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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.

##### MSC:
 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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