On the uniqueness of the Prym map

TL;DR

This paper proves the uniqueness of the Prym map for genus g curves, showing it is the only nonconstant holomorphic map from the moduli space of such curves to a moduli space of abelian varieties, confirming a conjecture of Farb.

Contribution

The authors establish the uniqueness of the Prym map for g ≥ 4, solving Farb's conjecture by classifying certain homomorphisms using geometric group theory and topology.

Findings

Prym map is unique for g ≥ 4 and h ≤ g-1.

Classification of orbifold fundamental group homomorphisms to symplectic groups.

Resolution of Farb's conjecture on Prym map uniqueness.

Abstract

The classical Prym construction associates to a smooth, genus $g$ complex curve $X$ equipped with a nonzero cohomology class $\theta \in H^1(X,\mathbb{Z}/2\mathbb{Z})$, a principally polarized abelian variety (PPAV) $\mbox{Prym}(X,\theta)$. Denote the moduli space of pairs $(X,\theta)$ by $\mathcal{R}_g$, and let $\mathcal{A}_h$ be the moduli space of PPAVs of dimension $h$. The Prym construction globalizes to a holomorphic map of complex orbifolds $\mbox{Prym}: \mathcal{R}_g \to \mathcal{A}_{g-1}$. For $g\geq 4$ and $h \leq g-1$, we show that $\mbox{Prym}$ is the unique nonconstant holomorphic map of complex orbifolds $F:\mathcal{R}_g \to \mathcal{A}_h$. This solves a conjecture of Farb. A main component in our proof is a classification of homomorphisms $\pi_1^{\mbox{orb}}(\mathcal{R}_g) \to \mbox{Sp}(2h,\mathbb{Z})$ for $h \leq g-1$. This is achieved using arguments from geometric group theory and low-dimensional topology.