Finiteness of fundamental groups

Tian, Zhiyu; Xu, Chenyang

doi:10.1112/s0010437x16007867

Cited by 14 publications

(48 citation statements)

References 33 publications

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“…In [Xu14] C. Xu showed that this holds for thé etale local fundamental group, in other words, for the profinite completion of π 1 (X \ {0}). Building on this result, [GKP13] proved the finiteness of theétale fundamental groups of the regular locus of KLT singularities (see also [TX15]). Over the past few decades, we have learned that KLT singularities are closely related to strongly F -regular singularities in characteristic p > 0, see [HW02,Har98].…”

Section: Introductionmentioning

confidence: 85%

Fundamental groups of F-regular singularities via F-signature

Carvajal-Rojas

Schwede²,

Tucker

2018

Ann. Sci. École Norm. Sup.

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We prove that the localétale fundamental group of a strongly F -regular singularity is finite (and likewise for theétale fundamental group of the complement of a codimension ≥ 2 set), analogous to results of Xu and Greb-Kebekus-Peternell for KLT singularities in characteristic zero. In fact our result is effective, we show that the reciprocal of the F -signature of the singularity gives a bound on the size of this fundamental group. To prove these results and their corollaries, we develop new transformation rules for the F -signature under finiteétale-in-codimension-one extensions. As another consequence of these transformation rules, we also obtain purity of the branch locus over rings with mild singularities (particularly if the F -signature is > 1/2). Finally, we generalize our F -signature transformation rules to the context of pairs and not-necessarilý etale-in-codimension-one extensions, obtaining an analog of another result of Xu.

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Section: Introductionmentioning

confidence: 85%

Fundamental groups of F-regular singularities via F-signature

Carvajal-Rojas

Schwede²,

Tucker

2018

Ann. Sci. École Norm. Sup.

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“…On the other hand, by [58,Prop. 3.6], we obtain the following direct consequence of Theorem 1, which can be seen as an infinite version of [34,Thm.…”

Section: Consequences (And Non-consequences) Of Our Main Theoremsmentioning

confidence: 98%

“…The regional fundamental group π reg 1 has-not under this name-already been considered in [44,51,58]. Of course, if x is isolated, both local and regional fundamental group coincide.…”

Section: Regional Fundamental Groupsmentioning

confidence: 99%

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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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“…In what follows, we will work with the algebraic local fundamental group instead, since the finiteness of local fundamental groups of klt 3-fold singularities is not known (see, e.g., [TX17] for the canonical case). The algebraic local fundamental group of a klt singularity is finite [Xu14,Sti17] .…”

Section: Degenerations Of Klt 3-fold Singularitiesmentioning

confidence: 99%