论文标题
关于Prym地图的独特性
On the uniqueness of the Prym map
论文作者
论文摘要
经典的Prym构造将配备了非零同胞级$ x $ x $与H^1（x，X，\ Mathbb {Z}/2 \ Mathbb {Z}）$相关联，主要是极性的Abelian Variety（PPAV）$ \ mbox {prym}（prym} $，用$ \ Mathcal {r} _g $表示Pairs $（x，θ）$的模量空间，然后让$ \ Mathcal {a} _H $是dimension $ h $的ppavs的Moduli空间。 Prym Construction将复杂的Orbifolds $ \ mbox {prym}的全体形态图全球化：\ Mathcal {r} _g \ to \ Mathcal {a} _ {g-1} $。对于$ g \ geq 4 $和$ h \ leq g-1 $，我们表明$ \ mbox {prym} $是复杂的orbifolds $ f的唯一非稳定的全态图：\ Mathcal {r} _G \ to \ Mathcal {A} _H $。这解决了Farb的猜想。我们证明的主要组成部分是同构的分类$π_1^{\ mbox {orb}}（\ Mathcal {r} _g）\ to \ mbox {sp}（2H，2H，\ Mathbb {Z}）$，for $ h \ leq G-1 $。这是使用几何群体理论和低维拓扑的参数来实现的。
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.
