## Geometry Seminar - Abstracts

### Talk

Tuesday 10 March 2020, 15:30-16:30 in HG00.065

**Simon Pepin Lehalleur** (RU)

*Mixed motives and 1-motives after Voevodsky*

### Abstract

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.

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