Pseudoeffective and nef classes on abelian varieties

Olivier Debarre; Lawrence Ein; Robert Lazarsfeld; Claire Voisin

doi:10.1112/S0010437X11005227

Pseudoeffective and nef classes on abelian varieties

Olivier Debarre
, Lawrence Ein
, Robert Lazarsfeld
, Claire Voisin

Mathematics

Ecole Normale Supérieure-PSL Research University
University of Illinois at Chicago
Institut de Mathématiques de Jussieu-Paris Rive Gauche

Research output: Contribution to journal › Article › peer-review

44 Scopus citations

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.

Original language	English
Pages (from-to)	1793-1818
Number of pages	26
Journal	Compositio Mathematica
Volume	147
Issue number	6
DOIs	https://doi.org/10.1112/S0010437X11005227
State	Published - Nov 2011

Keywords

abelian variety
pseudoeffective and nef cycles

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10.1112/S0010437X11005227

Cite this

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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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AU - Ein, Lawrence

AU - Lazarsfeld, Robert

AU - Voisin, Claire

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N2 - 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.

AB - 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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KW - pseudoeffective and nef cycles

UR - https://www.scopus.com/pages/publications/82955199980

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DO - 10.1112/S0010437X11005227

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Pseudoeffective and nef classes on abelian varieties

Abstract

Keywords

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