Hodge Theory

Hodge Theory

Complex Manifolds

by William M. Faucette

Adapted from lectures by Mark Andrea A. Cataldo

Structure of LectureConjugationsTangent bundles on a complex manifoldCotangent bundles on a complex manifoldStandard orientation of a complex manifoldAlmost complex structureComplex-valued formsDolbeault cohomology

Conjugations

ConjugationsLet us recall the following distinct notions

of conjugation.

First, there is of course the usual conjugation in C:

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γa γ

ConjugationsLet V be a real vector space and

be its complexification. There is a natural R-linear isomorphism given by

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VC := V ⊗R C

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c :VC → VC

u⊗ γ a u⊗ γ

Tangent Bundles on a Complex Manifold

Tangent Bundles on a Complex ManifoldLet X be a complex manifold of dimension

n, x2X and

be a holomorphic chart for X around x. Let zk=xk+iyk for k=1, . . . , N.

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(U; z1, K , zn )

Tangent Bundles on a Complex ManifoldXR) is the real tangent bundle on X.

The fiber TX,xR) has real rank 2n and it is the real span

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R ∂x1,K ,∂xn

,∂y1,K ,∂yn

Tangent Bundles on a Complex ManifoldXC):= XR)RC is the complex

tangent bundle on X. The fiber TX,xC) has complex rank 2n and it is the complex span

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C ∂x1,K ,∂xn

,∂y1,K ,∂yn

Tangent Bundles on a Complex ManifoldOften times it is more convenient to use a

basis for the complex tangent space which better reflects the complex structure. Define

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∂z j:= 1

2∂x j

− i∂y j( )

∂z j:= 1

2∂x j

+ i∂y j( )

Tangent Bundles on a Complex ManifoldWith this notation, we have

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∂x j= ∂z j

+ ∂z j

∂y j= i ∂z j

−∂z j( )

Tangent Bundles on a Complex ManifoldClearly we have

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TX ,x (C) = C ∂z1,K ,∂zn

,∂z1,K ,∂z n

Tangent Bundles on a Complex ManifoldIn general, a smooth change of

coordinates does not leave invariant the two subspaces

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R ∂x j{ } and R ∂y j{ }

Tangent Bundles on a Complex ManifoldHowever, a holomorphic change of

coordinates does leave invariant the two subspaces

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C ∂z j{ } and C ∂z j{ }

Tangent Bundles on a Complex ManifoldTX is the holomorphic tangent bundle on

X. The fiber TX,x has complex rank n and it is the complex span

TX is a holomorphic vector bundle.

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C ∂z1,K ,∂zn

Tangent Bundles on a Complex ManifoldTX is the anti-holomorphic tangent bundle

on X. The fiber TX,x has complex rank n and it is the complex span

TX is an anti-holomorphic vector bundle.

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C ∂z1,K ,∂z n

Tangent Bundles on a Complex ManifoldWe have a canonical injection and a

canonical internal direct sum decomposition into complex sub-bundles:

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TX (R)⊆TX (C) = ′ T X ⊕ ′ ′ T X

Tangent Bundles on a Complex ManifoldComposing the injection with the

projections we get canonical real isomorphisms

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T X ≅ TX (R) ≅ ′ ′ T X

Tangent Bundles on a Complex ManifoldThe conjugation map

is a real linear isomorphism which is not complex linear.

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c : TX (C) → TX (C)

Tangent Bundles on a Complex ManifoldThe conjugation map induces real linear

isomorphism

and a complex linear isomorphism

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c : ′ T X ≅R ′ ′ T X

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c : ′ T X ≅C ′ ′ T X

Cotangent Bundles on Complex Manifolds

Cotangent Bundles on Complex ManifoldsLet {dx1, . . . , dxn, dy1, . . . , dyn} be the

dual basis to {x1, . . . , xn, y1, . . . , yn}. Then

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dz j = dx j + i dy j dz j = dx j − i dy j

dx j = 12

dz j + dz j( ) dy j = 12i

dz j − dz j( )

Cotangent Bundles on Complex ManifoldsWe have the following vector bundles on

X: TX*(R), the real cotangent bundle, with

fiber

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TX ,x* (R) = R dx1,K ,dxn,dy1,K ,dyn

Cotangent Bundles on Complex Manifolds

TX*(C), the complex cotangent bundle, with fiber

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TX ,x* (C) = C dx1,K ,dxn,dy1,K ,dyn

Cotangent Bundles on Complex Manifolds

TX*(C), the holomorphic cotangent bundle, with fiber

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T X ,x*(C) = C dz1,K ,dzn

Cotangent Bundles on Complex Manifolds

TX*(C), the anti-holomorphic cotangent bundle, with fiber

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T X ,x*(C) = C dz1,K ,dzn

Cotangent Bundles on Complex ManifoldsWe have a canonical injection and a

canonical internal direct sum decomposition into complex sub-bundles:

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TX*(R)⊆TX

* (C) = ′ T X* ⊕ ′ ′ T X

*

Cotangent Bundles on Complex ManifoldsComposing the injection with the

projections we get canonical real isomorphisms

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T X* ≅R TX

*(R) ≅R ′ ′ T X*

Cotangent Bundles on Complex ManifoldsThe conjugation map

is a real linear isomorphism which is not complex linear.

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c : TX*(C) → TX

* (C)

Cotangent Bundles on Complex ManifoldsThe conjugation map induces real linear

isomorphism

and a complex linear isomorphism

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c : ′ T X* ≅R ′ ′ T X

*

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c : ′ T X* ≅C ′ ′ T X

*

Cotangent Bundles on Complex ManifoldsLet f(x1,y1,…, xn, yn)= u(x1,y1,…, xn, yn)+ i

v(x1,y1,…, xn, yn) be a smooth complex-valued function in a neighborhood of x. Then

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df = du + i dv = ∂f∂z j

dz jj∑ + ∂f∂z j

dz jj∑

The Standard Orientation of a Complex Manifold

Standard OrientationProposition: A complex manifold X

admits a canonical orientation.

If one looks at the determinant of the transition matrix of the tangent bundle of X, the Cauchy-Riemann equations immediately imply that this determinant must be positive.

Standard OrientationIf (U,{z1,…,zn}) with zj=xj+i yj, the real

2n-form

is nowhere vanishing in U.

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σU = dx1∧dy1∧K ∧dxn ∧dyn

= i 2( )n dz1∧dz1∧K ∧dzn ∧dzn

Standard OrientationSince the holomorphic change of

coordinates is orientation preserving, these non-vanishing differential forms patch together using a partition of unity argument to give a global non-vanish- ing differential form.

This differential form is the standard orientation of X.

The Almost Complex Structure

Almost Complex StructureThe holomorphic tangent bundle TX of a

complex manifold X admits the complex linear automorphism given by multiplication by i.

Almost Complex StructureBy the isomorphism

We get an automorphism J of the real tangent bundle TX(R) such that J2=Id. The same is true for TX* using the dual map J*.

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T X ≅ TX (R)

Almost Complex StructureAn almost complex structure on a real

vector space VR of finite even dimension 2n is a R-linear automorphism

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JR :VR ≅VR J 2 = −IdVR

Almost Complex StructureAn almost complex structure is equivalent

to endowing VR with a structure of a complex vector space of dimension n.

Almost Complex StructureLet (VR, JR) be an almost complex

structure. Let VC:= VRRC and JC:= JRIdC: VC VC be the complexification of JR.

The automorphism JC of VC has eigenvalues i and i.

Almost Complex StructureThere are a natural inclusion and a natural

direct sum decomposition

where the subspace VRVC is the fixed locus of the

conjugation map associated with the complexification.

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VR ⊆VC = ′ V ⊕ ′ ′ V

Almost Complex Structure V and V are the JCeigenspaces

corresponding to the eigenvalues i and i, respectively,

since JC is real, that is, it fixes VRVC, JC commutes with the natural conjugation map and V and V are exchanged by this conjugation map,

Almost Complex Structure there are natural Rlinear isomorphisms

coming from the inclusion and the projections to the direct summands

and complex linear isomorphisms

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V ≅R VR ≅R ′ ′ V

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V ≅C ′ ′ V

Almost Complex Structure The complex vector space defined by

the complex structure is Clinearly isomorphic to V.

Almost Complex StructureThe same considerations are true for the

almost complex structure (VR*, JR*). We have

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V * ≅R VR* ≅R ′ ′ V *

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V * ≅C ′ ′ V *

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VR* ⊆VC

* = ′ V * ⊕ ′ ′ V *

Complex-Valued Forms

Complex-Valued FormsLet M be a smooth manifold. Define the

complex valued smooth p-forms as

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A p (M) := E P (M)⊗R C ≅ C∞(M,TM (C))

Complex-Valued FormsThe notion of exterior differentiation

extends to complex-valued differential forms:

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d : A p (M) → A p +1(M)

Complex-Valued FormsLet X be a complex manifold of dimension

n, x2X, (p,q) be a pair of non-negative integers and define the complex vector spaces

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Λp,q (TX ,x* ) := Λp ( ′ T X ,x

* )⊗Λq ( ′ ′ T X ,x* )⊆ΛC

p +q (TX ,x* (C))

Complex-Valued FormsThere is a canonical internal direct sum

decomposition of complex vector spaces

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ΛCl (TX ,x

* (C)) = ⊕p +q= l Λp,q (TX ,x

* )

Complex-Valued FormsDefinition: The space of (p,q)forms on

X

is the complex vector space of smooth sections of the smooth complex vector bundle Λp,q(TX*).€

A p,q (X) := C∞(X,Λp,q (TX ,x* ))

Complex-Valued FormsThere is a canonical direct sum

decomposition

and

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A l (X) = ⊕p +q= l A p,q (X)

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d(A p,q )⊆ A p +1,q (X)⊕ A p,q +1(X)

Complex-Valued FormsLet l=p+q and consider the natural

projections

Define operators€

π p,q : A l (X) → A p,q (X)⊆ A l (X)

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d : A p,q (X ) → A p +1,q (X), ′ ′ d : A p,q (X) → A p,q +1(X)′ d = π p +1,q od, ′ ′ d = π p,q +1 od.

Complex-Valued FormsNote that

Also,

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d = ′ d + ′ ′ d , ′ ′ d 2= 0 = ′ d 2, ′ d ′ ′ d = − ′ ′ d ′ d

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d = ′ ′ d , ′ ′ d = ′ d

Dolbeault Cohomology

Dolbeault CohomologyDefinition: Fix p and q. The Dolbeault

complex is the complex of vector spaces

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0 → A p,0(M) ′ ′ d ⏐ → ⏐ A p,1(M) ′ ′ d ⏐ → ⏐ L

L ′ ′ d ⏐ → ⏐ A p,n−1(M) ′ d ⏐ → ⏐ A p,n (M) → 0

Dolbeault CohomologyThe Dolbeault cohomology groups are the

cohomology groups of the complex

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0 → A p,0(M) ′ ′ d ⏐ → ⏐ A p,1(M) ′ ′ d ⏐ → ⏐ L

L ′ ′ d ⏐ → ⏐ A p,n−1(M) ′ d ⏐ → ⏐ A p,n (M) → 0

Dolbeault CohomologyThat is,

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H ′ ′ d p,q := Ker ′ ′ d : A p,q (X) → A p,q +1(X)

Im ′ ′ d : A p,q−1(X) → A p,q (X)

Dolbeault CohomologyTheorem: (Grothendieck-Dolbeault Lemma)

Let q>0. Let X be a complex manifold and u2Ap,q(X) be such that du=0. Then, for every point x2X, there is an open neighborhood U of x in X and a form v2Ap,q-1(U) such that

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u U = ′ ′ d v.

Dolbeault CohomologyThe Grothendieck-Dolbeault Lemma

guarantees that Dolbeault cohomology is locally trivial.

Dolbeault CohomologyFor those familiar with sheaves and sheaf

cohomology, the Dolbeault Lemma tells us that the fine sheaves Ap,q

X of germs of C (p,q)-forms give a fine resolution of the sheaf p

X of germs of holomorphic p-forms on X. Hence, by the abstract deRham theorem

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H q (X,ΩXp ) ≅ H ′ ′ d

p,q (X).