Construction of automorphisms of hyperk\"ahler manifolds

TL;DR

The paper constructs deformations of hyperk"ahler manifolds with specific types of infinite order automorphisms, expanding understanding of their automorphism groups and geometric structures.

Contribution

It introduces methods to produce deformations of hyperk"ahler manifolds with hyperbolic and parabolic automorphisms of infinite order.

Findings

Existence of deformations with hyperbolic automorphisms for b2 ≥ 5.

Existence of deformations with parabolic automorphisms for b2 ≥ 14.

Automorphisms act hyperbolically on the space of real (1,1)-classes.

Abstract

Let $M$ be an irreducible holomorphic symplectic (hyperk\"ahler) manifold. If $b_2(M)\geq 5$, we construct a deformation $M'$ of $M$ which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real $(1,1)$-classes is hyperbolic. If $b_2(M) \geq 14$, similarly, we construct a deformation which admits a parabolic automorphism.