Fundamental group schemes of globally F-regular varieties

2026/07/29 by Charles Vial
Mathematics · #math.AG #msc:14F35 #msc:13A35 #msc:14G17 #msc:14L15 #msc:14J45 #msc:14B05

paper · pdf

22 pages, comments welcome

arxiv created 2026/07/29 · arxiv updated 2026/07/30

Abstract

We prove that every tower of connected quasi-torsor covers of a connected normal globally F-regular projective variety over an algebraically closed field of positive characteristic stabilizes, and that the Nori fundamental group scheme of its regular locus is finite and linearly reductive. The linear reductivity asserts that any quasi-torsor cover is tame, while the finiteness is a positive-characteristic analogue of theorems of Xu and Braun on the finiteness of the fundamental group of the smooth locus of a log Fano complex projective variety.

Citations

• On proper splinters in positive characteristic
• Perfectoid signature, perfectoid Hilbert-Kunz multiplicity, and an application to local fundamental groups
• Linearly Reductive Quotient Singularities
• The Nori fundamental gerbe of a fibered category
• The F-signature and strong F-regularity
• F-regular and F-pure rings vs. log terminal and log canonical singularities

Related

• The tame Nori fundamental group
• Fundamental groups of proper varieties are finitely presented
• The tame fundamental group schemes of curves in positive characteristic
• A smooth counterexample to Nori's conjecture on the fundamental group scheme
• Fundamental group-scheme of some rationally connected fibrations