Automorphisms of O'Grady's Manifolds Acting Trivially on Cohomology

TL;DR

This paper classifies automorphisms acting trivially on the second cohomology for O'Grady's hyperkähler manifolds, revealing triviality in ten dimensions and a specific finite group structure in six dimensions, along with fixed locus analysis.

Contribution

It precisely determines the subgroup of automorphisms acting trivially on cohomology for O'Grady's manifolds, including explicit group structures and fixed locus properties.

Findings

Trivial automorphism subgroup in 10-dimensional case.

Automorphism subgroup isomorphic to (Z/2Z)^8 in 6-dimensional case.

Analysis of fixed loci of automorphisms in the 6-dimensional case.

Abstract

We determine the subgroup of automorphisms acting trivially on the second integral cohomology for hyperkaehler manifolds which are deformation equivalent to O'Grady's sporadic examples. In particular, we prove that this subgroup is trivial in the ten dimensional case and isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{\times 8}$ in the six dimensional case. Furthermore we study the fixed loci of the automorphisms in the six dimensional case.