论文标题
三角形和正方形
Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares
论文作者
论文摘要
让$ s $为$ g $属的连接封闭式的封闭式表面。给定$ s $的三角剖分（分别四边形），将其每个顶点的索引定义为源自此顶点的边缘数量，$ 6 $（分别为$ 4 $）。调用记录非零索引的整数集（分别四边形）。如果$κ$是$ s $的三角形（分别四边形）的配置文件，则对于任何$ m \ in \ mathbb {z} _ {> 0} $，用$ \ mathscr {t}（t}（κ，m）$（ressscr $ \ $ \ mathscr {q}（q}（q}（κ，m）），（maths $）$ castectian（q）四边形）带有配置文件$κ$，最多包含$ m $ $三角形（分别是平方）。在本文中，我们将表明，如果$κ$是$ s $的三角形（分别为四个）的配置文件，以至于$κ$中的指数都不可除以$ 6 $（分别为$ 4 $），然后是$ \ mathscr {t}（t}（κ，m）（κ，M） $ \ mathscr {q}（κ，m）\ sim c_4（κ）m^{2g+|κ| -2} $），其中$ c_3（κ）\ in \ in \ mathbb {q} \ cdot（q} \ cdot（\ cdot） \ mathbb {q} \cdotπ^{2G+|κ| -2} $。证明的关键要素是J.Kollár在Hogde度量标准的曲率上与代数品种上霍奇结构变化的矢量分组之间的曲率与其扩展类别之间的联系。通过相同的方法，我们还获得了横向到内核叶面的翻译表面的Masur-Deech体积的合理性（超过$π$的某些功率）。
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.
