# On projective manifolds with pseudo-effective tangent bundle

## TL;DR

This paper develops a theory of singular hermitian metrics on vector bundles and provides a structure theorem for projective manifolds with pseudo-effective tangent bundles, classifying certain surfaces and exploring positively curved tangent bundles.

## Contribution

It introduces a structure theorem for projective manifolds with pseudo-effective tangent bundles and classifies minimal surfaces with this property.

## Key findings

- Existence of a smooth fibration to a flat projective manifold
- Classification of minimal surfaces with pseudo-effective tangent bundle
- Examples of positively curved tangent bundles

## Abstract

In this paper, we develop the theory of singular hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold $X$ with pseudo-effective tangent bundle: $X$ admits a smooth fibration $X \to Y$ to a flat projective manifold $Y$ such that its general fiber is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general non-minimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.