The Jordan property for local fundamental groups

Braun, Lukas; Filipazzi, Stefano; Moraga, Joaquín; Svaldi, Roberto

doi:10.48550/arxiv.2006.01253

Cited by 2 publications

(3 citation statements)

References 14 publications

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“…However, klt type singularities are better behaved than general rational singularities. For instance, we know that the fundamental group of a klt type singularity is finite [66,11] and it satisfies the Jordan property [12], while the fundamental group of a rational singularity can be an arbitrary Q-superperfect group [36]. In a similar vein, klt type singularities are local versions of Mori dream spaces [13], i.e., their local Cox ring is finitely generated, while this is no longer the case for general rational singularities.…”

Section: Introductionmentioning

confidence: 99%

Reductive quotients of klt singularities

Braun¹,

Greb²,

Langlois³

et al. 2021

Preprint

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We prove that the quotient of a klt type singularity by a reductive group is of klt type. In particular, given a klt variety X endowed with the action of a reductive group G and admitting a quasiprojective good quotient X Ñ X{{G, we can find a boundary B on X{{G so that the pair pX{{G, Bq is klt. This applies for example to GIT-quotients of klt varieties. Furthermore, our result has implications for complex spaces obtained as momentum map quotients of Hamiltonian Kähler manifolds and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing n-dimensional K-polystable Fano manifolds of volume v has klt type singularities.

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Section: Introductionmentioning

confidence: 99%

Reductive quotients of klt singularities

Braun¹,

Greb²,

Langlois³

et al. 2021

Preprint

Self Cite

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“…For instance, we can consider n = 3 and choose the polarization L E,5 on the 2-dimensional snc Calabi-Yau variety E. Then, we obtain the generator's relations γ −2 1 γ 2 γ 3 γ 4 , γ 1 γ −2 2 γ 3 γ 4 , γ 1 γ 2 γ −2 3 γ 4 , and γ 1 γ 2 γ 3 γ −2 4 . We conclude that π 1 (X 2,5 ; x) ≃ (Z/3Z) 3 . This example is interesting as it is closely related to toric singularities.…”

Section: Examples and Questionsmentioning

confidence: 65%

“…, n − 1}, measures the combinatorial complexity of the resolution of (X; x) (see Definition 2.9). The previous statement was obtained due to the work of many mathematicians [36,35,2,22,23,3]. In a few words, the previous theorem says that the fundamental group of a log terminal singularity behaves as the fundamental group of an orbifold singularity.…”

Section: Introductionmentioning

confidence: 97%

Fundamental groups of low-dimensional lc singularities

Figueroa¹,

Moraga²

2023

Preprint

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In this article, we study the fundamental groups of low-dimensional log canonical singularities, i.e., log canonical singularities of dimension at most 4. In dimension 2, we show that the fundamental group of an lc singularity is a finite extension of a solvable group of length at most 2. In dimension 3, we show that every surface group appears as the fundamental group of a 3-fold log canonical singularity. In contrast, we show that for r ≥ 2 the free group Fr is not the fundamental group of a 3-dimensional lc singularity. In dimension 4, we show that the fundamental group of any 3-manifold smoothly embedded in R 4 is the fundamental group of an lc singularity. In particular, every free group is the fundamental group of a log canonical singularity of dimension 4. In order to prove the existence results, we introduce and study a special kind of polyhedral complexes: the smooth polyhedral complexes. We prove that the fundamental group of a smooth polyhedral complex of dimension n appears as the fundamental group of a log canonical singularity of dimension n + 1. Given a 3-manifold M smoothly embedded in R 4 , we show the existence of a smooth polyhedral complex of dimension 3 that is homotopic to M . To do so, we start from a complex homotopic to M and perform combinatorial modifications that mimic the resolution of singularities in algebraic geometry. Contents 1. Introduction 1 2. Preliminaries 6 3. Fundamental groups in dimension 2 8 4. Smooth polyhedral complexes 23 5. Threefold log canonical singularities 31 6. Fourfold log canonical singularities 36 7. Examples and questions 42 Appendix A. Fundamental groups of lc surface singularities 44 References 46

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The Jordan property for local fundamental groups

Cited by 2 publications

References 14 publications

Reductive quotients of klt singularities

Reductive quotients of klt singularities

Fundamental groups of low-dimensional lc singularities

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