Aravind Asok (Washington)
Cohomology of Quotients Revisited
In her thesis, extending ideas of Atiyah-Bott, Frances Kirwan developed an inductive procedure for studying the ordinary cohomology of certain quotients of smooth projective algebraic varieties constructed by means of geometric invariant theory. Suppose X is a smooth projective variety over an algebraically closed field of characteristic zero equipped with an action of a connected reductive group. One can construct a natural ``instability" stratification for the G-action on X. The ``strata" are smooth locally closed subvarieties and the stratification is equivariantly perfect with rational coefficients; this latter condition means that the rational equivariant cohomology of X decomposes as a direct sum of the rational equivariant cohomology of the strata. It has been known for some time that the stratification can be defined over more general perfect fields k. We will discuss joint work with Brent Doran and Frances Kirwan extending the circle of results around Kirwan's thesis to more general cohomology theories for algebraic varieties including motivic cohomology in the sense of Voevodsky.
Andreas Rosenschon (Alberta)
Algebraic cycles on products of elliptic curves over p-adic fields
We give examples of smooth projective varieties X over p-adic fields such that for suitable l the Chow group in codimension 2 modulo l is infinite.
Sam Payne (Clay Mathematics Institute and Stanford)
Toric vector bundles and the resolution property
Is every coherent sheaf on an algebraic variety the quotient of a locally free sheaf of finite rank? I will discuss an investigation of this question via equivariant vectorbundles on toric varieties, and will give examples of complete (singular, nonprojective) toric threefolds with no nontrivial equivariant vector bundles of rank less than or equal to 3. It is not known whether these varieties have any nontrivial vector bundles at all.