Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares

Abstract

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 $\kappa$ is a profile for triangulations (resp. quadrangulations) of $S$, for any $m\in {\mathbb{Z}}_{>0}$, denote by $\mathcal{T}(\kappa ,m)$ (resp. $\mathcal{Q}(\kappa ,m)$) the set of (equivalence classes of) triangulations (resp. quadrangulations) with profile $\kappa$ which contain at most $m$ triangles (resp. squares). In this paper, we will show that if $\kappa$ is a profile for triangulations (resp. for quadrangulations) of $S$ such that none of the indices in $\kappa$ is divisible by $6$ (resp. by $4$), then $\mathcal{T}(\kappa ,m)\sim c_3(\kappa )m^{2g+|\kappa |-2}$ (resp. $\mathcal{Q}(\kappa ,m)\sim c_4(\kappa )m^{2g+|\kappa |-2}$), where $c_3(\kappa )\in \mathbb{Q}\cdot (\sqrt{3}\pi {)}^{2g+|\kappa |-2}$ and $c_4(\kappa )\in \mathbb{Q}\cdot {\pi }^{2g+|\kappa |-2}$. 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 $\pi$) of the Masur-Veech volume of arithmetic affine submanifolds of translation surfaces that are transverse to the kernel foliation.

24 pages, to appear in Journal de l'Ecole Polytechnique: Mathématiques