Harmonic cohomology of symplectic fiber bundles

TL;DR

This paper proves that in symplectic fiber bundles with closed Lefschetz fibers, every de Rham cohomology class has a Poisson harmonic representative, using a new characterization of Lefschetz manifolds.

Contribution

It introduces a novel characterization of closed Lefschetz manifolds and applies it to establish the existence of harmonic representatives in symplectic fiber bundles.

Findings

Every de Rham cohomology class admits a Poisson harmonic representative.

The proof relies on a new characterization of closed Lefschetz manifolds.

The results connect symplectic geometry with harmonic analysis on fiber bundles.

Abstract

We show that every de Rham cohomology class on the total space of a symplectic fiber bundle with closed Lefschetz fibers, admits a Poisson harmonic representative in the sense of Brylinski. The proof is based on a new characterization of closed Lefschetz manifolds.