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Unformatted text preview: SYMPLECTIC GEOMETRY, LECTURE 17 The Hodge decomposition stated last time places strong constraints on H of K 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 ,n . . . . . . . . (1) . . . . . . . . . h 1 , 1 h , 1 . h n, h 1 , h , is symmetric across the two diagonal axes. Moreover, note that [ p ] H p,p is nonzero, since [ n ] is the volume class. We have even stronger constraints, namely the hard Lefschetz theorem. Theorem 1. L n k = ( 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...
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This note was uploaded on 03/08/2010 for the course MATHEMATIC 570 taught by Professor Denisauroux during the Spring '10 term at Caltech.
- Spring '10