Hodge Theory Complex Manifolds

by William M. Faucette Adapted from lectures by Mark Andrea A. Cataldo

Structure of Lecture Conjugations Tangent bundles on a complex manifold Cotangent bundles on a complex manifold Standard orientation of a complex manifold Almost complex structure Complex-valued forms Dolbeault cohomology

Conjugations

Let us recall the following distinct notions of conjugation. First, there is of course the usual conjugation in C:

Conjugations Let V be a real vector space and be its complexification. There is a natural R-linear isomorphism given by

Tangent Bundles on a Complex Manifold

Let X be a complex manifold of dimension n, x2X and be a holomorphic chart for X around x. Let z k =x k +iy k for k=1,..., N.

Tangent Bundles on a Complex Manifold XR) is the real tangent bundle on X. The fiber T X,xR) has real rank 2n and it is the real span

Tangent Bundles on a Complex Manifold XC):= XR) R C is the complex tangent bundle on X. The fiber T X,xC) has complex rank 2n and it is the complex span

Tangent Bundles on a Complex Manifold Often times it is more convenient to use a basis for the complex tangent space which better reflects the complex structure. Define

Tangent Bundles on a Complex Manifold With this notation, we have

Tangent Bundles on a Complex Manifold Clearly we have

Tangent Bundles on a Complex Manifold In general, a smooth change of coordinates does not leave invariant the two subspaces

Tangent Bundles on a Complex Manifold However, a holomorphic change of coordinates does leave invariant the two subspaces

Tangent Bundles on a Complex Manifold T X is the holomorphic tangent bundle on X. The fiber T X,x has complex rank n and it is the complex span T X is a holomorphic vector bundle.

Tangent Bundles on a Complex Manifold T X is the anti-holomorphic tangent bundle on X. The fiber T X,x has complex rank n and it is the complex span T X is an anti-holomorphic vector bundle.

Tangent Bundles on a Complex Manifold We have a canonical injection and a canonical internal direct sum decomposition into complex sub- bundles:

Tangent Bundles on a Complex Manifold Composing the injection with the projections we get canonical real isomorphisms

Tangent Bundles on a Complex Manifold The conjugation map is a real linear isomorphism which is not complex linear.

Tangent Bundles on a Complex Manifold The conjugation map induces real linear isomorphism and a complex linear isomorphism

Cotangent Bundles on Complex Manifolds

Let {dx 1,..., dx n, dy 1,..., dy n } be the dual basis to { x 1,..., x n, y 1,..., y n }. Then

Cotangent Bundles on Complex Manifolds We have the following vector bundles on X: T X *(R), the real cotangent bundle, with fiber

Cotangent Bundles on Complex Manifolds T X *(C), the complex cotangent bundle, with fiber

Cotangent Bundles on Complex Manifolds T X *(C), the holomorphic cotangent bundle, with fiber

Cotangent Bundles on Complex Manifolds T X *(C), the anti-holomorphic cotangent bundle, with fiber

Cotangent Bundles on Complex Manifolds We have a canonical injection and a canonical internal direct sum decomposition into complex sub- bundles:

Cotangent Bundles on Complex Manifolds Composing the injection with the projections we get canonical real isomorphisms

Cotangent Bundles on Complex Manifolds The conjugation map is a real linear isomorphism which is not complex linear.

Cotangent Bundles on Complex Manifolds The conjugation map induces real linear isomorphism and a complex linear isomorphism

Cotangent Bundles on Complex Manifolds Let f(x 1,y 1,…, x n, y n )= u(x 1,y 1,…, x n, y n )+ i v(x 1,y 1,…, x n, y n ) be a smooth complex-valued function in a neighborhood of x. Then

The Standard Orientation of a Complex Manifold

Standard Orientation Proposition: 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 Orientation If (U,{z 1,…,z n }) with z j =x j +i y j, the real 2n-form is nowhere vanishing in U.

Standard Orientation Since 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 Structure The holomorphic tangent bundle T X of a complex manifold X admits the complex linear automorphism given by multiplication by i.

Almost Complex Structure By the isomorphism We get an automorphism J of the real tangent bundle T X (R) such that J 2 = Id. The same is true for T X * using the dual map J*.

Almost Complex Structure An almost complex structure on a real vector space V R of finite even dimension 2n is a R-linear automorphism

Almost Complex Structure An almost complex structure is equivalent to endowing V R with a structure of a complex vector space of dimension n.

Almost Complex Structure Let (V R, J R ) be an almost complex structure. Let V C := V R R C and J C := J R Id C : V C V C be the complexification of J R. The automorphism J C of V C has eigenvalues i and i.

Almost Complex Structure There are a natural inclusion and a natural direct sum decomposition where the subspace V R V C is the fixed locus of the conjugation map associated with the complexification.

Almost Complex Structure V and V are the J C eigenspaces corresponding to the eigenvalues i and i, respectively, since J C is real, that is, it fixes V R V C, J C commutes with the natural conjugation map and V and V are exchanged by this conjugation map,

Almost Complex Structure there are natural R linear isomorphisms coming from the inclusion and the projections to the direct summands and complex linear isomorphisms

Almost Complex Structure The complex vector space defined by the complex structure is C linearly isomorphic to V.

Almost Complex Structure The same considerations are true for the almost complex structure (V R *, J R *). We have

Complex-Valued Forms

Let M be a smooth manifold. Define the complex valued smooth p-forms as

Complex-Valued Forms The notion of exterior differentiation extends to complex-valued differential forms:

Complex-Valued Forms Let X be a complex manifold of dimension n, x2X, (p,q) be a pair of non-negative integers and define the complex vector spaces

Complex-Valued Forms There is a canonical internal direct sum decomposition of complex vector spaces

Complex-Valued Forms Definition: The space of (p,q) forms on X is the complex vector space of smooth sections of the smooth complex vector bundle p,q (T X *).

Complex-Valued Forms There is a canonical direct sum decomposition and

Complex-Valued Forms Let l=p+q and consider the natural projections Define operators

Complex-Valued Forms Note that Also,

Dolbeault Cohomology

Definition: Fix p and q. The Dolbeault complex is the complex of vector spaces

Dolbeault Cohomology The Dolbeault cohomology groups are the cohomology groups of the complex

Dolbeault Cohomology That is,

Dolbeault Cohomology Theorem: (Grothendieck-Dolbeault Lemma) Let q>0. Let X be a complex manifold and u2A p,q (X) be such that d u=0. Then, for every point x2X, there is an open neighborhood U of x in X and a form v2A p,q-1 (U) such that

Dolbeault Cohomology The Grothendieck-Dolbeault Lemma guarantees that Dolbeault cohomology is locally trivial.

Dolbeault Cohomology For those familiar with sheaves and sheaf cohomology, the Dolbeault Lemma tells us that the fine sheaves A p,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