Derived categories of hyper-Kähler manifolds: extended Mukai vector and integral structure

[摘要] We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-Kähler manifold using the extended Mukai lattice. This enables us to define a Mukai vector for certain objects in the derived category taking values inside the extended Mukai lattice which is functorial for derived equivalences. As applications, we obtain a structure theorem for derived equivalences between hyper-Kähler manifolds as well as an integral lattice associated to the derived category of hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of a K3 surface mimicking the surface case.

[学科分类] 数学（综合）

[关键词] Hyper-Kähler manifolds;derived categories;moduli spaces;Fourier–Mukai partners;Mukai vector;14J42;18G80;14J60;14C17