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Pseudoeffective and nef classes on abelian varieties

Part of: Cycles and subschemes Abelian varieties and schemes

Published online by Cambridge University Press:  01 June 2011

Olivier Debarre ,
Lawrence Ein ,
Robert Lazarsfeld  and
Claire Voisin
Olivier Debarre
Affiliation:
École Normale Supérieure, Département de Mathématiques et Applications, UMR CNRS 8553, 45 rue d’Ulm, 75230 Paris cedex 05, France (email: odebarre@dma.ens.fr)
Lawrence Ein
Affiliation:
Department of Mathematics, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607, USA (email: ein@math.uic.edu)
Robert Lazarsfeld
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA (email: rlaz@umich.edu)
Claire Voisin
Affiliation:
Institut de Mathématiques de Jussieu, Case 247, 4 Place Jussieu, 75005 Paris, France (email: voisin@math.jussieu.fr)
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Abstract

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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.

Keywords

pseudoeffective and nef cyclesabelian variety

MSC classification

Secondary: 14C25: Algebraic cycles 14K99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 6 , November 2011 , pp. 1793 - 1818
Copyright
Copyright © Foundation Compositio Mathematica 2011

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