[論文レビュー] Fundamental groups of compact Kahler varieties with nef anti canonical bundle

Xin Fu, Bin Guo+2 more|arXiv (Cornell University)|Feb 7, 2026

Geometry and complex manifolds被引用数 0

ひとこと要約

Extends Paun’s almost-Abelian fundamental group result to mildly singular compact Kähler varieties with nef anti-canonical data, via RCD space methods and Albanese map surjectivity; proves almost-Abelian π1 in lc/klt settings and 3-fold cases.

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)

研究の動機と目的

• Generalize Paun’s almost-Abelian fundamental group result to log canonical pairs (X, Δ) with nef −(KX+Δ).
• Extend the framework to compact Kähler manifolds with pseudo-effective −KX under suitable singularity conditions.
• Establish surjectivity of Albanese maps for Fujiki class C varieties with lc pairs and nef −(KX+Δ).
• Provide a strategy unifying complex geometry, PDEs, and metric measure space theory to study π1.
• Relate the lc case to broader conjectures on abelianity and special varieties.

提案手法

• Construct Ricci almost nonnegative Kähler currents in fixed Kähler classes to produce RCD spaces homeomorphic to X.
• Apply a metric measure space Margulis lemma for RCD spaces to derive virtual nilpotency of π1.
• Employ complex Monge-Ampère equations with singular data to regularize metrics and obtain convergence to RCD spaces.
• Prove surjectivity of the Albanese map in the lc setting by extending known projective and smooth results to Fujiki class C.
• Utilize asymptotically klt class theory to control singularities and enable analytic Nash-type regularizations.

実験結果

リサーチクエスチョン

• RQ1 Does π1(X) remain almost Abelian for compact klt Kähler varieties with lc pairs and nef −(KX+Δ)?
• RQ2 Can the Albanese map be shown to be surjective for Fujiki class C varieties with lc pairs and nef −(KX+Δ)?
• RQ3 Under what singularity assumptions does −KX being pseudo-effective force π1(X) to be almost Abelian?
• RQ4 How can RCD space theory be used to extend Paun-type results to mildly singular Kähler spaces?
• RQ5 What is the role of the lc case in relation to Kollár and Campana’s Abelianity conjectures?

主な発見

• π1(X) is almost Abelian under lc conditions and nef −(KX+Δ) provided Conjecture 1.4 on RCD spaces holds.
• π1(X) is virtually nilpotent via a construction of Ricci almost nonnegative RCD spaces homeomorphic to X.
• The Albanese map of X admits a smooth model that is a fibration in the lc Fujiki class C setting, generalizing known results.
• In dimension three, the nefness assumption on −(KX+Δ) yields almost-Abelian π1 without assuming projectivity.
• Theorem 1.3 extends surjectivity results for Albanese maps to singular X in Fujiki class C with lc pairs and nef −(KX+Δ).