Fundamental groups of compact K\"ahler manifolds with semi-positive holomorphic sectional curvature

TL;DR

This paper extends the structure theorem for compact K"ahler manifolds with semi-positive holomorphic sectional curvature, showing they admit a fibration with rationally connected fibers and a torus quotient base, and proves the abelianness of their fundamental group.

Contribution

It generalizes the structure theorem from projective to compact K"ahler manifolds and analyzes the foliation by flat tangent vectors to establish fundamental group properties.

Findings

Existence of a locally trivial fibration with rationally connected fibers and torus quotient base.

Fundamental group of such manifolds is abelian.

Extension of structure theorem from projective to K"ahler manifolds.

Abstract

In this paper, we prove that a compact K\"ahler manifold $X$ with semi-positive holomorphic sectional curvature admits a locally trivial fibration $\phi \colon X \to Y$, where the fiber $F$ is a rationally connected projective manifold and the base $Y$ is a finite \'etale quotient of a torus. This result extends the structure theorem, previously established for projective manifolds, to compact K\"ahler manifolds. A key part of the proof involves analyzing the foliation generated by truly flat tangent vectors and showing the abelianness of the topological fundamental group $\pi_{1}(X)$, with a focus on varieties of special type.