Derived equivalences of hyperkahler varieties of K3^[n] type - an introduction.

In this talk, I will do a brief introduction of the results about derived equivalences of hyperkahler varieties of K3^[n] type. First we will state that the Looijenga-Lunts-Verbitsky Lie algebra acting on the cohomology of a hyperkahler variety is a derived invariant using Hochschild homology and cohomology. Later, we will construct a rational "Mukai lattice" functorial for derived equivalences after introducing the Verbitsky component. And the compute (up to index 2) of the group of auto-equivalences on the cohomology of certain K3^[2] (more generally, K3^[n]). Some generalized results for derived equivalences of certain K3^[n]'s will be introduced. The talk is based on the paper from Lenny Taelman and Thorsten Beckmann.