Exploring Equivariant Chow Groups for Complexity One T-Varieties via Downgrading Techniques

Abstract

A T-variety is an algebraic variety X with an effective torus action T. The number c(X) = dim(X)−dim(T) is called complexity of T-variety X. Altmann, Hausen and Süss have described these spaces in terms of pp-divisor and divisorial fans. This description of a T-variety involves a variety of dimension c(X) and some combinatorial data encoded in the form of pp-divisors, i.e., divisors where the coefficients come from the Grothendieck group associated with the semigroup of polyhedra having a common tail cone. In case of complete T-variety with c(X) = 1, Ilten and Süss have described these spaces in terms of marked fancy divisor. Through this combinatorial description of a T-variety, Ustøl Nødland has provided an description of the Chow group of X. In the first part of the thesis we study the computation of the equivariant Chow group of T-variety X with c(X) = 1. This computation involves the following steps. For a complete complexity 1 T-variety X, we compute combinatorial description of X×ETN as a T-variety, where ETN is Nd-dimensional space approximating the contractible space on which T acts freely. Subsequently, through the application of a downgrading technique,we introduce a structure of a T-variety of complexity 1 on the quotient space X×ETN/T . By using combinatorial criterion of completeness of a T-variety, we have proved that if X is complete then the quotient space X×ETN/T is complete. Once it has a complete, complexity 1 T-variety structure, one can use Ustøl’s result to compute Chow group of X×ETN/T . For an affine T-variety X with the action of a torus T, denoted temporarily, by T ↷ X. Assume that T′ is a subtorus of T. Then X is a T-variety with respect to the action of T′, T′ ↷ X. T′ ↷ X is called a downgrading of T ↷ X. The second part provides a combinatorial description of T′ ↷ X, in terms of a T/T′-invariant pp-divisor. We also describe the corresponding GIT fan.