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- W2565605422 abstract "First recall some definitions and some results of Hodge theory. Let X be an oriented riemannian manifold and let d*:Ak(X)--*Ak_~(X) be the associated operator which is dual to the de Rham operator d (where A . ( X ) denotes the space of smooth complex valued forms). A form ~ is called harmonic if it satisfies d~ = d*~ = 0. One of the main results of Hodge theory states that when X is compact any cohomology class contains exactly one harmonic form. The aim of this paper is to investigate similar questions for symplectic manifolds (as opposed to riemannian on6s). Let us assume that we are given a symplectic manifold (X, oJ) of dimension 2rn. According to J. L. Kozsul [11] and J. L. Brylinski [4], one can similarily define the operator d* and the notion of harmonic form (however d* is denoted A or 6 in loc. cit.). Define the harmonic cohomology H*,r(X) to be the space of all cohomology classes which contain at least one harmonic form. Our result is the following characterization of H*ar(X ) a s a subspace of H*(X). Let G = SL(2) and let B be the subgroup of all upper triangular matrices. For a rational B-module M, there exists a unique maximal subrfiodule ~ M which is a quotient of a rational G-module (an explicit construction of it will be given in section 2). In fact H*(X) has a canonical structure of B-module. The corresponding infinitesimal action is generated by the cup-product by [co] and the operator deg m, where deg is the degree operator. We then prove." @default.
- W2565605422 created "2017-01-06" @default.
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- W2565605422 date "1995-01-01" @default.
- W2565605422 modified "2023-09-26" @default.
- W2565605422 title "Harmonic cohomology classes of symplectic manifolds" @default.
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- W2565605422 hasPublicationYear "1995" @default.
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