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Harmonic cohomology classes of symplectic manifolds. (English) Zbl 0831.58004
Hodge theory, originally defined for compact orientable Riemannian manifolds, now has a symplectic counterpart. Consider a symplectic manifold \((X, \omega)\) of dimension \(2m\). “According to J. L. Kozsul and J. L. Brylinski, one can define the operator \(d^*\) and the notion of harmonic form. Define the harmonic cohomology \(H^*_{\text{har}} (X)\) to be the space of all cohomology classes which contain at least one harmonic form. […] We prove
Theorem 1. We have \(H^*_{\text{har}} (X)= {\mathcal D} H^* (X)\).
Corollary. Assume that \(X\) is compact. Then the following two assertions are equivalent.
i) Any cohomology class contains at least one harmonic form.
ii) For any \(k\), \(m\) the cup-product \([\omega ]^k: H^{m- k(X)}\to H^{m+k} (X)\) is an isomorphism.”
The author also presents some examples where these conditions are not valid. In Theorem 1 \({\mathcal D} H^* (X)\) is the unique maximal submodule (with respect to the upper triangular \(2\times 2\) matrices), a quotient of a rational \(\text{SL} (2)\)-module. The canonical structure of modules comes from the cup product and the degree operator.

58A14 Hodge theory in global analysis
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
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