Aravind Asok (Washington)

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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)

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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)

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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.