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Annual EAGER Conference 2004
Barbieri Viale: Consider the derived category of 1-motives (up to isogeny) along with its fully-faithful embedding Tot into Voevodsky's triangulated category of motives. Regarding Tot as a universal realisation functor we show (jointly with B. Kahn) that it has a left adjoint LAlb, the motivic Albanese triangulated functor. Dually, composing with (motivic) Cartier duality, we obtain the functor RPic. Note that the counit also provide a universal map, the motivic Albanese map, which 'contains' the classical Albanese map.
These functors provide natural complexes of 1-motives (up to isogeny) LAlb(X) and RPic(X) of an algebraic variety X over a perfect field. Their 1-motivic homology and cohomology would recover the Picard and Albanese 1-motives (introduced jointly with V. Srinivas) as well as the 1-motives predicted by Deligne's conjecture (proven jointly with A. Rosenschon and M. Saito).
Beauville: The Chow ring CH(X) of a (smooth, projective) variety X is a fundamental invariant, unfortunately rather poorly understood. Some light is shed by the deep conjectures of Bloch and Beilinson, which predict the existence of a functorial ring filtration of CH(X) with a (conjectural) description of the associated graded ring. In some exemples the filtration actually splits, i.e. is the filtration associated to a graduation: this is the case for K3 surfaces and, conjecturally, for abelian varieties. I will discuss for what kind of varieties one can expect such behaviour, in particular why it might be the case for (holomorphic) symplectic manifolds.
Colliot-Thélčne: It is conjectured that the reduced zero-dimensional Chow group of a rationally connected variety over a p-adic field is a finite group. This is known in dimension 2, and in the good reduction case. For smooth compactifications of linear algebraic groups, I shall prove the finiteness up to p-torsion. A new algebraic tool is the notion of flasque resolution attached to a connected reductive group.
Deglise: In the theory of mixed motives of Voevodsky, the triangulated category of effective mixed motives is enlarged in a triangulated category of motivic complexes, following the approach of Beilinson to define motivic cohomology. We will show how this later category is related to the theory of cycle modules of Rost, a notion directly inspired by the work of Kato on unramified Milnor K-theory. More precisely, we will relate the category of cycle modules with the heart of the triangulated category of motivic complexes with respect to the natural t-structure (called the homotopy t-structure). As a subsequent development of this line of thought, we will then present the definition of a spectral sequence in motivic cohomology which looks like the Serre spectral sequence in classical singular cohomology.
Deninger: In joint work with Annette Werner we define functorial isomorphisms of parallel transport along etale paths for a class of vector bundles on a p-adic curve. All bundles of degree zero whose reduction is strongly semistable belong to this class. In particular, they give rise to representations of the algebraic fundamental group of the curve. This may be viewed as a partial analogue of the classical Narasimhan-Seshadri theory of vector bundles on compact Riemann
Esnault: The philosophy of motives, as developed by P. Deligne, predicts a link between the Hodge type of varieties defined over the field of complex numbers and congruences for the number of rational points of varieties defined over finite fields. We show that if a smooth projective variety defined over a p-adic field has its etale cohomology supported in codimension 1 (which according to the Hodge conjecture is equivalent to saying that the Hodge type is at least one), then the mod p reduction of a regular model has one rational point modulo the cardinality of the finite field.
Hanamura: Grothendieck's theory of pure motives studies smooth projective varieties over a field. This has been generalized to (a) the theory triangulated category of mixed motives over a field (by M. Levine, V. Voevodsky and myself) and (b) the theory of pure motives over a base variety. For my approach to (a), see M. Hanamura: Mixed motives and algebraic cycles II, Invent. Math, 2004. For (b) see A. Corti and M. Hanamura: Motivic decomposition and intersection Chow groups I, Duke Math. J. 103. The theory of relative motives has been applied to decomposition problems (Chow-K\"unneth decompositions proposed by Jacob Murre): (1) Hilbert modular varieties (work with B. Gordon and J. Murre), and (2) Lefschetz pencil of a surface.
In this talk I explain how (a) and (b) can be further generalized to the theory of mixed motivies over a base variety.
Oort: Some aspects of the scientific work of Jaap Murre will be presented. The contents of some papers from the period 1957 - 1975 will be discussed.
Tommasi: The moduli space of complex non-singular curves of genus 4 admits a stratification such that each stratum is the geometric quotient of the complement of a discriminant in complex projective space. This allows us to compute the rational cohomology of each stratum. The tools used are Vassiliev-Gorinov's method for the cohomology of the complement of a discriminant, and a theorem of Peters and Steenbrink on the cohomology of geometric quotients.
In this way we can determine the rational cohomology of the whole moduli space, with its mixed Hodge structure.