On Additive invariants of actions of additive and multiplicative groups

TL;DR

This paper develops a method to compute additive invariants of algebraic varieties with group actions, using fixed point formulas, and applies it to Chow varieties to derive various invariants and properties.

Contribution

It generalizes fixed point formulas to calculate additive invariants of Chow varieties and related structures, providing new explicit formulas and invariants.

Findings

Computed Hodge polynomial of Chow varieties in characteristic zero.

Determined the number of points of Chow varieties over finite fields.

Established vanishing of certain Hodge numbers for Chow and affine group varieties.

Abstract

The additive invariants of an algebraic variety is calculated in terms of those of the fixed point set under the action of additive and multiplicative groups, by using Bialynicki-Birula's fixed point formula for a projective algebraicset with a G_m-action or G_a-action. The method is also generalized to calculate certain additive invariants for Chow varieties. As applications, we obtain the Hodge polynomial of Chow varieties in characteristic zero and the number of points for Chow varieties over finite fields. As applications, we obtain the l-adic Euler-Poincare characteristic for the Chow varieties of certain projective varieties over an algebraically closed field of arbitrary characteristic. Moreover, we show that the virtual Hodge (p,0) and (0,q)-numbers of the Chow varieties and affine group varieties are zero for all p,q positive.