Minimal Cohomology Classes and Jacobians
Olivier Debarre

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 of a smooth curve of genus , any effective cycle in with cohomology class is a translate of or . We then use this result to prove that for , 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 for which (\resp ) is the class of an effective algebraic cycle. Moreover, on the intermediate Jacobian of a generic cubic threefold, is not the class of an effective algebraic cycle.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Commutative Algebra and Its Applications · Homotopy and Cohomology in Algebraic Topology
