Locally conformal Hermitian metrics on complex non-K\"ahler manifolds

TL;DR

This paper investigates complex non-Kähler manifolds with Hermitian metrics that are locally conformal to metrics with special cohomological properties, revealing conditions under which locally conformal holomorphic-tamed structures imply locally conformal K"ahler metrics.

Contribution

It establishes new links between locally conformal holomorphic-tamed and K"ahler structures on non-Kähler manifolds, providing explicit examples and conditions.

Findings

Existence of locally conformal holomorphic-tamed structures implies locally conformal K"ahler metrics.

Examples demonstrating the cohomological properties of these manifolds.

Conditions under which conformal structures lead to K"ahler metrics.

Abstract

We study complex non-K\"ahler manifolds with Hermitian metrics being locally conformal to metrics with special cohomological properties. In particular, we provide examples where the existence of locally conformal holomorphic-tamed structures implies the existence of locally conformal K\"ahler metrics, too.