Pseudoeffective and nef classes on abelian varieties

TL;DR

This paper investigates the positivity properties of higher codimension cycles on abelian varieties, revealing non-classical behaviors such as nef cycles that are not pseudoeffective, thus advancing understanding in algebraic geometry.

Contribution

It provides a detailed analysis of higher codimension cycle cones on specific abelian varieties, answering a longstanding question and highlighting new phenomena.

Findings

Nef cycles can fail to be pseudoeffective on abelian varieties.

Identifies non-classical behaviors in higher codimension cycle cones.

Addresses a question posed by Grothendieck in 1964.

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.