Fundamental groups of F-regular singularities via F-signature

Carvajal-Rojas, Javier; Schwede, Karl; Tucker, Kevin

doi:10.24033/asens.2370

Cited by 32 publications

(48 citation statements)

References 32 publications

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“…in the sense that they appear on minimal models of log smooth projective varieties. Although they originate in birational geometry, these singularities have turned out to be of great importance also in moduli theory [37], in the theory of algebraic K-stability [67], and in positive characteristic considerations [19]. The aim of this article is to prove that reductive quotient singularities are singularities of the minimal model program.…”

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

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“…Analogous statements for F-regular singularities and strongly F-regular schemes in characteristic p can be found in [7,17]. It is also possible to deduce the statements in characteristic zero from the ones in characteristic p [6].…”

Section: éTale Fundamental Groupsmentioning

confidence: 91%

The local fundamental group of a Kawamata log terminal singularity is finite

Braun

2021

Invent. math.

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We prove a conjecture of Kollár stating that the local fundamental group of a klt singularity x is finite. In fact, we prove a stronger statement, namely that the fundamental group of the smooth locus of a neighbourhood of x is finite. We call this the regional fundamental group. As the proof goes via a local-to-global induction, we simultaneously confirm finiteness of the orbifold fundamental group of the smooth locus of a weakly Fano pair.

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“…Next, local fundamental groups of F-regular singularities are finite by [CST18]. In particular, the local fundamental groups of F-regular RDPs are finite.…”

Section: 2mentioning

confidence: 99%

Torsors over the Rational Double Points in Characteristic $\mathbf{p}$

Liedtke¹,

Martin²,

Matsumoto³

2021

Preprint

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We study torsors under finite group schemes over the punctured spectrum of a singularity x ∈ X in positive characteristic.We show that the Dieudonné module of the (loc,loc)-part Picloc loc,loc X/k of the local Picard sheaf can be described in terms of local Witt vector cohomology, making Picloc loc,loc X/k computable. Together with the class group and the abelianised local étale fundamental group, Picloc loc,loc X/kcompletely describes the finite abelian torsors over X \ {x}.We compute Picloc loc,locfor every rational double point singularity, which complements results of Artin [Ar77] and Lipman [Li69], who determined π ét loc (X) and Cl(X). All three objects turn out to be finite. We extend the Flenner-Mumford criterion for smoothness of a normal surface germ x ∈ X to perfect fields of positive characteristic, generalising work of Esnault and Viehweg: If k is algebraically closed, then X is smooth if and only if Picloc loc,loc X/k , π ét loc (X), and Cl(X) are trivial. Finally, we study the question whether rational double point singularities are quotient singularities by group schemes and if so, whether the group scheme is uniquely determined by the singularity. We give complete answers to both questions, except for some D r n -singularities in characteristic 2. In particular, we will give examples of (F-injective) rational double points that are not quotient singularities.

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Fundamental groups of F-regular singularities via F-signature

Cited by 32 publications

References 32 publications

Reductive quotients of klt singularities

Reductive quotients of klt singularities

The local fundamental group of a Kawamata log terminal singularity is finite

Torsors over the Rational Double Points in Characteristic $\mathbf{p}$

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