# Pseudoeffective and nef classes on abelian varieties

@article{Debarre2011PseudoeffectiveAN, title={Pseudoeffective and nef classes on abelian varieties}, author={Olivier Debarre and Lawrence Ein and Robert Lazarsfeld and Claire Voisin}, journal={Compositio Mathematica}, year={2011}, volume={147}, pages={1793 - 1818} }

Abstract 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.

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