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Hodge Theory Complex Manifolds

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

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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

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Conjugations

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

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

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Tangent Bundles on a Complex Manifold

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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.

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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

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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

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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

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

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Tangent Bundles on a Complex Manifold Clearly we have

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

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

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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.

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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.

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

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

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

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

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Cotangent Bundles on Complex Manifolds

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

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

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

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

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

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

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

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

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

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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

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The Standard Orientation of a Complex Manifold

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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.

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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.

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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.

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The Almost Complex Structure

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

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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*.

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

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

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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.

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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.

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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,

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

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

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

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Complex-Valued Forms

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

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

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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

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

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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 *).

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

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

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Complex-Valued Forms Note that Also,

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Dolbeault Cohomology

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

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

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Dolbeault Cohomology That is,

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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

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

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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

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