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Authors
Affiliations
1 Laboratoire J.-A. Dieudonne, UMR 6621 Du CNRS, Universite De Nice, Parc Valrose,
F-06108 Nice Cedex 2, FR
Source
Journal of the Ramanujan Mathematical Society, Vol 25, No 3 (2010), Pagination: 253-263
Abstract
The title is somewhat paradoxical: we know that a linear group can only act trivially on an abelian variety. However we also know that there are not enough morphisms in algebraic geometry, a problem which may be fixed sometimes by considering correspondences between two varieties-that is, algebraic cycles on their product, modulo rational equivalence. Our main result is the construction of a natural morphism of the algebraic group SL2 into the group Corr(A)* of (invertible) self-correspondences of any polarized abelian variety A. As a consequence the group SL2 acts on the Q-vector space CH(A) parametrizing algebraic cycles (with rational coefficients) modulo rational equivalence, in such a way that this space decomposes as the direct sum of irreducible finite-dimensional representations. This gives various results of Lefschetz type for the Chow group.
