A note on compatibility of special Hermitian structures

TL;DR

This paper proves that compact Vaisman manifolds cannot support certain special Hermitian metrics, such as pluriclosed or balanced, and explores the relationship between symplectic forms and Hermitian structures.

Contribution

It establishes non-existence results for various special Hermitian metrics on compact Vaisman manifolds and examines their interaction with symplectic forms.

Findings

Vaisman manifolds cannot admit pluriclosed or balanced metrics.

Certain special Hermitian metrics are incompatible with Vaisman structures.

The interplay between locally conformally symplectic forms and Hermitian structures is analyzed.

Abstract

We prove that a compact Vaisman manifold $(M, J)$ cannot admit some type of special Hermitian metrics, such as special $k$-Gauduchon metrics, $p$-K\"ahler forms, Hermitian-symplectic or strongly Gauduchon metrics compatible to the same complex structure $J$. In particular, it cannot admit pluriclosed or balanced metrics. We also investigate the interplay between locally conformally symplectic forms taming the complex structure $J$ and special Hermitian structures.