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On the uniqueness of the Prym map

Carlos A. Serván

TL;DR

This work proves that the Prym map $\mathrm{Prym}: \mathcal{R}_g \to \mathcal{A}_{g-1}$ is the unique nonconstant holomorphic map from the Prym moduli $\mathcal{R}_g$ to any $\mathcal{A}_h$ with $h\le g-1$ for $g\ge 4$, resolving a conjecture of Farb. The authors develop a dual topological and holomorphic strategy: they classify low-dimensional representations of the relevant mapping class groups (via Prym representations and carefully analyzed lifts) to obtain rigidity obstructions, and then apply Farb’s holomorphic rigidity framework, augmented by level-$\psi$ structures and either rigid-curve arguments or variations of Hodge structures, to conclude that any nonconstant map must coincide with the Prym map in the $h=g-1$ case. A central technical tool is the Prym representation $\mathrm{Prym}_*$ arising from the $\mathbb{Z}/2\mathbb{Z}$-cover associated to $\theta$, along with a nonliftability result for $\widehat{\mathrm{Prym}}_*$ and a detailed analysis of the groups $\mathrm{Mod}(S_g,[\beta])$ and $\mathrm{Mod}(S_{2g-1},\sigma)$. The paper thus establishes holomorphic rigidity of Prym-type constructions and clarifies the role of orbifold structures in moduli problems related to abelian varieties and curves.

Abstract

The classical Prym construction associates to a smooth, genus $g$ complex curve $X$ equipped with a nonzero cohomology class $θ\in H^1(X,\mathbb{Z}/2\mathbb{Z})$, a principally polarized abelian variety (PPAV) $\mbox{Prym}(X,θ)$. Denote the moduli space of pairs $(X,θ)$ 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 $π_1^{\mbox{orb}}(\mathcal{R}_g) \to \mbox{Sp}(2h,\mathbb{Z})$ for $h \leq g-1$. This is achieved using arguments from geometric group theory and low-dimensional topology.

On the uniqueness of the Prym map

TL;DR

This work proves that the Prym map is the unique nonconstant holomorphic map from the Prym moduli to any with for , resolving a conjecture of Farb. The authors develop a dual topological and holomorphic strategy: they classify low-dimensional representations of the relevant mapping class groups (via Prym representations and carefully analyzed lifts) to obtain rigidity obstructions, and then apply Farb’s holomorphic rigidity framework, augmented by level- structures and either rigid-curve arguments or variations of Hodge structures, to conclude that any nonconstant map must coincide with the Prym map in the case. A central technical tool is the Prym representation arising from the -cover associated to , along with a nonliftability result for and a detailed analysis of the groups and . The paper thus establishes holomorphic rigidity of Prym-type constructions and clarifies the role of orbifold structures in moduli problems related to abelian varieties and curves.

Abstract

The classical Prym construction associates to a smooth, genus complex curve equipped with a nonzero cohomology class , a principally polarized abelian variety (PPAV) . Denote the moduli space of pairs by , and let be the moduli space of PPAVs of dimension . The Prym construction globalizes to a holomorphic map of complex orbifolds . For and , we show that is the unique nonconstant holomorphic map of complex orbifolds . This solves a conjecture of Farb. A main component in our proof is a classification of homomorphisms for . This is achieved using arguments from geometric group theory and low-dimensional topology.
Paper Structure (19 sections, 22 theorems, 76 equations, 5 figures)

This paper contains 19 sections, 22 theorems, 76 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Orbifold structures on $\mathcal{R}_g$
  3. Topological results
  4. Conjugation in $\mathop{\mathrm{Mod}}\nolimits(S_g,[\beta])$
  5. Complex of curves
  6. Low-dimensional representations of $\mathop{\mathrm{Mod}}\nolimits(S_g,[\beta])$
  7. Lifting relations to $\mathop{\mathrm{Mod}}\nolimits(S_{2g-1},\sigma)$
  8. Low dimensional representations of $\mathop{\mathrm{Mod}}\nolimits(S_{2g-1},\sigma)$
  9. Symplectic representations
  10. Holomorphic results
  11. Farb's proof strategy
  12. Case $h < g-1$
  13. Case $h = g-1$
  14. The Prym map
  15. $F$ homotopic to $\mathop{\mathrm{Prym}}\nolimits$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $g \geq 4$ and let $h \leq g-1$. Let $F:\mathcal{R}_g \to \mathcal{A}_h$ be a nonconstant holomorphic map of complex orbifoldsSee defn:orbi. Then $h = g-1$ and $F = \mathop{\mathrm{Prym}}\nolimits$.

Figures (5)

  • Figure 1: Path surgery.
  • Figure 2: Refining the path in $\mathcal{N}\mathcal{C}_1(S_g)$ to lie on $\mathcal{N}_1(S_g)$.
  • Figure 3: Complementary $k$-chains around $b$.
  • Figure 4: Top: Curve generators for $\mathop{\mathrm{Mod}}\nolimits(S_g,[\beta])$. Bottom: Torelli generators for $\mathop{\mathrm{Mod}}\nolimits(S_g,[\beta])$.
  • Figure 5: 2-chain relation.

Theorems & Definitions (49)

  • Theorem 1.1: Rigidity of $\Prym$
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3: Rigidity of $\mathop{\mathrm{Prym}}\nolimits_*$
  • Corollary 1.4
  • Definition 2.1: Orbifolds and maps between orbifolds
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • ...and 39 more