Hermitian ranks of compact complex manifolds

TL;DR

This paper explores the properties of degenerate special-Hermitian metrics, including degenerate Kähler and locally conformally Kähler metrics, on specific classes of non-Kähler compact complex manifolds.

Contribution

It introduces new insights into degenerate Hermitian structures on non-Kähler manifolds, expanding understanding of their geometric properties.

Findings

Characterization of degenerate Kähler metrics on non-Kähler manifolds

Conditions for locally conformally Kähler metrics in degenerate cases

Examples of special classes of non-Kähler manifolds with these metrics

Abstract

We investigate degenerate special-Hermitian metrics on compact complex manifolds, in particular, degenerate K\"ahler and locally conformally K\"ahler metrics on special classes of non-K\"ahler manifolds.