Fundamental groups of compact Kähler manifolds with semi-positive holomorphic sectional curvature
Shin-ichi Matsumura
TL;DR
The paper addresses the problem of extending structure theorems for varieties with semi-positive curvature to compact Kähler manifolds by showing that a compact Kähler manifold $X$ with semi-positive holomorphic sectional curvature admits a locally trivial fibration $X \to Y$ where $F$ is rationally connected and $Y$ is a finite étale quotient of a torus, with $\pi_1(X)$ virtually abelian. The authors leverage the Albanese map and a two-step strategy to control augmented irregularities, combined with Campana’s notion of varieties of special type to constrain the fundamental group and obtain a holomorphic splitting $T_X = V \oplus W$ with truly flat directions. A central contribution is the demonstration that $F$ must be rationally connected under these curvature hypotheses, which in turn yields a precise description of the fundamental group and the fibration structure. This work extends Yau-type conjectures to the Kähler setting, clarifying the geometric structure and potential uniformization of such manifolds and providing a bridge between projective and non-projective cases.
Abstract
In this paper, we prove that a compact Kähler manifold $X$ with semi-positive holomorphic sectional curvature admits a locally trivial fibration $φ\colon X \to Y$, where the fiber $F$ is a rationally connected projective manifold and the base $Y$ is a finite étale quotient of a torus. This result extends the structure theorem, previously established for projective manifolds, to compact Kähler 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 $π_{1}(X)$, with a focus on varieties of special type.