A motivic global Torelli theorem for isogenous K3 surfaces

TL;DR

This paper establishes a motivic Torelli theorem for K3 surfaces, showing that isogenous K3 surfaces have isomorphic Chow motives as Frobenius algebra objects, linking derived equivalence and Hodge isometry.

Contribution

It proves that Chow motives of twisted derived equivalent K3 surfaces are isomorphic as Frobenius algebra objects, extending the Torelli theorem in a motivic context.

Findings

Chow motives of isogenous K3 surfaces are isomorphic as Frobenius algebra objects.

Derived equivalence implies isomorphism of Chow motives as Frobenius algebra objects.

Existence of K3 surfaces with non-isomorphic motives as Frobenius algebra objects but isomorphic as algebra objects.

Abstract

We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two complex projective K3 surfaces are isogenous (i.e. their second rational cohomology groups are Hodge isometric) if and only if their Chow motives are isomorphic as Frobenius algebra objects; this can be regarded as a motivic Torelli-type theorem. We ask whether, more generally, twisted derived equivalent hyper-Kaehler varieties have isomorphic Chow motives as (Frobenius) algebra objects and in particular isomorphic graded rational cohomology algebras. In the appendix, we justify introducing the notion of "Frobenius algebra object" by showing the existence of an infinite family of K3 surfaces whose Chow motives are pairwise non-isomorphic as Frobenius algebra objects but isomorphic as algebra objects. In particular, K3 surfaces in that family are pairwise non-isogenous but have isomorphic rational Hodge algebras.