# University of Toronto's Symplectic Geometry Seminar

September 4, 2007, 3:30pm

BA 6183

## Aravind Asok

## GIT, solvable groups, and $A^1$-homotopy theor

**Abstract: **Geometric invariant theory mainly concerns itself with
formation and study of quotients of varieties by actions of reductive
groups. Recently, there has been much interest in studying quotients
of varieties by the actions of non-reductive groups (e.g., the group
of translations of affine space, automorphism groups of weighted
projective spaces), and in particular their cohomological or homotopic
invariants. In this talk, we will illustrate this study by focusing
on GIT quotients of (open subsets of) affine space by free actions of
solvable groups; familiar examples include all smooth projective toric
varieties.
Morel and Voevodsky's $A^1$-homotopy theory provides a firm foundation
for the study of ``generalized algebraic cohomology theories" on the
category of smooth varieties. In particular, it can be used to study
algebraic K-theory and (higher) Chow groups of such varieties. We
will explain some of the rich interplay between constructions of GIT
and $A^1$-homotopy theory and indicate how extremely refined
$A^1$-homotopic invariants are actually ``computable" in an appropriate
sense. Along the way, we will observe that i) quotients by free
actions of tori are often ``covering spaces" from the standpoint of
$A^1$-homotopy theory, ii) quotients of affine space by free actions of
unipotent groups are often contractible varieties from the standpoint
of $A^1$-homotopy theory. We will explain how these and related
observations/computations can be used to test many classical
conjectures in algebraic geometry.