On the finiteness of twists of irreducible symplectic varieties

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.