Secant sheaves and Weil classes

Location: 11th Floor Lecture Hall Speaker Eyal Markman, University of Massachusetts Amherst Session Chair Anthony Várilly-Alvarado, Rice University Abstract We will describe a general strategy for proving the algebraicity of the Hodge Weil classes on abelian varieties of Weil type. The latter are even dimensional abelian varieties admitting a suitable embedding of a CM number field in their rational endomorphism ring. We will describe the implementation of the strategy for abelian varieties of dimension 4 and 6, and why it implies the Hodge conjecture for abelian varieties of dimension at most 5. The cohomology of an abelian variety X is naturally the spin representation of Spin(V), where V is the first cohomology of the cartesian product X x Pic^0(X). We will emphasize the central role played by semi-regular coherent sheaves F on an abelian varieties X of dimension equal to half that of the abelian varieties of Weil-type and with a Chern character ch(F) lying on a line secant to the even spinorial variety (semi-regular secant sheaves).=