Computing discrete invariants of varieties in positive characteristic
Friday, 10.7.20, 10:30-11:30, virtueller Raum 404
For varieties (smooth projective, say) over fields of\npositive characteristic, we can define discrete invariants that have no\nnatural analogue in characteristic 0. A well-known example is that an\nelliptic curve in characteristic p is either ordinary or supersingular.\nI will first review in general terms how this can be generalized to\narbitrary varieties - there is in fact more than one natural\ngeneralization!\n\nAfter this, I will focus on one particular type of discrete invariant;\nfor abelian varieties this is known under the name 'Ekedahl-Oort type'.\nI will address the question how such discrete invariants can be\nconcretely computed. In particular, I will explain a new method that\nallows to explicitly compute the Ekedahl-Oort type of (the Jacobian of)\na complete intersection curve. For plane curves, a magma implementation\nof this method is now available, so if you have a favourite curve of\nwhich you want to know the E-O type, you can ask me and we can let\nmagma calculate the answer.\n\nAt the end of the talk I will try to say a few words about\ngeneralizations for higher-dimensional projective hypersurfaces. There\nis a simple pattern that emerges, but so far I can only prove that it's\ncorrect for varieties of low dimension.\n
Motives of moduli spaces of bundles over a curve
Friday, 17.7.20, 10:30-11:30, virtueller Raum 404
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 motives of certain algebraic stacks. I will then state and prove a formula 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 cohomology of small maps. If there is time, I will discuss how this result can be used to also study motives of moduli space of Higgs bundles. This is joint work with Simon Pepin Lehalleur.
K-Motives and Koszul duality
Friday, 24.7.20, 10:30-11:30, virtueller Raum 404
Koszul duality, as first conceived by Beilinson-Ginzburg-Soergel, is a remarkable symmetry in the representation theory of Langlands dual reductive groups. This talks argues that Koszul duality - in it's most natural form - stems from a duality between equivariant K-motives and monodromic sheaves. I will give a short guide to K-motives and monodromic sheaves and then discuss examples of Koszul duality in increasing difficulty: (1) Tori (2) Toric varieties (3) Reductive groups.