On L-equivalence for K3 Surfaces and Hyperkähler Manifolds

Abstract This paper explores the relationship between L -equivalence and D -equivalence for K3 surfaces and hyperkähler 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 hyperkähler fourfolds and moduli spaces of sheaves on K3 surfaces.