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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>K</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mn>3</m:mn> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:msup> </m:mrow> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>K</m:mi> <m:mo>⁢</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> K3 surface are derived equivalent if they are of the same dimension.