Modul: MAT770 Oberseminar: Algebraische Geometrie

## The motive of the moduli stack of vector bundles on a curve

Dr. Victoria Hoskins talk

Speaker invited by: Prof. Dr. Joseph Ayoub

**Date:** 26.11.18 **Time:** 13.15 - 14.45 **Room:** Y27H25

Following Grothendieck’s vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some properties of this category, I will explain how to define the motive of certain algebraic stacks. I will then state and sketch a proof for the motive of the moduli stack of vector bundles on a smooth projective curve; this formula is compatible with classical computations of invariants of this stack due to Harder, Atiyah-Bott and Behrend-Dhillon. The proof involves rigidifying this stack using Quot and Flag-Quot schemes parametrising Hecke modifications as well as a motivic version of an argument of Laumon and Heinloth on the relative cohomology of small maps. This is joint work with Simon Pepin Lehalleur.