Adrien Sauvaget

VOLUMES OF MODULI SPACES OF FLAT SURFACES

Keywords: Mathematics Subject Classification. 14H10, 14H51, 14C17, 30F45 Moduli spaces of differentials, intersection theory, flat surfaces. 1

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Volumes of moduli spaces of flat surfaces

Introduction

All stacks/schemes of the paper are defined over C. We use the following notation: U: the group of complex numbers of module 1 U k : the group of k th -roots of unity for k ≥ 1 (g, n): non-negative integers satisfying 2g -2 + n > 0 M g,n /M g,n : the moduli space of genus g, smooth/stable curves with n markings π : C g,n → M g,n : the universal curve σ i : M g,n → C g,n : sections of the markings ω log → C g,n : log-relative dualizing sheaf = ω Cg,n/Mg,n (σ 1 + . . . 1.1. Moduli spaces of flat surfaces. A marked flat surface with conical singularities (or flat surface for short in the text) is the datum of a marked compact surface (C, x 1 , . . . , x n ) and a flat metric η on C \ {x 1 , . . . , x n } such that the neighborhood of x i is isomorphic to a cone with angle 2πα i for some α i > 0, for all 1 ≤ i ≤ n.

+ σ n ) κ m : π * c 1 (ω log ) m+1 ∈ H 2m (M g,n , Q), for all m ≥ 0 ψ i : c 1 (σ * i ω Cg,n/Mg,n ) ∈ H 2 (M g,n , Q),
The genus of the surface satisfies the following Gauss-Bonnet formula:

2g(C) -2 + n = n i=1 α i .
We will say that two such surfaces (C, x 1 , . . . , x n , η) and (C , x 1 , . . . , x n , η ) are isomorphic if there exists an isometry up to a constant scalar φ : C → C such that φ(x i ) = x i for all 1 ≤ i ≤ n.

Given α ∈ ∆ + g,n , we denote by M(α) the moduli space of marked flat surfaces with angles 2πα i at x i for all 1 ≤ i ≤ n. This space is real-analytically isomorphic to the moduli space of curves M g,n (see [START_REF] Thurston | Shapes of polyhedra and triangulations of the sphere[END_REF] and [START_REF] Troyanov | Les surfaces euclidiennes à singularités coniques[END_REF] in genus 0, and [START_REF] Veech | Flat surfaces[END_REF] in general). Moreover, Veech showed that this moduli space is endowed with a natural volume form ν α (see Section 5 for conventions) thus defining the flat volume function:

Vol : ∆ + g,n → R ≥0 ∪ ∞ α → ν α (M g,n ).
Motivating Problem. Is Vol(α) finite? can we compute it?

We give the following partial answer to this problem.

Theorem 1.1. We assume that n ≥ 2. The function Vol is finite and lower semi-continuous. Moreover, there exits a finite continuous function Vol such that Vol(α) = Vol(α) for almost all α ∈ ∆ + g,n .

The function Vol can be explicitly computed. It will be defined at the end of this introduction. 1.2. Pluricanonical divisors. Let α ∈ ∆ g,n , and k ∈ Z >0 such that kα is integral. A k-canonical divisor of type α is a marked complex curve (C, x 1 , . . . , x n ), satisfying

ω ⊗k log O ((kα 1 ) • x 1 + . . . + (kα n ) • x n ) ,
where ω log = ω C (x 1 + . . . + x n ). We denote by M(α, k) the moduli space of kcanonical divisors of type α. It is a smooth sub-stack of M g,n of dimension    (2g -2 + n), if α ∈ Z n >0 , and k = 1 mixed dimension, if α ∈ Z n >0 , and k > 1 (2g -3 + n), otherwise in the second case, the space M(α, k) contains M(α, 1) which is of dimension (2g -2 + n), while all other components are of dimension (2g -3 + n) (see [START_REF] Schmitt | Dimension theory of the moduli space of twisted k-differentials[END_REF]). If kα is not integral, then we set M(α, k) to be the empty space by convention. If α is positive, then C \{x 1 , . . . , x n } is endowed with a canonical flat metric that has conical singularity of order α i at x i for all 1 ≤ i ≤ n. The holonomy character of this flat metric π 1 (C \ {x 1 , . . . , x n }, ) → U, (defined as the rotation part of the holonomy) has value in the set of kth-roots of unity. Conversely, any flat surface with finite holonomy character is obtained from a pluricanonical divisor. Therefore the moduli space M(α, k) may be defined as the subspace of M(α) of flat surfaces with holonomy valued in U k .

Figure 1. By gluing the couples of edges e i and e i on the two polygons above, we obtain equivalent flat surfaces in M((2/3, 4/3). In fact the holonomy characters have value in the set of 6th roots of unity, thus these surfaces sit in M((2/3, 4/3), 6).

Like the space of flat surfaces, the space M(α, k) is equipped with a natural volume form. We denote by Vol(α, k) the volume of the space for this form. This is the Masur-Veech volume of M(α, k) and it is finite (see [START_REF] Veech | Gauss measures for transformations on the space of interval exchange maps[END_REF], [START_REF] Masur | Interval exchange transformations and measured foliations[END_REF] for k = 1 and 2, and [START_REF] Nguyen | Volume form on moduli spaces of d-differentials[END_REF] in general). Along the proof of the main theorem 1.1, we will show the following result: Theorem 1.2. If α has no integral entry, then Vol(α, k) can be explicitly computed.

Strategy of proof.

Let Ω k g,n be the total space of the vector bundle π * ω ⊗k log . It is the space of tuples (C, x 1 , . . . , x n , η), where η is a k differentials with poles of order at most k at the markings. We denote by Ω(α, k) ⊂ Ω k g,n the subspace of k-differentials on smooth curves such that ord xi (η) = kα i for all i ∈ [ [1, n]]. The rescaling of the differentials provides a C * action on Ω(α, k), and

PΩ(α, k) is canon- ically isomorphic to M (α, k). We denote by M(α, k) (respectively PΩ(α, k)) the closure of M(α, k) in M g,n (respectively PΩ(α, k) in PΩ k g,n ). We have a morphism PΩ(α, k) → M(α, k) but this is not an isomorphism.
We denote by ξ ∈ H 2 (PΩ k g,n , Q) the Chern class of the tautological line bundle O(1). We will study the following intersection numbers

a(α, k) = PΩ(α,k) ξ 2g-3+n .
We will show that this number is computable. The computation relies on:

• the explicit expression of the Poincaré-dual class of M(α, k) in H * (M g,n , Q)
conjectured by Schmitt in [START_REF] Schmitt | Dimension theory of the moduli space of twisted k-differentials[END_REF] and proved recently in [BHP + ] (we recall these results in Section 2); • the expression of ξ in terms of boundary components of PΩ(α, k) (see Theorem 3.12). Then, in Section 5 we will show the following identity:

Vol(α, k) = (2π) 2g-2+n (2g -2 + n)! q(α) • a(α, k) k 2g-3+n , (1)
where

q(α) = (-1) g-1+n 2 2-n n i=1 sin(πα i ). (2)
This identity follows from the representation of ξ by a singular 2-form shown in [START_REF] Costantini | The area is a good metric[END_REF], and the existence of a U (p, q) structure on Ω(α, k) preserving an hermitian form with determinant q(α) that may be positive or negative (see Lemma 5.1, and lemma 5.2). This relation finishes the proof of Theorem 1.2.

In order to prove Theorem 1.3, we will define:

a : ∆ + g,n ∩ Q n → Q α → lim k→∞ kα∈Z n k -4g+3-n a(α, k)
(the limit is taken over the integers k such that kα is integral). We will show that this function is well defined and extends to a continuous piece-wise polynomial on ∆ + g,n that vanishes at vectors with integral values. The function Vol of Theorem 1.3 will be defined as

Vol : ∆ + g,n → R α → (2π) 2g-2+n (2g -2 + n)! q(α)
• a(α).

We will show that this function is well-defined and continuous at vectors with integral values. Then, Theorem 1.3 is the consequence of the following two facts:

• M(α) admits a natural foliation, the holonomy foliation (see [START_REF] Veech | Flat surfaces[END_REF]). If α is rational without integral entries, then the spaces M(α, k) are union of leaves of this foliation and equidistribute in M(α) for large values of k:

Vol(α) = lim k→∞ kα∈Z n k -2g Vol(α, k).
(see Formula (11)) • The function Vol is lower semi-continuous (see Lemma 5.3).

Flat recursion. We define a family of functions

v : ∆ + g,n → R recursively. The base of the induction is v(∆ + 0,3 ) = 1. 1.4.1. The function A i . Let 1 ≤ i ≤ n.
To define the functions v, we will require the following intersection numbers

A i (α, k) = M(α,k) ψ 2g-3+n i -α i ψ 2g-2+n i .
Using the recent results of [BHP + ]), we will show that A is polynomial in k of degree 2g (see Lemma 2). We denote by A i (α) the coefficient of k 2g in this polynomial. The function A is a rational polynomial of degree 2g in the α i 's that can be computed using the admcycle package in Sage (see [START_REF] Delecroix | admcycles -a sage package for calculations in the tautological ring of the moduli space of stable curves[END_REF]) In a forthcoming work with Costantini and Schmitt, we prove the following closed formula

A i (α) = [z 2g ] exp α i zS (z) S(z) n j =i S(α j z) S(z) 2g-2+n , where S(z) = sinh(z/2) z/2
, and the notation [z 2g ] stands for degree 2g coefficient in z in the formal series (see [CSS21] 1 ).. 1.4.2. The flat recursion. The recursion formula defining v is written as a sum on graphs. A star graph Γ is a type of stable graph (see Section 2 for definitions) determined by the following datum:

• a vector (g 0 , g 1 , . . . , g ) of non-negative integers of positive length ( + 1); • a vector of positive integers (e 1 , . . . , e ) that sum up to e 0 and such that g = e 0 -+ j≥0 g j . • a partition [[1, n]] = L 0 . . . L , with n j = |L j |, and satisfying 2g j -2 + n j + e j > 0 for all 0 ≤ j ≤ .

Given a star graph Γ and α ∈ ∆ + g,n , we denote by

∆(Γ, α) ⊂ R e >0 the set of vectors β = (β 1,1 , . . . , β 1,e1 , β 2,1 , . . . , β ,e ) satisfying i∈Lj α i + n i=1 β j,i = 2g j -2 + n j + e j
for all 1 ≤ j ≤ . Note that this domain is of dimension h 1 (Γ) = e 0 -. Let 1 ≤ i 0 ≤ n be an element of E 0 . We define the contribution of Γ relative to i 0 to be:

1 The proof of this explicit formula relies on the polynomiality of the function A i proved in Lemma 2 below.

α 2 α 3 3 β1,1 β1,2 2 β2,1 1 α 1 Figure 3. Example of star graph in Star 7,3 . The domain ∆(Γ, α) is the set of positive triples (β 1,1 , β 1,2 , β 2,1 ) satisfying β 2,1 = 4-α 3 , and β 1,1 + β 1,2 = 7 -α 2 . It is empty if α 2 > 7 or α 3 > 4. v i0 (Γ, α) = β∈∆(Γ,α) (-1) A i0 (α i ) i∈L0 , (-β j,i ) 1≤j≤ 1≤i≤ej × j=1   1≤i≤ej β j,i e j ! • v (α i ) i∈Lj , (β j,i ) 1≤i≤ej   .
We denote by Star g,n,i0 the set of star graphs such that the i 0 ∈ E 0 . The recursion formula for v is

v(α) = Γ∈Starg,n,i 0 v i0 (Γ, α) ! . (3)
This formula will be called the flat recursion relation (FR) by analogy with the topological recursion that computes in particular Weil-Petersson volumes (see [START_REF] Mirzakhani | Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces[END_REF]), and the volume recursion for Masur-Veech volumes (see [START_REF] Chen | Masur-Veech volumes and intersection theory on moduli spaces of Abelian differentials[END_REF]). The relation between the topological recursion and flat recursion will be investigated in a subsequent work. The following theorem makes the function Vol explicit.

Theorem 1.3. For all α ∈ ∆ + g,n , we have a(α) = v(α). 1.5. Previous works. If g = 0, and α ∈]0, 1[ n , then the volume form ν α had been introduced in the 80's by Deligne-Mostow and Thurston (see [START_REF] Deligne | Monodromy of hypergeometric functions and nonlattice integral monodromy[END_REF], [START_REF] Thurston | Shapes of polyhedra and triangulations of the sphere[END_REF], and [START_REF] Troyanov | Les surfaces euclidiennes à singularités coniques[END_REF]). In this case, the space M 0,n (α) has a complex hyperbolic structure and the volume is related to a weighted Euler characteristics of M 0,n . This Euler characteristics has been computed explicitly by McMullen (see [START_REF] Mcmullen | The Gauss-Bonnet theorem for cone manifolds and volumes of moduli spaces[END_REF]). An alternative proof of his formula has been given by Koziarz and Nguyen using intersection theory (see [START_REF] Koziarz | Complex hyperbolic volume and intersection of boundary divisors in moduli spaces of pointed genus zero curves[END_REF]). The volume of the moduli space is computed for all values of α in their domain of definition (and not "for almost all"). Note that the volume function that they compute is the function v.

Two facts simplifies the computation of the volumes in this range. First, Thurston described the metric completion of the moduli space in terms of cone manifolds, which has the same underlying topological space as M 0,n . Moreover, the holonomy foliation is trivial in genus 0. In particular, if α is rational and kα integral, then Koziarz and Nguyen may express the volume of M(α) as the top intersection of ξ in M(α, k) = M 0,n (does not depend on the choice of k). However, the line bundle O(1) → PΩ(α, k) is not a pull-back from M(α, k) in general (even in genus 0 but with general α).

We expect that the equality Vol(α) = Vol(α) is valid for all values of α. A way to prove this result would be to apply a version of the dominated convergence theorem. To do so, one would require a precise description of ν α along degenerating families of flat surfaces.

If (g, n) = (1, 2), then the total space the leaves of the holonomy foliation are complex hyperbolic surfaces. Ghazouani and Pirio computed the Euler characteristics of the quasi-projective leaves of this foliation. Then, they use the density of these special leaves in M(α) to interpret some limit of their Euler characteristics as a volume of M(α) (see [START_REF] Ghazouani | Moduli spaces of flat Tori and elliptic hypergeometric functions[END_REF], Section 6.4). This second part is generalized here to obtain the volumes of moduli spaces of flat surfaces as limit of volumes of moduli spaces of k-canonical divisors.

There is a long line of works relating the volumes of moduli spaces of metric surfaces to the intersection theory of M g,n . In the hyperbolic settings, the Weil-Petersson volumes were expressed in terms of intersection numbers by Wolpert (see [START_REF] Wolpert | Chern forms and the Riemann tensor for the moduli space of curves[END_REF]) and Mirzakhani for surfaces with geodesic boundaries (see [START_REF] Mirzakhani | Weil-Petersson volumes and intersection theory on the moduli space of curves[END_REF]). In the flat setting, we have mentionned the work of Koziarz-Nguyen in genus 0, and volumes of moduli space of canonical and 2-canonical divisors have been expressed in terms of intersection numbers in different ways (see [START_REF] Sauvaget | Volumes and Siegel-Veech constants of h(2g-2) and Hodge integrals[END_REF], [START_REF] Chen | Masur-Veech volumes and intersection theory on moduli spaces of Abelian differentials[END_REF], [CMS + 19], [START_REF] Charbonnier | Topological recursion for masur-veech volumes[END_REF], or [START_REF] Delecroix | Masur-veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves[END_REF]). We should emphasize that in all these cases, the volumes were first computed by other means and the expression of these volumes as intersection numbers brought new insight either on the combinatorics of either the intersection numbers or the volumes (see [START_REF] Mirzakhani | Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces[END_REF] for Weil-Petersson volumes, and [START_REF] Eskin | Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials[END_REF], [START_REF] Eskin | Pillowcases and quasimodular forms[END_REF], [START_REF] Engel | Hurwitz theory of elliptic orbifolds[END_REF] for Masur-Veech volumes). Here, the approach via interesection theory is the only way (until now) to compute the volumes Vol(α) or Vol(α, k) for k > 7.

Acknowledgment. This paper is the result of a question by Selim Ghazouani. He presented to me the equidistribution of pluricanonical divisors in moduli spaces of flat surfaces and pushed me to read the seminal paper of Veech despite its technicality. I would like to thank David Holmes both for useful discussions on the residue morphism, and for the hospitality of his office in Leiden University, as well as Johannes Schmitt for his comments and his help to to produce the above graphs with the admcycles Sage package presented in [START_REF] Delecroix | admcycles -a sage package for calculations in the tautological ring of the moduli space of stable curves[END_REF]. I would also like to thank Gaëtan Borot, Dawei Chen, Alessandro Giachetto, Martin Möller, Gabriele Mondello, and Dimitri Zvonkine for useful conversations and advices. The research was partially supported by the Dutch Research Council (NWO) grant 613.001.651.

Higher double ramification cycles

The purpose of the section is to prove the following lemma.

Lemma 2.1. For all i ∈ [[1, n]], the function A i is a polynomial of degree 2g in the variables kα 1 , . . . , kα n .

In order to prove this lemma, we recall the definition of higher DR cycles as a sum on twisted star graphs.

Twisted graphs.

A stable graph is the datum of

Γ = (V, H, g : H → N, i : H → H, φ : H → V, H i [[1, n]]),
where:

• The function i is an involution of H.

• The cycles of length 2 for i are called edges while the fixed points are called legs. We fix the identification of the set of legs with [[1, n]]. • An element of V is called a vertex. We denote by n(v) its valency, i.e. the cardinal of φ -1 {v}. • For all vertices v we have 2g(v) -2 + n(v) > 0.

• The genus of the graph is defined as

h 1 (Γ) + v∈V g(v), where h 1 (Γ) = |E| -|V | + 1 • The graph is connected.
We say that a stable graph is a star graph if it has a distinguished (central) vertex v 0 such that all edges are between v 0 and another (outer) vertex (this definition of star graph is equivalent to the one given in Section 1.4). We denote by Stab g,n and Star g,n the sets of stable and star graphs of genus g with n legs.

Definition 2.2. A twist on a stable graph Γ is a function β : H → R satisfying:

• For all v ∈ V , we have

h∈φ -1 (v) β(h) = 2g(v) -2 + n(v). • If (h, h ) is an edge of Γ, then we have β(h) = -β(h ).
• If (h 1 , h 1 ) and (h 2 , h 2 ) are edges between the same vertices v, v , then β(h 1 ) ≥ 0 ⇔ β(h 2 ) ≥ 0. In which case we denote v ≥ v . • The relation ≥ defines a partial order on the set of vertices. A twisted star graph, is a star graph with a twist such that the twists at half-edges adjacent to the central vertex are negative.

We denote by Tw g,n , and tStar g,n the sets of twisted graphs and twisted star graphs.

Definition 2.3. The multiplicity of a twisted graph is the number

m(Γ, β) = (h,h )∈Edges -β(h)β(h ). Definition 2.4. If k ∈ Z >0 , then a k-twist is a twist β such that the function kβ has integral values. Definition 2.5. If α ∈ ∆ g,n , then a twisted graph graph is compatible with α if β(i) = α i for all 1 ≤ i ≤ n.
Notation 2.6. If X is a type of twisted graph (i.e. Tw or tStar), then we denote by X k g,n , X(α), and X(α, k) the subsets of k-twisted graphs, graphs compatible with α, and k-twisted graphs compatible with k respectively.

Double ramification cycles via star graphs

. Let (Γ, β) ∈ tStar k g,n .
The stable graph Γ determines a stack

M Γ = v∈V M g(v),n(v) ,
and a morphism ζ Γ : M Γ → M g,n defined by compositions of gluing morphisms.

The twist β allows to define the following sub-stack of M Γ :

M Γ,β = M (α(v 0 ), k) × v∈V Out (Γ) M(α(v), 1) ⊂ M Γ,β
where Out(Γ) is the set of outer vertices of Γ.

Definition 2.7.

If α / ∈ Z n >0 , then the DR cycle associated to (α, k) is the class in H 2g (M g,n , Q) defined by DR(α, k) = (Γ,β)∈Star(α,k) m(Γ, β) • k |E(Γ)|-|V Out | |Aut(Γ, β)| • ζ Γ * [M Γ,β ] 2g , (4)
where [•] stands for the Poincaré-dual class in H * (M g,n , Q), and [•] 2g stands for its cohomological degree 2g part.

Remark 2.8. The above sum is well-defined as the set Star(α, k) is finite. Moreover, the summand determined by a twisted star graph is non-zero only if the twists at half-edges at outer vertices have positive integral values (see [START_REF] Schmitt | Dimension theory of the moduli space of twisted k-differentials[END_REF]).

This class was computed in [BHP + ] in terms of the so-called Pixton's classes. One of the main outcome of this result is the following proposition.

Proposition 2.9. (see [BHP + ] and [PZ]) The class DR(α, k) is a polynomial of degree 2g in the variables (kα i ) 1≤i≤n which can be explicitely computed in terms of generators of the tautological cohomology of M g,n .

Proof of Lemma 2. We show that A i (α, k) = DR(α,k) ψ 2g-3+n i . Then Lemma 2 is a straightforward consequence of Proposition 2.9.

If (α) / ∈ Z n >0 , then the class DR(α, k) is defined by the sum over twisted star graphs (4). The integral of ψ 2g-3+n i on the summand defined by (Γ, β) vanishes if Γ is a not the trivial graph. Thus, if α is not in (Z >0 ) n , then we indeed have the equality

DR(α,k) ψ 2g-3+n i = M(α,k) ψ 2g-3+n i .
Now we want to extend this equality to all α. We denote by α = (α 1 , . . . , α i + 1, . . . , α n , 0), and by π : M g,n+1 → M g,n the forgetful morphism of the marking n + 1. We also denote by δ the boundary divisor of M g,n+1 defined by the stable graph with the two vertices of genus 0 and g, one edge, and such that the vertex of genus 0 carries only the legs i and n + 1. If we assume that α is not integral, then:

π * (δ • DR(α , k)) = DR(α, k).
Thus, by polynomiality of DR-cycles, this equality holds for all values of α. If α ∈ Z n >0 , then we have:

ψ 2g-3+n i π * (δ • DR(α , k)) = M(α,k) ψ 2g-3+n i -α i M(α,1) ψ 2g-2+n i = A i (α, k).
This relation is obtained from the definition of DR(α k , k): the first term comes from the trivial graph, while the second one is obtained from δ i,n+1 with the twist of value α i at the unique edge.

Local structure of the boundary of PΩ(α, k)

In this section we describe the neighbourhood of a generic point in the boundary of PΩ(α, k) and use this description to compute a series of relations in the cohomology of this space. These results were proved in the case k = 1 in [START_REF] Sauvaget | Cohomology classes of strata of differentials[END_REF].

3.1. Incidence variety compactification. Let α ∈ ∆ g,n and k > 0. We decompose α as α = Z(α) -P (α), where Z(α) is the vector obtained by keeping all positive entries of α and sending the others to 0. If P is a vector of n nonnegative integers, then we denote by p : V Ω k g,n (P ) → M g,n the total space of the push forward of

ω ⊗k Mg,n/Cg,n n i=1 P i σ i under π : C g,n → M g,n . We denote by Ω(α, k) the sub-stack of V Ω k g,n (P ) of tuples (C, (x i ), η) such that: • C is smooth; • ord xi (η) = kα i for 1 ≤ i ≤ n; • η is not the k-th power of a differential.
We denote by PΩ(α, k) the closure of PΩ(α, k). This space is called the incidence variety compactification of PΩ(α, k)).

We denote by Ω(α, k) ab the space of k differentials obtained as k-th power of a 1differential with orders prescribed by α. We also denote by PΩ(α, k) ab its incidence variety compactification.

In the next sections, we recall the description of the boundary of these spaces by [START_REF] Bainbridge | Strata of k-differentials[END_REF].

Remark 3.1. We will pay a special attention to k differentials obtained as kth power of ordinary differentials (and not to powers of k differentials for some 1 < k < k) for two reasons: (1) the locus of such objects has an exceptional dimension as already mentioned in the introduction, (2) the global residue condition for limits of k-differentials on nodal curve is described by considering the irreducible components with such differentials (see [START_REF] Bainbridge | Strata of k-differentials[END_REF]).

3.2. Canonical cover. Let (C, (x i ), η) be a k-differential in Ω(α, k). There exists a canonical cyclic ramified cover of degree k, f : C → C. This covering is defined by

C = {(x, v) ∈ T ∨ C , such that v k = η} The covering curve C carries a natural differential v such that such that v k = η.
Each point x i with singularity of order m has gcd(m, k) preimages along which f ramifies with order k/gcd(m, k). Besides the order of v at each point is determined by α. Therefore a pair (α, k) determines a triple ( g, n, α) such that we have an embedding

Ω(α, k) → Ω( α, 1) U k ,
where the U k -action is defined by permuting the labels of preimages of a singularity. This morphism will be called the canonical cover morphism.

Residues. We denote by Pol

(α) ⊂ [[1, n]] the set of indices i such that α i ∈ Z ≤0 . Let (C, (x i ), η) be a k-differential in Ω(α, k), i ∈ Pol(α)
, and let f : C → C be the canonical cover.

The point x i has k preimages under f . These points are poles of order α i of the canonical differential v on C and the residues at two such points differ by a k-th root of unity. The residue at x i is the k th power of any of these residues and we denote it by res xi (η). We denote by res i : Ω(α, k) → C the ith residue morphism, i.e. the morphism defined by mapping η to res xi (η).

If E ⊂ Pol(α), then we denote by Ω(α, k, E) the sub-stack of Ω(α, k) of differentials with vanishing residues at x i for i ∈ E. We denote by PΩ(α, k, E) its projectivization and by PΩ(α, k, E) the closure of PΩ(α, k, E) in PΩ(α, k). Once again we call this space incidence variety compactification.

If i ∈ Pol(α) \ E, then the morphism res i is a section of the line bundle O(1) → PΩ(α, k, E) that extends to the boundary of the incidence variety compactification.

Lemma 3.2. The section res i vanishes with multiplicity k along PΩ(α, k, E ∪ {i}).

Proof. If k = 1, then the residue morphism is a submersion, thus the vanishing mulitiplicity of res i along PΩ(α, k, E ∪ {i}) is 1 (see Corollary 3.8 of [START_REF] Sauvaget | Cohomology classes of strata of differentials[END_REF]). For higher values of k, we use the canonical cover to embed locally Ω(α, k) → Ω( α, 1) (U k ). Then the residue at x i is the k-th power of the residue at any of the marked preimages of the canonical cover. The residue morphism is a submersion along the image of Ω(α, k) in Ω( α, 1) (U k ). Therefore the residue morphism at x i vanishes with multiplicity k.

k-decorated graphs.

In this section we define a refinement of the notion of k-twisted graphs called k-decorated graphs and some relevant subsets.

Definition 3.3. A level function on a k-twisted graph (Γ, β) is a function : V (Γ) → Z ≤0 such that (v ≤ v ) ⇒ ( (v) ≤ (v )) and such that -1 (0) is non- empty. Definition 3.4. A decorated graph is the datum of Γ = (Γ, β, , V (Γ) = V ab V nab ),
where:

• (Γ, β) is a twisted graph;

• : V (Γ) → Z ≤0 satisfying: for all vertices v and v , (v ≤ v ) ⇒ (v) ≤ (v ); • -1 (0) = ∅;
• all twists at half-edges adjacent to vertices in V ab are integral.

Definition 3.5. A k-bi-colored graph is the datum of a k-decorated graph such that:

• the image of the level function is {0, -1}; • all edges are between a level 0 vertex and a vertex of level -1.

Definition 3.6. A k-star graph is a bi-colored graph such that:

• The underlying graph is a star graph, the central vertex is of level -1, and the outer vertices are of level 0; • If a vertex of level 0 is in V ab , then it has only one edge to the central vertex.

• If an outer vertex v has an edge to the central vertex with an integral, then v ∈ V ab .

Notation 3.7. We denote respectively by dStar k g,n ⊂ Bic k g,n ⊂ Dec k g,n the sets of decorated star graphs, bi-colored graphs, and decorated graphs. As in Section 2.1, for all these type of graphs, we denote by X k g,n , X(α), or X(α, k), the sets of decorated graphs whose underlying twisted graph is respectively a k-twisted graph, a twisted graph compatible with α or a k-twisted graph compatible with α. Notation 3.8. Let Γ be a bi-colored graph. For all edge e = (h, h ), we denote β(e) = -β(h)β(h ). We introduce the notation

lcm(Γ, k) = lcm{kβ(e)} e∈E(Γ) , G(Γ, k) =   e∈E(Γ) U kβ(e)   U lcm(Γ,k) .
3.5. Strata associated to decorated graphs. Let E ⊂ Pol(α). Let Γ be a decorated graph in Dec(α, k). From such a datum one can construct a space Ω(Γ, k, E) whose projectivization sits in the boundary of the incidence variety compactification PΩ(α, k, E). The space PΩ(α, k, E) is the union of all such strata. This stratification is described in [START_REF] Bainbridge | Strata of k-differentials[END_REF].

Lemma 3.9. If we denote by P Ω(α, k, E) the union of the projectivized boundaries associated to

• decorated graphs with 1 level and 0 or 1 edge, • and bi-colored graphs,

then PΩ(α, k, E) \ P Ω(α, k, E) is of co-dimension 2 in PΩ(α, k, E).
Proof. This lemma follows from the dimension computation of Section 6 of [START_REF] Bainbridge | Strata of k-differentials[END_REF]. The co-dimension of the stratum associated to a decorated graph with N level is at least N -1. Besides, the horizontal nodes (nodes between two components of the same level) can be smoothed independently from the other nodes. Thus, if a graph (Γ) has N horizontal edges, then it defines a stratum of co-dimension at least N + N -1. Therefore a graph defining a stratum of PΩ(α, k) of co-dimension 1 has either one horizontal edge and 1 level, or 2 levels and no horizontal edges (bi-colored graph).

In all computations, we will only consider strata of codimension 1, thus we will only recall the definition of Ω(Γ, k, E) when Γ is a bi-colored graph.

Strata associated to bi-colored graphs. Let Γ be a decorated graph in Bic(α, k).

For i = 0 or -1, we denote by

Ω(Γ, k, E) i = v∈ -1 (i) v∈V nab Ω(α(v), k, E(v)) × v∈ -1 (i) v∈V ab Ω(α(v), k, E(v)) ab ,
where for all v ∈ V (Γ):

• α(v) is the vector of twists at the half-edges adjacent to v;

• E(v) is the subset of i ∈ E of indices adjacent to v.
Then we define Ω(Γ, k, E) 0 = Ω(Γ, k, E) 0 .

For the level -1, we define Ω(Γ, k, E) -1 as the sub-stack of Ω(Γ, k, E) -1 of kdifferentials satisfying the global resiude condition of ([BCGM19], Definition 1.4). We dot not state the precise definition of the global residue condition for general bi-colored graphs, as we will only need to know that Ω(Γ, k, E) -1 is a sub-stack of Ω(Γ, k, E) -1 . However, at the end of the section we describe it for k-star graphs, as it will be required further in the text.

Definition 3.10. The stratum Ω(Γ, k, E) is the product

Ω(Γ, k, E) 0 × PΩ(Γ, k, E) -1 . Moreover we denote by PΩ(Γ, k, E) = PΩ(Γ, k, E) 0 ×PΩ(Γ, k, E) -1 and by PΩ(Γ, k, E) its closure in i=0,-1 P   v∈ -1 (i) V Ω k g(v),n(v) (P (α(v)))   . Let (C 0 , (x 0 h ), η 0 ) × (C -1 , (x -1 h ), η -1 ) ∈ PΩ(Γ, k, E).
We construct a nodal marked curve (C, x i , η) by gluing markings of C 0 and C -1 as prescribed by Γ. We define a k-differential η on C by η |C0 = η 0 and η |C-1 = 0. This construction defines a morphism

ζ Γ,k : PΩ(Γ, k) → PΩ(α, k) which maps PΩ(Γ, k, E) to PΩ(α, k, E). The degree of any irreducible component D of PΩ(Γ, k, E) on its image is equal to: deg D/ζ Γ,k (D) = |Aut(Γ)| if dim(D) = dim(ζ Γ,k (D)), 0 otherwise.
3.5.2. Global residue condition for decorated star graphs. Let Γ be a k-bi-colored graph. We denote by V ab (E) the set of vertices of Γ such that:

• v ∈ V ab ∩ -1 (0); • E(v) = Pol(α(v)
). The dimension count of Section 6 of [START_REF] Bainbridge | Strata of k-differentials[END_REF] gives the following inequalities:

|V 0 | -1 -|V ab (E)| ≤ dim(D) -dim(ζ Γ (D)) ≤ |V 0 | -1 (5)
If Γ is a k-decorated star graph, with central vertex v -1 . Then, we define the set E -1 as the set of half-edges adjacent to v -1 and part of an edge to a vertex in

V

ab (E). Besides, we still denote by E(v -1 ) the set of legs in E adjacent to v -1 .

Finally we denote by

E -1 = E -1 ∪ E(v -1
). With this notation, we may define

Ω(Γ, k, E) -1 Ω(α(v -1 ), k, E -1 ).
3.6. Relations in the cohomology of PΩ(α, k). We will consider the class

ξ = c 1 (O(1)) ∈ H * (PΩ(α, k), Q).
Notation 3.11. For all 1 ≤ i ≤ n, we denote Bic(α, k, i) and dStar(α, k, i) the sets of graphs such that the label i is adjacent to a vertex of level -1.

If E ⊂ Pol(α) and i ∈ Pol(α) \ E, then we denote by Bic(α, k, E, i) and dStar(α, k, E, i) the set of graphs such that i is adjacent to either:

• a vertex of level -1;

• or a vertex v ∈ V ab such that: for all i ∈ Pol(α) \ {i}, if i is adjacent to v then i ∈ E.

The main purpose of the section is to prove the following Theorem.

Theorem 3.12. Let E ⊂ Pol(α). For all Γ ∈ Bic(α, k, i), and all irreducible components D of PΩ(Γ, k, E), there exists an integer m D such that for all 1 ≤ i ≤ n, we have:

ξ + (kα i )ψ i = Γ∈Bic(α,k,i) D∈Irr(PΩ(Γ,k,E)) m D |Aut(Γ)| • ζ Γ,k * [D]; (6) if i ∈ Pol(α) \ E, then we have: ξ = k[PΩ(α, k, E ∪ {i})] + Γ∈Bic(α,k,E,i) D∈Irr(PΩ(Γ,k,E)) m D |Aut(Γ)| • ζ Γ,k * [D]. ( 7
)
If D is an irreducible component of PΩ(Γ, k, E) for Γ ∈ dStar(α, k),

and α has only positive entries, then m

D = k |E(Γ)| m(Γ, β).
Proof. We assume that kα is integral.

Step 1: relation for fixed value of i. Let 1 ≤ i ≤ n, we denote by m i = kα i . We consider the line bundle O(1) ⊗ L mi i → PΩ(α, k, E). This line bundle has a natural section defined by s i : η → m i th order of η at x i .

This section does not vanish

• on PΩ(α, k, E);

• on strata associated to decorated graphs with one level 0;

• on strata associated to bi-colored graphs in Bic(α, k) \ Bic(α, k, i).

Therefore, up to co-dimension 2 loci of PΩ(α, k), the vanishing locus of s i is the union of the irreducible component D ⊂ PΩ(Γ, k, E) for Γ in Bic(α, k, i). Thus, for each such D, there exists an integer m i D such that:

ξ + m i ψ i = Γ∈Bic(α,k,i) D∈Irr(PΩ(Γ,k,E)) m i D |Aut(Γ)| • ζ Γ,k * [D]. (8)
If i ∈ Pol(α) \ E, then we consider the line bundle O(1) and its section given by the i-th residue morphism. This section vanishes along PΩ(α, k, E) with multiplicity k (Lemma 3.2). Besides, this section does not vanish identically on boundary components associated to k-decorated graphs with one level neither on boundary components associated to bi-colored graphs in Bic(α, k) \ Bic(α, k, E, i). Therefore, we have:

ξ = k[PΩ(α, k, E ∪ {i})] + Γ∈Bic(α,k,E,i) D∈Irr(PΩ(Γ,k,E)) m i D |Aut(Γ)| • ζ Γ,k * [D], (9)
where the m i D are integers.

Step 2: independence of 1 ≤ i ≤ n. We will show that the numbers m i D can be chosen independently of i ∈ [[1, n]]. If D is of dimension smaller than dim(PΩ(α, k))-1, then we can set m D = 0 thus, from now on we will only consider D of dimension dim(PΩ(α, k)) -1.

Let 1 ≤ i = i ≤ n. Let D be an irreducible component of PΩ(Γ, k, E) such that Γ ∈ Bic(α, k, i) ∩ Bic(α, k, i ). Let ∆ be an open disk of C parametrized by . Let ∆ → PΩ(α, k, E) be a family of differentials such that the image of ∆ \ {0} lies in PΩ(α, k, E) while 0 is mapped to a generic point of D.

Up to a choice of a smaller disk, there exists an integer and holomorphic functions f and f that do not vanish ∆ such that s i = f and s i = f (see the "necessary" part of Theorem 1.5 of [START_REF] Bainbridge | Strata of k-differentials[END_REF]). Thus s i and s i vanish with the same multiplicity along = 0. Therefore the vanishing multiplicity of s i and s i along D are equal and the integers m i D can be chosen independently of 1 ≤ i ≤ n.

Step 3: vanishing of residues. Let 1 ≤ i ≤ n, and i ∈ Pol(α) \ E (not necessarily different). Let D be an irreducible component of PΩ(Γ, k) such that Γ ∈ Bic(α, k, i) ∩ Bic(α, k, E, i ). We will show that m i D = m i D . We chose a family ∆ → PΩ(α, k, E) such that the image of = 0 is a generic point of D \ PΩ(α, k, E ∪ {i }) (this is a generic point of D). Once again we can find an integer and holomorphic functions f and f that do not vanish ∆ such that s i = f and res i = f . Thus the two functions vanish to the same order. • For all γ ∈ G(Γ, k), the morphism ι induces an isomorphism

V × {0} × g with V . • The image of V × (∆ \ {0}) × G(Γ, k) lies in PΩ(α, k, E). • The section s i vanishes with multiplicity lcm(Γ, k) along V × {0} × G(Γ, k).
• The morpihsm ι is a degree 1 parametrization of a neighborhood of y in PΩ(α, k, E).

Proof of Lemma 3.13. The proof is similar to the proof of Lemma 5.6 of [START_REF] Sauvaget | Cohomology classes of strata of differentials[END_REF].

In the case of k-star graph, the morphism p -1 : PΩ(Γ, k, E) -1 → M g(v0),n(v0) is an embedding. In particular we can identify: k,E) -1 (see Section 3.5 for the notation). Thereore we can decompose the point y into

PΩ(Γ, k, E) = PΩ(Γ, k, E) 0 × PΩ(Γ,
y = y 0 × y -1 = (C 0 , [η 0 ]) × (C -1 , [η -1 ]),
where η i is a k-differential up to a scalar (we omit the notation of the markings).

For i = 0 and -1, we chose a neighborhood U i of y i in PΩ(Γ, k, E) i together with a trivialization σ i of O(-1) → PΩ(Γ, k, E) i . We assume that U = U 0 × U -1 has coordinate u = (u 0 , u -1 ) and that y = {u = (0, 0)}. We can rephrase the choice of trivialization of the line bundle as: we chose a family of k-differentials

(C i (u j ), η j (u j )) for u i ∈ U i such that (C i (0), [η i (0)]) = y i for i = 0 or -1.

Constructing a smoothing of η. Let e = (h, h ) be an edge of Γ with twist kβ(e).

Let σ 0 : U → C 0 and σ -1 : U → C -1 be the sections corresponding to the branch of the node associated to e. For i = 0, -1, there exists a neighborhood

V i of σ i in C i of the form U i × ∆ e,i
where ∆ e,i is disk of the plane parametrized by z e,i , and such that

η i (u i , z e,i ) = z ±kβ(e) e,i dz e,i z e,i k ,
where the sign is positive for i = 0 and negative for i = -1. Note that no residue is involved because we assumed that (Γ, k) is a k-star graph and that α is positive. The coordinates z e,i are only defined up to a kβ(e)-th root of unity. We fix such a choice for all edges e and i = 0, -1.

For all e ∈ E(Γ), we fix

ι : P(U ) × ∆ × G(Γ, k) → PΩ(α, k) (u, , γ) → (C γ (u, ), η γ (u, ))
is of degree 1 on its image. To check that this morphism parametrizes a neighboorhod of y, we can show as in the case of abelian differentials that there exists a retraction η V : V → V , where V is a neighborhood of y in PΩ(α, k, E). Besides, all points y in V lies in the image of {η(y )} × ∆ × G(Γ, k) under ι (see "Proof of the fourth point" of Lemma 5.6 in [START_REF] Sauvaget | Cohomology classes of strata of differentials[END_REF]).

Lemma 3.14. We assume that α has at least one negative entry. If D is an irreducible component of a component PΩ(Γ, k, E) for a bi-colored graph with two vertices, then m

D ≤ k |E(Γ)| m(Γ),.
Proof. We refer to [START_REF] Costantini | The area is a good metric[END_REF]. We define PΩtot(α, k, E) = PΩ ab (α, k, E) PΩ(α, k, E), and by PΩ tot (α, k, E) its incidence variety compactification. There exists a smooth compactification PΞ tot (α, k, E) of PΩ tot (α, k, E) together with a forgetful morphism

PΞ(α, k, E) → PΩ tot (α, k)
The functions s i can be defined on PΞ(α, k, E) and vanish with order lcm(Γ, k) along PΞ(Γ, k) if the marking i is adjacent to the vertex of level -1. Therefore, the multiplicity m D of any irreducible component of PΩ(Γ, k, E) in the divisor defined as the vanishing locus of

s i is at most gcd(Γ, k) × lcm(Γ, k) = m(Γ)k |E(Γ)| .

Flat recursion

The purpose of the section is to prove the following proposition which implies directly Theorem 1.3. Proposition 4.1. For all (α, k) ∈ ∆ + g,n ×Z >0 , the number a(α, k) = PΩ(α,k) ξ 2g-3+n can be explicitly computed. Moreover, there exists a constant K > 0 such that

k -4g+3-n a(α, k) -v(α) < K/k, if kα is integral.
4.1. Growth of sums on k-star graphs. Let α ∈ ∆ g,n , and (Γ, v 0 ) ∈ Star(α). We denote by Twist(Γ, α) the set of twists on Γ compatible with α. This set is the quotient of the open domain ∆(Γ, α) ⊂ R h1(Γ) (defined in the introduction) by the action of Aut(Γ, v 0 ). This action is free on an open dense subset of ∆(Γ, α).If k ≥ 2, then we denote by Twist(Γ, α, k) the set of k-twists on Γ compatible with α. Lemma 4.2. We assume that α is rational. Let f : Twist(Γ, α) → R and f k : Twist(Γ, k, α) → R be functions such that:

• f is continuous;

• there exists K > 0 such that, for all k and β ∈ Twist(Γ, k, α), we have

|f k (β) -f (β)| < K/k.
Then we have

lim k→∞ kα∈Z n 1 k h1(Γ) • β∈Twist(Γ,α,k) f (β) |Aut(Γ, β)| = 1 |Aut(Γ)| ∆(Γ,α) f (β),
where f is the composition ∆(Γ, α) → Twist(Γ, α)

f → R.
Proof. For all k ≥ 2, we denote by ∆(Γ, α, k) ⊂ Z

E(Γ)

>0 the set of vectors β such that β ∈ ∆(Γ, α) and kβ is integral. Then Twist(Γ, α, k) is the quotient of ∆(Γ, α, k) by Aut(Γ) and we can rewrite

β∈Twist(Γ,α,k) f k (β) |Aut(Γ, β)| = β∈∆(Γ,α,k) f k (β) |Aut(Γ)| where f k is the composition ∆(Γ, α, k) → Twist(Γ, α, k) f → R.
Then, the lemma follows from the the convergence of Riemann sums:

lim k→∞ kα∈Z n 1 k h1(Γ) • β∈∆(Γ,α,k) f (β) = ∆(Γ,α) f ∞ (β).

Recursion relations for fixed k.

We begin by writing a recursion relation for the a(α, k) with a fixed value of k > 1. In order to state it we will denote by

a ab g = PΩg(2g-1,1) ab ξ 2g-2 .
These intersection numbers are determined by the following formula:

[z 2g ]F(z) 2g = (2g)! [z 2g ]S(z) -1 ,
where F(z) = 1 + >0 (2g -1)a ab g z 2g (see [START_REF] Sauvaget | Volumes and Siegel-Veech constants of h(2g-2) and Hodge integrals[END_REF]).

Lemma 4.3. We assume that α is non-negative. Let 0 ≤ j ≤ 2g -4 + n be an integer. Let Γ be a non-trivial bi-colored graph in Bic(α, k, i) and let D be an irreducible component of PΩ(Γ) such that

D ψ j i ξ 2g-4+n-j = 0, then: a) Γ ∈ dStar(α, k, i); b) all legs are adjacent to vertices of V \ V ab . c) the central vertex satisfies j = 2g(v 0 ) -3 + n(v 0 ) -|V ab |. If (Γ) is k-
star graph satisfying these three conditions, then we have

a(Γ, i) def = PΩ(Γ) (-α i ψ i ) j ξ 2g-4+n-j (10) =     v| (v)=0, v / ∈V ab a(α(v), k) × v| (v)=0, v∈V ab k 2g(v)-1 a ab g(v)     × PΩ(Γ,t)-1 (-α i ψ i ) j .
We denote by dStar(α, k, i) * ⊂ dStar(α, k, i) the set of k-star graphs such that no legs is adjacent to a vertex in V ab .

Proof. Let Γ be a non-trivial bi-colored graph in Bic(α, k, i) and let D be an irreducible component of PΩ(Γ). We decompose:

ξ 2g-4+n-j ψ j i • [D] = ξ 2g-4+n-j • [D 0 ] × ψ j i • [D -1 ] ,
where D = D 0 × D -1 and D i is an irreducible component of PΩ(Γ) i for i = 0, -1.

In particular this integral vanishes if j = dim PΩ(Γ) -1 . We assume that this relation holds. Then we further decompose the first term as

ξ 2g-4+n-j • [D 0 ] = v∈ -1 (i) v / ∈V ab ξ 2g(v)-3+n(v) [D(v)] × v∈ -1 (i) v∈V ab ξ 2g(v)-2+n(v) [D(v)] ,
where D = P( v (O(-1)| Dv ) (the product of the total spaces of the line bundles

O(-1) → D v )) and D v is an irreducible component of PΩ(α(v), k) or PΩ(α(v), k) ab . It was proved in [Sau18] (Proposition 3.3) that ξ 2g-2+n • [PΩ(α, 1) ab ] = a ab g if α = (2g -1) 0 otherwise.
Moreover, the argument used in [START_REF] Sauvaget | Volumes and Siegel-Veech constants of h(2g-2) and Hodge integrals[END_REF] implies that ξ 2g-2+n • [D] = 0 for any irreducible component of PΩ(α, 1) ab with α = (2g -1).

Let k ≥ 2, and D be an irreducible component of PΩ(α, k), where α has at least one entry divisible by k. Then the integral D ξ 2g-3+n vanishes. The argument is given for k = 2 in the proof of Theorem 1.6 of [CMS + 19]: the tangent space to a point PΩ(α, k), seen as a subspace of PΩ(α, 1) U k has directions in the strictly relative cohomology of the covering curve. However, the class ξ can be realized as a 2-form involving only absolute periods of the covering curve (see Lemma 5.2 below).

Therefore, the contribution of a bi-colored graph vanishes if the upper-vertices contain at least one vertex in V ab with more than two adjacent edges, or a vertex in V \ V ab that has a twist divisible by k.

The final condition that we need to check is that there is exactly one vertex of level -1. Indeed, if we assume that the a graph has at least two vertices of level -1 then a simple dimension computation shows that PΩ(Γ) is of co-dimension at least 2 in PΩ(α, k) (see dimension computation of [START_REF] Bainbridge | Strata of k-differentials[END_REF]).

Putting everything together, we proved that ξ 2g-4+n-j ψ j i • [D] = 0 for any irreducible component D of PΩ(Γ) if Γ is not in dStar(α, k, i) * . Besides, we have also proved that if Γ is in Star(α, k, i) * then a(Γ) is given by the formula (10).

An immediate corollary of Lemma 4.3 is the following lemma.

Lemma 4.4. For all α and 1 ≤ i ≤ n, we have:

a(α, k) = Mg(α) (-α i ψ i ) 2g-3+n + Γ∈dStar(α,k,i) * k |E(Γ)| m(Γ) |Aut(Γ, t)| • a(Γ, i).
Proof. We write

ξ 2g-3+n =   j≥0 ξ 2g-4+n-j (-kα i ψ i ) j   (ξ + kα i ψ i ).
Then we use formula (6) to express (ξ +kα i ψ i ) in terms of classes [PΩ(Γ, k)] for Γ in dStar(α, k, i) * up to a term δ supported on the union of the PΩ k (Γ, k) for (Γ, k) ∈ Bic(α, k, i) \ dStar(α, k, i) * . The integral of ξ 2g-4+n-j (-m i ψ i ) j on δ vanishes for all j by Lemma 4.3. Besides, the integral of ξ 2g-4+n-j ψ j i on PΩ k (Γ, k) for a k-star graph is also given by Lemma 4.3.

Growth of intersection numbers on strata with residue conditions.

Let B > 0. We define the set ∆ B g,n ⊂ ∆ g,n as the set of vectors α ∈ Q n such that α i > -B for all i, and either:

(1) at most two entries of α are positive;

(2) or

α i / ∈ Z >0 for all i > 1. Let E ⊂ [[2, n]
] be a subset of cardinal r. We consider the following function

A g,E : ∆ B g,n → Q (α, k) → M(α,k,E) ψ 2g-3+n+p-r 1
The purpose of the section is to prove the following statement.

Lemma 4.5. There exists a real constant K B > 0, such that for all (α, k) ∈ ∆ B g,n

we have:

|A g,E (α, k)| < K B • k 2g .
If Γ ∈ Bic(α, 1) \ Bic(α, i 0 , E). Then i 0 is necessarily adjacent to the vertex of level 0. We have

D ψ 2g-3+n+p-r 1 = D0 1 × D-1 ψ 2g-3+n+p-r 1 .
Therefore, this contribution vanishes if the space D 0 is positive dimensional. This imposes that this vertex has to be of one of the types of Lemma 4.6. Besides, this vertex can only be of type 1 or 3 as these are the only cases for which the residue map is not trivial.

• If D 0 is of type 1, then n = 1 and there is at most one vertex of level -1.

• If D 0 is of type 3, then |V ab | = 0. Thus, there can be only vertex of level -1. Finally, as the upper vertex is of type 3, the contact orders between the two vertices are not divisible by k (and Γ is a k-star graph).

If Γ ∈ Bic(α, i 0 , E) \ Bic(α, 1). Then we have:

D ψ 2g-3+n+p-r 1 = D0 ψ 2g-3+n+p-r 1 × D-1 1 .
The fact that Γ belongs to Bic(α, i 0 , E) leads to two possibilities:

(1) If i 0 is adjacent to the vertex of level 0. Then this vertex is in V ab and all indices of [[n + 1, n + p]] \ {i 0 } adjacent to the upper vertex are in E 0 (the condition that residue vanishes at i 0 follows from the fact that the sum of residues of a holomorphic 1-form vanishes). Then |V ab | = 0 and thus there is one vertex of level -1. This vertex has to be of type 1 or 3 in Lemma 4.6 as the residue condition is empty. In the first case, the graph has two vertices in V ab and in the second it is a k-star graph. (2) If i 0 is adjacent to a vertex of level -1, then the condition dim(D -1 ) = 0 implies that all vertices of level -1 have to be of type 1, 2 or 3 of Lemma 4.6. Then we use the conditions of the definition of ∆ B g,n :

• if α has at most two positive entries which are integral, then there can be only one vertex of level -1 (in V ab ). • if all positive entries different from α 1 are not divisible by k, then all vertices of level -1 are of type 3. Then the edge from the vertex carrying i 0 to the level 0 has necessarily a vanishing residue. Thus the condition dim(D -1 ) = 0 imposes that there is only one vertex of level -1 (not in V ab in this case). Besides this vertex has only one edge to the upper vertex which has to be in V ab . Therefore, this graph is a k-star graph.

Proof of Lemma 4.5. The proof will be done by induction on r and g. The base of the induction (r = 0) is a direct consequence of Lemma 2. Thus, we assume that r > 0. Let B > 0 and let E be a subset of [[n + 1, n + p]] of cardinal r -1, and

i 0 ∈ [[n + 1, n + p]] \ E.
We chose α ∈ ∆ B g,n . Taking the difference between the equation (6) for i = 1 and the equation (7) for i = i 0 , we get the following relation:

(kα 1 )ψ 1 = k[PΩ(α, k, E ∪ {i 0 })] + Γ∈Bic(α,1)∆Bic(α,i0,E) D∈Irr(PΩ(Γ,E)) ±m D • ζ Γ * ([D]),
where the ± depends on whether Γ belongs to Bic(α, 1) or Bic(α, i 0 , E). If we multiply this expression by ψ 2g-3+n+p-r 1 , we get:

(kα 1 )A g,E (α, k) -kA g,E∪{i0} (α, k) = Γ∈Bic(α,1)∆Bic(α,i0,E) D∈Irr(PΩ(Γ,E)) ±m D • D ψ 2g-3+n+p-r 1 .
Using both Lemma 3.14 and Lemma 4.7 to obtain the following inequality:

|kA g,E∪{i0} (α, k)| ≤ |(kα 1 )A g,E (α, k)| + Γ∈Bic(α,1)∆Bic(α,i0,E) D∈Irr(PΩ(Γ,E)) m(Γ) • PΩ(Γ,E) k |E(Γ)|+r-1 ψ 2g-3+n+p-r 1 .
There are only a finite number of underlying star graphs in Bic(α, 1) and Bic(α, i 0 , E).

Besides, the fact that the entries α belong to the domain ∆ B g,n , imposes that the vectors α 0 and α -1 belongs to domains of the form ∆ Bi gi,ni for some B i > 0 (independent of the choice of the graph).

As the graphs appearing in the sum have two vertices (Lemma 4.7), we can decompose these integrals as a product of two integrals at the vertices of level i = 0 and -1.

• If the vertex v i is not in V ab , then the integral is equal to A 1 gi,Ei (α i , k) for i = 0, or -1. • As α is bounded and the number of star graphs is finite, there are finitely may values for the tuples (g i , α i , E i ). Besides the contribution of the integral at a vertex in V ab depends only on these tuples. Thus the integrals at vertices in V ab are bounded by a common constant. Now using the induction hypothesis, there exists a constant K B such that:

A g,E∪{i0} (α, k) ≤ K B •   α 1 • k 2g + Γ∈Bic(α,1)∆Bic(α,i0,E) m(Γ) • k |E(Γ)|+2g-2h1(Γ)  
The boundedness of the twists implies that α 1 ≤ B, and m(Γ) < B for some B > 0. Putting everything together, there exists a constant K B such that

A 1 g,E∪{i0} (α, k) ≤ K B • k 2g + Γ∈Bic(α,1)∆Bic(α,i0,E) k 2g-h1(Γ) .
There are finitely many underlying star graphs in the last sum and for each such star graph the number of compatible k-twist is bounded by a constant times k h1 (Γ) . Therefore we obtain the desired estimate.

4.4. Proof of Proposition 4.1. We prove Proposition 4.1 by induction on g and n. The base of the induction is valid. Indeed, if g = 0 and n = 3, then the function a 0,n (α, k) = 1. Then, we fix some g, n ≥ 0. We define the following set of vectors

∆ g,n = ∆ + g,n ∩ (R × (R \ Z) n ) , if n ≥ 3 ∆ + g,n , otherwise.
Step 1. Let α ∈ ∆ g,n be a rational vector. Let Γ be star graph in Star g,n,1 and let V ab ⊂ V be a subset of the outer vertices such that for all v ∈ V ab there is only one half-edge adjacent to v.

For all k ≥ 2, A twist β ∈ Twist(Γ, α, k) determines at a unique structure of bi-colored graph. We define the following function:

f Γ,V ab : Twist(Γ, α, k) → R β → m(Γ, β)a((Γ, V ab , β), 1) k 4g-3+n-|E(Γ)|
(extended by 0 if β does not determine a k-star graph). There exists a constant K Γ,V ab such that for all β ∈ Twist k (Γ, α), we have

a((Γ, V ab , β), 1) < K Γ,V ab ×     v| (v)=0, v / ∈V ab k 4g(v)-3+n(v)     ×     v| (v)=0, v∈V ab k 2g(v)-2     × k 4g(v0)-3+n(v0)-|V ab | ≤ K Γ,V ab • k 4g-3+n-h1(Γ)-|E(Γ)|-|V ab | .
Here we have used the expression (10) to decompose a((Γ, V ab , β), 1) into a product of 3 terms. We bounded the first term by the induction hypothesis and the third by applying Lemma 4.5. In particular there exists a K Γ,V ab such that

β∈Twist(Γ,α,k) m(Γ, β)k |E(Γ)| a((Γ, V ab , β), 1) < K Γ,V ab k 4g-3+n-|V ab |
for all k ≥ 2. Now, if V ab is empty, then we can show by the same arguments that there exist a constant K Γ such that

v(Γ, α, i) - β∈Twist(Γ,α,k) m(Γ, β) a((Γ, V ab , β), 1) k 4g-3+n-|E(Γ)| < K Γ /k.
(here we have used the second part of Lemma 4.5 and Lemma 4.2 to bound the the sum over the twists). As the number of star graphs appearing in the expression of the a(α, k) is finite, there exists a constant K such that for all α ∈ ∆ g,n , we have:

v(α) - a(α, k) k 4g-3+n ≤ Γ∈Starg,n,i V ab =∅ v(Γ, α) - β∈Twist k (Γ,α) m(Γ, β) a((Γ, V ab , β), 1) k 4g-3+n-|E(Γ)| + Γ∈Starg,n V ab =∅ β∈Twist(Γ,α,k) m(Γ, β) a((Γ, V ab , β), 1) k 4g-3+n-|E(Γ)| < K/k.
Here we have used Lemma 4.4 to decompose a(α, k).

Step 2. For all values of k, the function a

(•, k) is S n invariant by definition. Therefore, v is S n invariant on ∆ g,n . As v is continuous, it is S n -invariant on ∆ + g,n in general.
If α 1 ∈ Z >0 , then v(α) = 0. Indeed, a(α, k) vanishes if one the entries of α is integral, and the first point of the theorem (restricted to ∆ g,n ) implies that v g,n (α) is the limit of trivial sequence.

Finally, the result of Step 1 is valid for all α in ∆ + g,n as |a(α) -v(α)| vanishes if at least one entry of α is integral. 4.5. Wall-crossing properties of the flat recursion. By the flat recursion relation (3), the function v is continuous and piece-wise polynomials on ∆ + g,n of degree at most 4g -3 + n. The chambers of polynomiality are delimited by walls of the form: i∈S α i = κ for a strict and non-empty subset S of [[1, n]], and an integer κ. The purpose of this section is to characterize the level of discontinuity of the functions v g,n along the walls. The results will be used further to prove Theorem 1.1 using Theorem 1.3 Lemma 4.8. For all g ≥ 1, we have lim

α1 →0 v(α 1 , 2g -α 1 ) = 0.
Proof. We use the fact that the only terms in the flat recursion formula (3) which are not divisible by α 1 are those for which the central component is a vertex of genus 0 with 3 half-edges. For small values of α 1 , this condition is satisfied only by the graph with the markings 1, and 2, adjacent to a central vertex of genus 0 and with one edge. Indeed, if α 1 is smaller than 1/2, then the second markings belongs to the lower vertex as 2g -2 -α 1 > 2g -1. Finally the contribution of this graph is equal to (2g -1)v(2g -1) = 0, as (2g -1) is integral.

Proposition 4.9. Let κ ∈ Z >0 . In the neighborhood of a generic point of the wall α i = κ, the function v g,n is of the form

(α i -κ) v, if n ≥ 3 (α i -κ) 2 v, if n = 2 ,
where v is a continuous piece-wise polynomial.

Proof. We prove the statement by induction on g and n. For (g, n) = (0, 3) the statement is empty as ∆ 0,3 does not contain vectors with integral values.

Let (g, n) = (0, 3). By S n -invariance we can assume that i = 2. We begin by writing the flat recursion formula (3):

v(α) = Γ∈Starg,n,1 v i (Γ, α)
Let Γ be a star graph in Star g,n,1 . The function α → v 1 (Γ, •) is a piece-wise polynomial on the domain ∆(Γ) bounded by the walls:

i →v α i = 2g v -2 + n v
for all vertices v of level 0. It is extended by 0 outside the domain ∆(Γ). In order to understand the behavior of v Γ in the neighborhood of a generic point of the wall α 2 = κ, we distinguish 3 cases: the label i = 2 is adjacent to the central vertex, a vertex with more than one leg, or an outer vertex with only the leg i = 2.

If the marking 2 belongs to the vertex of level -1 then v 1 (Γ, •) is polynomial on a domain containing a generic point of any wall of the form α 2 = κ.

If the label i = 2 is adjacent to a leg with at least one other marking, then a generic point of the wall α i = κ is in the interior of ∆(Γ). Indeed, otherwise it would be at the intersection of two wall α i = κ and α i = κ for all i adjacent to the same vertex as i = 2 (non generic configuration). In the interior of ∆(Γ), the function v 1 (Γ, •) is defined as the partial integration of a product of a polynomial and functions of the form v for smaller values of g and n. Thus by induction hypothesis, v 1 (Γ, α) = (α 2 -κ) v Γ for some continuous piece-wise polynomial v Γ .

Finally, if i = 2 is the unique leg adjacent to its outer vertex v, then the wall α 2 = 2g v -2 + n v is a boundary of the domain ∆(Γ). From the flat recursion:

v 1 (Γ, α) = ∆(Γ,α) v(α 2 , β 1 , . . . , β nv-1 ) • (β 1 . . . β nv-1 ) • Q(α, β).
where Q is a continuous piecewise polynomial. Therefore, v 1 (Γ, α) is of the form (α 2 -(2g v -2 + n v )) 2 v Γ for some continuous piece-wise polynomial v Γ . Indeed for n v ≥ 3, this follows from the fact that v Γ is the integral of a polynomial with valency at least one in each

β i for 1 ≤ i ≤ n v -1. If n v = 2, it follows from the fact that v(α 2 , (2g v -2 + n v ) -α 2 ) tends to 0 as α 2 goes to 2g v -2 + n v .
Using these results we can write:

v = Q 1 + (α 2 -κ)v + (α 2 -κ) 2 v
where Q 1 is a polynomial (contribution of graphs with i = 2 adjacent to the central vertex), and v , v are continuous piecewise polynomials (respectively contribution of graphs with i = 2 adjacent to vertex with other legs or not). The polynomial Q 1 vanishes along α 2 = κ as v does, thus if n ≥ 2, we can indeed factorize v by (α 2 -κ).

If n = 2, then term v = 0 (as there are no graphs with at least two legs on the outer vertices for n = 2). Thus we need to show that the derivative of α 2 → Q 1 (2g -α 2 , α 2 ) vanishes at κ. This follows from Theorem 1.2. Indeed, the function Vol is non-negative for all rational entries and the sign of sin(πα 1 )sin(πα 2 ) is constant when n = 2. Thus, by (2), the sign of v is constant on ∆ g,2 . This implies that Q 1 vanishes to the order at least 2.

From intersection theory to volumes

In this section we recall the convention for the normalisation of volumes of moduli spaces of flat surfaces and we complete the proof of Theorems 1.1 and 1.2. 5.1. U (p, q) structures. Let h be an hermitian metric on C p+q of signature p + q. We denote by C h ⊂ C p+q the positive cone for h, i.e. the set of vectors x such that h(x, x) > 0 and by proj : C h \ {0} → PC h its projectivization. We can define two measures (in fact volume forms) on PC h . The first one is defined by ν 1 (U ) = Lebesgue measure proj -1 (U ) ∩ {x|h(x, x) ≤ 1} .

The second is defined by considering the line bundle O(-1) → P C h . Indeed this line bundle is endowed with the hermitian metric equal h as we identify O(-1) * C * h . We denote by -ω h the curvature form of this metric h. Then we define the volume form ν 2 = ω p+q-1 h . Lemma 5.1. We have ν 1 = π p+q (p+q)!det(h) ν 2 .

Proof. The proof is similar to Lemma 2.1 of [START_REF] Sauvaget | Volumes and Siegel-Veech constants of h(2g-2) and Hodge integrals[END_REF] and Lemma 2.1 of [CMS + 19]. We can assume that h is diagonal and given by h(x, x) = 1≤i≤p+q h i |x i | 2 with h i > 0 for 1 ≤ i ≤ p. Using the action of the group U (p + q) ∩ U (p, q) it is sufficient to compare the form on the set of vectors of the form (x 1 , 0, . . . , 0, x p+1 , 0, . . .).

We consider the chart of P C h defined by x 1 = 1. In this chart the measure ν 1 is the measure associated to the differential form:

2π h(x, x) p+q dim R (C h ) • p+q i=2 ( i 2 dx i ∧ dx i ).
In this same chart the form ω h is given by

ω h = (h 1 + h p+1 |x p+1 | 2 ) • ( p+q i=2 h i dx i ∧ dx i ) -h 2 p+1 |x p+1 | 2 dx p+1 ∧ dx p+1 2iπ(h 1 + h p+1 |x p+1 | 2 ) 2 .
From this expression, we deduce the equality

ω p+q-1 h = (p + q -1)! p+q i=1 h i (2iπ) p+q-1 h(x, x) p+q • p+q i=2 dx i ∧ dx i . = (p + q)!det(h) π p+q ν 2 .
5.2. The holonomy map. We fix a reference oriented marked surface (S, s 1 , . . . , s n ) of genus g. Given α ∈ ∆ + g,n , we denote by T (α) the moduli space of flat surfaces (C, x 1 , . . . , x n , η) with conical singularities prescribed by α together with an isomorphism C → S preserving the markings. This is the Teichmüller moduli space of flat surfaces of type α.

In [START_REF] Veech | Flat surfaces[END_REF], Veech showed that there exists a map: hol : T (α) → U 2g , the holonomy character map. This map is a submersion for any value of α / ∈ N n , and the leaves are complex manifolds. For any value of λ ∈ U 2g , we denote by T λ α = hol -1 (λ) the level set associated to λ.

There exists a C ∞ -complex line bundle proj : L(α) → T (α) equipped with an hermitian metric h α . This line bundle is defined by fixing a choice of orientation and normalization of a flat surface. The restriction of this line bundle to any leaf of the holonomy foliation is holomorphic. The metric h α is the area of the flat surface.

For all λ ∈ U 2g , the leaf T λ α has an atlas of charts of the form ϕ : U → PC h λ,U ⊂ P 2g-3+n

for some hermitian form h λ,U depending on λ and U . Besides L(α) |U ϕ * O(-1).

The hermitian metric h α is the pull-back of h λ,U (seen as a metric on O(-1)) and the transition maps are given by elements in U (p, q). Finally, the determinant and the signature of h λ,U are independent of both λ and U .

5.3. Measure on M(α). Let λ ∈ K(α). Using the U (p(α), q(α)) structure on L(α) |T λ α , we define a measure ν λ α on T λ α by ν λ α (U ) = Lebesgue measure proj -1 (U ) ∩ {x|h α (x, x) ≤ 1} , (this is well-defined as U (p, q) transition maps are in U (p, q)). As in the previous section we can also consider -ω λ α , the curvature form of the line bundle L(α) |T λ α ) for the hermitian metric h α . Lemma 5.2. We have the equality: Proof. Using Lemma 5.1 and the U (p, q) structure on L(α) |T λ α ), we get the equality:

ν λ α = π 2g-2+n det(h α )(2g -2 + n)! (ω λ α ) 2g-3+n ,
where det(h α ) is the determinant of h λ,U for any chart U of T λ α . This determinant has been computed by Veech (see [START_REF] Veech | Flat surfaces[END_REF], Lemmas 14.10, 14.17, and 14.32): det(h α ) = Q(α) 4 2g-2+n , where the function Q(α) is defined by (-1) a S n-2-2a ((cotan(πα i ) 1≤i≤n-1 ) .

Q(α) = (2i) 2g n-1 i=1 1 -e 2iπαi 2 •

  for all 1 ≤ i ≤ n | • |: size of a vector, or cardinal of a set ∆ g,n /∆ + g,n ⊂ R n : vectors/positive vectors of size 2g -2 + n Several families of graphs will be defined in the text, here is a diagram summarizing their place of definition as well as their interplay: are maps defined by forgetting part of the data defining a class of graphs).

Figure 2 .

 2 Figure 2. Graphs of the functions α 1 → (-1) g v(α 1 , 2g -α 1 ) (top), and α 1 → Vol(α 1 , 2g -α 1 ) (bottom), for g = 1 (left), and g = 2 (right).

Step 4 :

 4 Computation of m D for k-star graphs. The fact that m D = k |E(Γ)| m(Γ, β) for an irreducible component of the stratum associated to a k-star graph is a direct consequence of the following Lemma. Lemma 3.13. Let E ⊂ Pol(α), and 1 ≤ i ≤ n. Let Γ ∈ Star k g,n,i . If y is a point of PΩ(Γ, k, E), then there exists an open neighborhood V in PΩ(Γ, k, E), a disk ∆ in C containing 0 and a morphism ι : V × ∆ × G(Γ, t) → PΩ(α, k, E) satisfying:

  ζ e a kβ(e)-th root of unity. This determine an element ζ ∈ e∈E(Γ) U kβ(e) . With this datum, we construct a family of curves C ζ → ∆ × U (where ∆ is a disk parametrized by ) as follows. Around a node corresponding to e ∈ E(Γ), we define C ζ ( , u) as the solution ofz e,0 • z e,-1 = ζ e • lcm(Γ,k)/(kβ(e))in ∆ e,0 × ∆ e,-1 . Outside a neighborhood of the nodes, we define C ζ (u, ) C 0 (u) or C -1 (u). On this family of curves, we can define a k-differential byη ζ = z z e,0 z e,-1 = ζ e • lcm(Γ,t) . Then this differential is extended by η 0 or -lcm η -1 outside a neighborhood of the nodes. Neighborhood of the boundary. Two deformations (C ζ , η ζ ) and (C ζ , η ζ ) are isomorphic if and only if ζ = ρζ for some lcm(Γ, t)-th root of unity ρ. Therefore the morphism:

  -1) g+n-1 (2π) 2g-2+n n i=1 2 sin(πα i ) • (2g -2 + n)! (ω λ α ) 2g-3+n .

  a S n-2-2a ((cotan(πα i ) 1≤i≤n-1 ) , and S is the th symmetric function. Then we use the following identity (-1) n-1 sin(πα n ) = sin(πα 1 + . . . + πα n-1

If E is empty then there exists K B such that

Finally, if α has no positive integral entry, then A g,E (α, k) can be explicitly computed.

We begin by stating two lemmas.

Lemma 4.6. A space PΩ(α, k, E) • with • ∈ {∅, ab} is of dimension 0 if and only if one of the following situation holds:

(1) • = ab, g = 0, n = 1, r = p -2;

(2) • = ab, g = 0, n = 2, r = p -1;

(3) • = ∅, g = 0, n = 2, r = p -1.

(4) • = ∅, g = 0, n = 3, r = p. In the third and fourth cases, the entries of α are not integral. In the second and fourth cases, the residue map is trivial on the total space. These four spaces are irreducible.

Proof. We first assume that • = ∅. The dimension of PΩ(α, k, E) is equal to 2g -3 + n + p -r. However, p -r ≥ 0 implies that g = 0 or 1. We can see that the case g = 1 cannot occur as n = 1 would imply that k = 1. If g = 0 then 0 ≤ n ≤ 3 and p = r + n. The cases n = 0 is impossible from the condition |α| = 2g -3 + p -r. The case n = 1 is not possible either as it would imply that α is divisible by k. This let the two remaining cases.

The case of • = ab is treated in the same way. The vanishing of the residue map for the second case follows from the fact that the residues of an holomorphic 1-form sum up to 0.

then Γ is a bi-colored with two vertices satisfying either:

Proof. Let Γ be a graph satisfying the hypothesis of the Lemma. We begin by remarking that ψ 1 is a pull-back from the moduli space of curves. Therefore this integral vanishes if the push-forward of [D] along the forgetful morphism PΩ(α, k) → M g,n+p vanishes. This is the case if there are at least 2 vertices of level 0 (as in this case, the fibers of [D] on its image have positive dimension).

If we use the notation of the paragraph 3.5.2, then |V ab (E)| = 0 or 1. As we require dim(D) = dimζ Γ (D), inequality (5) implies that

We will finish the proof of the Lemma by studying separately all possibilities of configuration: 1 is adjacent to a vertex of level 0 or -1, and the same for i 0 .

Combining this identity with the fact that 1 -e 2iπαi 2 = 4 sin(πα i ) 2 , we deduce that

In order to define a volume form on T (α), we will use the holonomy character. First we assume that α / ∈ N n . The form ν λ α depends continuously on the parameters λ. Thus, it defines a form in 2(2g-3+n) Ω(T (α) hol * Ω(U 2g ) .

Therefore the form

(where ν U 2g is the Haar volume form) is a volume form on T (α). This form is invariant under the action of the mapping class group on T (α) (see [START_REF] Veech | Flat surfaces[END_REF], Theorem 13.14) and thus defines a volume form on the moduli space M(α).

Case of integral α. If α ∈ N n , then we denote by T (α, 1) the pre-image of M(α, 1) by the quotient morphism T (α) → M(α). Veech showed that the holonomy character morphism hol α is a submersion outside T (α, 1). Therefore the construction of the volume form ν α for non-integral values of α also gives a continuous volume form ν α on M(α) \ M(α, 1). Therefore, we define the volume of M(α) as the volume of M(α) \ M(α, 1) for integral values of α.

Reducing to moduli spaces of k-differentials. Let α ∈ ∆ +

g,n ∩ \Z n . To compute the volume V g,n (α), we chose a sequence of sets (E ) ∈N ⊂ U 2g that equidistributes (for the Haar measure of U 2g ) as goes to infinity. Then the sequence of measures

weakly converges to ν α as hol is a submersion. Now, we assume that α is in (Q\Z) n , and that kα is integral for some k 0 > 1. We set E = (U k ) 2g . Then for all k, we have h -1 (E k ) PΩ(α, k), and the identification of line bundles:

By [START_REF] Costantini | The area is a good metric[END_REF], we have the equality:

where ω α is the curvature form of h α . In particular

may be explicitely computable by Proposition 4.1, thus finishing the proof of Theorem 1.2. Now using Lemma 5.2 and Theorem 1.3 we get the equality:

v(α). (11) 5.5. Finiteness of the volume function. We finish here the proof of Theorem 1.1. Proposition 4.9 implies that the function Vol admits a continuous extension to ∆ + g,n (that we denote by the same letter).

Lemma 5.3. The function Vol is lower semi-continuous, and Vol ≤ Vol.

Proof. Let α 0 be a point of ∆ + g,n . Let K be a compact in M g,n . The function α → ν α (K) is continuous as ν α is a volume form that depends continuously on α. Thus, we have:

End of the proof of Theorem 1.1. We have seen that Vol = Vol on a dense set of of values (see formula (11)) and that V g,n is continuous.

Let > 0. We denote by U ⊂ ∆ + g,n the set of vectors α such that Vol(α) > Vol-. This set is open (as Vol is lower semi-continuous) and dense (as it contains a dense subset). Now if we denote by U 0 the set of vectors α such that Vol(α) = Vol(α), then we have

which is a countable intersection of sets whose complement is of measure 0. Therefore the complement of U 0 is of measure 0.