Abstract We conjecture that a natural twisted derived category of any hyper-Kähler variety of K 3 n K3^n -type is controlled by its Markman–Mukai lattice. We prove the conjecture under numerical constraints, and our proof relies on Markman’s projectively hyperholomorphic bundle and a recently proven twisted version of the D-equivalence conjecture. In particular, we prove a conjecture of Huybrechts, stating that any two fine moduli spaces of stable sheaves on a K 3 K3 surface are derived equivalent if they are of the same dimension.
Ruxuan Zhang (Mon,) studied this question.