Equivalences of derived categories and K3 surfaces

TL;DR

The paper develops a general kernel-based description of derived-equivalence between smooth projective varieties, proving that any full, faithful exact functor with adjoints is a Fourier–Mukai transform with a unique kernel on the product. Building on this, it specializes to K3 surfaces to show that two complex K3s have equivalent derived categories precisely when their transcendental Mukai lattices are Hodge isometric. The approach combines resolution of the diagonal, ample-sequence technology, and Mukai lattice isometries, linking categorical equivalences to geometric moduli data and Torelli-type results. This provides a complete criterion for derived equivalence of K3 surfaces via lattice-theoretic data.

Abstract

We consider derived categories of coherent sheaves on smooth projective varieties. We prove that any equivalence between them can be represented by an object on the product. Using this, we give a necessary and sufficient condition for equivalence of derived categories of two K3 surfaces.