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

Koziarz, Vincent; Nguyen, Duc-Manh

Journal de l’École polytechnique - Mathématiques, Tome 8 (2021), pp. 831-857.

Soit $S$ une surface connexe fermée orientée de genre $g$. Étant donnée une triangulation (resp. quadrangulation) de $S$, on définit l’indice d’un sommet comme étant le nombre d’arêtes partant de ce sommet moins $6$ (resp. moins $4$). On appelle profil de la triangulation (resp. quadrangulation) l’ensemble des indices non nuls. Si $\kappa$ est le profil de triangulations (resp. quadrangulations) de $S$, pour tout $m\in {\mathbb{Z}}_{>0}$, on note $𝒯(\kappa ,m)$ (resp. $𝒬(\kappa ,m)$) l’ensemble des (classes d’équivalence de) triangulations (resp. quadrangulations) de profil $\kappa$ qui contiennent au plus $m$ triangles (resp. carrés). Dans cet article, nous montrons que si $\kappa$ est un profil de triangulations (resp. quadrangulations) de $S$ tel qu’aucun des indices de $\kappa$ n’est divisible par $6$ (resp. par $4$), alors $𝒯(\kappa ,m)\sim c_3\left(\kappa \right)m^{2g+\left|\kappa \right|-2}$ (resp. $𝒬(\kappa ,m)\sim c_4\left(\kappa \right)m^{2g+\left|\kappa \right|-2}$), où $c_3\left(\kappa \right)\in \mathbb{Q}·{\left(\sqrt{3}\pi \right)}^{2g+\left|\kappa \right|-2}$ et $c_4\left(\kappa \right)\in \mathbb{Q}·{\pi }^{2g+\left|\kappa \right|-2}$. La preuve repose sur un résultat de J. Kollár [24] qui fait le lien entre la courbure de la métrique de Hogde sur les sous-fibrés vectoriels d’une variation de structure de Hodge sur une variété algébrique, et les classes de Chern de leurs extensions. Par la même méthode, nous obtenons également la rationalité (à une puissance de $\pi$ près) du volume de Masur-Veech des sous-variétés affines arithmétiques de surfaces de translation transverses au feuilletage noyau.

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 $𝒯(\kappa ,m)$ (resp. $𝒬(\kappa ,m)$) 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 $𝒯(\kappa ,m)\sim c_3\left(\kappa \right)m^{2g+\left|\kappa \right|-2}$ (resp. $𝒬(\kappa ,m)\sim c_4\left(\kappa \right)m^{2g+\left|\kappa \right|-2}$), where $c_3\left(\kappa \right)\in \mathbb{Q}·{\left(\sqrt{3}\pi \right)}^{2g+\left|\kappa \right|-2}$ and $c_4\left(\kappa \right)\in \mathbb{Q}·{\pi }^{2g+\left|\kappa \right|-2}$. The key ingredient of the proof is a result of J. Kollár [24] on the link between the curvature of the Hodge 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.

Classification : 30F30, 32G15, 52C20
Keywords: Tilings of surfaces, differentials on Riemann surfaces, moduli spaces of flat surfaces, Masur-Veech volume, variation of Hodge structure
Mot clés : Pavages de surfaces, différentielles sur les surfaces de Riemann, espaces de modules de surfaces plates, volume de Masur-Veech, variation de structure de Hodge

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