Fourier-stable subrings in the Chow rings of abelian varieties

DOI10.4310/MRL.2008.V15.N4.A9zbMATH Open1171.14007arXiv0705.0772MaRDI QIDQ2518156

Alexander Polishchuk

Publication date: 15 January 2009

Published in: Mathematical Research Letters (Search for Journal in Brave)

Abstract: We study subrings in the Chow ring $C{H}^{*}(A{)}_{BbbQ}$ 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 $leq1$ one gets such a subring. We also show how to construct finite-dimensional Fourier-stable subrings in $C{H}^{*}(A{)}_{BbbQ}$. Another result concerns the relation between the Pontryagin product and the usual product on the $C{H}^{*}(A{)}_{BbbQ}$. 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 $C{H}^{*}(A{)}_{BbbQ}$.

Full work available at URL: https://arxiv.org/abs/0705.0772

zbMATH Keywords

Fourier transformabelian varietyChow ring

Mathematics Subject Classification ID

Algebraic cycles (14C25) Algebraic theory of abelian varieties (14K05) (Equivariant) Chow groups and rings; motives (14C15) Abelian varieties and schemes (14Kxx)

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