Fourier-stable subrings in the Chow rings of abelian varieties

Details Fourier-stable subrings in the Chow rings of abelian varieties Fourier-stable subrings in the Chow rings of abelian varieties

2007-05-05

arxiv.org/abs/0705.0772v3

Mathematics

Algebraic Geometry

8 pages, in v.2 the order of the differential operator given by the product is computed exactly, v.3 minor typos corrected

Scientific paper

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 $\CH^*(A)_{{\Bbb Q}}$.

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