Fundamental groups of compact Kahler varieties with nef anti canonical bundle

TL;DR

This paper extends the understanding of the fundamental groups of compact Kahler varieties, including singular cases, showing they are almost Abelian under certain nef conditions on the anti-canonical bundle, using advanced geometric analysis.

Contribution

It generalizes previous results to mildly singular varieties and log canonical pairs, establishing almost Abelian fundamental groups under broader nef conditions.

Findings

Fundamental groups are almost Abelian for certain singular Kahler varieties.

Extended Paun's results to log canonical pairs with nef anti-canonical divisors.

Proved surjectivity of Albanese maps for varieties with nef anti-log canonical divisors.

Abstract

It is proved by M. Paun (1997, 2017) that the fundamental group of a compact Kahler manifold X is almost Abelian if the anti-canonical bundle -KX is nef. In this paper, we apply the recent geometric analytic theory of Kahler spaces developed by Guo-Phong-Song-Sturm to study fundamental groups of mildly singular compact Kahler varieties. We first extend Paun's result to log canonical pairs (X,Delta) with smooth X and nef -(KX+Delta) as well as to compact Kahler manifolds X with pseudo-effective -KX under a suitable assumption on the singularities of c1(-KX). We further prove that, for a 3-dimensional log canonical pair (X,\Delta) with X being klt, pi 1(X) is almost Abelian if -(KX+\Delta) is nef. Moreover, as one of the main ingredients for the proof of these results, we establish the surjectivity of the Albanese maps of compact normal complex varieties X in Fujiki class C that admits an effective R-divisor \Delta such that the pair (X,\Delta) is log canonical with nef anti-log canonical divisor -(KX+\Delta).This generalizes the corresponding theorems for projective varieties (Zhang, 2005), for klt pairs (Matsumura-Wang-Wu-Zhang, 2025) and for log smooth case (Fu-Han-Zou, 2025)