Tuesday 10 March 2020, 15:30-16:30 in HG00.065
Simon Pepin Lehalleur (RU)
Mixed motives and 1-motives after Voevodsky
Complex algebraic varieties can be profitably studied via the methods of algebraic topology. In particular their singular cohomology carries a mixed Hodge structure, which is a very rich invariant and leads to strong restrictions on the topology. In parallel, algebraic varieties over general fields have l-adic cohomology groups with Galois actions, which are again fundamental invariants in arithmetic geometry. There are still other cohomology theories (de Rham, rigid,...) with strikingly similar properties. The theory of motives aims at understanding these common features and connecting them with the geometry of subvarieties (or algebraic cycles). I will introduce the framework of Voevodsky motives, which gives a partial answer in that direction. I will then discuss the case of motives attached to curves and families of curves (1-motives), where much more is understood and where we can construct abelian categories of 1-motives.