University of Toronto's Symplectic Geometry Seminar

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.