zbMATH — the first resource for mathematics

Hodge structure on symplectic manifolds. (English) Zbl 0872.58002
The article deals with symplectic harmonic forms. A recent result by O. Mathieu states: For a symplectic manifold of \(\dim 2n\) the following properties are equivalent: (1) There exists a harmonic cocycle in every cohomology class and (2) the cup product is surjective.
This article provides an alternative, more direct proof for this fact using a particular \(\text{sl}(2,C)\) representation. An application in symplectic differential topology concludes the paper.

58A14 Hodge theory in global analysis
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57R19 Algebraic topology on manifolds and differential topology
14F40 de Rham cohomology and algebraic geometry
Full Text: DOI