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

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.

The valency e v of a vertex v of Γ is the number of directed edges originating from v. Note that a loop at v counts twice in e v . If Γ is a triangulation, define the index of v to be κ(v) := e v − 6. If Γ is a quadrangulation then the index of v is κ(v) := e v − 4. The vertices with nonzero index are said to be singular. By computing the Euler characteristic of S, one readily finds , if Γ is a triangulation, , if Γ is a quadrangulation.
We will call the sequence of numbers κ(v), v singular vertex the profile of Γ. Let κ = (κ 1 , . . . , κ n ) be a sequence of integers. We say that κ is an admissible profile for triangulations of S if • κ i > −6 and κ i = 0, for all i = 1, . . . , n, • κ 1 + · · · + κ n = 12(g − 1). Similarly, we will say that κ is an admissible profile for quadrangulations of S if • κ i > −4 and κ i = 0, for all i = 1, . . . , n, • κ 1 + · · · + κ n = 8(g − 1). Given an admissible profile κ for triangulations of S, for any m ∈ Z >0 , we denote by T (κ, m) the set of equivalence classes of triangulations of S with profile κ and number of faces at most m. In the same manner, if κ is an admissible profile for quadrangulations of S, we denote by Q(κ, m) the set of equivalence classes of quadrangulations of S with profile κ and number of faces at most m. In this article, we will show: J.É.P. -M., 2021, tome 8 Theorem 1. 1 (i) Let κ = (κ 1 , . . . , κ n ) be an admissible profile for triangulations of S. If (κ 1 , . . . , κ n ) satisfies κ i ∈ 6 · Z for all i = 1, . . . , n, then we have where c 3 (κ) is a constant in Q · ( √ 3π) 2g+n−2 . (ii) Let κ = (κ 1 , . . . , κ n ) be an admissible profile for quadrangulations of S. If (κ 1 , . . . , κ n ) satisfies κ i ∈ 4 · Z for all i = 1, . . . , n, then we have Remark 1. 2 -In [36], Thurston studied triangulations of the sphere where the valency of every vertex is at most 6. He relates the asymptotics of the number of such triangulations with the volume of the moduli space of pointed genus zero curves with respect to some complex hyperbolic metric. Those volumes have been computed by different methods in [28] and [26]. The problem of enumerating tilings of surfaces by triangles and squares has also been addressed in [14,12,13].
-The existence of the limits in (1.2) and (1.3) is a consequence of the main theorem of [31]. In [12], Engel shows that the limits in (1.2) and (1.3), if finite, must belong to the ring K[π], where K is either Q or Q( √ 3) (K = Q for quadrangulations). The main content of Theorem 1.1 is that these constants belong to Qπ 2g+n−2 or to Q( √ 3π) 2g+n−2 in the case κ satisfies the hypothesis of (i) and (ii).

Enumerating square-tiled surfaces in affine invariant submanifolds
Translation surfaces are pairs (X, ω) where X is a compact Riemann surface and ω a non-zero holomorphic 1-form on X. The 1-form ω defines a flat metric with conical singularities at its zeros. A square-tiled surface is a pair (X, ω) where ω is the pullback of the 1-form dz on the standard torus T = C/(Z ⊕ Zı) via a ramified cover f : X → T, which is branched over a unique point.
The space of translation surfaces of genus g 2 is naturally stratified by the orders of the zeros of ω. Given an n-tuple of positive integers k = (k 1 , . . . , k n ) such that k 1 + · · · + k n = 2g − 2, denote by ΩM g (k) the set of translation surfaces (X, ω) such that ω has exactly n zeros with orders given by (k 1 , . . . , k n ). It is well-known that ΩM g (k) is a complex orbifold of dimension 2g + n − 1. For any (X, ω) ∈ ΩM g (k), a neighborhood of (X, ω) can be identified with an open subset of H 1 (X, {zeros of ω}; C) by local charts called period mappings.
There is an action of GL + (2, R) on ΩM g (k) defined as follows: let (z 1 , . . . , z d ) be some local coordinates of ΩM g (k) given by a period mapping, and A a matrix in GL + (2, R). Then the action of A is given by A : (z 1 , . . . , z d ) → (A(z 1 ), . . . , A(z d )), where A acts on C via the standard identification C R 2 . The dynamics of this action of GL + (2, R) has deep connections with various domains such as billiards in rational polygons, interval exchange transformations, Teichmüller dynamics in moduli space (see for instance [27,40]).
The properties of the GL + (2, R) action, in particular the structure of the orbit closures, are the subject of a fast growing literature in the last few decades. It follows from the groundbreaking results of Eskin-Mirzakhani [15] and Eskin-Mirzakhani-Mohammadi [16] that any GL + (2, R)-orbit closure is an immersed suborbifold of ΩM g (k), which is locally defined by linear equations with real coefficients in local charts by period mappings. Such suborbifolds are commonly known as invariant affine submanifolds (or affine submanifolds for short) of ΩM g (k).
An affine submanifold M is said to be arithmetic if it is locally defined by linear equations with coefficients in Q. It is shown in [39] that M is arithmetic if and only if it contains a square-tiled surface. Our second main result concerns the enumerating of square-tiled surfaces in arithmetic affine submanifolds. Before giving the statement of the second theorem, let us recall some relevant features of affine submanifolds. Each stratum of translation surfaces carries naturally two local systems H 1 rel and H 1 whose fibers over (X, ω) are respectively H 1 (X, {zeros of ω}; C) and H 1 (X, C). Let p : H 1 (X, {zeros of ω}; C) → H 1 (X, C) denote the natural projection. Then p gives rise to a morphism p : H 1 rel → H 1 of local systems over ΩM g (k). By definition, the tangent bundle T M of an affine submanifold M is a local subsystem of H 1 rel over M. Moreover, we have a foliation of M, called the kernel foliation, whose tangent space at every point is identified with ker(p). We will say that M is absolutely rigid if the restriction of p to T M is injective. Equivalently, M is absolutely rigid if it is transverse to the kernel foliation of ΩM g (k). Examples of such affine submanifolds include strata of translation surfaces having a single singularity (minimal strata), double covers of quadratic differentials which only have zeros of odd order, and closed orbits generated by square-tiled surfaces (this list is not exhaustive).
The limit on the left hand side of (1.4) is often referred to as the Masur-Veech volume of M. The rationality of this volume has been known for several classes of arithmetic absolutely rigid affine submanifolds. If dim M = 2, then M is the GL + (2, R)-orbit of a square-tiled surface. In this case, it is a well-known fact that the projection of M in the moduli space of Riemann surfaces is a Teichmüller curve (see for instance [35]). Up to a universal constant in Q · π 2 , the Masur-Veech volume of M is equal to the Euler characteristic of this Teichmüller curve. In the case M is a minimal stratum (which consists of Abelian differentials with a single zero), the rationality of the Masur-Veech volume was proved in the work [17] (see also [32] for related formulas). In the case M consists of double covers of quadratic differentials with odd order zeros and simple poles, this rationality was shown in [1] for genus zero case, and in [7] for the general case (see also [18,21]). An unexpected arithmetic absolutely rigid affine submanifold of dimension 4 in genus four was discovered in [29]. Its Masur-Veech volume was computed in [37]. However, to the authors' knowledge, for general arithmetic absolutely rigid affine submanifolds, the rationality of the Masur-Veech volume was not already known.

1.3.
Outline and remarks on the proof of the main theorems. -The proofs of Theorem 1.1 and Theorem 1.3 go as follows: we first relate the asymptotics we are interested in to the Masur-Veech volumes of some moduli spaces of projectivized pluridifferentials on Riemann surfaces. The moduli spaces under consideration belong to a special class of subvarieties, which will be called linear submanifolds, of the projectivizations of strata of Abelian differentials (cf. Definition 2.1). By construction, these moduli spaces carry a tautological line bundle, which comes equipped with a natural Hermitian metric.
Under some appropriate hypothesis, that is, the linear submanifolds are supposed to be polarized and absolutely rigid (cf. Section 2.5), we then show that up to a rational constant, the Masur-Veech volume form is pointwise equal to some power of the curvature form of the natural metric on the tautological line bundle. To show the rationality of the Masur-Veech volumes (up to multiplication by some power of π), instead of constructing specific compactifications for the linear submanifolds, we will make use of the variation of (mixed) Hodge structure over these submanifolds.
To fix ideas, let us denote by M a linear submanifold of a stratum of Abelian differentials, and by PM its projectivization. By definition, there is a variation of Z-mixed Hodge structure over PM. The tautological line bundle is actually a holomorphic line subbundle of the vector bundle associated with the Z-local system of this variation of Hodge structure (VHS). Up to taking some finite cover, and some modification of an arbitrary compactification of PM with normal crossing boundary, one can show that the tautological line bundle extends as a line subbundle of the canonical extension of the VHS. Since the Hermitian metric on the tautological line bundle coincides with the Hodge metric of the VHS, it follows from a result of J. Kollár [24] that any power of the curvature of this metric is a representative in the sense of currents of the corresponding power of the first Chern class of the extended line bundle. Since the Masur-Veech volume of PM is equal to the integral of the maximal power of this curvature form multiplied by a rational number, this enables us to conclude.
A few comments on the strategy of the proof are in order. First, the relation between the Masur-Veech volumes and the asymptotics of the counting problems was known since the work [17]. Second, that the Masur-Veech volume form is proportional to the top power of the curvature of the tautological line bundle (on the associated projectivized moduli spaces) was known to experts in the field. For minimal strata of Abelian differentials and quadratic differentials with odd order zeros, their ratios were computed in [32,7]. However, to the authors' knowledge, for moduli spaces of k-differentials (with k ∈ {2, 3, 4, 6}) this ratio has not been explicitly calculated in the literature. In this paper, we limit ourselves to showing that this ratio is a rational number in these cases (cf. Proposition 5.2 and Proposition 6.6).
Finally, for Theorem 1.1 an alternative method to show that the integral of the top power of the curvature form gives a rational number is to use the compactifications of the corresponding moduli spaces of k-differentials constructed in [5,8]. Indeed, the results in [8] imply that the integral under consideration is equal to the integral of some power of the first Chern class of a line bundle over a compact complex orbifold, thus must be a rational number. In [7], this method was used to calculate the Masur-Veech volumes of some moduli spaces of quadratic differentials. The main novelty of the current paper consists in the use of variation of Hodge structure and Kollár's result, which bypasses the involving construction of the compactifications in [5,8], and provides a uniform treatment for strata of k-differentials (Theorem 1.1) and absolutely rigid linear submanifolds (Theorem 1.3). The drawback is that our approach does not provide an effective way to compute the corresponding Masur-Veech volumes. Nevertheless, one may expect that in some situations, where the construction of the finite cover mentioned above can be carried out explicitly, it is possible to obtain computable formulas for the Masur-Veech volumes by this method.

1.4.
Organization. -The paper is organized as follows: in Section 2 we recall basic properties of strata of Abelian differentials and introduce the notion of linear submanifolds as well as the variation of mixed Hodge structure over these varieties. In Section 3, we give the definition of several natural volume forms on linear submanifolds and the relations between them. In Section 4, we prove the rationality of the volumes of the projectivized linear submanifolds that are polarized and absolutely rigid (cf. Theorem 4.1). The proofs of Theorem 1.3 and of Theorem 1.1 are given in Section 5 and Section 6 respectively.
Acknowledgements. -We are grateful to Yohan Brunebarbe for explaining the ingredients of the proof of Theorem 4.1 to us, and for sharing with us his point of view on the construction of the volume form. We thank the referees for the careful reading and their useful comments.

Moduli spaces of Abelian differentials and linear subvarieties
2.1. Moduli spaces of Abelian differentials. -The moduli space ΩM g of pairs (C, ω) where C is a smooth complex curve of genus g and ω is a non trivial Abelian differential (i.e., a holomorphic 1-form) is the total space of the Hodge bundle over the moduli space M g of smooth curves of genus g, with the zero section removed.
The space ΩM g is an orbifold which is naturally stratified by the multiplicities of zeroes of the corresponding Abelian differentials. For any partition k = (k 1 , . . . , k n ) of 2g −2 by positive integers, the associated stratum is a locally closed subset of ΩM g for the Zariski topology. The strata are always non-empty but not necessarily connected, though each has no more than three connected components [25]. Each stratum is a complex algebraic variety with a complex orbifold structure that will be denoted by ΩM g (k). To lighten the notation, throughout this paper, we use ΩM g (k) to denote a connected component of the corresponding stratum.
If (C, ω) ∈ ΩM g (k), we will use the notation Z(ω) = {x 1 , . . . , x n }, where the point x i ∈ C is a zero of ω of order k i . We have a preferred atlas on C − Z(ω) given by the local primitives of the closed form ω. Two charts in this atlas differ by a translation, so that this atlas defines on C − Z(ω) a flat metric structure with cone singularities at Z(ω). Its area is given by A(C, ω) = ı 2 C ω ∧ ω. In some situations, it is relevant to consider Abelian differentials (C, ω) together with some marked points on C that are not zeros of ω. These marked points can be considered as zeros of order 0 of ω. Let k = (k 1 , . . . , k n ) be a vector of non-negative integers such that k 1 + · · · + k n = 2g − 2. We denote by ΩM g (k) the space of triples (C, ω, Z), where (C, ω) ∈ ΩM g , and Z = {x 1 , . . . , x n } is a finite subset of C such that div(ω) = k 1 x 1 + · · · + k n x n . Note that we do not fix any preferred numbering on the points in Z, the only requirement is that x i is a zero of order k i . Since the elements of ΩM g do not record the location of the marked points, in the case some of the k i 's are 0, ΩM g (k) is not a subvariety of ΩM g . Nevertheless, it is well-known that ΩM g (k) still enjoys the same properties as the strata of ΩM g , in particular, ΩM g (k) is an algebraic variety and has an orbifold structure as a complex analytic space. In what follows we will call ΩM g (k) a stratum of Abelian differentials indifferently whether k has some entries equal to 0 or not.

2.2.
Period coordinates and linear submanifolds. -Since each stratum is an orbifold, a local chart on an open subset U ⊂ ΩM g (k) will have to be understood as a chart over a finite (ramified) covering U of U, where U can be chosen simply connected, endowed with a linear action of a finite group Γ U such that U = U/Γ U . Objects over ΩM g (k) will be defined locally over U and endowed with an action of Γ U . Alternately, one can define ΩM g (k) as a Deligne-Mumford stack, but we will not use this point of view.
We fix a stratum S = ΩM g (k) and let U = U/Γ U be a neighborhood of (C, ω), where U is as above. Then, for all (C , ω ) ∈ U one can canonically identify the relative homology group H 1 (C , Z(ω ); Z) and its dual H 1 (C , Z(ω ); Z) with H 1 (C, Z(ω); Z) and H 1 (C, Z(ω); Z) respectively. Integrating the form ω along relative cycles, we obtain a class in H 1 (C, Z(ω); C) = H 1 (C, Z(ω); C) ∨ . A fundamental result of Veech [38] asserts that the resulting map U → H 1 (C, Z(ω); C) (usually called a period mapping) is a local biholomorphism. In this way, we define a linear structure on ΩM g (k), meaning that we get an atlas with linear changes of coordinates. A choice of a Z-basis of H 1 (C, Z(ω); Z) will provide us with local coordinates around (C, ω) in ΩM g (k) that will be called period coordinates. Note that changes of period coordinates are given by matrices with integral coefficients.
There is a natural C * -action on ΩM g (k) by multiplying the Abelian differential by a scalar. We will denote by PΩM g (k) = ΩM g (k)/C * the projectivization of ΩM g (k).
If (C, ω) is an element of ΩM g (k), its projectivization in PΩM g (k) will be denoted by (C, [ω]). By definition, PΩM g (k) is a locally closed subset of PΩM g , and ΩM g (k) can be interpreted as the total space of the tautological line bundle over PΩM g (k) with the zero section removed.
For our purpose, we will be particularly interested in the following class of subvarieties of ΩM g (k).
is a complex algebraic subvariety such that the local irreducible components of M are defined by linear equations in period coordinates. If M is a linear submanifold of ΩM g (k), by a slight abuse of language, we will call PM := M/C * a linear submanifold of PΩM g (k).
Remark 2.2. -By definition, every local branch of M (considered as a complex analytic space) corresponds to a vector subspace in local charts by period mappings of ΩM g (k). Recall that ΩM g (k) has a structure of a complex orbifold. It follows that the normalization of M (where all the local branches are separated) also has a structure of complex orbifold.
Originally linear submanifolds arose from the study of GL + (2, R)-action on ΩM g (k). It follows from the works [15,16,20] that every GL + (2, R)-orbit closure in ΩM g (k) is a linear submanifold locally defined by linear equations with real coefficients. These are commonly known as invariant affine submanifolds. Other samples of linear subvarieties arise from strata of moduli spaces of pluridifferentials (cf. Section 1.1). In this case, the subvarieties are locally defined by linear equations with coefficients in Q(ζ), where ζ is root of unity. In this paper, we are essentially concerned with these two families of linear submanifolds. Note also that in [30, Def. 6.4], Möller introduced a notion of linear manifold which is similar to ours, but somewhat more restrictive.

2.3.
Numbered zeros and marked points. -Let M g,n denote the moduli space of n-pointed genus g smooth curves. Given a vector k = (k 1 , . . . , k n ) of non-negative integers such that k 1 +· · ·+k n = 2g−2, we denote by M g,n (k) the set of (C, x 1 , . . . , x n ) ∈ M g,n such that k 1 x 1 + · · · + k n x n is the zero divisor of a holomorphic 1-form on C. The locus M g,n (k) is a subvariety of M g,n . There is a natural algebraic line bundle over M g,n (k) defined as follows: let p : C g,n → M g,n be the universal curve over M g,n , and σ 1 , . . . , σ n be the sections of p associated with the marked points.
Let K Cg,n/Mg,n be the relative canonical line bundle associated with p, and K := K Cg,n/Mg,n ⊗ L ⊗k1 By definition, the restriction of K to the fiber of p over every point in M g,n (k) is the trivial line bundle. Hence L := p * (K |p −1 (Mg,n(k)) ) is a line bundle over M g,n (k) which will be called the tautological line bundle. The complement of the zero section in the total space of L is the set of tuples (C, x 1 , . . . , x n , ω), where (C, x 1 , . . . , x n ) ∈ M g,n (k), and ω is a holomorphic 1-form on C such that div(ω) = k 1 x 1 + · · · + k n x n . We denote this set by ΩM g,n (k). By construction ΩM g,n (k) is a C * -bundle over M g,n (k).
There is a natural map F : ΩM g,n (k) → ΩM g (k) which consists in forgetting the numbering of the marked points. This is actually a finite morphism of algebraic varieties, which is also an orbifold covering between the underlying complex analytic spaces. Given a point x = (C, ω) ∈ ΩM g (k), we will often implicitly endow the set Z(ω) with a compatible numbering, which means that we actually consider a point in ΩM g,n (k) that projects to x. That F is an orbifold covering implies that this choice of numbering can be made consistently in an orbifold local chart.
By a linear submanifold of ΩM g,n (k) we will mean a subvariety having the property described in Definition 2.1. By extension, we will also call any algebraic variety which admits a finite morphism into ΩM g,n (k) whose image has the property of Definition 2.1 a linear submanifold of ΩM g,n (k).

2.4.
Variation of Hodge structure associated with a stratum. -Let us now fix a stratum S = ΩM g (k). As we mentioned earlier, each stratum S is an orbifold, as well as PS. Actually, there exists a manifold P S which is an orbifold covering of PS. To see this, we first recall that the forgetful map F : ΩM g,n (k) → S is an orbifold covering of finite degree. This map induces an orbifold covering F : PΩM g,n (k) → PS. Since the complex line generated by a (non-trivial) Abelian differential is uniquely determined by its divisor, we can identify PΩM g,n (k) with M g,n (k). This means that M g,n (k) is an orbifold covering of PS.
By definition M g,n = T g,n /Γ g,n , where T g,n is the Teichmüller space of smooth curves of genus g with n marked points, and Γ g,n is the corresponding modular group. Orbifold points of M g,n (k) correspond to fixed points of finite subgroups of Γ g,n . It is well-known that Γ g,n contains torsion free finite index subgroups. The preimage of M g,n (k) in some orbifold covering of M g,n associated with such subgroups is actually a complex manifold. In conclusion, we see that PS admits an orbifold covering P S of finite degree which is a smooth quasi-projective variety, and over which we have a universal family of n-pointed genus g smooth curves. Passing to a larger covering, one can even assume that there exists a finite group Γ acting holomorphically on P S such that PS = P S/Γ. Objects over PS are defined over P S and endowed with an action of Γ.
Let L be the pullback to P S of the tautological line bundle over PS. Denote by S the total space of L with the zero section removed. By construction P S = S/C * . We will denote by π : S → P S the natural projection.
Over P S, the relative homology groups H 1 (C, Z(ω); Z) assemble in a Z-local system. We denote by H 1 rel the dual Z-local system, whose fiber is identified with H 1 (C, Z(ω); Z). Note that the integration of the form ω along relative cycles as defined in Section 2.2 can be seen as a holomorphic section τ over S of the holomorphic vector bundle π −1 H 1 rel ⊗ Z O S . Since S and P S are locally identified with S and PS respectively, we will often identify elements of S (resp. of P S) with elements of S (resp. S). For any (C, [ω]) in P S, denoting by Z = Z(ω) the zeroes of ω, the relative cohomology group H 1 (C, Z; Z) fits in an exact sequence On H 0 (C, C) and H 0 (Z, C) we have canonical (positive) polarizations defined over Z.
We endow H 0 (Z, C) with the quotient polarization, which will be denoted by h 0 . As for H 1 (C, C), it is endowed with the Hermitian pseudo-metric h 1 of signature (g, g) defined by where (a 1 , . . . , a g , b 1 , . . . , b g ) is any symplectic basis of H 1 (C, Z). Observe that the corresponding skew-symmetric form is defined over Z.
Denoting by F the complex vector subspace H 0 (C, Ω 1 ) of H 1 (C, C) formed by cohomology classes of holomorphic 1-forms, the Hodge filtration is the decreasing If we let (C, [ω]) vary in P S, then W := H 0 (Z, Z) and H 1 (C, Z) assemble in Z-local systems W and H 1 that fit in an exact sequence of Z-local systems Remark that with our assumption on P S, W is actually constant. The Hermitian forms h 0 and h 1 induce a flat (constant) Hermitian metric h 0 on W C and a flat Hermitian pseudo-metric h 1 on H 1 C respectively. Correspondingly, the Z-local system H 1 (resp. W) supports a variation of polarized Z-pure Hodge structure of weight 1 (resp. of weight 0 associated with the trivial filtration).
The group H 1 (C, Z; Z) is endowed with a graded-polarized Z-mixed Hodge structure, and the exact sequence (2.1) expresses it as an extension of pure polarized Z-Hodge structures. The weight filtration W • of H 1 (C, Z; Z) is defined by and the Hodge filtration of H 1 (C, Z; C) is the pullback of the Hodge filtration on H 1 (C, C). This defines on H 1 rel a variation of graded-polarized Z-mixed Hodge structure.

2.5.
Linear submanifolds revisited. -Let PM ⊂ PS be a linear submanifold (cf. Definition 2.1). The preimage P M of PM in P S is also a linear submanifold of P S. The normalization of P M is then a smooth algebraic variety equipped with an immersive finite generically one-to-one morphism into P S. We abusively denote this smooth variety by P M. Define M to be the pullback of the tautological C * -bundle over P S The trivial bundles with fibers V 0 , V, V 1 over open subsets as above patch together to form three local systems over P M, which will be denoted by V 0 , V, V 1 respectively. Note that V 0 , V, V 1 are sub-local systems of W C , (H 1 rel ) C , H 1 C respectively. We have the following exact sequence By Deligne [10], the Q-local system (H 1 Q ) |P M is semi-simple and hence: We still denote by h 0 resp. h 1 their restriction to V 0 resp. V 1 . As they are flat, (det h 0 ⊗ |det h 1 |) ∨ defines a flat Hermitian form (1) on the canonical bundle K M pr −1 (det V) ∨ . However, if h 1 is degenerate (in restriction to V 1 ), this Hermitian form vanishes, something we would like to avoid. This leads us to the following: is a linear submanifold such that V 1 is a subvariation of Hodge structure of (H 1 C ) |P M . In particular, the restriction of h 1 to V 1 is non degenerate.
Note that this condition is actually automatically satisfied if we assume that no other local subsystem of (H 1 C ) |P M is isomorphic to V 1 . Indeed in that case, V 1 inherits from (H 1 C ) |P M a structure of complex variation of Hodge structures, see [11, §1.12-13]. Concretely, this means that the following decomposition holds Remark 2.6. -Definition 2.5 is independent of the choice of M.
(1) For any finite dimensional C-vector space W endowed with a non degenerate Hermitian form h, we define the Hermitian form det h on det W = Λ dim C W W by ((det h)(w 1 ∧ · · · ∧ wr)) 2 := det(h(w i , w j )) 1 i,j r , if (w 1 , . . . , wr) is any basis of W .  Let us now give a description of µ in more concrete terms. Let (C, ω) be a point in M, and (C, [ω]) be its projection in P M. By some period mapping φ, a neighborhood of (C, ω) in M is identified with an open subset of a linear subspace V ⊂ H 1 (C, Z(ω); C), which is the fiber of V over (C, [ω]). Note that if v = φ(C, ω) then Thus the image of φ is contained in the cone Let P + (V ) := C + /C * ⊂ P(V ) and pr : C + → P + (V ) be the natural projection.
1}. For any subset A ⊂ P + (V ), let C 1 (A) := pr −1 (A) ∩ C + 1 . Since the period mapping φ is equivariant with respect to the C * -actions on M and V , it induces a biholomorphism φ from a neighborhood U of (C, [ω]) in P M onto an open subset Ω of P + (V ). Moreover, φ induces an isomorphism of Hermitian line bundles L |U L P + (V )|Ω , where L P + (V ) is the tautological line bundle over P + (V ) endowed with the Hermitian metric h 1 . In this setting, the restriction of the measure µ to U is given as follows: for any Borel subset B ⊂ U , we have µ(B) = vol(C 1 ( φ(B))).
From this description, it is not difficult to see that µ is actually induced by a volume form dµ on P M (see Lemma 3.1 below).

3.2.
Alternative definition of dµ. -Let us describe the construction of dµ from another point of view. We denote by G ⊂ GL(V ) the subgroup consisting in automorphisms that act as the identity on V 0 and whose induced automorphism on V 1 preserves h 1 . The group G fits in an exact sequence One easily checks that P + (V ) is an orbit for the action of G on P(V ). Since M is locally modeled on (G, C + ), objects on M and P M can be defined as G-invariant objects on C + and P + (V ).
A volume form on a complex manifold X can be viewed as a section of the bundle K X ⊗ K X where K X is the canonical bundle of X. A Hermitian metric on the canonical bundle of X induces a metric | . | on volume forms. A complex manifold being orientable, it always admits a non vanishing global volume form dV . We will say that the volume form dV /|dV | is the volume form associated with the metric.
Since P + (V ) is an homogeneous manifold for the action of G, its tangent bundle, and hence its canonical bundle, is naturally a G-equivariant bundle. This is also the case for the restriction to P + (V ) of the tautological line bundle L over P(V ). Note also that h 1 induces on L a G-invariant Hermitian metric that we still denote by h 1 . On P + (V ) we have the Euler exact sequence of bundles [22,Th. II.8.13]). It follows that the canonical line bundle Specifically, let (e 1 , . . . , e r ) be an orthonormal basis of V 0 with respect to h 0 , and (e r+1 , . . . , e d ) an orthonormal basis of V 1 with respect to h 1 . For i = r + 1, . . . , d, let e i be a vector in V that projects to e i . Then (e 1 , . . . , e d ) is a basis of V , and e 1 ∧ · · · ∧ e d ∈ det V has norm 1 with respect to det h 0 ⊗ |det h 1 |. Let (z 1 , . . . , z d ) be the coordinates of V in the basis (e 1 , . . . , e d ). Then the associated volume form on V is Let v be a vector in C + (V ). Since h 1 (v, v) > 0, there is some i ∈ {r + 1, . . . , d} such that z i (v) = 0. We can assume that z d (v) = 0. In a neighborhood of [v] in P + (V ), we have the local coordinates w := (w 1 , . . . , w d−1 ) = (z 1 /z d , . . . , z d−1 /z d ). Using the isomorphism provided by the Euler exact sequence, the volume form associated with where h(w) := h 1 w 1 , . . . , w d−1 , 1 . Recall that in this setting the measure µ is the push forward of the volume form dvol on C + 1 by the natural projection pr |C + 1 : It follows that which implies that µ is induced by the volume form 3.3. Pure case.
-We now turn to the case where the term V 0 in (2.2) is trivial.
Proof. -Recall that for polarized absolutely rigid submanifolds, the map p : V → V 1 in (2.2) is an isomorphism. Thus h 1 is a non-degenerate Hermitian form on V , which means that d = p + q.
Proof. -We can identify V with C p+q endowed with the Hermitian form where z = (z 1 , . . . , z p+q ). In these coordinates, we have Observe that U(p, q) U(h 1 ) preserves µ and the curvature form Θ (since U(p, q) preserves the Hermitian metric h 1 on L). It follows that dµ/(ıΘ) p+q−1 is a function on P + (V ) invariant under the action of U(p, q). Since U(p, q) acts transitively on P + (V ), dµ/(ıΘ) p+q−1 is actually constant. To evaluate this constant, it suffices to compute the ratio dµ/(ıΘ) p+q−1 at the point [1 : 0 : · · · : 0] ∈ P + (V ).
By the computations in Lemma 3.1, we have To compute Θ, we will use the following section of L over B σ : Set h(w) := h 1 (σ(w), σ(w)), we then have Thus Remark 3.4. -We remind the reader that our volume form dvol is defined only by using h 1 . In fact, if d ν is any volume form on V that is proportional to the Lebesgue measure, then the push forward measure on P + (V ), denoted by dν, is proportional to Θ p+q−1 by the invariance under the action of U(h 1 ) = U(p, q). In the cases where d ν is the Masur-Veech volume form on minimal strata of Abelian differentials H(2g − 2), or on strata of quadratic differentials with only zeros of odd order, the ratio dν/(ıΘ) p+q−1 has been computed in [32] and [7]. See also [33,Lem. 5.1] for a related calculation.

Computation of the volume in terms of characteristic classes
Our goal now is to show where Θ is the curvature of the Hodge metric . on L. If there is a compact complex manifold X together with a normal crossing divisor ∂X such that -P M X ∂X, -L extends to a holomorphic line bundle L on X, and Z). Unfortunately, the existence of such a compact manifold has not been proved in general. Nevertheless, we have: Proof of the claim. -Recall that over P M, we have a VHS {H 1 , F }, where F is the holomorphic subbundle of H 1 C = H 1 ⊗ Z C whose fiber over a point (C, ω) is H 1,0 (C). By definition, the tautological line bundle L is a subbundle of F . Moreover the Hermitian metric . on L is precisely the restriction of the Hodge metric of H 1 to L.
Let X be a smooth compactification of P M with normal crossing boundary divisor ∂X. It is a well-known fact that the monodromy of H 1 about each component of ∂X is quasi-unipotent. Thus there is a finite covering q : Y → X ramified over ∂X such that the pullback of H 1 to Y := Y ∂Y has unipotent monodromies about ∂Y := q −1 (∂X) (see [23,Th. 17] and [6,Lem. 3.5]). It follows from the work of Deligne [9] and Schmid [34] that the pullback of the filtration {H 1 C , F } to Y extends canonically to a filtration {H 1 C , F } of holomorphic vector bundles over Y . However, the line bundle L does not necessarily extends to Y . To fix this issue, we construct a modification of Y as follows: the pullback of L to Y provides us with a section σ of the projective bundle  By the claim, let N be the degree of q, we have Since P M is a finite cover of PM, the theorem follows. There is a surjective birational morphism from the space Y in Claim 4.2 onto PM. A natural question one may ask is whether L is isomorphic to the pullback of O(−1) to Y . If this is true, it would simplify the computation of the integral Y c d−1 1 ( L), which is equal to the self-intersection number of the divisor associated to L. It seems to us that this should be the case, but we do not have a proof of this fact.

Proof of Theorem 1.3
Let M now be an arithmetic affine submanifold of dimension d in some stratum ΩM g (k). By definition, M is locally defined by linear equations with rational coefficients in period coordinates. By a result of [2], M is always polarized.
Let (C, ω) be a surface in M. As usual, denote by Z(ω) the zero set of ω. We identify H 1 (C, Z(ω); C) with C 2g+n−1 using a basis of H 1 (C, Z(ω); Z). The image of a neighborhood of (C, ω) in M is an open subset of a linear subspace V ⊂ C 2g+n−1 which is defined over Q. It follows that Λ Z V := V ∩ (Z ⊕ ıZ) 2g+n−1 is a lattice in V . Note that the square-tiled surfaces in M and close to (C, ω) are mapped to points in Λ Z V . There is a unique volume form on V which is proportional to the Lebesgue measure such that the covolume of Λ Z V is equal to one. This volume form gives rise to a welldefined volume form on M, which will be called the Masur-Veech volume form and denoted by dvol * . The volume form dvol * is important to us because of the following folkloric lemma (see [17,Prop. 1.6]).
We can identify U i with a subset of a linear subspace V ⊂ C 2g+n−1 as above. Let C(U i ) be the infinite cone R * + · U i ⊂ V . Let ST(U i , m) be the set of square-tiled surfaces of area at most m whose image is contained in C(U i ). For all s > 0, denote by Let ∆ be a fundamental domain for the action of Λ Z V in V . We can suppose that ∆ contains 0 in its interior. Set By definition, vol * (∆) = 1, thus vol * (W (U i , m)) = #ST(U i , m). We now remark that . Summing up over the family {U i | i ∈ N} we get the desired conclusion.
The Masur-Veech volume form on M is not necessarily equal to the volume form dvol defined in Section 3.1. However, we have: Proof. -By construction, dvol * /dvol is locally constant. Since M is irreducible, it follows that dvol * = α dvol where α is constant on M. It is enough to show that α is rational in some local chart of M.
Let (C, ω) be an element of M. We fix a basis of H 1 (X, Z(ω), Z) and identify H 1 (X, Z(ω); C) with C 2g+n−1 using this basis. Let z = (z 1 , . . . , z 2g+n−1 ) be the coordinates of C 2g+n−1 . Up to a renumbering of these coordinates, the Hermitian form h 1 is given by Since dim C V = d, there is a sequence I = (i 1 , . . . , i d ) such that the projection φ I restricts to an isomorphism from V onto C d . Let w = (w 1 , . . . , w d ) be the coordinates of C d . The inverse of the map φ I|V : V → C d is an injective linear map ψ : C d → C 2g+n−1 , such that ψ(C d ) = V , and φ I • ψ = id C d . Since V is defined over Q, the matrix of ψ in the canonical bases of C d and C 2g+n−1 has rational entries. This means that if ψ(w 1 , . . . , w d ) = (z 1 , . . . , z 2g+n−1 ) then z j is a linear function of w with rational coefficients. As a consequence, the pullback of the Hermitian form h 1 to C d is given by where γ ij ∈ Q, and γ ij = −γ ji . By definition, we have Observe that the covolume of (Z ⊕ ıZ) d is 1 with respect to the volume form (ı/2) d dw 1 dw 1 . . . dw d dw d . Thus the covolume of φ I (Λ Z V ) is r with respect to this volume form. Now, by definition, Λ Z V has covolume 1 with respect to the Masur-Veech volume form dvol * . Thus φ I (Λ Z V ) has covolume form 1 with respect to ψ * dvol * . This means that Since det(γ ij ) ∈ Q, we have ψ * dvol * /ψ * dvol ∈ Q, and the proposition follows. where α is a rational constant by Proposition 5.2. By assumption, M is absolutely rigid. It follows from a result of [2] that M is polarized. Thus Theorem 4.1 allows us to conclude.

Counting triangulations and quadrangulations
6.1. Flat metrics and k-differentials. -Let us now fix a finite subset Σ = {s 1 , . . . , s n } ⊂ S. Consider a triangulation Γ of S with profile κ. We can always assume that Σ is the set of singular vertices of Γ. There is a diffeomorphism from each face of Γ onto the equilateral triangle = (0, 1, e ıπ/3 ) mapping the edges of Γ to the sides of . We endow each face of Γ with the Euclidean metric on ∆ via such a map. This metric extends smoothly across the vertices whose valency is equal to 6. At each vertex whose valency is not 6 (that is, a singular vertex), we get a conical singularity with cone angle πe/3, where e is the valency of the vertex. Let S := S Σ. The linear holonomy of the flat metric provides us with a homomorphism ρ : π 1 (S , * ) → SO (2). The image of ρ is contained in Thus there exists k ∈ {1, 2, 3, 6} such that Im(ρ) = U k , where U k is the group of k-th roots of unity. This means that the flat metric on S is induced by a meromorphic k-differential ξ. Note that ξ is only determined up to a constant in S 1 . The set of zeros and poles of ξ is precisely the set of singularities of the flat metric, that is, Σ. Moreover, the orders of zeros and poles of ξ are completely determined by the profile of Γ. Namely, the order of ξ at a singular vertex v is given by k(e v − 6)/6. In particular, v is a pole of ξ if and only if e v < 6, and v is a zero of ξ if and only if e v > 6.
Similarly, if Γ is a quadrangulation, then by endowing each face of Γ with the Euclidean metric on the unit square = (0, 1, 1 + ı, ı), we also get a flat metric on S with conical singularities at Σ. This metric is defined by a meromorphic k-differential ξ on S determined up to constant in S 1 , where k ∈ {1, 2, 4}. The set of zeros and poles of ξ is equal to Σ, and the order of ξ at a vertex v ∈ Σ is given by k(e v − 4)/4.

6.2.
Canonical covering. -Let ξ be a meromorphic k-differential on a compact Riemann surface X. In what follows, we will always assume that ξ is not a power of some k -differential with k < k, and that the poles of ξ have order at most k − 1. It follows from a classical construction (see for instance [19,4,31]) that there is a cyclic covering : X → X of degree k, a holomorphic 1-form ω on X, and an automorphism τ of X of order k such that (a) * ξ = ω k , (b) τ * ω = ζ ω, where ζ is a primitive k-th root of unity, (c) X/ τ X.
Let Z denote the set of zeros and poles of ξ, and Z = −1 (Z). Then, by construction, Z contains the zero set of ω and all the branched points of . Moreover, the genus g of X, the cardinality n of Z, and the action of τ on Z are completely determined by the orders of the zeros and poles of ξ.
The following proposition characterizes the set of k-differentials arising from triangulations (resp. quadrangulations) on S. Moreover, the triangulation is uniquely determined by the triple ( X, ω, τ ).
(ii) The k-differential ξ arises from a quadrangulation of S if and only if up to multiplication by some constant in S 1 , ω satisfies Moreover, the quadrangulation is uniquely determined by the triple ( X, ω, τ ).
Proof. -Let us first suppose that the triple ( X, ω, τ ) is obtained from a triangulation on S. By construction, we have a triangulation of X by equilateral triangles of unit side that is the pullback of the triangulation on S. Since any cycle c in H 1 ( X, Z; Z) can be represented by a path composed by some edges of this triangulation, up to rotation, we have c ω ∈ Z ⊕ Ze ıπ/3 . Assume now that ( X, ω) satisfies (6.1). Fix a point x 0 ∈ Z, and define a map ϕ : X → C/(Z ⊕ Ze ıπ/3 ) as follows where the integral is taken along any path from x 0 to x. It is straightforward to check that ϕ is a ramified covering with all the branched points contained in Z, and satisfies ϕ( Z) = {0}. The torus C/(Z ⊕ Ze ıπ/3 ) admits a triangulation Γ 0 composed by 2 equilateral triangles with 0 being the unique vertex. The pullback of Γ 0 by ϕ gives a triangulation of X. Since τ * ω = ζ ω, where ζ ∈ U 6 , it follows that there exists an automorphism j of the torus C/(Z⊕Ze ıπ/3 ) which fixes 0 and satisfies ϕ•τ = j •ϕ. Note that j preserves the triangulation Γ 0 . Therefore, τ preserves the triangulation of ( X, ω), which means that this triangulation descends to a triangulation of X/ τ S. It is clear that each triple ( X, ω, τ ) provides us with a unique triangulation of S up to homeomorphisms.
The case where ( X, ω, τ ) arises from a quadrangulation of S follows from similar arguments.
Let Ω k M g (k) denote the space of pairs (X, ξ) where X is a Riemann surface of genus g, and ξ is a meromorphic k-differential on X whose zeros and poles have orders prescribed by k. Since every k-differential is uniquely determined by its canonical covering, we can identify Ω k M g (k) with the space of triples ( X, ω, τ ) satisfying the conditions (a), (b), (c) in Section 6.2. Denote by PΩ k M g (k) the projectivization of Ω k M g (k), that is, PΩ k M g (k) = Ω k M g (k)/C * .
Given (X, ξ) ∈ Ω k M g (k), denote by Z the set of zeros and poles of ξ. Let Z be the inverse image of Z in X. By construction, all the zeros of ω are contained in Z. However, some of the points in Z may not be zeros of ω. We will consider these points as zeros of order 0 of ω, and subsequently Z as the zero set of ω. Let k := ( k 1 , . . . , k n ) be the sequence of non-negative integers recording the orders of the zeros in Z. Recall that ΩM g ( k) is the stratum of Abelian differentials consisting of triples ( X, ω, Z), where X is a Riemann surfaces of genus g, ω an Abelian differential on X, and Z = { x 1 , . . . , x n } is a finite subset of X such that div( ω) = k 1 x 1 + · · · + k n x n .
is a finite cover of a polarized linear submanifold of ΩM g ( k). Moreover, if k i ∈ k · Z, for all i = 1, . . . , n, then this linear submanifold is absolutely rigid.
Proof. -Let ( X, ω, τ ) be an element of Ω k M g (k), and Z be the zero set of ω. Let ψ 1 : Ω k M g (k) → ΩM g ( k) be the map which sends the triple ( X, ω, τ ) to the triple ( X, ω, Z), that is, we forget about the automorphism τ . This map is actually a finite morphism of algebraic varieties. In particular ψ 1 (Ω k M g (k)) is an algebraic subvariety of ΩM g ( k). Recall that a neighborhood of ( X, ω, Z) in ΩM g ( k) can be identified with an open subset of H 1 ( X, Z, C). There is a neighborhood U of ( X, ω, τ ) in Ω k M g (k) such that ψ 1 (U) is an open subset of V ζ ⊂ H 1 ( X, Z, C), where V ζ is the eigenspace associated with the eigenvalue ζ of the action of τ on H 1 ( X, Z, C) (see for instance [4,31]). Thus ψ 1 (Ω k M g (k)) is a linear submanifold of ΩM g ( k).
Thus if k i ∈ k · Z for all i = 1, . . . , n, then p(V ζ ) is absolutely rigid.
Proof. -We only give the proof for the case x, x ∈ Ω k M g (k). Assume that x α· x, where α 6 = 1. This means that there is an isomorphism f : X → X such that f * ω = α ω and τ = f −1 • τ • f . Since both ω and ω satisfy (6.1), f * ω and ω give two triangulations of X by unit equilateral triangles (see Proposition 6.1). Since f * ω = α ω, with α ∈ U 6 , the two triangulations coincide. Thus, they induce the same triangulation of S. The proof of the converse is left to the reader.
For any m ∈ Z >0 , denote by Ω k M g (k, m) the set of elements of Ω k M g (k) whose canonical triangulation is composed by at most km triangles. Since the area of an equilateral triangle with unit side is √ 3/4, x ∈ Ω k M g (k, m) if and only if x satisfies (6.1), and By definition, the triangulations of S that are induced by elements of Ω k M g (k, m) have at most k triangles. Similarly, denote by Ω k M g (k, m) the set of x ∈ Ω k M g (k) such that the canonical quadrangulation of x has at most km squares, or equivalently ω 2 km. The quadrangulations of S induced by elements of Ω k M g (k, m) have at most m squares.
-For any k ∈ {1, 2, 4} such that (4/k)| gcd(κ 1 , . . . , κ n ), we have Proof. -By Lemma 6.3, we have a bijection between T (k) (κ, m) and the set of U 6 -orbits in Ω k M g (k, m). By construction, the stabilizer of a point x ∈ Ω k M g (k) for the U 6 action contains U k . Thus generically, an U 6 -orbit contains 6/k elements. There may exist x ∈ Ω k M g (k) such that U 6 · x contains less than 6/k elements, in which case x is an orbifold point of Ω k M g (k). Since the set of orbifold points of Ω k M g (k) is a (finite) union of proper subvarieties, the number of elements of Ω k M g (k, m) that are orbifold points is negligible compared to #Ω k M g (k, m) as m → +∞. Therefore, we have lim m→∞ #Ω k M g (k, m) #T (k) (κ, m) = 6 k .
The proof of (6.4) follows the same lines.