On L-equivalence for K3 surfaces and hyperk\"ahler manifolds

TL;DR

This paper investigates the connection between L-equivalence and D-equivalence in K3 surfaces and hyperk"ahler manifolds, proving that very general L-equivalent K3 surfaces are also D-equivalent using Hodge theory and Torelli theorems.

Contribution

It establishes a link between L- and D-equivalence for K3 surfaces and extends some results to hyperk"ahler fourfolds, introducing new relations between lattice structures and Hodge endomorphisms.

Findings

Very general L-equivalent K3 surfaces are D-equivalent.

Two lattice structures on an integral Hodge structure are related by a rational endomorphism.

Partial extension of results to hyperk"ahler fourfolds and moduli spaces.

Abstract

This paper explores the relationship between L-equivalence and D-equivalence for K3 surfaces and hyperk\"ahler manifolds. Building on Efimov's approach using Hodge theory, we prove that very general L-equivalent K3 surfaces are D-equivalent, leveraging the Derived Torelli Theorem for K3 surfaces. Our main technical contribution is that two distinct lattice structures on an integral, irreducible Hodge structure are related by a rational endomorphism of the Hodge structure. We partially extend our results to hyperk\"ahler fourfolds and moduli spaces of sheaves on K3 surfaces.