Minimal Cohomology Classes and Jacobians

TL;DR

This paper characterizes effective cycles with specific cohomology classes on Jacobians, proving their geometric nature and implications for the structure of certain abelian varieties and intermediate Jacobians.

Contribution

It establishes a precise geometric description of effective cycles with given cohomology classes on Jacobians and explores their role in the structure of abelian varieties and cubic threefolds.

Findings

Effective cycles with class $ heta^d/d!$ are translates of $W_{g-d}(C)$ or their negatives.

The Jacobian locus is an irreducible component of abelian varieties with specific effective cycle classes.

On generic intermediate Jacobians of cubic threefolds, certain classes are not effective.

Abstract

We show that on the Jacobian $(JC,\theta)$ of a smooth curve $C$ of genus $g$, any effective cycle in $JC$ with cohomology class $\theta^d/d!$ is a translate of $W_{g-d}(C)$ or $-W_{g-d}(C)$. We then use this result to prove that for $1<d<g$, the Jacobian locus (\resp the locus of intermediate Jacobians of cubic threefolds) is an irreducible component of the set of principally polarized abelian varieties of dimension $g$ for which $\theta^d/d!$ (\resp $\theta^3/3!$) is the class of an effective algebraic cycle. Moreover, on the intermediate Jacobian of a generic cubic threefold, $\theta^2/2!$ is not the class of an effective algebraic cycle.