Pure and Applied Mathematics Quarterly

Volume 9 (2013)

Number 4

Special Issue: In Memory of Andrey Todorov, Part 1 of 3

Pseudo-Effective Classes and Pushforwards

Pages: 643 – 664

DOI: http://dx.doi.org/10.4310/PAMQ.2013.v9.n4.a3

Authors

Olivier Debarre (Département de Mathématiques et Applications, CNRS UMR 8553, École Normale Supérieure, Paris, France)

Zhi Jiang (Département de Mathématiques, CNRS UMR 8628, Université Paris-Sud, Orsay, France)

Claire Voisin (Centre de math´ematiques Laurent Schwartz, CNRS UMR 7640, École Polytechnique, Palaiseau, France)

Abstract

Given a morphism between complex projective varieties, we make several conjectures on the relations between the set of pseudo-effective (co)homology classes which are annihilated by pushforward and the set of classes of varieties contracted by the morphism. We prove these conjectures for classes of curves or divisors. We also prove that one of these conjectures implies Grothendieck’s generalized Hodge conjecture for varieties with Hodge coniveau at least 1.

Keywords

pseudo-effective classes, Hodge conjecture

2010 Mathematics Subject Classification

14C25, 14C30

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