# Numerically flat holomorphic bundles over non K\"ahler manifolds

## TL;DR

This paper investigates the properties of numerically flat holomorphic vector bundles over non-Kähler manifolds, establishing their equivalence to several geometric and stability conditions, thus answering a question posed by Demailly, Peternell, and Schneider.

## Contribution

It proves the equivalence of various notions of flatness, effectiveness, and stability for these bundles on non-Kähler manifolds, extending known results.

## Key findings

- Numerically flat bundles are equivalent to numerically effective bundles with vanishing first Chern number.
- They are also characterized by semistability with vanishing first and second Chern numbers.
- The paper confirms the existence of filtrations with Hermitian flat quotients.

## Abstract

In this paper, we study numerically flat holomorphic vector bundles over a compact non-K\"ahler manifold $(X, \omega)$ with the Hermitian metric $\omega$ satisfying the Gauduchon and Astheno-K\"ahler conditions. We prove that numerically flatness is equivalent to numerically effectiveness with vanishing first Chern number, semistablity with vanishing first and second Chern numbers, approximate Hermitian flatness and the existence of a filtration whose quotients are Hermitian flat. This gives an affirmative answer to the question proposed by Demailly, Peternell and Schneider.