Mar 9 at 17:30 in 507

Title: On non-abelian Fourier tranforms for compact hyper-Kähler varieties.

Abstract: Compact hyper-Kähler varieties are higher dimensional analogs of K3 surfaces. It has been expected that, from many perspectives, the geometry of compact hyper-Kähler varieties "behaves like" that of abelian varieties (e.g. the Beavuille-Voisin conjectures predict that this is the case for motives and algebraic cycles.) In this talk, I will discuss this phenomenon from the perspective of Fourier transforms. In particular, I will explain that for hyper-Kähler varieties of K3[n]-type, the analogues of Makai's Poincaré line bundles for abelian varieties should be given by certain hyperholomorphic bundles constructed by Markman recently. Thess ideas lead to a proof of the Bondal-Orlov D-equivalence conjeture for K3[n]-type varieties (joint with Davesh Maulik, Qizheng Yin, and Ruxuan Zhang), and a proof of Fu-Vial's multiplicative version of the Orlov conjecture for homological motives (joint work with Davesh Maulik and Qizheng Yin). Conjectures regarding this Fourier transform package will be discussed.