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

Vincent Koziarz; Duc-Manh Nguyen

doi:10.5802/jep.159

Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares
[Variation de structure de Hodge et énumération de pavages de surfaces par des triangles et des carrés]

Vincent Koziarz ¹ ; Duc-Manh Nguyen ¹

¹ Université de Bordeaux, IMB, CNRS, UMR 5251 F-33400 Talence, France

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

Résumé
Abstract

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 $κ$ est le profil de triangulations (resp. quadrangulations) de $S$ , pour tout $m \in ℤ_{> 0}$ , on note $𝒯 (κ, m)$ (resp. $𝒬 (κ, m)$ ) l’ensemble des (classes d’équivalence de) triangulations (resp. quadrangulations) de profil $κ$ qui contiennent au plus $m$ triangles (resp. carrés). Dans cet article, nous montrons que si $κ$ est un profil de triangulations (resp. quadrangulations) de $S$ tel qu’aucun des indices de $κ$ n’est divisible par $6$ (resp. par $4$ ), alors $𝒯 (κ, m) \sim c_{3} (κ) m^{2 g + | κ | - 2}$ (resp. $𝒬 (κ, m) \sim c_{4} (κ) m^{2 g + | κ | - 2}$ ), où $c_{3} (κ) \in ℚ \cdot {(\sqrt{3} π)}^{2 g + | κ | - 2}$ et $c_{4} (κ) \in ℚ \cdot π^{2 g + | κ | - 2}$ . 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 $π$ 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 $κ$ is a profile for triangulations (resp. quadrangulations) of $S$ , for any $m \in ℤ_{> 0}$ , denote by $𝒯 (κ, m)$ (resp. $𝒬 (κ, m)$ ) 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 $𝒯 (κ, m) \sim c_{3} (κ) m^{2 g + | κ | - 2}$ (resp. $𝒬 (κ, m) \sim c_{4} (κ) m^{2 g + | κ | - 2}$ ), where $c_{3} (κ) \in ℚ \cdot {(\sqrt{3} π)}^{2 g + | κ | - 2}$ and $c_{4} (κ) \in ℚ \cdot π^{2 g + | κ | - 2}$ . 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 $π$ ) of the Masur-Veech volume of arithmetic affine submanifolds of translation surfaces that are transverse to the kernel foliation.

Reçu le : 2020-11-05
Accepté le : 2021-03-16
Publié le : 2021-03-30

DOI : 10.5802/jep.159

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

Affiliations des auteurs :

Vincent Koziarz ¹ ; Duc-Manh Nguyen ¹

¹ Université de Bordeaux, IMB, CNRS, UMR 5251 F-33400 Talence, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Variation of {Hodge} structure and enumerating tilings of surfaces by triangles~and squares},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Vincent Koziarz; Duc-Manh Nguyen. Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 831-857. doi : 10.5802/jep.159. https://jep.centre-mersenne.org/articles/10.5802/jep.159/

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