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# lect17 - SYMPLECTIC GEOMETRY LECTURE 17 Prof Denis Auroux...

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SYMPLECTIC GEOMETRY, LECTURE 17 The Hodge decomposition stated last time places strong constraints on H of ahler manifolds, e.g. dim H k = H q,p is even for k odd because C conjugation gives isomorphisms H p,q (note that this is false for symplectic manifolds in general). The Hodge star gives isomorphisms H p,q → H n q,n p and the Hodge diamond structure on the the ranks of the Dolbeault cohomology groups, i.e. h n,n h 0 ,n · · · · · · . . . . . . . . (1) . . . . . . . . . h 1 , 1 h 0 , 1 . h n, 0 h 1 , 0 h 0 , 0 · · · is symmetric across the two diagonal axes. Moreover, note that [ ω p ] ∈ H p,p is nonzero, since [ ω n ] is the volume class. We have even stronger constraints, namely the “hard Lefschetz theorem”. Theorem 1. L n k = ( · ∧ ω n k ) : H k ( X, R ) H 2 n k ( X, R ) is an isomorphism. This is false for many symplectic manifolds. Moreover, combining this with Poincar´ e duality gives that, for k n, H k × H k R , α, β �→ α β ω n k is a nondegenerate bilinear pairing

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