Non-Archimedean Geometry中的Riemann-Hurwitz公式
A Riemann-Hurwitz Formula for Skeleta in Non-Archimedean Geometry
摘要：
设$ϕ: C' \to C$是定义在非阿基米德值代数闭域$k$上的光滑、射影、不可约曲线之间的有限态射。这个态射在曲线的解析之间引入了一个态射。我们构造了$C'^{an}$和$C^{an}$的相容变形收缩对，其像$Υ_{C'^{an}}$和$Υ_{C^{an}}$是$C'^{an}$和$C^{an}$的闭子空间，它们同胚于有限度量图.我们把这样的闭子空间称为“闭子空间”。此外，子空间$C'^{an}$和$C^{an}$使得它们在两个解析中的补空间分解为Berkovich开球和环的不交并.我们可以把这些黑猩猩与一个属联系起来。这对相容的变形收缩迫使态射$ϕ^{an}$限制为映射$Υ_{C'^{an}} \to Υ_{C^{an}}$。我们将研究如何使用态射$ϕ^{an}: Υ_{C'^{an}} \to Υ_{C^{an}}$计算$Υ_{C'^{an}}$的亏格。
英文摘要：
Let $ϕ: C' \to C$ be a finite morphism between smooth, projective, irreducible curves defined over a non-archimedean valued, algebraically closed field $k$. This morphism induces a morphism between the analytifications of the curves. We will construct a compatible pair of deformation retractions of $C'^{an}$ and $C^{an}$ whose images $Υ_{C'^{an}}$ and $Υ_{C^{an}}$ are closed subspaces of $C'^{an}$ and $C^{an}$ which are homeomorphic to finite metric graphs. We refer to such closed subspaces as skeleta. In addition, the subspaces $C'^{an}$ and $C^{an}$ are such that their complements in the two analytifications decompose into the disjoint union of Berkovich open balls and annuli. To these skeleta we can associate a genus. The pair of compatible deformation retractions forces the morphism $ϕ^{an}$ to restrict to a map $Υ_{C'^{an}} \to Υ_{C^{an}}$. We will study how the genus of $Υ_{C'^{an}}$ can be calculated using the morphism $ϕ^{an}: Υ_{C'^{an}} \to Υ_{C^{an}}$.
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