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On Compact Kähler Manifolds Let (X, G, J) Be a Hermtian Manifold and Let

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On Compact Kähler Manifolds Let (X, G, J) Be a Hermtian Manifold and Let On compact K¨ahler manifolds Let (X,g,J) be a Hermtian manifold and let ω be its fundamental form. If dω =0, [ω] H2(X, R) by the definition of the de Rham cohomology ∈ group, and ω is called a K¨ahler form. In this case we call (X,g,J) a K¨ahler manifold. If (X,g,J) is an Hermitian manifold, the following statements are equivalent. 1. (X,g,J) is K¨ahler, i.e. dω = 0. 2. The Levi-Civita connection is almost-complex. 3. The Chern connection is torsion-free. 4. For each p X, there exist local holomorphic coordinates centered at p such that ∈ 2 gij = δij + O( z ). | | 5. For each p X, there is a neighborhood U of p and a smooth, real-valued function f : U R,∈ such that ω = √ 1∂∂f on U. → − Theorem 6.3 Let (X,g) be a compact K¨ahler manifold of dimension n. Then there isnat- ural orthogonal decomposition k p,q H (X, C)= p+q=kH (X). ⊕ This decomposition does not depend on the chosen K¨ahler structure. Moreover, with respect to complex conjugation on H∗(X, C)= H∗(X, R) C, one has Hp,q(X)= Hq,p(X) and Serre duality yields Hp,q(X) Hn−p,n−q(X)∗. ⊗ ≃ Corollary 6.4 Let M be a compact K¨ahler manifold. Then the odd Betti numbers b2k+1 of M are even. 2k+1 Proof By the de Rham theorem, b2k+1 = dimC H (M, C), which is equal to k hp,q =2 hp,2k+1−p p+q=2k+1 p=0 X X p,q q,p p,q p,q because h = h by the Hodge Decomposition Theorem, where h = dimC H (X). 54 Corollary 6.5 C, if 0 p = q n, Hq(CPn, Ωp)= Hp,q(CPn)= ≤ ≤ (0, otherwise. Proof: Recall Z, ifp =0, 2, ..., 2n, Hp(CPn, Z)= (0, otherwise. Similarly by considering Ceckˇ cohomology of the sheaf of germs of constant complex-valued functions, we obtain C, ifp =0, 2, ..., 2n, Hp(CPn, C)= (0, if otherwise. 2p i,2p−i p,p By the Hodge Decomposition Theorem, 1 = b2p = h . Thus h =1for0 p n, i=0 ≤ ≤ and all other Hodge numbers vanish. P n Corollary 6.6 Every holomorphic line bundle on CP is of the form CPn (k) for some integer k. O Proof: By the exact cohomology sequence c1 H1(CPn, ) H1(CPn, ∗) H2(CPn, Z) O → O −→ arising from the exponeential sheaf sequence on CPn and by the vanishing of H1(Pn, ), we see O P ic(CPn) H2(CPn, Z) Z. (24) ≃ ≃ On the other hand, [ (k)] P ic(CPn), integer k 1, generates a subgroup Z so that O n ∈ ∀ ≥ ≃ [ (k)] k∈Z,k>0 = P ic(CP ). . { O } 55 7 Hodge Conjecture Hodge conjecture Let M be a compact complex manifold of (complex) dimension n. For any complex submanifold Z of complex dimension p, there is a fundamental class [Z] H2p(M, Z) and a Poincar´eduality isomorphism ⊂ 2n−2p H2p(M, Z) H (M, Z) ≃ given by the intersection paring 2n−2p H2p(M, Z) H (M, Z) Z × → ([Z] , α) α Z → Z | Thus, [Z] can be regarded as an element in H2p(X, Z). Moreover,R one sees that [Z] ∈ Hp,p(X, Z) and it can also be defined for singular analytic subvarieties Z X. ⊂ Hodge conjecture [Hodge52] Let X be a projective complex manifold. Then every Hodge class in the group H2k(X, Q) Hk,k(X) is a linear combination with rational coefficients of the cohomology classes of complex∩ subvarieties of X. Remarks 1. Hodge’s original conjecture was: Let X be a projective complex manifold. Then every cohomology class in H2k(X, Z) Hk,k(X) is the cohomology class of an algebraic cycle ∩ with integral coefficients on X. This is now known to be false. The first counterexample was constructed by M. Atiyah and F. Hirzebruch (1962). [AH62] 2. When k = 1, Hodge conjecture was proved by Lefschetz theorem on (1,1)-classes 20: Any element of H2(X, Q) H1,1(X) is the cohomology class of a Q-divisor on X. ∩ 3. By the Hard Lefschetz theorem, one can prove: If the Hodge conjecture holds for Hodge classes of degree p,p<n, then the Hodge conjecture holds for Hodge classes of degree 2n p. − 4. The Hodge conjecture holds for sufficiently general abelian varieties, for products of elliptic curves, and for simple abelian varieties. 5. We may formulate Hodge conjecture for K¨ahler varieties, vector bundle version. 6. S.T. Yau wrote [Yau05]: 20Lefschetz theorem on (1,1)-classes holds on compact K¨ahler manifolds 56 “The work of Hodge on the Hodge structures of K¨ahler manifolds was also used extensively by Kodaira. At the same time it puts the old theory of Picard and Lefschetz on a new setting. The conjecture of Hodge on algebraic cycles is perhaps the most elegant and important question in algebraic ge- ometry. Due to its relation to arithmetic question, a lot of number theorists made contribution to it.” 7. Since Lefschetz’s theorem on (1,1) classes holds on compact K¨ahler manifold, one may ask for similar conjectures on compact K¨ahler manifold: Let X be a compact K¨ahler manifold. Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of vector bundles on X. or Let X be a compact K¨ahler manifold. Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of coherent sheaves on X. Claire Voisin proved that the Chern classes of coherent sheaves give strictly more Hodge classes than the Chern classes of vector bundles and that the Chern classes of coherent sheaves are insufficient to generate all the Hodge classes. Consequently, the only known formulations of the Hodge conjecture for K¨ahler varieties are false. 8. The Hodge conjecture is a major unsolved problem in algebraic geometry. The Hodge conjecture is one of the Clay Mathematics Institute’s Millennium Prize Problems, so there is a US $1,000,000 prize for proving the Hodge conjecture. Michael Atiyah said: 21 “Famous Theorems define the mathematical landscape. They beckon from afar, rising dimly in the mists, an elusive challenge to the mathematical com- munity. Are they accessible or is there some vast ravine or raging torrent that has to be traversed ?” 21Comments for Award of the Millennium Prize to Grigoriy Perelman For Resolution of the Poincar´e Conjecture, Laudations, Paris, June 8, 2010. 57.
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