Cycles of small codimension on a~simple $2p$- or $4p$-dimensional Abelian variety

S. G. Tankeev

Izvestiya. Mathematics , Tome 63 (1999) no. 6, pp. 1221-1262

Résumé

Let $J$ be a simple $2p$- or $4p$-dimensional Abelian variety over the field of complex numbers, where $p\ne 5$ is a prime number. Assume that one of the following conditions holds: 1) $\operatorname{Cent\,End}^0(J)$ is a totally real field of degree 1, 2 or 4 over $\mathbb Q$; 2) $J$ is a simple $2p$-dimensional Abelian variety of CM-type $(K,\Phi)$ such that $K/\mathbb Q$ is a normal extension; 3) $J$ is a simple $2p$-dimensional Abelian variety such that $\operatorname{End}^0(J)$ is an imaginary quadratic extension of $\mathbb Q$. Then for every positive integer $r$ the $\mathbb Q$-space $H^{2r}(J,\mathbb Q)\cap H^{r,r}$ is spanned by cohomology classes of intersections of divisors.