# lect15 - SYMPLECTIC GEOMETRY LECTURE 15 Prof Denis Auroux 1...

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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) ∀ α, β ∈ Ω k , α ∧ ∗ β = α, β . vol Definition 2. The codifferential is the map (2) d ∗ = ( − 1) n ( k − 1)+1 ∗ d ∗ : Ω k ( M ) Ω k − 1 ( M ) → Proposition 1. d ∗ is the L 2 formal adjoint to the deRham...
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lect15 - SYMPLECTIC GEOMETRY LECTURE 15 Prof Denis Auroux 1...

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