A twisted derived category of hyper-Kähler varieties of $K3^{[n]}$-type

We conjecture that a natural twisted derived category of any hyper-Kähler variety of $K3^{[n]}$-type is controlled by its Markman-Mukai lattice. We prove the conjecture under numerical constraints, and our proof relies heavily on Markman's projectively hyperholomorphic bundle and a recently proven twisted version of the D-equivalence conjecture. In particular, we prove that any two fine moduli spaces of stable sheaves on a $K3$ surface are derived equivalent if they have the same dimension.