Cohomological properties of Hermitian symplectic threefolds

TL;DR

This paper proves a $dd^c$-lemma for certain forms on compact Hermitian symplectic threefolds, establishing conditions under which these manifolds are Kähler, thereby advancing understanding of their geometric structure.

Contribution

It demonstrates the $dd^c$-lemma for 1- and (1,1)-forms on compact Hermitian symplectic threefolds and explores implications for their Kählerness.

Findings

$dd^c$-lemma holds for 1- and (1,1)-forms

Albanese map is well-defined for these manifolds

Kählerness follows when the Albanese image dimension is maximal

Abstract

A Hermitian symplectic manifold is a complex manifold endowed with a symplectic form $\omega$, for which the bilinear form $\omega(I\cdot,\cdot)$ is positive definite. In this work we prove $dd^c$-lemma for 1- and (1,1)-forms for compact Hermitian symplectic manifolds of dimension 3. This shows that Albanese map for such manifolds is well-defined and allows one to prove K\"ahlerness if the dimension of the Albanese image of a manifold is maximal.