The local fundamental group of a Kawamata log terminal singularity is finite
type: Item
instance of: scholarly article
title: The local fundamental group of a Kawamata log terminal singularity is finite
zbMATH Open document ID: 1479.14029
author: Lukas Braun
published in: Inventiones Mathematicae
publication date: 2021-11-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/2004.00522
review text: Let \(X\) be a (normal) complex algebraic variety and \(x \in X\) a (closed) point. The \textit{local fundamental group} \(\pi_1^{\mathrm{loc}}(X, x)\) is defined to be \(\pi_1(B \setminus \{x\})\), where \(B\) is the intersection of \(X\) with a sufficiently small euclidean ball around \(x\) (with respect to a local embedding \(X \hookrightarrow \mathbb C^n\)). Even if \(X\) has only log canonical singularities, it is known that the local fundamental group can be arbitrarily complicated. On the other hand, if \((X, \Delta)\) has only Kawamata log terminal (klt) singularities, then Kollár has conjectured that \(\pi_1^{\mathrm{loc}}(X, x)\) is finite. This epic paper proves (generalizations of) this conjecture. More precisely, in the above setting one can also define the \textit{regional fundamental group} \(\pi_1^{\mathrm{reg}}(X, x)\) as \(\pi_1(B \setminus X_{\mathrm{sg}})\) (local and regional fundamental group coincide for isolated singularities, but in general finiteness of the regional fundamental group is stronger). Then Theorem~1 states that the regional fundamental group of a klt singularity is finite. There is also a more general version concerning the \textit{orbifold} regional fundamental group. The proof of Theorem~1 relies on the following Theorem~2, which generalizes and proves another conjecture: let \((X, \Delta = \Delta' + \Delta'')\) be a klt weakly Fano pair (i.e.~\(-(K_X + \Delta)\) is big and nef). Then the orbifold fundamental group of the smooth locus is finite. The simultaneous proof of the two theorems is contained in Theorems~7 and~8: Theorem~1 in dimension \(n\) implies Theorem~2 in dimension \(n\), and Theorem~2 in dimension \(n\) implies Theorem~1 in dimension \(n + 1\). Previous results on the topic (on which this paper partly builds) include [\textit{C. Xu}, Compos. Math. 150, No. 3, 409--414 (2014; Zbl 1291.14057); \textit{Z. Tian} and \textit{C. Xu}, Compos. Math. 153, No. 2, 257--273 (2017; Zbl 1453.14014); \textit{D. Greb} et al.,…
reviewed by: Patrick Graf
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 32S50
zbMATH DE Number: 7433732
zbMATH Keywords: klt singularity
zbMATH Keywords: weakly Fano pair
zbMATH Keywords: local fundamental group
zbMATH Keywords: regional fundamental group
MaRDI profile type: Publication
OpenAlex ID: W3187365476
arXiv ID: 2004.00522
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DOI: 10.1007/S00222-021-01062-0
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