Joint Seminar Dijon-Freiburg-Strasbourg |
The moduli space of Fano threefolds of genus 10 is a tenfold that maps onto the moduli space of genus-2 curves via the period map. I will describe the fibre of this map as an open dense subset of the dual of the Coble cubic in P^8.
The stable motivic homotopy category SH(S) over a scheme S is characterized by an 1-categorical universal property: each homology theory defined on the category of smooth S-schemes satisfying algebraic analogues of the Eilenberg-Steenrod axioms factors essentially uniquely through SH(S). Ayoub has equipped the categories SH(S) with a six-functor formalism, i.e., pullback, pushforward, and tensor operations satisfying the same formal properties as derived categories of l-adic sheaves. In this talk, we will promote the universal property of SH(S) to a universal property of the associated six-functor formalism: each six-functor formalism S \mapsto D(S) satisfying a reasonable list of axioms admits an essentially unique family of functors SH(S) \mapsto D(S) compatible with the six functors. As an application, we construct various motivic realization functors.
Let S be a Shimura variety of abelian type, associated to a reductive group G. Each algebraic representation V of G gives rise to a mixed sheaf \mu(V) on S (both in the Hodge-theoretic and in the l-adic sense). When S is non-compact, the weight filtrations on the cohomology of \mu(V), and on the cohomology of its degeneration at the boundary of a compactifiation of S, carry important arithmetic information. In this talk, we will look at the case of G = GSp_4/F (for F a totally real number field), corresponding to genus 2 Hilbert-Siegel varieties, and we will explain how to characterize the presence of certain weights in terms of an invariant of irreducible representations of G, called corank. In particular, thanks to Wildeshausâ€™ theory, this description allows one to construct, in many cases, homological motives associated to cuspidal automorphic representations of G.