Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 831-857.

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 $m\in {ℤ}_{>0}$, denote by $𝒯\left(\kappa ,m\right)$ (resp. $𝒬\left(\kappa ,m\right)$) 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 $𝒯\left(\kappa ,m\right)\sim {c}_{3}\left(\kappa \right){m}^{2g+|\kappa |-2}$ (resp. $𝒬\left(\kappa ,m\right)\sim {c}_{4}\left(\kappa \right){m}^{2g+|\kappa |-2}$), where ${c}_{3}\left(\kappa \right)\in ℚ·{\left(\sqrt{3}\pi \right)}^{2g+|\kappa |-2}$ and ${c}_{4}\left(\kappa \right)\in ℚ·{\pi }^{2g+|\kappa |-2}$. The key ingredient of the proof is a result of J. Kollár  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.

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 $m\in {ℤ}_{>0}$, on note $𝒯\left(\kappa ,m\right)$ (resp. $𝒬\left(\kappa ,m\right)$) 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 $𝒯\left(\kappa ,m\right)\sim {c}_{3}\left(\kappa \right){m}^{2g+|\kappa |-2}$ (resp. $𝒬\left(\kappa ,m\right)\sim {c}_{4}\left(\kappa \right){m}^{2g+|\kappa |-2}$), où ${c}_{3}\left(\kappa \right)\in ℚ·{\left(\sqrt{3}\pi \right)}^{2g+|\kappa |-2}$ et ${c}_{4}\left(\kappa \right)\in ℚ·{\pi }^{2g+|\kappa |-2}$. La preuve repose sur un résultat de J. Kollár  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.

Accepted:
Published online:
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
Vincent Koziarz 1; Duc-Manh Nguyen 1

1 Université de Bordeaux, IMB, CNRS, UMR 5251 F-33400 Talence, France
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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, Volume 8 (2021), pp. 831-857. doi : 10.5802/jep.159. https://jep.centre-mersenne.org/articles/10.5802/jep.159/

 J. S. Athreya, A. Eskin & A. Zorich - “Right-angled billiards and volumes of moduli spaces of quadratic differentials on $ℂ{\mathrm{P}}^{1}$, Ann. Sci. École Norm. Sup. (4) 49 (2016) no. 6, p. 1311-1386, With an appendix by Jon Chaika | Article | MR: 3592359

 A. Avila, A. Eskin & M. Möller - “Symplectic and isometric $\mathrm{SL}\left(2,ℝ\right)$-invariant subbundles of the Hodge bundle”, J. reine angew. Math. 732 (2017), p. 1-20 | Article | MR: 3717086

 M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky & M. Möller - “Compactification of strata of Abelian differentials”, Duke Math. J. 167 (2018) no. 12, p. 2347-2416 | Article | MR: 3848392 | Zbl: 1403.14058

 M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky & M. Möller - “Strata of $k$-differentials”, Algebraic Geom. 6 (2019) no. 2, p. 196-233 | Article | MR: 3914751

 M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky & M. Möller - “The moduli space of multi-scale differentials”, 2019 | 1910.13492

 Y. Brunebarbe - “Symmetric differentials and variations of Hodge structures”, J. reine angew. Math. 743 (2018), p. 133-161 | Article | MR: 3859271

 D. Chen, M. Möller & A. Sauvaget - “Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials”, 2019, with an appendix by G. Borot, A. Giacchetto & D. Lewanski | 1912.02267

 M. Costantini, M. Möller & J. Zachhuber - “The area is a good enough metric”, 2019 | 1910.14151

 P. Deligne - Équations différentielles à points singuliers réguliers, Lect. Notes in Math., vol. 163, Springer-Verlag, Berlin-New York, 1970 | Zbl: 0244.14004

 P. Deligne - “Théorie de Hodge. II”, Publ. Math. Inst. Hautes Études Sci. (1971) no. 40, p. 5-57 | Article | Numdam

 P. Deligne - “Un théorème de finitude pour la monodromie”, in Discrete groups in geometry and analysis (New Haven, Conn., 1984), Progress in Math., vol. 67, Birkhäuser Boston, Boston, MA, 1987, p. 1-19 | Article | Zbl: 0656.14010

 P. Engel - “Hurwitz theory of elliptic orbifolds, I”, Geom. Topol. 25 (2021) no. 1, p. 229-274 | Article | MR: 4226230

 P. Engel - “Hurwitz theory of elliptic orbifolds, II”, 2018 | 1809.07434

 P. Engel & P. Smillie - “The number of convex tilings of the sphere by triangles, squares, or hexagons”, Geom. Topol. 22 (2018) no. 5, p. 2839-2864 | Article | MR: 3811773

 A. Eskin & M. Mirzakhani - “Invariant and stationary measures for the $\mathrm{SL}\left(2,ℝ\right)$ action on moduli space”, Publ. Math. Inst. Hautes Études Sci. 127 (2018), p. 95-324 | Article | MR: 3814652 | Zbl: 06914160

 A. Eskin, M. Mirzakhani & A. Mohammadi - “Isolation, equidistribution, and orbit closures for the $\mathrm{SL}\left(2,ℝ\right)$ action on moduli space”, Ann. of Math. (2) 182 (2015) no. 2, p. 673-721 | Article | MR: 3418528 | Zbl: 1357.37040

 A. Eskin & A. Okounkov - “Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials”, Invent. Math. 145 (2001) no. 1, p. 59-103 | Article | MR: 1839286 | Zbl: 1019.32014

 A. Eskin & A. Okounkov - “Pillowcases and quasimodular forms”, in Algebraic geometry and number theory, Progress in Math., vol. 253, Birkhäuser Boston, Boston, MA, 2006, p. 1-25 | Article | MR: 2263191 | Zbl: 1136.14039

 H. Esnault & E. Viehweg - Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkhäuser Verlag, Basel, 1992 | Article | MR: 1193913 | Zbl: 0779.14003

 S. Filip - “Splitting mixed Hodge structures over affine invariant manifolds”, Ann. of Math. (2) 183 (2016) no. 2, p. 681-713 | Article | MR: 3450485 | Zbl: 1342.14015

 E. Goujard - “Volumes of strata of moduli spaces of quadratic differentials: getting explicit values”, Ann. Inst. Fourier (Grenoble) 66 (2016) no. 6, p. 2203-2251 | Article | Numdam | MR: 3580171 | Zbl: 1368.30020

 R. Hartshorne - Algebraic geometry, Graduate Texts in Math., vol. 52, Springer-Verlag, New York-Heidelberg, 1977

 Y. Kawamata - “Characterization of abelian varieties”, Compositio Math. 43 (1981) no. 2, p. 253-276 | Numdam | MR: 622451

 J. Kollár - “Subadditivity of the Kodaira dimension: fibers of general type”, in Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, p. 361-398 | Article | MR: 946244

 M. Kontsevich & A. Zorich - “Connected components of the moduli spaces of Abelian differentials with prescribed singularities”, Invent. Math. 153 (2003) no. 3, p. 631-678 | Article | MR: 2000471

 V. Koziarz & D.-M. Nguyen - “Complex hyperbolic volume and intersection of boundary divisors in moduli spaces of pointed genus zero curves”, Ann. Sci. École Norm. Sup. (4) 51 (2018) no. 6, p. 1549-1597 | Article | MR: 3940904 | Zbl: 1422.32020

 H. Masur & S. Tabachnikov - “Rational billiards and flat structures”, in Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, p. 1015-1089 | Article | Zbl: 1057.37034

 C. T. McMullen - “The Gauss-Bonnet theorem for cone manifolds and volumes of moduli spaces”, Amer. J. Math. 139 (2017) no. 1, p. 261-291 | Article | MR: 3619915 | Zbl: 1364.53038

 C. T. McMullen, R. E. Mukamel & A. Wright - “Cubic curves and totally geodesic subvarieties of moduli space”, Ann. of Math. (2) 185 (2017) no. 3, p. 957-990 | Article | MR: 3664815 | Zbl: 06731862

 M. Möller - “Linear manifolds in the moduli space of one-forms”, Duke Math. J. 144 (2008) no. 3, p. 447-487 | Article | MR: 2444303

 D.-M. Nguyen - “Volume form on moduli spaces of $d$-differentials”, 2019 | 1902.04830

 A. Sauvaget - “Volumes and Siegel-Veech constants of $ℋ\left(2G-2\right)$ and Hodge integrals”, Geom. Funct. Anal. 28 (2018) no. 6, p. 1756-1779 | Article | MR: 3881834

 A. Sauvaget - “Volumes of moduli spaces of flat surfaces”, 2018 | 2004.03198

 W. Schmid - “Variation of Hodge structure: the singularities of the period mapping”, Invent. Math. 22 (1973), p. 211-319 | Article | MR: 382272

 J. Smillie & B. Weiss - “Minimal sets for flows on moduli space”, Israel J. Math. 142 (2004), p. 249-260 | Article | MR: 2085718 | Zbl: 1052.37025

 W. P. Thurston - “Shapes of polyhedra and triangulations of the sphere”, in The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, 1998, p. 511-549 | Article | MR: 1668340 | Zbl: 0931.57010

 D. Torres-Teigell - “Masur-Veech volume of the gothic locus”, J. London Math. Soc. (2) 102 (2020) no. 1, p. 405-436 | Article | MR: 4143738

 W. A. Veech - “Moduli spaces of quadratic differentials”, J. Analyse Math. 55 (1990), p. 117-171 | Article | MR: 1094714

 A. Wright - “The field of definition of affine invariant submanifolds of the moduli space of abelian differentials”, Geom. Topol. 18 (2014) no. 3, p. 1323-1341 | Article | MR: 3254934

 A. Zorich - “Flat surfaces”, in Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, p. 437-583 | Article | MR: 2275673

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