A note on automorphisms and birational transformations of holomorphic symplectic manifolds

TL;DR

This paper characterizes when automorphisms of Hilbert schemes of points on K3 surfaces come from surface automorphisms and proves the finite generation of the birational transformation group of certain holomorphic symplectic manifolds.

Contribution

It provides a necessary and sufficient condition for automorphisms of Hilbert schemes to originate from surface automorphisms and establishes the finite generation of birational transformation groups.

Findings

Automorphisms of Hilbert schemes are characterized by surface automorphisms.

The birational transformation group of projective irreducible holomorphic symplectic manifolds is finitely generated.

The results hold for non-algebraic K3 surfaces.

Abstract

We give a necessary and sufficient condition for an automorphism of the Hilbert scheme of points on a K3 surface (non necessarily algebraic) to be induced by an automorphism of the surface. We prove furthermore that the group of birational transformations of a projective irreducible holomorphic symplectic manifold is finitely generated.