[Paper Review] Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares

Vincent Koziarz, Duc-Manh Nguyen|arXiv (Cornell University)|Jul 8, 2020

Algebraic Geometry and Number Theory48 references10 citations

TL;DR

This paper establishes asymptotic enumeration formulas for triangulations and quadrangulations of closed oriented surfaces by linking their combinatorics to variation of Hodge structure. Using results on curvature and Chern classes in Hodge theory, it proves that the number of such tilings with a given profile grows as $ m^{2g + | ho| - 2} $, with constants in $ \mathbb{Q} \cdot (\sqrt{3}\pi)^{2g + | ho| - 2} $ for triangulations and $ \mathbb{Q} \cdot \pi^{2g + | ho| - 2} $ for quadrangulations, under non-divisibility conditions on vertex indices.

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 $\\mathscr{T}(\\kappa,m)$ (resp. $\\mathscr{Q}(\\kappa,m)$) 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 $\\mathscr{T}(\\kappa,m)\\sim c_3(\\kappa)m^{2g+|\\kappa|-2}$ (resp. $\\mathscr{Q}(\\kappa,m) \\sim c_4(\\kappa)m^{2g+|\\kappa|-2}$), 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. Koll\\'ar 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.

Motivation & Objective

• To determine the asymptotic growth rate of equivalence classes of triangulations and quadrangulations of a genus $ g $ surface with a fixed profile of vertex indices.
• To establish that the growth constants lie in $ \mathbb{Q} \cdot (\sqrt{3}\pi)^{2g + n - 2} $ for triangulations and $ \mathbb{Q} \cdot \pi^{2g + n - 2} $ for quadrangulations under non-divisibility conditions on indices.
• To connect the enumeration problem to variation of Hodge structure and the geometry of affine invariant submanifolds in moduli spaces of translation surfaces.
• To prove rationality (up to powers of $ \pi $) of the Masur-Veech volume of arithmetic affine submanifolds transverse to the kernel foliation.

Proposed method

• Leverages J. Kollár’s result on curvature of Hodge metrics and Chern classes of vector bundle extensions in variation of Hodge structure.
• Relates the enumeration of tilings to the geometry of $ k $-differentials on Riemann surfaces, particularly in strata $ \Omega^k\mathcal{M}_g(\underline{k}) $.
• Uses the Masur-Veech volume forms induced from Lebesgue measures on cohomology lattices to define volume invariants on moduli spaces.
• Applies a volume comparison argument via the ratio $ \frac{d\mu^*_s}{d\mu} \in \mathbb{Q} $ under transversality to the kernel foliation.
• Applies Theorem 4.1 to show that the volume of the projectivized stratum lies in $ \mathbb{Q} \cdot \pi^{2g + n - 2} $.
• Combines asymptotic counting of tilings with volume asymptotics via scaling in $ m $, leading to the final growth rate and constant structure.

Experimental results

Research questions

• RQ1 What is the asymptotic growth rate of the number of triangulations of a genus $ g $ surface with a fixed profile $ \kappa $ of non-zero vertex indices?
• RQ2 How do the growth constants for such triangulations depend on the genus and the number of singular vertices?
• RQ3 Under what conditions on the vertex indices does the growth constant lie in $ \mathbb{Q} \cdot (\sqrt{3}\pi)^{2g + n - 2} $?
• RQ4 Can the Masur-Veech volume of arithmetic affine submanifolds in translation surface moduli spaces be shown to be rational multiples of $ \pi^{2g + n - 2} $?
• RQ5 What is the role of variation of Hodge structure in connecting combinatorial tiling counts to geometric volumes in moduli spaces?

Key findings

• The number of triangulations $ \mathscr{T}(\kappa, m) $ with profile $ \kappa $ and at most $ m $ triangles grows asymptotically as $ c_3(\kappa) m^{2g + n - 2} $, where $ c_3(\kappa) \in \mathbb{Q} \cdot (\sqrt{3}\pi)^{2g + n - 2} $, provided no index in $ \kappa $ is divisible by 6.
• Similarly, the number of quadrangulations $ \mathscr{Q}(\kappa, m) $ grows as $ c_4(\kappa) m^{2g + n - 2} $, with $ c_4(\kappa) \in \mathbb{Q} \cdot \pi^{2g + n - 2} $, if no index is divisible by 4.
• The asymptotic constants arise from the Masur-Veech volume of a projectivized stratum of $ k $-differentials, which lies in $ \mathbb{Q} \cdot \pi^{2g + n - 2} $.
• The proof relies on the rationality of the ratio of volume forms on affine invariant submanifolds transverse to the kernel foliation, established via Hodge-theoretic curvature bounds.
• The result extends to the rationality (up to $ \pi $) of the Masur-Veech volume of arithmetic affine submanifolds in $ \Omega\mathcal{M}_g(\underline{k}) $, under the same non-divisibility condition.
• The exponent $ 2g + n - 2 $ in the growth rate matches the complex dimension of the corresponding stratum $ \Omega^k\mathcal{M}_g(\underline{k}) $, confirming geometric consistency.