A stronger derived Torelli theorem for K3 surfaces

TL;DR

This paper refines the understanding of derived equivalences for K3 surfaces, showing that preserving ample cones implies isomorphism, strengthening the Torelli theorem in this context.

Contribution

It demonstrates that filtered derived equivalences preserving ample cones correspond to isomorphisms, providing a stronger Torelli-type result for K3 surfaces.

Findings

Filtered derived equivalences preserving ample cones induce isomorphisms.

Strengthens the Torelli theorem for K3 surfaces.

Connects cohomological actions with geometric isomorphisms.

Abstract

In an earlier paper the notion of a filtered derived equivalence was introduced, and it was shown that if two K3 surfaces admit such an equivalence then they are isomorphic. In this paper we study more refined aspects of filtered derived equivalences related to the action on the cohomological realizations of the Mukai motive. It is shown that if a filtered derived equivalence between K3 surfaces also preserves ample cones then one can find an isomorphism that induces the same map as the equivalence on the cohomological realizations.