On the uniqueness of the Prym map
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866918161074356224 |
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| author | Serván, Carlos A. |
| author_facet | Serván, Carlos A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_01704 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the uniqueness of the Prym map Serván, Carlos A. Algebraic Geometry Geometric Topology 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. |
| title | On the uniqueness of the Prym map |
| topic | Algebraic Geometry Geometric Topology |
| url | https://arxiv.org/abs/2207.01704 |