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

2021/03/24 by Thorsten Beckmann, Beckmann, Thorsten · 6 citations
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Geometry and complex manifolds #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2103.13382

Abstract

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.

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