On compact K\"ahler manifolds with pseudo-effective tangent bundle

TL;DR

This paper extends the structure theorem for smooth projective varieties to compact K"ahler manifolds with pseudo-effective tangent bundles, showing they admit a rationally connected fibration onto a quotient of a complex torus.

Contribution

It proves that compact K"ahler manifolds with pseudo-effective tangent bundles have a specific fibration structure similar to projective cases, broadening the understanding of their geometry.

Findings

Existence of a rationally connected fibration for such manifolds

Fibration maps onto a finite étale quotient of a complex torus

Extension of structure theorem from projective to K"ahler manifolds

Abstract

In this paper, we prove that a compact K\"ahler manifold $X$ with pseudo-effective (resp. singular positively curved) tangent bundle admits a smooth (resp. locally constant) rationally connected fibration $\phi \colon X \to Y$ onto a finite \'etale quotient $Y$ of a compact complex torus. This result extends the structure theorem previously established for smooth projective varieties to compact K\"ahler manifolds.