Hodge结构的变异与曲面的三角形和正方形计数镶嵌
Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares
摘要：
设$S$是亏格$g$的连通闭定向曲面.一个三角形（四边形），将其每个顶点的索引定义为源自该顶点的边的数量减去$6$（分别为减去$4$）。将记录非零索引的整数集合称为三角剖分的轮廓（分别为：四边形）。如果$κ$是三角测量的配置文件（四边形），对于任何$S$，用$\mathscr{T}(κ,m)$表示（分别$\mathscr{Q}(κ,m)$）三角剖分（的等价类）的集合（分别为四边形），其轮廓$κ$包含至多$m$三角形（分别正方形）。在本文中，我们将证明，如果$κ$是一个三角剖分（分别）的轮廓。对于四边形）的$S$，使得$κ$中的索引都不能被$6$整除（相应地，通过$4$），然后$\mathscr{T}(κ,m)\sim c_3(κ)m^{2g+|κ|-2}$（分别$\mathscr{Q}(κ,m) \sim c_4(κ)m^{2g+|κ|-2}$），其中$c_3(κ) \in \mathbb{Q}\cdot(\sqrt{3}π)^{2g+|κ|-2}$和$c_4(κ)\in \mathbb{Q}\cdotπ^{2g+|κ|-2}$。证明的关键成分是J. Kollár关于代数簇上Hodge结构的变体的向量子丛上的Hogde度量的曲率与其扩展的Chern类之间的联系的结果。用同样的方法，我们还得到了平移曲面的算术仿射子流形的Masur-Veech体积的合理性（直到$π$的某个幂）。
英文摘要：
Let $S$ be a connected closed oriented surface of genus $g$. Given a triangulation (resp. quadrangulation) of $S$, define the index of each of its vertices to be the number of edges originating from this vertex minus $6$ (resp. minus $4$). Call the set of integers recording the non-zero indices the profile of the triangulation (resp. quadrangulation). If $κ$ is a profile for triangulations (resp. quadrangulations) of $S$, for any $m\in \mathbb{Z}_{>0}$, denote by $\mathscr{T}(κ,m)$ (resp. $\mathscr{Q}(κ,m)$) the set of (equivalence classes of) triangulations (resp. quadrangulations) with profile $κ$ which contain at most $m$ triangles (resp. squares). In this paper, we will show that if $κ$ is a profile for triangulations (resp. for quadrangulations) of $S$ such that none of the indices in $κ$ is divisible by $6$ (resp. by $4$), then $\mathscr{T}(κ,m)\sim c_3(κ)m^{2g+|κ|-2}$ (resp. $\mathscr{Q}(κ,m) \sim c_4(κ)m^{2g+|κ|-2}$), where $c_3(κ) \in \mathbb{Q}\cdot(\sqrt{3}π)^{2g+|κ|-2}$ and $c_4(κ)\in \mathbb{Q}\cdotπ^{2g+|κ|-2}$. The key ingredient of the proof is a result of J. Kollár on the link between the curvature of the Hogde metric on vector subbundles of a variation of Hodge structure over algebraic varieties, and Chern classes of their extensions. By the same method, we also obtain the rationality (up to some power of $π$) of the Masur-Veech volume of arithmetic affine submanifolds of translation surfaces that are transverse to the kernel foliation.
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