Rationality of motivic Chow series modulo A^1-homotopy

TL;DR

This paper investigates the rationality of motivic Chow series, a formal power series encoding algebraic cycles, demonstrating rationality in many cases using Picard product formula and torus actions, and comparing with motivic zeta series.

Contribution

It establishes the rationality of motivic Chow series in various cases and includes explicit computations for comparison with motivic zeta series.

Findings

Motivic Chow series are rational in many cases.

Explicit computations of motivic zeta series are provided.

Comparison between motivic Chow series and zeta series is demonstrated.

Abstract

Consider the formal power series $\sum [C_{p, \alpha}(X)]t^{\alpha}$ (called Motivic Chow Series), where $C_p(X)=\disjoint C_{p, \alpha}(X)$ is the Chow variety of $X$ parametrizing the $p$-dimensional effective cycles on $X$ with $C_{p, \alpha}(X)$ its connected components, and $[C_{p, \alpha}(X)]$ its class in $K(ChM)_{A^1}$, the $K$-ring of Chow motives modulo $A^1$ homotopy. Using Picard product formula and Torus action, we will show that the Motivic Chow Series is rational in many cases. We have added the computation of the motivic zeta series in some of our examples so the reader can compare both series in each case.