Pseudoeffective and nef classes on abelian varieties

Abstract

The cones of divisors and curves defined by various positivity conditions on a smooth projective variety have been the subject of a great deal of work in algebraic geometry, and by now they are quite well understood. However the analogous cones for cycles of higher codimension and dimension have started to come into focus only recently. The purpose of this paper is to explore some of the phenomena that can occur by working out the picture fairly completely in a couple of simple but non-trivial cases. Specifically, we study cycles of arbitrary codimension on the self-product of an elliptic curve with complex multiplication, as well as two dimensional cycles on the product of a very general abelian surface with itself. Already one finds various non-classical behavior, for instance nef cycles that fail to be pseudoeffective: this answers a question raised in 1964 by Grothendieck in correspondence with Mumford. We also propose a substantial number of open problems for further investigation.

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• Algebraic capacities
• Decomposable theta divisors and generic vanishing
• Volume-type functions for numerical cycle classes
• Effective cycles on some linear blowups of projective spaces
• Betti numbers and pseudoeffective cones in 2-Fano varieties
• Pseudo-effective cones of projective bundles and weak Zariski decomposition
• The indeterminacy locus of the Voisin map
• A note about connectedness theorems a la Barth
• Higher Codimension Cycles on the Hilbert Scheme of Three Points on the Projective Plane