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Unformatted text preview: SYMPLECTIC GEOMETRY, LECTURE 15 Prof. Denis Auroux 1. Hodge Theory Theorem 1 (Hodge) . For M a compact K ahler manifold, the deRham and Dolbeault cohomologies are related = H q,p by H k ( M, C ) = H p,q ( M ) , with H p,q . dR p,q Before we discuss this theorem, we need to go over Hodge theory for a compact, oriented Riemannian manifold ( M, g ). k Definition 1. For V an oriented Euclidean vector space, the Hodge operator is the linear map V n k V which, for any oriented orthonormal basis e 1 , . . . , e n , maps e 1 e k e k +1 e n . Example. For any V , (1) = e 1 e n , and = ( 1) k ( n k ) . Applying this to T x M , we obtain a map on forms. Remark. Note that, (1) , k , = , . vol Definition 2. The codifferential is the map (2) d = ( 1) n ( k 1)+1 d : k ( M ) k 1 ( M ) Proposition 1. d is the L 2 formal adjoint to the deRham...
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