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 $κ$ 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.