On the uniqueness of the Prym map

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) $\mbox{Prym}(X,θ)$. 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 $g\ge 4$ and $h\le g-1$, 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 $π_1^{\mbox{orb}}(\mathcal{R}_g) \to \mbox{Sp}(2h,\mathbb{Z})$ for $h\le g-1$. This is achieved using arguments from geometric group theory and low-dimensional topology.

arXiv comment26 pages. Corrected typos, incorporated feedback from referees. To appear in JDG