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Hodge Theory

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

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Hodge Theory

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  1. Hodge Theory Complex Manifolds

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

  3. 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

  4. Conjugations

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

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

  7. Tangent Bundles on a Complex Manifold

  8. 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 zk=xk+iyk for k=1, . . . , N.

  9. Tangent Bundles on a Complex Manifold TXR) is the real tangent bundle on X. The fiber TX,xR) has real rank 2n and it is the real span

  10. Tangent Bundles on a Complex Manifold TXC):= TXR)RC is the complex tangent bundle on X. The fiber TX,xC) has complex rank 2n and it is the complex span

  11. 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

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

  13. Tangent Bundles on a Complex Manifold Clearly we have

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

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

  16. Tangent Bundles on a Complex Manifold TX 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.

  17. Tangent Bundles on a Complex Manifold TX 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.

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

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

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

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

  22. Cotangent Bundles on Complex Manifolds

  23. Cotangent Bundles on Complex Manifolds Let {dx1, . . . , dxn, dy1, . . . , dyn} be the dual basis to {x1, . . . , xn, y1, . . . , yn}. Then

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

  25. Cotangent Bundles on Complex Manifolds • TX*(C), the complex cotangent bundle, with fiber

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

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

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

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

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

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

  32. Cotangent Bundles on Complex Manifolds Let f(x1,y1,…, xn, yn)= u(x1,y1,…, xn, yn)+ iv(x1,y1,…, xn, yn) be a smooth complex-valued function in a neighborhood of x. Then

  33. The Standard Orientation of a Complex Manifold

  34. 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.

  35. Standard Orientation If (U,{z1,…,zn}) with zj=xj+i yj, the real 2n-form is nowhere vanishing in U.

  36. 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.

  37. The Almost Complex Structure

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

  39. Almost Complex Structure By 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*.

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

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

  42. Almost Complex Structure Let (VR, JR) be an almost complex structure. Let VC:= VRRC and JC:= JRIdC: VC VCbe the complexification of JR. The automorphism JC of VC has eigenvalues i and -i.

  43. Almost Complex Structure There are a natural inclusion and a natural direct sum decomposition where • the subspace VRVC is the fixed locus of the conjugation map associated with the complexification.

  44. Almost Complex Structure • V and V are the JCeigenspaces corresponding to the eigenvalues i and -i, respectively, • since JC is real, that is, it fixes VRVC, JC commutes with the natural conjugation map and V and V are exchanged by this conjugation map,

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

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

  47. Almost Complex Structure The same considerations are true for the almost complex structure (VR*, JR*). We have

  48. Complex-Valued Forms

  49. Complex-Valued Forms Let M be a smooth manifold. Define the complex valued smooth p-forms as

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

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