亚纯微分的平移曲面和周期
Translation surfaces and periods of meromorphic differentials
摘要：
设$S$是亏格$g$和$n$穿刺的定向曲面。$S$上的任何亚纯微分的周期，相对于复结构的选择，确定了一个表示$χ:Γ_{g,n} \to\mathbb C$，其中$Γ_{g,n}$是$S$的第一同调群。我们刻画了这样产生的表象，即存在于周期图$\textsf{Per}:Ω\mathcal{M}_{g,n}\to \textsf{Hom}(Γ_{g,n},\mathbb{C})$的图像中的表象。这推广了Haupt在全纯情形下的一个经典结果。此外，我们确定的形象，这个时期的地图时，限制到任何阶层的亚纯微分，规定的订单的零和极点。我们的证明是几何的，因为他们的目的是构建一个翻译结构的$S$与规定的holonomy $χ$。一路上，我们描述了一个连接与赫尔维茨问题的存在与规定的分支数据的分支覆盖。
英文摘要：
Let $S$ be an oriented surface of genus $g$ and $n$ punctures. The periods of any meromorphic differential on $S$, with respect to a choice of complex structure, determine a representation $χ:Γ_{g,n} \to\mathbb C$ where $Γ_{g,n}$ is the first homology group of $S$. We characterize the representations that thus arise, that is, lie in the image of the period map $\textsf{Per}:Ω\mathcal{M}_{g,n}\to \textsf{Hom}(Γ_{g,n},\mathbb{C})$. This generalizes a classical result of Haupt in the holomorphic case. Moreover, we determine the image of this period map when restricted to any stratum of meromorphic differentials, having prescribed orders of zeros and poles. Our proofs are geometric, as they aim to construct a translation structure on $S$ with the prescribed holonomy $χ$. Along the way, we describe a connection with the Hurwitz problem concerning the existence of branched covers with prescribed branching data.
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