Cycles positifs dans les vari\'et\'es ab\'eliennes

TL;DR

This paper investigates the structure of the algebra generated by Hodge classes on self-products of very general abelian varieties and compares different positivity notions for higher codimension cycles, revealing classes that are nef but not pseudoeffective.

Contribution

It provides a detailed analysis of the algebra of Hodge classes and demonstrates the existence of nef but not pseudoeffective classes in higher codimensions, generalizing previous results.

Findings

The R-algebra generated by Hodge classes has a specific structure.

Existence of nef but not pseudoeffective classes in certain codimensions.

Generalization of earlier results by Debarre, Ein, Lazarsfeld, and Voisin.

Abstract

In the first part, we study the structure of the R-algebra generated by the Hodge classes on the self-product A^e of a very general principally polarized abelian variety A. In the second part, we compare various notions of positivity for cycles of higher codimension in A^e. In particular, we prove that, in every codimension 1 < k < en-1, there exist classes that are numerically effective but not pseudoeffective, which generalises a result of Debarre, Ein, Lazarsfeld and Voisin.