On the finiteness of twists of irreducible symplectic varieties

TL;DR

This paper proves the finiteness of twists of irreducible symplectic varieties over fixed finite field extensions, utilizing the cone conjecture, and explores related finiteness results and cases over positive characteristic fields.

Contribution

It establishes the finiteness of twists for irreducible symplectic varieties using the cone conjecture and extends the discussion to non-closed fields and positive characteristic cases.

Findings

Finiteness of twists over fixed finite field extensions proved.

Application to finiteness of derived equivalent twists.

Discussion of cone conjecture over non-closed fields and positive characteristic.

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.