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Finite abelian groups acting on rationally connected threefolds I: Groups of product type

Konstantin Loginov

TL;DR

The paper classifies finite abelian groups acting faithfully and biregularly on 3-dimensional rationally connected varieties by partitioning actions into three types: product type, K3 type, and a third-type potential scenario with $|-K_X|=\varnothing$. It establishes that product-type groups are precisely subgroups of $Cr_1(\mathbb{C})\times Cr_2(\mathbb{C})$, leveraging Blanc’s results, while K3-type groups are finite in number, with a bounded extension structure over abelian groups acting on K3 surfaces. The central conjecture is that the third type does not occur beyond the first two, and the paper proves a boundedness result for K3-type groups, laying groundwork for a broader dimension-independent framework. The approach combines a $G$-equivariant minimal model program, detailed analysis of the anti-canonical linear system, and dual-complex topology to constrain group actions on rationally connected 3-folds and to reduce to $G\mathbb{Q}$-Fano threefolds in which the action is tightly controlled.

Abstract

We initiate the study of finite abelian groups that faithfully act on 3-dimensional rationally connected varieties. We show that these groups can be naturally divided into three types: the groups of product type are finite abelian groups that are products of two groups which belong to the Cremona group of rank 1 and 2, respectively; the group of K3 type consists of cyclic extensions of finite abelian groups acting on a K3 surface; the third type consists of groups that act on terminal Fano threefolds with empty anti-canonical linear system. The classification of groups of product type follows from a result of J. Blanc. For the groups of K3 type, we show that there are only finitely many of them. We also formulate a conjecture regarding the groups of the third type.

Finite abelian groups acting on rationally connected threefolds I: Groups of product type

TL;DR

The paper classifies finite abelian groups acting faithfully and biregularly on 3-dimensional rationally connected varieties by partitioning actions into three types: product type, K3 type, and a third-type potential scenario with . It establishes that product-type groups are precisely subgroups of , leveraging Blanc’s results, while K3-type groups are finite in number, with a bounded extension structure over abelian groups acting on K3 surfaces. The central conjecture is that the third type does not occur beyond the first two, and the paper proves a boundedness result for K3-type groups, laying groundwork for a broader dimension-independent framework. The approach combines a -equivariant minimal model program, detailed analysis of the anti-canonical linear system, and dual-complex topology to constrain group actions on rationally connected 3-folds and to reduce to -Fano threefolds in which the action is tightly controlled.

Abstract

We initiate the study of finite abelian groups that faithfully act on 3-dimensional rationally connected varieties. We show that these groups can be naturally divided into three types: the groups of product type are finite abelian groups that are products of two groups which belong to the Cremona group of rank 1 and 2, respectively; the group of K3 type consists of cyclic extensions of finite abelian groups acting on a K3 surface; the third type consists of groups that act on terminal Fano threefolds with empty anti-canonical linear system. The classification of groups of product type follows from a result of J. Blanc. For the groups of K3 type, we show that there are only finitely many of them. We also formulate a conjecture regarding the groups of the third type.
Paper Structure (22 sections, 73 theorems, 109 equations)

This paper contains 22 sections, 73 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Contractions
  4. Pairs and singularities
  5. Log Fano and log Calabi--Yau pairs
  6. Mori fiber space
  7. Dual complex
  8. Group action on spheres
  9. Group actions on algebraic varieties.
  10. Groups of product type
  11. Group action on curves
  12. Group action on surfaces
  13. Simple elliptic and cusp singularities
  14. Warm-up: finite abelian groups in $\mathrm{Cr}_2(\mathbb{C})$
  15. Terminal singularities
  16. ...and 7 more sections

Key Result

Proposition 1.2

Let $G$ be a finite abelian subgroup of $\mathrm{Aut}(\mathbb{P}^1)=\mathrm{PGL}(2, \mathbb{C})$. Then $G$ is isomorphic to one of the following groups:

Theorems & Definitions (149)

  • Proposition 1.2
  • Theorem 1.3: Bl07
  • Theorem 1.4: Beau07
  • Theorem 1.5: Pr11, Pr14, PS17, Kuz20, Xu20, Lo22
  • Theorem 1.6: KZh24
  • Theorem 1.7
  • Conjecture 1.8
  • Proposition 1.9
  • Corollary 1.10
  • Proposition 1.11: =Proposition \ref{['prop-abstact-extension']}
  • ...and 139 more