Deformations of minimal cohomology classes on abelian varieties

TL;DR

This paper investigates how minimal cohomology classes on abelian varieties deform, showing their connection to curve deformations and providing evidence for a conjecture on the classification of ppavs with such classes.

Contribution

It establishes a correspondence between deformations of Brill--Noether loci and curves, and proves hyperellipticity of Jacobians deforming with minimal cohomology classes.

Findings

Deformations of Brill--Noether loci correspond to deformations of the underlying curve.

Jacobians deforming with minimal cohomology classes are hyperelliptic.

Studied simultaneous deformations of Fano surfaces and intermediate Jacobians.

Abstract

We show that the infinitesimal deformations of the Brill--Noether locus $W_d$ attached to a smooth non-hyperelliptic curve $C$ are in one-to-one correspondence with the deformations of $C$. As an application, we prove that if a Jacobian $J$ deforms together with a minimal cohomology class out the Jacobian locus, then $J$ is hyperelliptic. In particular, this provides an evidence to a conjecture of Debarre on the classification of ppavs carrying a minimal cohomology class. Finally, we also study simultaneous deformations of Fano surfaces of lines and intermediate Jacobians.