = Z(ω(W )
)−W
(ω(Z )
)−ω([Z ,W ]).
Using statement (iii), we decompose the exterior derivatived : Ωk M → Ωk+1M as d = ∂+ ∂, where ∂ : Ωp,qM → Ωp+1,qM,∂ : Ωp,qM → Ωp,q+1M.
Lemma
∂2 = 0, , ∂2 = 0, ∂∂+ ∂∂ = 0.
Proof.
0 = d2 = (∂+ ∂)2 = ∂2 + ∂2 +(∂∂+ ∂∂) and the three operators in thelast term take values in different subbundles of Λ∗M⊗C.
The Dolbeault operator
Definition
The operator ∂ : Ωp,qM → Ωp,q+1M is called the Dolbeault operator. Ap-form ω of type (p,0) is called holomorphic if ∂ω = 0.
One uses ∂ to define Dolbeault cohomology groups of a complexmanifold, analogously to the de Rham cohomology:
Z p,q∂
(M) = ω ∈ Ωp,q(M); ∂ω = 0 − ∂-closed forms.
Hp,q∂
(M) =Z p,q
∂(M)
∂Ωp,q−1(M).
A holomorphic map f : M → N between complex manifolds induces amap
f ∗ : Hp,q∂
(N)→ Hp,q∂
(M).
Warning: Dolbeault cohomology is not a topological invariant: itdepends on the complex structure.
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The ordinary Poincare lemma says that every closed form on Rn isexact, and, hence, that the de Rham cohomology of Rn or a ballvanishes. Analogously:
Lemma (Dolbeault Lemma)
For B a ball in Cn, Hp,q∂
(B) = 0, if p +q > 0.
For a proof, see Griffiths and Harris, p. 25.
Example (Dolbeault cohomology of P1=CP1)
First of all Ω0,2(P1) = Ω2,0(P1) = 0, since dimC P1 = 1. HenceH0,2
∂(P1) = H2,0
∂(P1) = 0. Also H0,0
∂(P1) = C.
Secondly, ∂Ω1,0 = dΩ1, and, hence H1,1∂
(P1) = C.
Thirdly, if ω ∈ Z 1,0∂
(P1) = 0, then dω = 0 and using the vanishing of
H1(S2), we conclude that H1,0∂
(P1) = 0. This also shows that there
are no global holomorphic forms on P1, i.e. Z 1,0∂
(P1) = 0.
Finally, we compute H0,1∂
(P1) = 0:
Complex and holomorphic vector bundles
Let M be a smooth manifold. A (smooth) complex vector bundle (ofrank k ) on M consists of a family of Exx∈M of (k -dimensional)complex vector spaces parameterised by M, together with a C∞
manifold structure on E =S
x∈M Ex , such that:
(1) The projection map π : E →M, π(Ex) = x , is C∞, and
(2) Any point x0 ∈M has an open neighbourhood, such that thereexists a diffeomorphism
φU : π−1(U)−→ U×Ck
taking the vector space Ex isomorphically onto x×Ck for eachx ∈ U.
The map φU is called a trivialisation of E over U. The vector spaces Ex
are called the fibres of E . A vector bundle of rank 1 is called a linebundle.Examples: The complexified tangent bundle T CM, T 1,0M, T 0,1M,bundles Λp,qM.
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For any pair φU ,φV of trivialisations, we have the C∞-mapgUV : U ∩V → GL(k ,C) given by
gUV (x) =(φU φ
−1V
)|x×Ck .
These transition functions satisfy
gUV (x)gVU(x) = I, gUV (x)gVW (x)gWU(x) = I.
Conversely, given an open cover U = Uα of M and C∞-mapsgαβ : Uα∩Uβ → GL(k ,C) satisfying these identities, there is a uniquecomplex vector bundle E →M with transition functions gαβ.All operations on vector spaces induce operations on vector bundles:dual bundle E∗, direct sum E ⊕F , tensor product E ⊗F , exteriorpowers Λr E . The transition functions of these are easy to write down,e.g. if E ,F are vector bundles of rank k and l with transition functionsgαβ and hαβ, respectively, then the transition functions of E ⊕Fand E ⊗F are(
gαβ(x) 00 hαβ(x)
)∈GL
(Ck⊕Cl), gαβ(x)⊗hαβ(x)∈GL
(Ck⊗Cl).
An important example is the determinant bundle Λk E of E(k = rankE). It is a line bundle with transition functions
detgαβ(x) ∈ GL(1,C) = C∗.
A subbundle F ⊂ E of a vector bundle E is a collection Fx ⊂ Exx∈M
of subspaces of the fibres Ex such that F =S
x∈M Fx is a submanifoldof E . This means that there are trivialisations of E , relative to whichthe transition functions look like
gUV (x) =
(hUV (x) kUV (x)
0 jUV (x)
).
The bundle F has transition functions hUV , while jUV are transitionfunctions of the quotient bundle E/F , given by Ex/Fx .A homomorphism between vector bundles E and F on M is given by aC∞-map f : E → F , such that fx = f|Ex : Ex → Fx and fx is linear. Wehave the obvious notions of ker(f ) - a subbundle of E , and Im(f ) - asubbundle of F . Also, f is an isomorphism, if each fx is anisomorphism.A vector bundle E on M is trivial, if E is isomorphic to the productbundle M×Ck .
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Given a C∞-map f : M → N and a vector bundle Eπ→ N, we define the
pullback bundle f ∗E on M by
f ∗E = (x ,e) ∈M×E ; f (x) = π(e),
i.e. (f ∗E)x = Ef (x).Finally, a section s of a vector bundle E →M is a C∞-map s : M → Esuch that s(x) ∈ Ex for all x ∈M (just like a vector field). The space ofsections is denoted by Γ(E).Observe that trivialising a rankk bundle E over U is equivalent togiving k sections s1, . . . ,sk , which are linearly independent at everypoint of U. Such a collection (s1, . . . ,sk) is called a frame for E over U.
Now, let M be a complex manifold. A holomorphic vector bundleE
π→M is defined as a complex vector bundle, except that C∞ isreplaced by “holomorphic” everywhere.Examples: T 1,0M, Λp,0M (but not Λp,qM if q 6= 0). The line bundleΛn,0M, n = dimC M is called the canonical bundle of M, and is denotedby KM . Its sections are holomorphic n-forms. The dualK ∗
M = Λn(T 1,0M
)is the anti-canonical bundle.
Example (The tautological and canonical bundles on CPn)
The tautological line bundle J → CPn is defined by setting JA = A,where A is a line in Cn+1 defining a point of CPn.We have KCPn = Jn+1.
For every holomorphic bundle E →M we define the bundlesΛp,qE = Λp,qM⊗E of E-valued forms of type (p,q). The space ofsections is denoted by Ωp,qE . If we choose a trivialisation of E , i.e. alocal frame, then an element σ of Ωp,qE is, in this trivialisation,σ = (ω1, . . . ,ωk), ωi - a local (p,q)-form on M.Now observe that wehave a well-defined operator ∂ : Ωp,qE → Ωp,q+1E , given by:
∂σ = (∂ω1, . . . , ∂ωk)
in any trivialisation.Note that ∂ satisfies ∂2 = 0 and the Leibniz rule∂(fσ) = ∂f ⊗σ+ f (∂σ), for any f ∈ C∞(M) and σ ∈ Ωp,qE .The existence of a such a natural operator ∂ on sections of E = Λ0,0Eis a remarkable property of holomorphic vector bundles. On a generalcomplex vector bundle there is no canonical way of differentiatingsections. It has to be introduced ad hoc, via the concept of connection:
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Connections and their curvature
DefinitionA connection (or covariant derivative) on a complex vector bundleE →M is a map D : Γ(E)→ Ω1E satisfying the Leibniz rule:
D(fs) = (df )⊗ s + f (Ds),
for any f ∈ C∞(M) and s ∈ Γ(E).
If we choose a frame (local basis) (e1, . . . ,ek) for E over U, then
Dei = ∑j
θijej ,
where the matrix of 1-forms is called the connection matrix withrespect to e.The data e and θ determine D. The connection matrix depends on thechoice of e: if (e′1, . . . ,e
′k) is another frame with e′(z) = g(z)e(z),
g(z) ∈ GL(k ,C), then:
θe′ = gθeg−1 +dgg−1.
Curvature
One can extend any connection D to act on ΩpE , i.e. on E-valuedp-forms, by forcing the Leibniz rule:
D(ω⊗σ) = dω⊗σ+(−1)pω∧Dσ, ∀ω ∈ ΩpM, σ ∈ Γ(E).
In particular, we have the operator D2 : Γ(E)→ Ω2E . It is tensorial,i.e. linear over C∞(M):
D2(fσ) = D(df ⊗σ+ f (Dσ)
)=−df ∧Dσ+df ∧Dσ+ fD2
σ = fD2σ.
Consequently, D2 is induced by a bundle map E → Λ2M⊗E , i.e. by asection of Λ2M⊗Hom(E ,E). This Hom(E ,E)-valued 2-form RD iscalled the curvature of the connection D.The curvature matrix. In a local frame, we had Dei = ∑j θijej , and,hence:
D2ei = D(∑θijej
)= ∑
(dθij −∑θip ∧θpj
)⊗ej .
Therefore, we get the Cartan structure equations for the curvaturematrix with respect to a frame e:
Θe = dθe−θe∧θe.
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Hermitian metrics
DefinitionLet E →M be a complex vector bundle. A hermitian metric on E is asmoothly varying hermitian inner product on each fibre Ex , i.e. ifσ = (s1, . . . ,sk) is a frame for E , then the functionshij(x) = 〈si(x),sj(x)〉 are C∞.
A complex vector bundle equipped with a hermitian metric is called ahermitian vector bundle.Example: E = T 1,0M - a hermitian metric on M.On a hermitian vector bundle over a complex manifold, we have acanonical connection D by requiring:
D is compatible with the metric, i.e.
d〈s1,s2〉= 〈Ds1,s2〉+ 〈s1,Ds2〉,i.e. the metric is parallel for D, andD is compatible with the complex structure, i.e. if we writeD = D1,0 +D0,1 with D1,0 : Γ(E)→ Ω1,0E andD0,1 : Γ(E)→ Ω0,1E , then D0,1 = ∂.
TheoremIf E →M is a hermitian vector bundle over a complex manifold, thenthere is unique connection D (called the Chern connection) compatiblewith both the metric and the complex structure.
Proof.
Let (e1, . . . ,ek) be a holomorphic frame for E and put hij = 〈ei ,ej〉. IfD is compatible with the complex structure, then Dei is of type (1,0),and, as D is compatible with the metric:
dhij = 〈Dei ,ej〉+ 〈ei ,Dej〉= ∑p
θiphpj +∑p
θpjhip.
The first term is of type (1,0) and the second one of type (0,1) and,hence:
∂hij = ∑p
θiphpj , ∂hij = ∑p
θpjhip.
Therefore ∂h = θh and ∂h = hθT and θ = ∂hh−1 is the unique solutionto both equations.
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Remark on the curvature of the Chern connection. If D iscompatible with the complex structure, then the (0,2)-component ofthe curvature vanishes:
R0,2 = D0,1 D0,1 = ∂2 = 0.
If D is also compatible with the hermitian metric, then the(2,0)-component vanishes as well, since
0 = d2〈ei ,ej〉= 〈D2ei ,ej〉+2〈Dei ,Dej〉+ 〈ei ,D2ej〉
and so the (2,0)-component of D2 is the negative hermitian transposeof the (0,2)-component.Thus, the curvature of the Chern connection is of type (1,1).
Curvature of a Chern connection in a holomorphic frame
Recall that the curvature matrix of a connection D with respect to anyframe e = (e1, . . . ,en) is given by
Θe = dθe−θe∧θe = dθe− [θe,θe],
where θe is the connection matrix w.r.t. e.If D is the Chern connection of a hermitian holomorphic vector bundle,and e is a holomorphic frame, then we computed: θe = ∂hh−1, wherehij = 〈ei ,ej〉. We compute dθe:
(∂+ ∂)θe = ∂θe +∂(∂hh−1) = ∂θe−∂h∧∂(h−1) = ∂θe +∂hh−1∧∂hh−1.
Therefore, the curvature matrix of the Chern connection w.r.t. aholomorphic frame is given by
Θ = ∂θ.
In the case of a line bundle, a local non-vanishing holomorphic sections and h = 〈s,s〉:
θ = ∂ logh, Θ = ∂∂ logh.
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Example (Curvature of the tautological bundle of CPn)
J is a subbundle of the trivial bundle Pn×Cn+1:
J = l× z ∈ Pn×Cn+1; z ∈ l.
We have a hermitian metric on Pn×Cn+1 given by the standardhermitian inner product on Cn+1. It induces a hermitian metric on theline bundle J. We compute the curvature of the associated Chernconnection in the patch U0 (z0 6= 0), in which J is trivialised viaφ : Cn×C→ J: (
(z1, . . . ,zn),u)7−→ u(1,z1, . . . ,zn).
We have a non-vaninshing section s(z1, . . . ,zn) = (1,z1, . . . ,zn), and,consequently, h = 〈s,s〉= 1+∑
ni=1 |zi |2. Thus, the curvature is
∂∂ logh. =n
∑i=1
1+∑r 6=i |zr |2
(1+∑nr=1 |zr |2)2 dzi ∧dzi .
Hermitian metrics on complex manifolds
We now look at the particular case E = T 1,0M (M - a complexmanifold). A choice of holomorphic frame is given by a choice of localholomorphic coordinates z1, . . . ,zn (ei = ∂
∂zi), and a hermitian metric
on T 1,0M can be written as
g = ∑i,j
hijdzi ⊗dzj .
where hij is hermitian matrix, h∗ = h. Notice that this also defines aHermitian metric on T CM, and by restriction a C-valued hermitianmetric on TM.Its real part is a real-valued symmetric bilinear form, which we alsodenote by g, and refer to as hermitian metric on M. Observe thatg(JX ,JY ) = g(X ,Y ) ∀X ,Y ∈ TxM.The negative of the imaginary part is a skew-symmetric bilinear form ω
called the fundamental form of g. We have ω(X ,Y ) = g(JX ,Y ), andin local coordinates:
g =12
Re∑i,j
hijdzidzj ,
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ω =
√−12 ∑
i,jhijdzi ∧dzj .
Observe that ω ∈ Ω1,1M.
Example (Connection and curvature in dimension one)
z = x + iy - a local coordinate, ∂
∂z - a local holomorphic frame, and ahermitian metric on T 1,0M is written as hdz⊗dz, for a local functionh > 0. The connection matrix of the Chern matrix is ∂hh−1 = ∂ logh
∂z dz,and the curvature matrix is
Θ = ∂∂ logh =∂2 logh∂z∂z
dz ∧dz =(−1
4∆ logh
)dz ∧dz.
Now, the fundamental form on M is√−12 hdz ∧dz and hence,
Θ =−√−1K ω,
where K = (−∆ logh)/2h is the usual Gaussian curvature of asurface.
Recall the different behaviour of positively and negatively curvedsurfaces. Similarly, we say that the curvatureRE ∈ Γ
(Λ1,1M⊗Hom(E ,E)
)of a Chern connection on a hermitian
vector bundle is positive at x ∈M (notation RE(x) > 0), if thehermitian matrix RE(x)(v , v) ∈ Hom(Ex ,Ex) is positive definite. Inlocal coordinates, this means that Θ(x)(v , v) is positive definite ∀v ,e.g. for surfaces with positive Gaussian curvature.Similarly RE(x)≥ 0,RE(x)≤ 0,RE(x) < 0. We write RE > 0 etc. ifthe curvature is positive everywhere. Observe that the curvature of thetautological bundle JPn (with induced metric) is negative.Let us compute the curvature RF of the Chern connection of aholomorphic subbundle F ⊂ E (with the induced hermitian metric). LetN = F⊥. This is a C∞ complex subbundle of E .If s ∈ Γ(F) and t ∈ Γ(N), then:
0 = d〈s, t〉= 〈Ds, t〉+ 〈s,Dt〉,
so that in a frame for E consisting of frames for F and N, theconnection matrix of D = DE is
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θE =
(θF A−A∗ θN
).
Moreover, if the frame on F is holomorphic, then A (matrix of 1-forms)is of type (1,0). Now, the curvature matrix is
ΘE = dθE −θE ∧θE =
(dθF −θF ∧θF +A∧A∗ ∗
∗ ∗
),
and so ΘF = ΘE |F −A∧A∗. Therefore
RF ≤ RE |F ,
so that the curvature decreases in holomorphic subbundles. Inparticular, if E is a trivial bundle with Euclidean metric (so that RD ≡ 0)and F ⊂ E is a holomorphic subbundle with the induced metric, thenRD
F ≤ 0.Applying this to a submanifold M of Cn and F = T 1,0M ⊂ T 1,0Cn
∣∣M
with the induced hermitian metric, we see that the curvature of T 1,0Mis always non-positive. In particular, if M is a Riemann surface, then itsGaussian curvature K ≤ 0.
Observe that the same calculation for the quotient bundle Q = E/Fand conclude that
RQ ≥ RE |F ,
i.e. the curvature increases in holomorphic quotient bundles.As an application, consider a holomorphic vector bundle E →M“spanned by its sections”, i.e. there exist holomorphic sectionss1, . . . ,sk ∈ Γ(E), such that s1(x), . . . ,sk(x), k ≥ rankE , generate Ex
for every x ∈M. Then we have a surjective map M×Ck → E ,(x ,λ) 7→ ∑
ki=1 λisi(x).
Thus, E is a quotient bundle of a trivial bundle and, if we give E themetric induced from the Euclidean metric on M×Ck , then RQ ≥ 0.
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The first Chern class
We go back to a non-holomorphic setting and consider complex vectorbundles over smooth manifolds. We equip such a bundle E →M witha connection D. Recall that the curvature RD of D is a section ofΛ2M⊗Hom(E ,E), so basically a matrix of 2-forms. We can speak ofthe trace of the curvature: trRD, which is a 2-form.Recall the formula for the curvature matrix in a local frame:Θ = dθ− [θ,θ]. It follows that trΘ = trdθ = d trθ and, hence, trRD isa closed 2-form. It is called the Ricci form of D.
Lemma
The cohomology class[trRD
]∈ H2
DR(M) does not depend on D.
Proof.
A = D−D′ is a section of Λ1M⊗Hom(E ,E), so trRD − trRD′= d trA,
which is an exact form.
The cohomology class c1(E) =√−1
2π
[trRD
]∈ H2
DR(M) is called the1st Chern class of E . It is a topological invariant.
Example
We compute c1(JP1). Recall that for the Chern connection inducedfrom P1×C2, we computed the curvature matrix in the chart U0 withholomorphic coordinate z as
1
(1+ |z|2)2 dz ∧dz.
Now, H2DR(P1) is identified with C via integration: ω 7→
RP1 ω. Thus
c1(JP1) =
√−1
2π
ZC
1
(1+ |z|2)2 dz∧dz =1
2π
Z[0,2π]×[0,∞)
r
(1+ r2)2 dθ∧dr ,
where z = reiθ. Taking the orientation into account we obtain
c1(JP1) =−12π
Z +∞
0
Z 2π
0
r
(1+ r2)2 dθdr =−1.
It follows from this that c1(JPn) =−1 for any n.
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Let E and F be bundles of ranks m and n, respectively. Then:
(i) c1(ΛmE) = c1(E),
(ii) c1(E ⊕F) = c1(E)+ c1(F),
(iii) c1(E ⊗F) = nc1(E)+mc1(F),
(iv) c1(E∗) =−c1(E),
(v) c1(f ∗E) = f ∗c1(E).
Let M be a complex manifold. The first Chern class c1(M) of M isc1(T 1,0M
)= c1(K ∗), i.e. the first Chern class of the anti-canonical
bundle of M.
Example
c1(Pn) = c1(K∗) = c1
((J∗)⊗(n+1)
)= (n +1)c1(J
∗) = n +1.
For n = 1, we get c1(P2) = 2.
This is just the Gauss-Bonnet theorem: for any hermitian metric on acompact surface S :
c1(S) =1
2π
ZS
K ω = χ(S).
Examples of manifolds with c1(M) = 0:Observe that c1(M) = 0, if the canonical bundle KM is trivial, i.e. thereexists a non-vanishing holomorphic n-form on M (n = dimC M).
Cn. Other boring examples include any M with H2(M) = 0.
Quotients of Cn by finite groups of holomorphic isometries, e.gby lattices: abelian varieties (complex tori).
The quadric Q = (z1,z2,z3) ∈ C3; z21 + z2
2 + z23 = 1. This is a
complexification of S2 and so H2(Q) 6= 0. The followingholomorphic 2-form is non-vanishing on Q and trivialises KQ :
z1dz2∧dz3 + z2dz3∧dz1 + z3dz1∧dz2.
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The famous K3-surface (one of them, anyway):S = [z0,z1,z2,z3] ∈ P3; z4
0 + z41 + z4
2 + z43 = 0.
We’ll show how to compute c1 of hypersurfaces. Let V ⊂M be acomplex submanifold of dimension 1. We have the exact sequence ofholomorphic vector bundles over V :0→ N∗
V → Λ1,0M|V → Λ1,0V → 0.Taking the maximal exterior power gives: KM |V ' KV ⊗N∗
V , so
KV = KM |V ⊗NV .
Now, observe that, if f = 0 is a local (say, on U ⊂ V ) defining equationfor V , then, by definition, df generates N∗
V over U.If, on U ′ ⊂ V , the equation of V is f ′ = 0, then on U ∩U ′:
df = d( f
f ′f ′)
= d( f
f ′)f ′+
ff ′
df ′ =ff ′
df ′,
so the transition functions for N∗V are f/f ′.
We can now easily finish the computation for K3: N∗S ' KP3 |S .
Higher Chern classes
The first Chern class of a complex vector bundle was defined using thetrace of the curvature RD ∈ Γ
(Λ2M⊗Hom(E ,E). We can use other
invariant polynomials defined on matrices, e.g. the determinant or thesum of the squares of eigenvalues.For a k × k matrix M, let us write
det(t +M) =k
∑i=0
Pi(M)tk−i .
so that P1(M) = trM, Pk(M) = detM, etc. Each Pi is a homogeneouspolynomial of degree k in the entries of A.We can allow matrices over any commutative algebra - the Pi(M) arewell defined - in particular over the even part of the exterior algebra ofa smooth manifold. Applying this to RD or its curvature matrix, weobtain closed forms Pi(RD) = Pi(Θ) ∈ Ω2iM.
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Definition
The i-th Chern class of a complex vector bundle E over a smoothmanifold M is the cohomology class
ci(E) =
[Pi
(√−1
2πRD)]
∈ H2iDR(M), i = 1, . . . ,k .
Once again, it does not depend on the connection D (exercise). Also,for a complex manifold M, we define the Chern classes of M to be theChern classes of T 1,0M.The Gauss-Bonnet theorem generalise to higher dimensions asfollows: if M is a compact complex manifold of dimension n, then
cn(M) = χ(M).
There are of course purely topological ways of defining Chern classes!
Chern classes of a holomorphic bundles
Now E is hermitian holomorphic vector bundle over a complexmanifold M. Recall that the curvature of a Chern connection D is oftype (1,1) and that its matrix is skew-hermitian, so that Pi
(√−12π
RD)
isa real (i, i)-form, and consequently, for any holomorphic bundle
ci(E) ∈ H i,i(M)∩H2iDR(M;R).
Recall also that we defined notions of the curvature being> 0,< 0,≥ 0,≤ 0. For example, nonnegativity meant that RD(v , v)has all eigenvalues nonnegative, for any v ∈ T 1,0M. This means that
Pi(RD)(v1, v1, . . . ,vi , vi))≥ 0 ∀v1, . . . ,vi ∈ T 1,0M,
and consequently that ZZ
ci(E)≥ 0
for any i-dimensional complex submanifold Z of M. In particular, thisholds for any holomorphic vector bundle generated by its sections.Note that these conditions depend only on ci(E), not on theconnection or its curvature.
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Prescribing the Ricci curvature of Chern connections
Thus, we say that c1(E)≥ 0 (resp. > 0,< 0,≤ 0), if the cohomologyclass c1(E) ∈ H1,1(M) can be locally represented by a closed real2-form φ, such that in local complex coordinates
φ =
√−1
2π∑φ
αβdzα∧dzβ,
and the hermitian matrix [φαβ
] has all eigenvalues nonnegative (resp.positive, negative, nonpositive).E.g., any holomorphic vector bundle generated by its sections satisfiesc1(E)≥ 0.Let now a closed real (1,1)-form φ represent c1(E). We ask
Is there a connection D on E such that trRD =−2π√−1φ?
For example, suppose that c1(E) = 0. It is clearly represented byφ≡ 0 and we would like to know whether there is a hermitian metric,such that the associated Chern connection is Ricci-flat (i.e. trRD ≡ 0).
Let 〈,〉 be a hermitian metric on E . Recall that in a local holomorphicframe (e1, . . . ,en) with the associated matrix hij = 〈ei ,ej〉, we had thefollowing formula for RD:
Θ = ∂(∂hh−1).
Therefore, the Ricci form trRD is represented in this local frame by
∂∂(logdeth).
Now we modify 〈,〉 by taking ef/k〈,〉, where k = rankE and f is asmooth real function on M. Let h′ij = ef/k〈ei ,ej〉. Then deth′ = ef dethand the Ricci forms of the two Chern connections are related by
trRD′− trRD = ∂∂f .
Therefore we find a hermitian metric with trRD′= φ, providing we can
solve the equation∂∂f = φ− trRD.
The right-hand side is a closed imaginary (1,1)-form cohomologous to0:[φ− trRD
]=−2π
√−1(c1(E)− c1(E)).
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Therefore the answer to our question: can we find a hermitian metricon E , the Ricci form of which is a given form φ, is yes on manifolds Mfor which the following condition (called the global ∂∂-lemma) holds:
Any exact real (1,1)-form β on M is of the form√−1∂∂f , for a
smooth function f : M → R.
Lemma
Let M be a complex manifold with H0,1(M) = 0. Then the global∂∂-lemma holds on M.
Proof. Since β is exact, there exists a real 1-form α such that dα = β.We decompose α as τ+ τ′, where τ is (1,0) and τ′ is (0,1). We musthave τ′ = τ.Now, β = (∂+ ∂)(τ+ τ) = ∂τ+(∂τ+∂τ)+ ∂τ. Comparing the types,we get:
β = ∂τ+∂τ, ∂τ = 0, ∂τ = 0.
Since H0,1(M) = 0, there exists a function u : M → C such thatτ = ∂u. Therefore τ = ∂u, andβ = ∂τ+∂τ = ∂∂u +∂∂u = ∂∂(u− u) = 2i∂∂(Imu).
The result follows with f = 2Imu.
Kahler metricsWe now consider the case E = TM, where M is a complex manifoldwith a complex structure J. TM is a complex vector bundle - J allowsus to identify TxM with Cn, for every n. In other words we identify
TM ' T 1,0M, X 7−→ X − iJX .
Recall that a hermitian metric on M is a smoothly varying inner productg on each TxM, such that g(JX ,JY ) = g(X ,Y ) for all X ,Y ∈ TxM.We will now consider connections on TM. First, of all observe, that forsuch a connection D : Γ(TM)→ Ω1(M)⊗Γ(TM), the (covariant)directional derivative DZ (X) := D(X)(Z ) of a vector field is again avector field.We now reinterpret the two properties defining the Chern connectionD of g:1. D being metric, i.e. dg(X ,Y ) = g(DX ,Y )+g(X ,DY ), is equivalentto
Z .g(X ,Y ) = g(DZ X ,Y )+g(X ,DZ Y ) ∀X ,Y ,Z ∈ Γ(TM).
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2. D being compatible with the complex structure, i.e. D0,1 = ∂, isequivalent to
DJ = 0, i.e. DZ (JX) = JDZ (X) ∀X ,Z ∈ Γ(TM).
i.e. J is parallel for D (follows from the fact that the connection matrixin the holomorphic frame has type (1,0)).On the other hand, since g is a Riemannian metric, i.e. a smoothlyvarying inner product, there exists a unique connection (theLevi-Civita connection) ∇ on TM, which is again metric and it has zerotorsion, i.e.
∇X Y −∇Y X = [X ,Y ], ∀X ,Y ∈ Γ(TM).
Thus, on a complex manifold with a hermitian metric, we have twonatural connections: Chern and Levi-Civita. We ask for whichhermitians metrics the two coincide.
Theorem
Let g be a hermitian metric on a complex manifold (M,J). Thefollowing conditions are equivalent:
(i) J is parallel for the Levi-Civita connection ∇.
(ii) The Chern connection D has zero torsion.
(iii) The Levi-Civita and the Chern connections coincide.
(iv) The fundamental form ω of g is closed, dω = 0, (recall thatω(X ,Y ) = g(JX ,Y )).
(v) For each point p ∈M, there exists a smooth real function in aneighbourhood of p, such that ω = i∂∂f .
(vi) For each point p ∈M, there exist holomorphic coordinates wcentred at p, such that g(w) = 1+O(|w |2).
Proof. (i), (ii), and (iii) are clearly equivalent from the uniquenessproperties. We show (i) =⇒ (iv) =⇒ (v) =⇒ (vi) =⇒ (i).
(i) =⇒ (iv). Since ∇g = 0 and ∇J = 0, ∇ω = 0. For any p-form andany torsion-free connection ∇ one has (exercise):
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dω(X0, . . . ,Xp) =p
∑i=0
(−1)i(∇Xi ω
)(X0, . . . , Xi , . . . ,Xp),
which implies that parallel forms are closed.
(iv) =⇒ (v). This is the ∂∂-lemma applied to a ball B in Cn
(Dolbeault Lemma says that H0,1(B) = 0).
(v) =⇒ (vi). We can choose local coordinates around p0, so thatω = i ∑l,m ωlmdzl ∧dzm, where
ωlm =12
δlm +∑j(ajlmzj +bjlmzj)+O(|z|2).
Reality implies that ajlm = bjml . On the other hand, using (v), we have
ajlm =∂3f
∂zj∂zl∂zm,
so ajlm = aljm. We now put
wm = zm +∑j,l
ajlmzjzl , =⇒ dwm = dzm +2∑j,l
ajlmzjdzl ,
and compute:
12
i ∑m
dwm∧dwm =12
i ∑m
dzm∧dzm + i ∑j,l,m
ajlmzjdzl ∧dzm
+ i ∑j,l,m
ajlmzjdzm∧dzl +O(|z|2)
= i ∑l,m
ωlmdzl ∧dzm +O(|z|2) = ω+O(|z|2) = ω+O(|w |2).
(vi) =⇒ (i) If we write zi = xi +√−1yi for the coordinates obtained
in (v), then the connection matrix of the Levi-Civita connection(Christoffel symbols) is equal to zero at p0. Consequently, ∇Jvanishes at p0, and, since p0 is arbitrary, it vanishes everywhere.
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A hermitian metric on a complex manifold satisfying the equivalentconditions (i)− (vi) is called a Kahler metric.
The fundamental form ω of a Kahler metric is called a Kahler form andthe local function f in (v) is called a local Kahler potential.
Examples:1. The standard Euclidean metric on Cn.
g =12
Ren
∑s=1
dzs⊗dzs, ω =i2
Ren
∑s=1
dzs ∧dzs =i2
∂∂|z|2.
Thus f (z) = 12 |z|
2 is a global Kahler potential on Cn. The pair (g,J) isU(n)-invariant.
2. The Fubini-Study metric on CPn.For a z = (z0, . . . ,zn) ∈ Cn+1−0, we put
ω = i∂∂ log(|z|2).
This is invariant under rescaling by C∗ and, so, it induced a real closed(1,1)-form on Pn. We need to check that the correspondingsymmetric form g(X ,Y ) = ω(X ,JY ) is positive-definite. We computefirst ω in the chart U0 = z0 6= 0, where the local coordinates arewi = zs
z0, s = 1, . . . ,n:
ω = i∂∂ log(1+ |w |2) = ∂
(i
1+ |w |2n
∑s=1
wsdws
)=
i1+ |w |2
n
∑s=1
dws∧dws−i
(1+ |w |2)2
(n
∑s=1
wsdws
)∧
(n
∑s=1
wsdws
)
Observe that ω is U(n +1)- invariant, and so it is enough to checkpositivity at any point p ∈ Pn, e.g p = [1,0, . . . ,0], i.e. w = 0. OK!
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Remark. The Fubini-Study metric is the quotient metric onCPn ' Sn+1/S1 obtained from the standard round metric on Sn+1.
These two basic examples (Cn and CPn) yield plenty of others,because:
LemmaA complex submanifold of a Kahler manifold, equipped with theinduced metric, is Kahler.
Proof. Let (N,J,gN)⊂ (M,J,gM) as in the statement. Thefundamental form ωN of gN is just the pullback (restriction) of thefundamental form ωM of gM , hence it is closed. Observe also that a product of two Kahler manifolds is Kahler.
On the other hand, many complex manifolds do not admit any Kahlermetric, because:
Lemma
If M is a compact Kahler manifold, then H2qDR(M) 6= 0, q ≤ n = dimC M.
Proof.
Let ω be the Kahler form. Then ωq is a closed form, which is not exact.Indeed, had we ωq = dψ, then ωn = d(ψ∧ωn−q), and, so:
vol(M) =Z
Mω
n =Z
Md(ψ∧ω
n−q) = 0,
which is impossible. Thus, for example, there is no Kahler metric on S1×S2n−1 for n ≥ 2(this is a complex manifold - see Examples Set 1).Another easily provable topological restriction is
Lemma
Let M be a compact Kahler manifold. Then the identity map on ΩqMinduces an injective map
Hq,0∂
(M) → HqDR(M,C),
i.e. every holomorphic (q,0)-form is closed and never exact.
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Proof. Let η be a holomorphic (q,0)-form. Write η in a local unitaryframe φi as η = ∑ fIφI (I runs over subsets of cardinality q and φI isthe wedge product of φj with j ∈ I). Then η∧ η = ∑ fI fJφI ∧ φJ .
On the other hand, ω =√−12 ∑φi ∧ φi , and (n = dimC M):
ωn−q = cq ∑
#K=n−q
φK ∧ φK .
Henceη∧ η∧ω
n−q = c′q ∑#I=q
|fI |2ωn,
since the only non-zero wedge-products arise for K disjoint from I andJ (so for I = J). In particular, if η 6= 0, then the integral over M of theLHS is non-zero. If, however, η = dψ, then :
η∧ η∧ωn−q = d
(ψ∧ η∧ω
n−q) ,since dη = 0 and dω = 0, and we obtain a contradiction.Thus, a non-zero holomorphic (q,0)-form is never exact. To show thatη is closed, note: dη = (∂+ ∂)η = ∂η, and, so, dη is a holomorphic(q +1,0)-form, hence dη = 0.
The last result is a particular case of a much stronger fact, which goesunder the name of Hodge relations or Hodge decomposition:
Theorem (Hodge)
The complex cohomology of a compact Kahler manifold satisfies:
H r (M,C) =M
p+q=r
Hp,q(M), Hp,q(M) = Hq,p(M).
In particular: for r odd, dimH r (M) is even.
To give even an idea of a proof of this theorem of Hodge requires asubstantial detour.
Before, let us see how badly this theorem fails for non-compact Kahlermanifolds:
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Example
M = C2−0, so that H1DR(M) = 0. We shall show that
dimH0,1∂
(M) = +∞. Let U1 = z1 6= 0= C∗×C,U2 = z2 6= 0= C×C∗, so that U1∪U2 = M, U1∩U2 = C∗×C∗.Let λ1,λ2 : M → R be a partition of unity subordinated to U1,U2 andlet f be a holomorphic function on U1∩U2. Then g1 = λ2f defines asmooth function on U1 and g2 =−λ1f defines a smooth function onU2. Observe that on U1∩U2, ∂(g1−g2) = ∂f = 0, and so we candefine a (0,1)-form ω on M, with ∂ω = 0, by
ω =
∂g1 = f ∂λ2 on U1
−∂g2 =−f ∂λ1 on U2.
Suppose that ω = ∂h for some h ∈ C∞(M). Then ∂(g1−h) = 0 on U1
and ∂(g2−h) = 0, and, so g1−h is holomorphic on U1, g2−h isholomorphic on U2. Then f = g1−g2 = (g1−h)− (g2−h), and,hence, f = u1 +u2, where ui is holomorphic on Ui , i = 1,2.But any convergent Laurent series f on U1∩U2, which contains zm
1 zn2 ,
m,n < 0, is not u1 +u2, so ω defined by such an f is not exact.
An idea of a proof of the Hodge theorem
Let V be a vector space, equipped with an inner product 〈,〉. Thenthere is an induced inner product on the tensor powers V⊗k . Byrestriction, there is an inner product on the exterior powers Λk V . Itcan be described by choosing an orthonormal basis (e1, . . . ,en) anddecreeing the following basis of Λk V :
ei1 ∧·· ·∧eik ; 1≤ i1 < · · ·< ik ≤ n
to be orthonormal. Now, if (M,g) is an oriented Riemannian manifold,then we can do the above on each tangent space TxM. Since a dualof a vector space with an inner product also has an inner product, weget an inner product on Λk T ∗
x M, and, if (M,g) is oriented, i.e. we havea nonvanishing volume form dV , and compact, then we can define aninner product on differential forms in Ωk M:
〈α,β〉=Z
M〈α|x ,β|x〉dV .
The idea is that in every cohomology class in HkDR(M) we seek a
representative α with the smallest norm.
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We also show that such a representative must be unique (an elementwith the smallest norm in a closed affine subspace of a Hilbert space).We do the same for the Dolbeault cohomology classes Hp,q
∂(M).
Finally, we show that, if M is Kahler, then, if we decompose theα ∈ Hk
DR(M,C) with minimal norm into forms of type (p,q), p +q = k ,then each component has the minimal norm in Hp,q
∂(M).
How to find the element of smallest norm in a cohomology class[ψ] ∈ Hk
DR(M)?We are looking for the element of smallest norm in an affine subspaceP = ψ+dη; η ∈ Ωk−1M. For M compact, Ωk(M) with the innerproduct 〈,〉 is a pre-Hilbert space (essentially an L2-space), and wereP a closed subspace, we could find an element of the smallest normby the orthogonal projection (Ωk(M) = dΩk−1M⊕ (dΩk−1M)⊥).The orthogonal projection, on the other hand, can be expressed viathe adjoint operator to d :
‖ψ+dη‖2 = ‖ψ‖2 +‖dη‖2 +2〈ψ,dη〉= ‖ψ‖2 +‖dη‖2 +2〈d∗ψ,η〉.
Thus, if d∗ψ = 0, then ψ has the smallest norm in P.
Hodge dual
So cohomology classes should be represented by forms ψ such thatdψ = 0 and d∗ψ = 0, once we define d∗ : Ωk+1 → Ωk .
To define d∗, we go back to the situation described earlier: V is avector space, dimV = n, equipped with an inner product 〈,〉 and anorientation, i.e. a preferred element of ΛnV , e.g. e1∧·· ·∧en.Any α ∈ Λn−k V defines a linear functional φ on Λk V by:
ω∧α = φ(ω)e1∧·· ·∧en.
Such a functional must be represented by ω 7→ 〈ω,τ〉 for someτ ∈ Λk V . This way, we get a 1-1 correspondence between Λk V andΛn−k V :
∗ : Λk V → Λn−k V , τ 7→ α,
ω∧∗τ = 〈ω,τ〉e1∧·· ·∧en.
The operator ∗ is called the Hodge dual. Notice, in particular:
∗1 = e1∧·· ·∧en, 〈∗τ1,∗τ2〉= 〈τ1,τ2〉, ∗2 = (−1)k(n−k) on Λk V .
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Again, we obtain such a ∗ on differential forms of an orientedRiemannian manifold (M,g).Example: Take M = R3 with the Euclidean metric g = ∑
3i=1 dxi ⊗dxi
and the standard orientation dx1∧dx2∧dx3. We get:
∗dx1 = dx2∧dx3, ∗dx2 = dx3∧dx1, ∗dx3 = dx1∧dx2.
Now, observe that ∗d∗ : Ωk+1 → Ωk and we have (dV denotes thevolume form determining orientation):
〈dα,β〉dV = dα∧∗β = d(α∧∗β)− (−1)kα∧d ∗β, so
〈dα,β〉dV −d(α∧∗β) =−(−1)kα∧d ∗β
=−(−1)k(−1)k(n−k)α∧∗2d ∗β =−(−1)kn〈α,∗d ∗β〉dV .
Therefore, the operator d∗ = (−1)kn+1 ∗d∗ : Ωk+1 → Ωk is the adjointof d , and is called the codifferential. A form ω, such that d∗ω = 0 iscalled co-closed.
We need to show that on a compact oriented manifold (M,g), anycohomology class has a representative ψ with d∗ψ = 0 (and dψ = 0).
The Riemannian Laplacian
Observe that such a form also satisfies (dd∗+d∗d)ψ = 0. Theoperator
∆ = dd∗+d∗d : Ωk M → Ωk M
is called the Riemannian Laplacian or the Laplace-Beltrami operator.
Let’s check that we really get the usual Laplacian on functions on Rn:
(dd∗+d∗d)f = d∗df =−∗d ∗(
∑∂f∂xj
dxj
)=−∗d
(∑
∂f∂xj
∗dxj
)=−∗
(∑
∂2f∂xi∂xj
dxi ∧∗dxj
)=−∗
(∑
∂2f∂x2
i
)dV = ∆f .
In general, let a Riemannian metric be given in local coordinates byg = ∑ij dxi ⊗dxj . Let [g ij ] be the inverse matrix of [gij ] (this definesthe metric on T ∗M), and |g|= det[gij ]. Then
∆g f =− 1√|g|∑i,j
∂
∂xi
(√|g|g ij ∂f
∂xj
).
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A form ψ such that ∆ψ = 0 is called harmonic. Thus, a formsatisfying dψ = d∗ψ = 0 is harmonic. On a compact manifold, wealso have the converse:
Lemma
If M is compact, then any harmonic form ψ satisfies dψ = d∗ψ = 0.
Proof. “Integration by parts”:
0 =Z
M〈∆ψ,ψ〉dV =
ZM〈dd∗ψ+d∗dψ,ψ〉dV =
ZM
(|d∗ψ|2 + |dψ|2
)dV .
Corollary
A harmonic function on a compact Riemannian manifold is constant.
Let H k(M) be the vector space of harmonic k -forms, i.e.
H k(M) = ψ ∈ Ωk M; ∆ψ = 0.
For M compact and oriented, let 〈,〉 denote the global inner product onΩk M (given by integration).
Theorem (Hodge-de Rham)
On a compact oriented manifold (M,g):
Ωk M = H k(M)⊕dΩk−1M⊕d∗Ωk+1M,
where the summands are orthogonal with respect to the global innerproduct 〈,〉.
Before discussing the proof, let’s see some applications:
Corollary (Hodge isomorphism)
The natural map f : H k(M)→ HkDR(M), given by ψ 7→ [ψ] is an
isomorphism.
Proof. We known that dψ = 0, so the map is well defined. Since H isorthogonal to exact forms, the kernel of f is zero. Finally, let[ω] ∈ Hk
DR(M) and decompose ω = ωH +dλ+d∗φ with ωH harmonic.Then
0 = 〈dω,φ〉= 〈dd∗φ,φ〉= 〈d∗φ,d∗φ〉.Therefore d∗φ = 0 and [ω] = [ωH +dλ] = ωH , so f is surjective.
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Corollary (Poincare duality)
On a compact oriented n-dimensional manifold M,Hk
DR(M)' Hn−kDR (M).
Proof. Put any metric on M. The Hodge dual ∗ gives an isomorphismH k(M)'H n−k(M).
On the proof of the Hodge-de Rham TheoremIt is clear that the three summands H k(M), dΩk−1M, d∗Ωk+1M aremutually orthogonal: if ω is harmonic, then 〈ω,dφ〉= 〈d∗ω,φ〉= 0.Similarly 〈ω,d∗ψ〉= 0, and, finally:
〈dφ,d∗ψ〉= 〈ddφ,ψ〉= 0.
The hard part is to show that their sum is all of Ωk M.First of all, the space of smooth differential forms with the innerproduct 〈,〉 is not a Hilbert space, i.e. it is not complete. This is just asfor C∞ functions on an interval - they do not form a complete lineartopological space. In both cases, the completed space (equivalenceclasses of Cauchy sequences) consists of L2-integrable objects. Butthen, we cannot differentiate them.
Put in another way: after completing Ωk M, we can find element ofsmallest norm in the closure of every cohomology class. But whyshould such an element be a smooth differentiable form?
The solution is to complete Ωk M not with respect to the norm‖φ‖= 〈φ,φ〉1/2, but one that involves also integration of (covariant)derivatives up to high order s. Such a completion is called a Sobolevspace. Let us denote it by W k
s M. It is a Hilbert space, and theLaplacian extends to a Fredholm operator ∆s : W k
s (M)→W ks−2(M)
(Fredholm = dimKer∆ < +∞, dimCoker∆ < +∞). MoreoverKer∆s = Ker∆, so that every “Sobolev class” harmonic form isactually smooth.We have now a well-defined Y = (Ker∆s)
⊥ and what we need is toshow that Y ∩Ωk M = dΩk−1M⊕d∗Ωk+1M. We haveY = ℑ∆∗
s : W ks−2(M)→W k
s (M), but on smooth forms, ∆∗s = ∆ (∆ is
self-adjoint) so any smooth form ψ orthogonal to Ker∆ is in the imageof ∆, i.e. ψ = ∆u = (dd∗+d∗d)u = d(d∗u)+d∗(du).I hope that this rough outline gives some idea why the Hodge-deRham theorem is true!
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We now wish to an analogous decomposition on a compact Hermitianmanifold (M,h,J) using the operator ∂. Thus, we define the formaladjoint ∂∗ : Ωp,q+1M → Ωp,qM of ∂ and the complex Laplacian∆
∂= ∂∗∂+ ∂∂∗. The main points are:
We have a hermitian inner product on Ωp,qM.There is a natural orientation on a complex manifold.The Hodge star maps Ωp,qM to Ωn−q,n−pM.As n is even, ∗2 = (−1)p+q .∂∗ =−∗∂∗.
The formula for ∆∂
in local coordinates is similar to the one given forthe Riemannian Laplacian: if h = Re∑hijdzi ⊗dzj , then (on functions)
∆∂f =− 1√
|h|∑i,j∂
∂zi
(√|h|hij ∂f
∂zj
).
But note the crucial difference!
A differential form φ, such that ∆∂φ = 0 is called ∂-harmonic.
Just like for ∆g : if M is compact, then φ is ∂-harmonic iff ∂φ = 0 and∂∗φ = 0.
We denote by H p,qM the space of ∂-harmonic forms of type (p,q).
Theorem (Dolbeaut decomposition theorem)
On a compact Hermitian manifold (M,h,J):
Ωp,qM = H p,q(M)⊕ ∂Ωp,q−1M⊕ ∂∗Ωp,q+1M,
where the summands are orthogonal with respect to the globalhermitian product 〈,〉.
The proof is similar to that of the Hodge-de Rham theorem.; Again:
Corollary (Dolbeaut isomorphism)
The natural map H p,q(M)→ Hp,q(M), given by ψ 7→ [ψ] is anisomorphism.
Corollary (Serre duality)
On a compact complex n-dimensional manifold M,Hp,q(M)' Hn−p,n−q(M).
Proof. ∗H p,q(M)→H n−p,n−q(M) is an isomorphism.
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Thus, on a Hermitian manifold (M,g, I) we have two Laplacians: theRiemannian ∆h and the complex one ∆
∂. In general, there is no
relation between them; hence no relation between harmonic and∂-haarmonic forms; hence no relation between De Rham andDolbeault cohomology.A miracle of Kahler geometry:
Proposition
If (M,h,J) is a Kahler manifold, then ∆h = 2∆∂.
Proof. Both Laplacians, written in local coordinates, involve only firstderivatives of the metric. Thus, in normal Kahler coordinates (complexcoordinates in which the metric is Euclidean + O(|z|2) at a point p, thetwo Laplacians are ∆EUCL +O(|z|) and ∆
∂−EUCL +O(|z|). Since theProposition holds in Cn, we have ∆h|p = 2∆
∂|p and, hence,everywhere.
On a compact Kahler manifold we obtain now Hodge relations fromthis Proposition, the Hodge-De Rham and the Dolbeaultdecomposition theorems.