Pseudoeffective and nef classes on abelian varieties

International audienceAbstract We study the cones of pseudoeffective and nef cycles of higher codimension on the self product of an elliptic curve with complex multiplication, and on the product of a very general abelian surface with itself. In both cases, we find for instance the existence of nef classes that are not pseudoeffective, answering in the negative a question raised by Grothendieck in correspondence with Mumford. We also discuss several problems and questions for further investigation