# Verra fourfolds, twisted sheaves and the last involution

## TL;DR

This paper explores the geometry of moduli spaces of twisted sheaves on K3 surfaces, introduces automorphisms, and proves unirationality of certain holomorphic symplectic manifolds with specific involutions, revealing new geometric structures.

## Contribution

It introduces induced automorphisms on moduli spaces of twisted sheaves and proves unirationality for specific irreducible holomorphic symplectic manifolds with non-symplectic involutions.

## Key findings

- Proves unirationality of certain moduli spaces of IHS manifolds.
- Shows IHS fourfolds contain non-nodal Enriques surfaces.
- Establishes automorphisms induced from K3 surfaces on moduli spaces.

## Abstract

We study the geometry of some moduli spaces of twisted sheaves on K3 surfaces. In particular we introduce induced automorphisms from a K3 surface on moduli spaces of twisted sheaves on this K3 surface. As an application we prove the unirationality of moduli spaces of irreducible holomorphic symplectic manifolds of $K3^{[2]}$-type admitting non symplectic involutions with invariant lattices $U(2)\oplus D_4(-1)$ or $U(2)\oplus E_8(-2)$. This complements the results obtained in [Mongardi and Wandel 2015], [Bossiere et al 2016], and the results from [arXiv:1603.00403] about the geometry of IHS fourfolds constructed using the Hilbert scheme of $(1,1)$ conics on Verra fourfolds. As a byproduct we find that IHS fourfolds of $K3^{[2]}$-type with Picard lattice $U(2)\oplus E_8(-2)$ naturally contain non-nodal Enriques surfaces.