Fourier-stable subrings in the Chow rings of abelian varieties

We study subrings in the Chow ring $\CH^*(A)_{\Q}$ 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 $\le 1$ one gets such a subring. We also show how to construct finite-dimensional Fourier-stable subrings in $\CH^*(A)_{\Q}$. Another result concerns the relation between the Pontryagin product and the usual product on the $\CH^*(A)_{\Q}$. 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 $\CH^*(A)_{\Q}$.

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Published 1 January 2008