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On the local étale fundamental group of KLT threefold singularities

Javier Carvajal-Rojas, Axel Stäbler, János Kollár

TL;DR

This work proves that for a threefold Kawamata–log terminal singularity $S$ over an algebraically closed field with characteristic $p>5$, the local étale fundamental group of any big open $S^{\circ}$ is tame and finite, and in particular the prime-to-$p$ part accounts for the whole group. The authors develop generalized PLT blowups extracting Kollár components that are purely $F$-regular, enabling a local-to-global analysis of tame covers via $ au$-modules and Frobenius actions on local cohomology. A key result is an Abhyankar-type fundamental theorem for tame covers, reducing global tameness questions to local data on Kollár components and their strict transforms. Consequently, the paper deduces finiteness and tameness of $\pi_1^{\et}(S^{\circ})$, shows that quasi-étale covers of $S$ with degree divisible by $p$ are étale, and proves rationality-based surjectivity statements for torsors with unipotent structure groups on big opens, with implications for local Picard schemes and local divisor class groups. Overall, the work extends tameness and finiteness results beyond strongly $F$-regular cases to KLT threefold singularities in positive characteristic, using a blend of birational techniques, $F$-singularity methods, and Abhyankar-type ramification control.

Abstract

Let $S$ be KLT threefold singularity over an algebraically closed field of positive characteristic $p>5$. We prove that its local étale fundamental group is tame and finite. Further, we show that every finite unipotent torsor over a big open of $S$ is realized as the restriction of a finite unipotent torsor over $S$.

On the local étale fundamental group of KLT threefold singularities

TL;DR

This work proves that for a threefold Kawamata–log terminal singularity over an algebraically closed field with characteristic , the local étale fundamental group of any big open is tame and finite, and in particular the prime-to- part accounts for the whole group. The authors develop generalized PLT blowups extracting Kollár components that are purely -regular, enabling a local-to-global analysis of tame covers via -modules and Frobenius actions on local cohomology. A key result is an Abhyankar-type fundamental theorem for tame covers, reducing global tameness questions to local data on Kollár components and their strict transforms. Consequently, the paper deduces finiteness and tameness of , shows that quasi-étale covers of with degree divisible by are étale, and proves rationality-based surjectivity statements for torsors with unipotent structure groups on big opens, with implications for local Picard schemes and local divisor class groups. Overall, the work extends tameness and finiteness results beyond strongly -regular cases to KLT threefold singularities in positive characteristic, using a blend of birational techniques, -singularity methods, and Abhyankar-type ramification control.

Abstract

Let be KLT threefold singularity over an algebraically closed field of positive characteristic . We prove that its local étale fundamental group is tame and finite. Further, we show that every finite unipotent torsor over a big open of is realized as the restriction of a finite unipotent torsor over .

Paper Structure

This paper contains 16 sections, 36 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Generalized PLT blowups and Tameness
  4. Generalized PLT blowups
  5. Tameness
  6. Étale fundamental groups of big opens of log del Pezzo surfaces
  7. $\tau$-modules
  8. Main results concerning tameness
  9. Questions of interest
  10. On the structure of tame covers via local tame fundamental groups
  11. Stratifications
  12. Abhyankar's lemma
  13. Proof of \ref{['thm.FundamentalTheorem']}
  14. Finiteness and tameness of local étale fundamental groups
  15. Local Picard schemes and local divisor class groups
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $x\in (X,\Delta)$ be a KLT closed point of a log pair of dimension $d\geq 2$ and characteristic $p$. Then, a Kollár component at $x$ exists if either $p=0$ or $d \leq 3$.

Theorems & Definitions (91)

  • Remark 1
  • Remark 2
  • Definition 1: PLT blowups and Kollár components
  • Remark 3
  • Theorem 2.1
  • Theorem 2.2: Generalized PLT blowups
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • ...and 81 more