Fourier-stable subrings in the Chow rings of abelian varieties

TL;DR

This paper investigates Fourier-stable subrings within the Chow ring of an abelian variety, demonstrating how Pontryagin products generate such subrings and exploring their algebraic properties and relations.

Contribution

It introduces a method to construct Fourier-stable subrings using Pontryagin products and analyzes the interaction between Pontryagin and usual products in the Chow ring.

Findings

Pontryagin products of classes of dimension ≤ 1 form Fourier-stable subrings.

Constructs finite-dimensional Fourier-stable subrings in the Chow ring.

Shows the usual product acts as a differential operator with respect to the Pontryagin product.

Abstract

We study subrings in the Chow ring $\CH^*(A)_{{\Bbb Q}}$ of an abelian variety $A$, stable under the Fourier transform with respect to an arbitrary polarization. We prove that by taking Pontryagin products of classes of dimension $\leq 1$ one gets such a subring. We also show how to construct finite-dimensional Fourier-stable subrings in $\CH^*(A)_{{\Bbb Q}}$. Another result concerns the relation between the Pontryagin product and the usual product on the $\CH^*(A)_{{\Bbb Q}}$. We prove that the operator of the usual product with a cycle is a differential operator with respect to the Pontryagin product and compute its order in terms of the Beauville's decomposition of $\CH^*(A)_{{\Bbb Q}}$.