Infinitesimal Derived Torelli Theorem for K3 surfaces

TL;DR

This paper establishes a new infinitesimal Torelli theorem for K3 surfaces, linking first order deformations to derived equivalences via special cohomological isometries, extending classical Torelli results.

Contribution

It proves a novel infinitesimal Torelli theorem for K3 surfaces, connecting derived equivalences with specific cohomological isometries that preserve additional structures.

Findings

Derived equivalence corresponds to special cohomology isometries.

Compatibility of Hochschild homology and singular cohomology actions.

Generalization of the classical Torelli theorem to an infinitesimal setting.

Abstract

We prove that the first order deformations of two smooth projective K3 surfaces are derived equivalent under a Fourier--Mukai transform if and only if there exists a special isometry of the total cohomology groups of the surfaces which preserves the Mukai pairing, an infinitesimal weight-2 decomposition and the orientation of a positive 4-dimensional space. This generalizes the derived version of the Torelli Theorem. Along the way we show the compatibility of the actions on Hochschild homology and singular cohomology of any Fourier--Mukai functor.