Fundamental groups of compact Kahler varieties with nef anti canonical bundle
Xin Fu, Bin Guo, Jian Song, Juanyong Wang
TL;DR
This work extends M. P n’s result on the almost-Abelian nature of fundamental groups from smooth compact Kähler manifolds to mildly singular settings, treating lc pairs $(X,\Delta)$ with nef $-(K_X+\Delta)$ and compact Kähler manifolds with pseudo-effective $-K_X$ under controlled singularities. The authors combine nonlinear PDE techniques on Kähler spaces with the metric–measure theory of Ricci curvature, via the Kähler–RCD framework, to construct Ricci almost nonnegative spaces and apply a Margulis-type lemma to derive virtual nilpotence of $\pi_1(X)$; surjectivity of Albanese maps then upgrades this to almost abelianity. A central novelty is the generalization to lc pairs and to Fujiki class $\mathscr C$ varieties, including a 3-dimensional log canonical case with $X$ klt, and a detailed development of the Albanese surjectivity, positivity of twisted relative canonical divisors, and the notion of asymptotically klt classes. The results connect analytic and geometric techniques (complex Monge–Ampère equations, RCD theory) with topological conclusions about fundamental groups, and extend key conjectures and specialness properties to broader Kähler settings, providing a unified framework for understanding fundamental groups in the presence of nef anti-canonical data. The work thus advances the Abelianity program for fundamental groups in complex geometry, with implications for the structure theory of Kähler varieties and their Albanese fibrations.
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,Δ) with X being klt, pi 1(X) is almost Abelian if -(KX+Δ) 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 Δsuch that the pair (X,Δ) is log canonical with nef anti-log canonical divisor -(KX+Δ).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)