Fundamental groups of $F$-regular singularities via $F$-signature

Abstract

We prove that the local etale fundamental group of a strongly $F$-regular singularity is finite (and likewise for the étale fundamental group of the complement of a codimension $\ge 2$ 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 finite etale-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-necessarily etale-in-codimension-one extensions, obtaining an analog of another result of Xu.

22 pages, comments welcome

References in corpus (2)

• New examples of terminal and log canonical singularities
• Finiteness of fundamental groups

Cited by in corpus (20)

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• Étale Covers and Local Algebraic Fundamental Groups
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• The F-signature Function on the Ample Cone
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• Inversion of adjunction for $F$-signature
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• Étale coverings in codimension 1 with applications to Mori Dream Spaces