By Gutierrez J., Shpilrain V., Yu J.-T. (eds.)
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19. 21, RD is of type (1, 1), hence dD dD α is of type (p + 1, q + 1). By definition, dD dD α = ∂ D ∂ D α + ∂∂α + (∂ D ∂ + ∂∂ D )α. The first two forms on the right hand side vanish since they have type (p + 2, q) and (p, q + 2), respectively. 33). Let (X1 , Y1 , . . , Xm , Ym ) be a local orthonormal frame of M with JXj = Yj . 30) and let Z1∗ , . . , Zn∗ , Z1∗ , . . , Zn∗ be the corresponding dual frame of TC∗ M . With our conventions, we have Zj♭ := Zj , · = 1 ∗ Z 2 j and Zj♭ := Zj , · = 1 ∗ Z .
1. References for the basic theory of symplectic forms and manifolds are [Ar, Chapter 8] and [Du, Chapter 3]. 53. 43)). , that 1 1 ω m−j . ∗ ωj = j! (m − j)! 23 Theorem. Let M be a K¨ ahler manifold as above. 1) If M is closed, then the cohomology class of ω k in H 2k (M, R) is non-zero for 0 ≤ k ≤ m. In particular, H 2k (M, R) = 0 for such k. 2) If N ⊂ M is a compact complex submanifold without boundary of complex dimension k, then the cohomology class of ω k in H 2k (M, R) and the homology class of N in H2k (M, R) are non-zero.
If ω is a symplectic form on M , then the real dimension of M is even, dim M = 2m, and ω m is non-zero at each point of M . The K¨ahler form of a K¨ ahler manifold is symplectic. 1. References for the basic theory of symplectic forms and manifolds are [Ar, Chapter 8] and [Du, Chapter 3]. 53. 43)). , that 1 1 ω m−j . ∗ ωj = j! (m − j)! 23 Theorem. Let M be a K¨ ahler manifold as above. 1) If M is closed, then the cohomology class of ω k in H 2k (M, R) is non-zero for 0 ≤ k ≤ m. In particular, H 2k (M, R) = 0 for such k.