On some applications of Gauduchon metrics

TL;DR

This paper explores applications of Gauduchon metrics in complex geometry, including implications for holomorphic sectional curvature, non-existence of certain holomorphic sections, and restrictions on specific forms on compact complex manifolds.

Contribution

It demonstrates new applications of Gauduchon metrics to problems in complex geometry, linking metric properties to algebraic and differential geometric conditions.

Findings

Hermitian metrics with nonnegative holomorphic sectional curvature imply algebro-geometric properties.

Under certain conditions, holomorphic sections on Hermitian vector bundles do not exist.

Restrictions are established for the $ ext{ extbackslash d extbackslash d}$-closedness of certain real forms on compact complex manifolds.

Abstract

We apply the existence and special properties of Gauduchon metrics to give several applications. The first one is concerned with the implications of algebro-geometric nature under the existence of a Hermitian metric with nonnegative holomorphic sectional curvature. The second one is to show the non-existence of holomorphic sections on Hermitian vector bundles under certain conditions. The third one is to give a restriction on the $\partial\bar{\partial}$-closedness of some real $(n-1,n-1)$-forms on compact complex manifolds.