# Complex manifolds with negative curvature operator

## TL;DR

This paper investigates compact complex manifolds with negative Chern curvature operator, establishing conditions under which they admit certain metrics or are Kähler with ample canonical bundle, and provides a full classification for complex surfaces.

## Contribution

It proves a dichotomy for such manifolds and offers a complete classification of complex surfaces with these properties, utilizing pluriclosed flow techniques.

## Key findings

- Manifolds either admit a dd^c-exact positive (1,1) current or are Kähler with ample canonical bundle.
- Complete classification achieved for complex surfaces with negative Chern curvature operator.
- Global existence and convergence of pluriclosed flow are key tools in the proofs.

## Abstract

We prove that compact complex manifolds with admitting metrics with negative Chern curvature operator either admit a $dd^c$-exact positive (1,1) current, or are K\"ahler with ample canonical bundle. In the case of complex surfaces we obtain a complete classification. The proofs rely on a global existence and convergence result for the pluriclosed flow.