# Nonnegative Hermitian vector bundles and Chern numbers

## TL;DR

This paper proves that holomorphic vector bundles with nonnegative Hermitian metrics have nonnegative Chern numbers, providing bounds and new results for complex manifolds with specific geometric properties.

## Contribution

It establishes that certain linear combinations of Chern forms are strongly nonnegative for bundles with nonnegative Hermitian metrics, leading to bounds on Chern numbers and new geometric insights.

## Key findings

- All Chern numbers are nonnegative for such bundles.
- Derived bounds relate Chern numbers to special Chern numbers.
- New results on complex manifolds with homogeneous or torus-immersible structures.

## Abstract

We show in this article that if a holomorphic vector bundle has a nonnegative Hermitian metric in the sense of Bott and Chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of Chern forms are strongly nonnegative. This particularly implies that all the Chern numbers of such a holomorphic vector bundle are nonnegative and can be bounded below and above respectively by two special Chern numbers. As applications, we obtain a family of new results on compact connected complex manifolds which are homogeneous or can be holomorphically immersed into complex tori, some of which improve several classical results.