{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}