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On the finiteness of twists of irreducible symplectic varieties

Teppei Takamatsu

TL;DR

This work proves finiteness of twists $ ext{Tw}_{k'/k}(X)$ for key classes of varieties over a fixed finite extension, notably K3 surfaces in characteristic not equal to $2$, non-supersingular K3 surfaces in characteristic $2$, and irreducible symplectic varieties in characteristic $0$, with uniform bounds depending on $[k':k]$, the deformation class, and the isometry class of the geometric Néron–Severi lattice. The authors leverage the cone conjectures for irreducible symplectic varieties (birational and automorphism cones) proven by Markman and Amerik–Verbitsky, and adapt Bright–Logan–van Luijk’s method to non-closed fields, establishing fundamental polyhedral domains and Galois-descent arguments to bound twists. They also give applications to the finiteness of derived-equivalent twists $ ext{Tw}^D(X)$ for several standard ISV families (K3$^{[n]}$, generalized Kummer, OG$_6$, OG$_{10}$) and discuss Enriques surfaces in positive characteristic. The results contribute a Shafarevich-type finiteness phenomenon for ISVs, illuminate the role of automorphism groups in twist finiteness, and provide tools for understanding derived-equivalent models over non-closed fields.

Abstract

Irreducible symplectic varieties are higher-dimensional analogues of K3 surfaces. In this paper, we prove the finiteness of twists of irreducible symplectic varieties via a fixed finite field extension of characteristic $0$. The main ingredient of the proof is the cone conjecture for irreducible symplectic varieties, which was proved by Markman and Amerik--Verbitsky. As byproducts, we also discuss the cone conjecture over non-closed fields by Bright--Logan--van Luijk's method. We also give an application to the finiteness of derived equivalent twists. Moreover, we discuss the case of K3 surfaces or Enriques surfaces over fields of positive characteristic.

On the finiteness of twists of irreducible symplectic varieties

TL;DR

This work proves finiteness of twists for key classes of varieties over a fixed finite extension, notably K3 surfaces in characteristic not equal to , non-supersingular K3 surfaces in characteristic , and irreducible symplectic varieties in characteristic , with uniform bounds depending on , the deformation class, and the isometry class of the geometric Néron–Severi lattice. The authors leverage the cone conjectures for irreducible symplectic varieties (birational and automorphism cones) proven by Markman and Amerik–Verbitsky, and adapt Bright–Logan–van Luijk’s method to non-closed fields, establishing fundamental polyhedral domains and Galois-descent arguments to bound twists. They also give applications to the finiteness of derived-equivalent twists for several standard ISV families (K3, generalized Kummer, OG, OG) and discuss Enriques surfaces in positive characteristic. The results contribute a Shafarevich-type finiteness phenomenon for ISVs, illuminate the role of automorphism groups in twist finiteness, and provide tools for understanding derived-equivalent models over non-closed fields.

Abstract

Irreducible symplectic varieties are higher-dimensional analogues of K3 surfaces. In this paper, we prove the finiteness of twists of irreducible symplectic varieties via a fixed finite field extension of characteristic . The main ingredient of the proof is the cone conjecture for irreducible symplectic varieties, which was proved by Markman and Amerik--Verbitsky. As byproducts, we also discuss the cone conjecture over non-closed fields by Bright--Logan--van Luijk's method. We also give an application to the finiteness of derived equivalent twists. Moreover, we discuss the case of K3 surfaces or Enriques surfaces over fields of positive characteristic.

Paper Structure

This paper contains 16 sections, 44 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and notations
  4. Almost abelian grouops
  5. The case of surfaces
  6. K3 surfaces
  7. Enriques surfaces
  8. Cone conjecture for irreducible symplectic varieties
  9. Birational cone conjecture
  10. Automorphism cone conjecture
  11. Proof of the main theorem
  12. Uniform bounds
  13. Bounds of automorphism groups
  14. Proof of Theorem \ref{['effective']} (1)
  15. Proof of Theorem \ref{['effective']} (2)
  16. ...and 1 more sections

Key Result

Theorem 1.0.1

Let $k'/k$ be a finite extension of fields, $X$ a variety over $k$. Then $\mathrm{Tw}_{k'/k} (X)$ is a finite set in the following cases.

Theorems & Definitions (100)

  • Theorem 1.0.1: Theorem \ref{['fintwistK3']}, Theorem \ref{['hypkfintwist']}
  • Theorem 1.0.2: Theorem \ref{['fintwistEnr']}
  • Corollary 1.0.3: Theorem \ref{['findertwist']}
  • Corollary 1.0.4
  • Theorem 1.0.5: Theorem \ref{['bircone']}, Theorem \ref{['automcone']}
  • Definition 2.1.1
  • Remark 2.1.2
  • Definition 2.1.3
  • Lemma 2.1.4
  • proof
  • ...and 90 more