# Derived equivalences of hyperk\"ahler varieties

## TL;DR

This paper proves that certain algebraic structures acting on the cohomology of hyperk"ahler varieties are preserved under derived equivalences, leading to new invariants and insights into their cohomological behavior.

## Contribution

It establishes the invariance of the Looijenga--Lunts--Verbitsky Lie algebra under derived equivalences and derives several consequences for the cohomology of hyperk"ahler varieties.

## Key findings

- Looijenga--Lunts--Verbitsky Lie algebra is a derived invariant.
- Derived equivalent hyperk"ahler varieties have isomorphic $ extbf{Q}$-Hodge structures.
- Constructed a rational 'Mukai lattice' functorial for derived equivalences.

## Abstract

We show that the Looijenga--Lunts--Verbitsky Lie algebra acting on the cohomology of a hyperk\"ahler variety is a derived invariant, and obtain from this a number of consequences for the action on cohomology of derived equivalences between hyperk\"ahler varieties. This includes a proof that derived equivalent hyperk\"ahler varieties have isomorphic $\mathbf{Q}$-Hodge structures, the construction of a rational `Mukai lattice' functorial for derived equivalences, and the computation (up to index 2) of the image of the group of auto-equivalences on the cohomology of certain Hilbert squares of K3 surfaces.