The Action of SL2 on Abelian Varieties

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.