VVOLUMES OF MODULI SPACES OF FLAT SURFACES
ADRIEN SAUVAGET
Abstract.
We study the moduli spaces of flat surfaces with prescribed conicalsingularities. Veech showed that these spaces are diffeomorphic to the modulispaces of marked Riemann surfaces, and endowed with a natural volume formdepending on the orders of the singularities. We show that the volumes ofthese spaces are finite. Moreover we show that they are explicitely computableby induction on the Euler characteristics of the punctured surface for almostall orders of the singularities. The proof relies on the computation of thelarge k asymptotics of intersection numbers on moduli spaces of k -canonicaldivisors. This analysis was made possible by recent progress in the study ofthe intersection theory of the universal Jacobian (see [BHPSS20]). Contents
1. Introduction 12. Higher double ramification cycles 53. Local structure of the boundary of P Ω kg,n ( µ ) 94. Flat recursion 175. From intersection theory to volumes 26References 301. Introduction
Moduli spaces of flat surfaces. A marked flat surface with conical singular-ities (or flat surface for short in the text) is the datum of a marked compact surface( C, x , . . . , x n ) and a flat metric η on C \ { x , . . . , x n } such that the neighborhoodof x i is isomorphic to a cone with angle 2 πα i for some α i >
0, for all 1 ≤ i ≤ n .The genus of the surface satisfies the following Gauss-Bonnet formula:2 g ( C ) − n = n X i =1 α i . Two such surfaces (
C, x , . . . , x n , η ) and ( C , x , . . . , x n , η ) are isomorphic if thereexists a diffeomorphism φ : C → C such that: φ ( x i ) = x i for all 1 ≤ i ≤ n, and φ ∗ η is proportional to η .Let g, and n be non-negative integers such that 2 g − n >
0. We denoteby b ∆ g,n ⊂ R n the set of vectors whose entries sum up to 2 g − n and by∆ g,n = b ∆ g,n ∩ R n> . For any α ∈ ∆ g,n , we denote by M g,n ( α ) the moduli space of Date : April 8, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Moduli spaces of differentials, intersection theory, flat surfaces. a r X i v : . [ m a t h . AG ] A p r ADRIEN SAUVAGET marked flat surfaces with angles 2 πα 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 [Thu98] and [Tro86]in genus 0, and [Vee93] in general). Besides, Veech showed that this moduli spaceis endowed with a natural volume form ν α (see Section 5 for conventions) thusdefining the flat volume function : V g,n : ∆ g,n → R ≥ ∪ ∞ α ν α ( M g,n ) . The main problem of the text is: is the the function V g,n finite? can we computeit? (see Theorems 1.1 and 1.2).1.2. Flat surfaces with finite holonomy.
In the paper, the notation U standsfor U (1), the group of complex vectors of module 1. For k ≥
1, the notation U k stands for the group of k th -roots of unity.Let k ∈ Z > , and µ = ( m , . . . , m n ) ∈ Z n , be such that | µ | = k (2 g − n ). A k -canonical divisor of type µ is a marked curve ( C, x , . . . , x n ) satisfying: ω ⊗ k log ’ O ( m x + . . . + m n x n ) , where ω log = ω C ( x + . . . + x n ). We denote by M kg,n ( µ ) the moduli space of k -canonical divisors. It is an algebraic sub-stack of M g,n .If we assume that µ is positive, then C \{ x , . . . , x n } is endowed with a canonicalflat metric that has conical singularity πm i k at x i for all 1 ≤ i ≤ n . The holonomycharacter of this flat metric π ( C \ { x , . . . , x n } , ? ) → U has value in the set of k th-roots of unity. Conversely any flat surface with finiteholonomy character is obtained from a pluricanonical divisor. Therefore the modulispace M kg,n ( µ ) can also be defined as the subspace of M g,n ( µ/k ) of flat surfaceswith holonomy valued in U k . Figure 1.
By gluing the couples of edges e i and e i onthe two polygons above, we obtain equivalent flat surfaces in M , (2 / , / M , (2 , M kg,n ( µ ) is also equipped with a volume form, and we denote by V ( M kg,n ( µ )) the volume of the total space. In order to compute the function V g,n ,we will use two facts. OLUMES OF MODULI SPACES OF FLAT SURFACES 3 (1) Any flat surface can be approximated by flat surfaces with finite holonomy.I.e. for all k ∈ Z > , the moduli space M k kg,n ( k µ ) sits in M ( µ/k ), and thesemoduli spaces equidistribute for large values of k , i.e.lim k →∞ V ( M kk g,n ( k µ ))( kk ) g = V g,n ( µ/k ) . (2) The volume of M kg,n ( µ ) can be computed as an intersection number on analgebraic compactification of this moduli space.A large part of the paper is devoted to moduli spaces of moduli spaces of k -canonical divisors. We will describe the local structure of the boundary of standardcompactification of these moduli spaces in Section 3. We use this description to com-pute relations between certain intersection numbers of moduli spaces of k -canonicaldivisors for fixed values of k (Theorem 3.11). These relations will involve doubleramification cyles (see the next sub-section and Section 2). In Section 4, we showthat these intersection numbers converge for large values of k .In Section 5, we use the above two points to compute the function V g,n for ratio-nal conical singularities. The first point reduces the computation of the volumes ofmoduli spaces of flat surfaces to volumes of moduli spaces of k -canonical divisors.The second point and the description of the large k limit of the intersection theoryof moduli spaces of k -canonical divisors provide a procedure to compute the func-tion V g,n by induction on g and n for rational conical singularities (Theorem 1.1).The partial extension of these results to real conical singularities (Theorem 1.2)follows from the lower semi-continuity of the function V g,n and the study of thewall and chamber structure of the function V g,n for rational conical singularities.1.3. Double ramification cycles and flat recursion.
We denote by M g,n the moduli space of stable curves . Let ( k, µ ) be a vector in ∈ Z > × Z n such that µ/k ∈ b ∆ g,n . In [Sch16], Schmitt defined a class DR g,n ( k, µ ) ∈ H g ( M g,n , Q ),satisfying: DR g,n ( k, µ ) |M g,n = [ M kg,n ( µ )] ∈ H g ( M g,n ) , if µ/k / ∈ Z n> (here [ · ] stands for the Poincaré-dual class). This class is called a higher double ramification cycle . It can be constructed via the intersection theoryof the universal Jacobian on M g,n (see [HS19]), and it has recently been computedin terms of the generators of the tautological rings of the moduli spaces of curves(see [BHPSS20]).For fixed values of g and n , the double ramification cycles are polynomials ofdegree 2 g in k and the entries of µ (see [PZ]). Therefore, we define the followingfunction A ∞ , g,n : b ∆ g,n → R α Z M g,n ( − α ψ ) g − n · g DR g,n (1 , α ) , where g DR g,n is the degree 2 g part of the polynomial DR g,n and ψ is the Chernclass of the cotangent line at the first marked point of the curve. The function A ∞ , g,n is a polynomial of degree 4 g − n . ADRIEN SAUVAGET α α β , β , β , α Figure 2.
Example of star graph in Star , . The domain ∆(Γ , α )is the set of positive triples ( β , , β , , β , ) satisfying β , = 4 − α , and β , + β , = 7 − α . It is empty if α > α > Star graphs. A star graph Γ is a specific type of stable graph (see Section 2for definitions) determined by the following datum: • a vector ( g , g , . . . , g ‘ ) of non-negative integers of positive length ( ‘ + 1); • a vector of positive integers ( e , . . . , e ‘ ) that sum up to e and such that g = e − ‘ + P i ≥ g i . • a partition [[1 , n ]] = L t . . . t L ‘ , with n i = Card( L i ), and satisfying.2 g i + n i + e i > ≤ i ≤ ‘ .We denote by Star g,n the set of star graphs and by Star g,n (1) the set of star graphssuch that 1 ∈ L . Given a star graph Γ and α ∈ ∆ g,n , we denote by ∆(Γ , α ) ⊂ R e> the set of vectors β = ( β , , . . . , β ,e , β , , . . . , β ‘,e ‘ ) satisfying X i ∈ L j α i + n X i =1 β j,i = 2 g j + n j + e j for all 1 ≤ j ≤ ‘. Note that this domain is of dimension h (Γ) = e − ‘ .1.3.2. Flat Recursion.
Here we define a family of piecewise polynomials v g,n :∆ g,n → R ≥ inductively on g and n . The base of the induction is given by v , ( α ) = 1.If 2 g − n >
0, and Γ ∈ Star g,n (1), then we define the contribution of Γ to v g,n by: v (Γ , α ) = Z β ∈ ∆(Γ ,α ) A ∞ , g ,n + e (( α i ) i ∈ L , − β , , . . . , − β ‘,e ‘ ) × (cid:16) Y ≤ j ≤ n ≤ i ≤ e j β i (cid:17) × (cid:16) ‘ Y j =1 v g i ,n i + e i (( α i ) i ∈ L j , β j, , . . . , β j,e j ) (cid:17) . Then we define v g,n ( α ) = X Γ ∈ Star g,n (1) v (Γ , α )Aut(Γ) . (1)This formula will be called the flat recursion relation (FR) by analogy with the topo-logical recursion that computes in particular Weil-Petersson volumes (see [Mir07]),and the volume recursion for Masur-Veech volumes (see [CMSZ19]). The relation OLUMES OF MODULI SPACES OF FLAT SURFACES 5 between of topological recursion and flat recursion will be investigated in a subse-quent work.The two main theorems of the paper relate the functions v g,n and V g,n . Theorem 1.1.
For all g ≥ , n ≥ , and α ∈ ∆ g,n ∩ ( Q \ Z ) n , the following equalityholds: V g,n ( α ) = 4 · ( − g + n − (2 π ) g − n (cid:0) Q ni =1 πα i ) (cid:1) · (2 g − n )! · v g,n ( α ) . (2) Theorem 1.2.
For all g ≥ , n ≥ , the right-hand side of formula (2) can beextended to a continuous function b V g,n on ∆ g,n . The function V g,n is finite, lowersemi-continuous, and for almost all α ∈ ∆ g,n , we have V g,n ( α ) = b V g,n ( α ) . Previous works. If g = 0 , and α ∈ ]0 , n , then the volume form ν α had beenintroduced in the 80’s by Deligne-Mostow and Thurston (see [DM86], [Thu98],and [Tro86]). In this case, the space M ,n ( α ) has a hyperbolic structure and thevolume is related to the Euler characteristics. The Euler characteristics has beencomputed explicitly by McMullen (see [McM17]). An alternative proof of his for-mula has been given by Koziarz and Nguyen using intersection theory (see [KN18]).In both papers, the volume of the moduli space is computed for all values of α intheir domain of definition (and not “for almost all”). This follows from the descrip-tion by Thurston of the metric completion of the moduli space in terms of conemanifolds.We expect that Theorem 1.2 is valid for all values of α . A way to prove thisresult, would be to apply a version of the dominated convergence theorem. To doso, one would require a precise description of ν α along degenerating families of flatsurfaces.If g = 1 and n = 2, then the total space M , ( α ) is not hyperbolic. However,it admits a foliation by hyperbolic surfaces. Ghazouani and Pirio computed theEuler characteristics of the quasi-projective leaves of this foliation. They use thedensity of these special leaves in M , ( α ) to interpret some limit of their Eulercharacteristics as a volume of M , ( α ) (see [GP16], Section 6.4). Acknowledgment.
This paper is the result of a question by Selim Ghazouani. Hepresented to me the equidistribution of pluricanonical divisors in moduli spaces offlat surfaces and pushed me to read the seminal paper of Veech despite its technical-ity. I would like to thank David Holmes both for useful discussions on the residuemorphism, and for the hospitality of his office in Leiden University. I would alsolike to thank Gaëtan Borot, Dawei Chen, Alessandro Giachetto, Martin Möller,Gabriele Mondello, Johannes Schmitt, and Dimitri Zvonkine for useful conversa-tions and advices. The research was partially supported by the Dutch ResearchCouncil (NWO) grant 613.001.651.2.
Higher double ramification cycles
We recall here the notion of twisted k -canonical divisors as well as the definitionof higher DR cycles as a sum on simple star graphs. Besides, we present theconjectural expression of DR cycles in terms of Pixton’s classes. Finally we showthat the non-trivial star graphs contribute trivially to the large k limit of DR cycles. ADRIEN SAUVAGET
Twisted graphs.
Let g and n be such that 2 g − n > Definition 2.1. A stable graph is the datum ofΓ = ( 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. thecardinal of φ − { v } . • For all vertices v we have 2 g ( v ) − n ( v ) > • The genus of the graph is defined as h (Γ) + P v ∈ V g ( v ), where h (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 such that all edges are between v and another ( outer ) vertex (this definition ofstar graph is equivalent to the one given in Section 1.3). Definition 2.2. A twist on a stable graph Γ is a function t : H → R satisfying: • For all v ∈ V , we have X h ∈ φ − ( v ) t ( h ) = 2 g ( v ) − n ( v ) . • If ( h, h ) is an edge of Γ, then we have I ( h ) = − I ( h ). • If ( h , h ) and ( h , h ) are edges between the same vertices v , v , then h ≥ ⇔ h ≥
0. In which case we denote v ≥ v . • The relation ≥ defines a partial order on the set of vertices.If k ∈ Z > , then a k -twist is a function t : H → Z such that t/k is a twist. Wedenote by TW g,n and TW kg,n the set of twisted graphs and k -twisted graphs , i.e. theset of graphs endowed with a ( k )-twist. Definition 2.3.
The multiplicity of a ( k )-twisted graph is the real number m (Γ , t ) = Y ( h,h ) ∈ Edges p − t ( h ) t ( h ) . Definition 2.4.
We say that a k -twisted graph (Γ , t ) is a simple star graph if Γ isa star graph and all twists half-edges adjacent to an outer vertex are positive anddivisible by k . We denote by SStar kg,n ⊂ TW kg,n the set of simple star graphs.2.2. Twisted pluricanonical divisors.
Let µ = ( m , . . . , m n ) be a vector ofintegers satisfying | µ | = k (2 g − n ). Definition 2.5. A k -twisted graph is compatible with µ if t ( i ) = m i for all 1 ≤ i ≤ n . We denote by TW kg,n ( µ ) and SStar kg,n ( µ ) the set of k -twisted and simple stargraphs compatible with µ .Let (Γ , t ) ∈ TW kg,n ( µ ). The stable graph Γ determines the stack M Γ = Y v ∈ V M g ( v ) ,n ( v ) , OLUMES OF MODULI SPACES OF FLAT SURFACES 7 and M Γ its compactification by stable curves. Besides Γ determines morphism ζ Γ : M Γ → M g,n defined by compositions of gluing morphisms.The k -twist t determines a sub-stack M Γ ,t of M Γ defined by M Γ ,t = Y v ∈ V M kg ( v ) ,n ( v ) ( µ ( v )) , where µ ( v ) is the vector of twists of half-edges adjacent to v , i.e. µ ( v ) = ( t ( h )) h ∈ φ − ( v ) .If (Γ , t ) is a simple star graph, then we denote M s Γ ,t = M kg ( v ) ,n ( v ) ( µ ( v )) × Y v ∈ V Out (Γ) M g ( v ) ,n ( v ) (cid:18) µ ( v ) k (cid:19) , where Out(Γ) is the set of outer vertices of Γ. Definition 2.6.
The
DR cycle associated to ( k, µ ) is the class in ∈ H ∗ ( M g,n , Q )defined by DR g,n ( k, µ ) = X (Γ ,t ) ∈ SStar kg,n ( µ ) m (Γ , t ) | Aut(Γ) | · k | V Out | · ζ Γ ∗ [ M s Γ ,t ] , (3)where M s Γ ,t is the closure of M s Γ ,t in M Γ .2.3. Expression of higher DR cycles.
Let r >
0. A r/k -spin structure of type µ on a marked curve ( C, x , . . . , x n ) is a line bundle L → C with an identification L ⊗ r ’ ω ⊗ kC ( − m x . . . − m n x n ) . We denote by M k/rg,n ( µ ) the moduli space of such r/k -spin structures.This moduli space admits a standard compactification M r/kg,n ( µ ) which is clas-sically constructed by using the notion of orbifold curve. There exists a universalcurve π : C r/kg,n ( µ ) → M r/kg,n ( µ ) and a universal line bundle L → C r/kg,n ( µ ). Besides,we have a forgetful map of the spin structure (cid:15) : M r/kg,n ( µ ) → M g,n . We will beinterested in the following classes c g,n ( k, µ, r ) = r · (cid:15) ∗ (cid:0) c g ( R π ∗ L − R π ∗ L ) (cid:1) ∈ H g ( M g,n , Q ) . These classes can be computed by using the Grothendieck-Riemann-Roch Formula(see [Chi08]).
Proposition-Definition 2.7. (see [JPPZ17] ) The class c g,n ( k, µ, r ) depends poly-nomialy on r , for r large enough. The Pixton class P g,n ( k, µ ) is defined as thedegree of this polynomial. The most important property of the Pixton class will be its polynomial depen-dence in the parameters.
Proposition 2.8. (see [PZ] ) The class P g,n ( k, µ ) is a polynomial of degree g in k and the entries of µ . A corner stone of the present work is the following result.
Proposition 2.9. (Theorem 60 of [BHPSS20] ) Let ( k, µ ) be pair such that µ/k is not a postive integral vector. The equality P g,n ( k, µ ) = DR g,n ( k, µ ) holds in H g ( M g,n , Q ) . ADRIEN SAUVAGET
Large k limits. We show here that the classes [ P g,n ( k, µ )] and [ M g,n ( k, µ )]are equivalent for large values of k . Notation 2.10.
The set D g,n (respectively b D g,n ) is defined as the subset of Z > × Z n of vectors ( k, µ ) such that µk ∈ ∆ g,n (respectively in b ∆ g,n ). Lemma 2.11.
The function diff g,n : b D g,n → H ∗ ( M g,n , Q )( k, µ ) DR g,n ( k, µ ) − [ M kg,n ( µ )] is a piece-wise polynomial on the domain of pairs ( k, µ ) satisfying k ≥ and | µ | = k (2 g − n ) . Moreover, the local polynomials are of degree smaller than g .Proof. We proceed in two steps. In the first step, we assume that µ/k / ∈ Z n> , andin the second step we extend the result to all pairs ( k, µ ). Step 1.
We prove this step by induction on g . The fact that this statement holdsfor g = 0 is obvious as [ M k ,n ( µ )] = DR ,n ( k, µ ) = 1 for all choices of n, k, and µ .Let g, n ≥ g,n be the set of all simple star graphs (for all choicesof k ). Let (Γ , t ) and (Γ , t ) be k - and k -simple star graphs. We say that thesegraphs are equivalent if: • the underlying graphs are isomorphic; • for all outer vertices v , we have µ ( v ) /k = µ ( v ) /k .The set SStar g,n / ∼ is finite. Indeed to define such a class of equivalence, one needto chose a star graph and a partition of 2 g ( v ) − n ( v ) for all outer vertices v .Conversely, a simple star graph (Γ , t ) is fully determined by its equivalence classin Star g,n / ∼ , the value of k , and the value of the twist of the legs on the centralvertex.All simple star graphs occurring in the expression 3 are two-by-two non-equivalent.Thus we can write DR g,n ( k, µ ) = X [Γ ,t ] ∈ SStar g,n / ∼ F [Γ ,t ] ( k, µ ) , where the function F [Γ ,t ] has value 0 if the class [(Γ , t )] does not occur in theexpression of DR g,n ( k, µ ) and m (Γ ,t ) | Aut(Γ) | k | V Out | ζ Γ ∗ ( M s Γ ,t ) otherwise.For all [Γ , t ], the function F [Γ ,t ] depends only on k and the entries m i of µ suchthat the label i is adjacent to the central vertex. Indeed, for any outer vertex v ,the class [ M g ( v ) ,n ( v ) ( µ ( v ) /k )] is a constant in the equivalence class of the aboveequivalence relation as µ ( v ) /k is constant by definition. Thus the function F [Γ ,t ] depends only on the value of m (Γ , t ) k | V Out | [ M kg ( v ) ,n ( v ) ( µ ( v ))] , where it is non zero. Now m (Γ , t ) = (constant · k | Edges | ) in a fixed equivalence class.Thus the coefficient m (Γ ,t ) k | V Out | is of the form: constant · k h (Γ) .Finally we remark that the outer vertices of a simple star graph have positivegenus. Thus, for any non-trivial simple star graph, we have g ( v ) + h (Γ) < g .Therefore, we can apply the induction hypothesis. It implies that F [Γ ,t ] is piece-wise polynomial of degree h (Γ) + 2 g ( v ) < g . All in all, we get thatDR kg,n ( µ ) = [ M kg,n ( µ )] + F OLUMES OF MODULI SPACES OF FLAT SURFACES 9 where F is a piece-wise polynomial of degree smaller than 2 g . Proposition 2.9 andProposition 2.8 imply that DR kg,n ( µ ) is a polynomial of degree 2 g . Thus [ M kg,n ( µ )]is piece-wise polynomial of degree 2 g and its degree 2 g part is the same as the oneof P g,n . Step 2.
In order to finish the proof of the lemma, we work in M g,n +1 . Wedenote by π : M g,n +1 → M g,n the forgetful morphism of the marking n + 1. Wealso denote by δ ,n +1 the boundary divisor of M g,n +1 defined by the stable graphwith the two vertices of genus 0 and g , one edge, and such that the vertex of genus0 carries only the legs 1 and n + 1. We will use the following identity:[ M kg,n ( m , . . . , m n )] = π ∗ (cid:16) δ ,n +1 · [ M kg,n +1 ]( m − , m . . . , k + 1)] (cid:17) . We define the following polynomialDR g,n ( k, µ ) = π ∗ ( δ ,n +1 · DR g,n ( k, m − , m , . . . , k + 1)) . Using the first step of the proof, the difference [ M kg,n ( µ )] − DR g,n ( k, µ ) is a piecewisepolynomial with degree smaller than 2 g for all values of k and µ . Thus DR g,n ( k, µ ) − DR g,n ( k, µ ) is a polynomial of degree smaller than 2 g . Therefore the difference[ M kg,n ( µ )] − DR g,n ( k, µ ) is a piece-wise polynomial of degree smaller than 2 g for allvalues of ( k, µ ). (cid:3) We have seen that the class P g,n ( k, µ ) ∈ is a polynomial in k and µ of degree 2 g .We define the polynomial P ∞ g,n in Q [ α , . . . , α n ] ⊗ H g ( M g,n , Q ) by P ∞ g,n = e P g,n (1 , α ) , where e P g,n is the homogeneous part of P g,n of degree 2 g . Lemma 2.12.
Let ( k ‘ , µ ‘ ) ‘ ∈ N be a sequence of vectors in Z > × Z n , such that | µ ‘ | = k ‘ (2 g − n ) , k ‘ tends to infinity as ‘ goes to infinity, and µ ‘ /k ‘ converges.Then, we have k g‘ [ M k ‘ g,n ( µ ‘ )] , k g‘ [ P g,n ( k ‘ , µ ‘ )] = P ∞ g,n ( µ ‘ /k ‘ ) + O ‘ →∞ (cid:16) k ‘ (cid:17) . Proof.
This is a corollary of Lemma 2.11. (cid:3) Local structure of the boundary of P Ω kg,n ( µ )In this section we extend partially the result of [Sau19] for strata of abeliandifferentials to strata of k -differentials. We describe the neighboorhod of a genericpoint in the boundary of these moduli spaces and use this description to computea series of relations in the cohomology of these moduli spaces.3.1. Incidence variety compactification.
Let g and n be nonnegative integerssuch that 2 g − n >
0. Let k >
0, and µ ∈ Z n be such that | µ | = k (2 g − n ). The moduli space of k -differentials of type µ is the moduli space of objects( C, η, x , . . . , x n ) where η is a (possibly meromorphic) k -differentials on C suchthat ord x i ( η ) = m i − k for all 1 ≤ i ≤ n and such that η is not the k th power of ameromorphic 1-differential. We decompose the vector µ as: µ = Z ( µ ) − P ( µ ) , where the entries of Z ( µ ) is the vector obtained by keeping all positive entries of µ and sending the others to 0. We denote by π : C g,n → M g,n the universal curvetogether with the sections σ , . . . , σ n associated to the markings. If P is a vectorof n nonnegative integers, then we denote by p : V Ω kg,n ( P ) → M g,n the total spaceof the vector bundle π ∗ ω ⊗ k M g,n / C g,n (cid:0) n X i =1 p i σ i ) ! . There is a natural embedding Ω kg,n ( µ ) , → V Ω kg,n ( P ( µ )). This embedding preservesthe C ∗ -action defined by scaling the k -differential, thus it defines an embedding P Ω kg,n ( µ ) , → P V Ω kg,n ( P ( µ )). Definition 3.1.
The incidence variety compactification is the Zariski closure of P Ω kg,n ( µ ) in P V Ω kg,n ( P ). It is denoted by P Ω kg,n ( µ ).Finally, if µ is divisible by k , then we define by Ω k, ab g,n ( µ ) , → V Ω kg,n ( P ( µ )) thespace of k -differentials obatined as k th power of a abelian differentials with singu-larities prescribed by µ . Besides, we denote by P Ω k, ab g,n ( µ ) its projectivization and P Ω k, ab g,n ( µ ) the incidence variety compactification.In the next sections, we recall the description of the boundary of the incidencevariety compactification by [BCGGM16].3.2. Canonical cover.
Let ( C, ( x i ) , η ) be a k -differential in Ω kg,n ( µ ). There existsa canonical cyclic ramified cover of degree k , f : b C → C . This covering is definedby b C = { ( x, v ) ∈ T ∨ C , such that v k = η } The covering curve b C carries a natural differential v such that such that v k = η .Each point 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 determinedby µ . Therefore a quadruple ( g, n, k, µ ) determines a triple ( b g, b n, b µ ) such that wehave an embedding Ω kg,n ( µ ) , → Ω b g, b n ( b µ ) . 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 .3.3. Residues.
We denote by Pol k ( µ ) ⊂ [[1 , n ]] the set of indices i such that m i ≤ k | m i . Let ( C, ( x i ) , η ) be a k -differential in Ω kg,n ( µ ) and i ∈ Pol k ( µ ). Let f : b C → C be the canonical cover.The point x i has k preimages under f . These points are poles of order ( m i − k ) /k of the canonical differential v on b C and the residues at two such points differ bya k -th root of unity. The residue at x i is the k th power of any of these residuesand we denote it by res x i ( η ). We denote by res i : Ω kg,n ( µ ) → C the i th residuemorphism , i.e. the morphism defined by mapping η to res x i ( η ).If E ⊂ Pol k ( µ ), then we denote by Ω kg,n ( µ, E ) the substack of Ω kg,n ( µ ) of dif-ferentials with vanishing residues at x i for i ∈ E . We denote by P Ω kg,n ( µ, E ) itsprojectivization and by P Ω kg,n ( µ, E ) the closure of P Ω kg,n ( µ, E ) in P Ω kg,n ( µ ). Onceagain we call this space incidence variety compactification. OLUMES OF MODULI SPACES OF FLAT SURFACES 11 If i ∈ Pol k ( µ ) \ E , then the morphism res i is a section of the line bundle O (1) → P Ω kg,n ( µ, E ) that extends to the boundary of the incidence variety compactification. Lemma 3.2.
The section res i vanishes with multiplicity k along P Ω kg,n ( µ, E ∪ { i } ) .Proof. If k = 1, then the residue morphism is a submersion, thus the vanishingmulitiplicity of res i along P Ω kg,n ( µ, E ∪ { i } ) is 1 (see Corollary 3.8 of [Sau19]).For higher values of k , we use the canonical cover to embed locally Ω kg,n ( µ ) , → Ω b g, b n ( b µ ) (cid:14) ( U k ). Then the residue at x i is the k -th power of the residue at any of themarked preimages of the canonical cover. The residue morphism is a submersionalong the image of Ω kg,n ( µ ) in Ω b g, b n ( b µ ) . ( U k ). Therefore the residue morphism at x i vanishes with multiplicity k . (cid:3) 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 , t ) is a function ‘ : V (Γ) → Z ≤ such that ( v ≤ v ) ⇒ ( ‘ ( v ) ≤ ‘ ( v )) and such that ‘ − (0) is non-empty. Definition 3.4. A k -decorated graph is the datum of(Γ , t, ‘, V ab ) , where (Γ , t, ‘ ) is a k -twisted graph with a level function and V ab is a subset of V (Γ)such that: k | t ( h ) for all half-edges h adjacent to some v ∈ V ab .We note by Dec kg,n the set of k -decorated graphs and by Dec kg,n ( µ ) the set of k -decorated graphs compatible with µ (i.e. the underlying twisted graph is compatiblewith µ ). Definition 3.5. A k -bi-colored graph is the datum of a k -decorated graph suchthat: • the image of the level function is { , − } ; • all edges are between a level 0 vertex and a vertex of level − 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, andthe outer vertices are of level 0; • If a vertex of level 0 is in V ab , then it has only one edge to the centralvertex. • If an outer vertex v has an edge to the central vertex with a twist divisibleby k , then v ∈ V ab .We denote respectively by Star kg,n ⊂ Bic kg,n ⊂ Dec kg,n the sets of k -star graphsand of k -bi-colored graphs. We also denote by Star kg,n ( µ ) and Bic kg,n ( µ ) the setof such graphs compatible with µ . Note that bi-colored graphs are completelydetermined by (Γ , t, V ab ) and star graphs are determined by (Γ , t ). Thus we willsimplify the notation in this sense. Notation 3.7.
Let (Γ , t, V ab ) be a k -bi-colored graph. For all edge e = ( h, h ), wedenote t ( e ) = p − t ( h ) t ( h ). We introduce the notationlcm(Γ , t, V ab ) = lcm { t ( e ) } e ∈ E (Γ) ,G (Γ , t, V ab ) = (cid:16)Y e ∈ E (Γ) Z (cid:14) t ( e ) Z (cid:17). ( Z (cid:14) lcm(Γ , t ) Z ) . Stratum associated to a k -decorated graph. Let E ⊂ Pol k ( µ ). In [BCGGM16],the authors showed that the spaces P Ω g,n ( µ, E ) admit a natural stratification in-dexed by k -decorated graph. We recall part of their construction here.3.5.1. Strata associated to bi-colored graphs.
Let (Γ , t, ‘, V ab ) be a k -decorated graphin Dec kg,n ( µ ). From such a datum one can construct a space Ω(Γ , t, ‘, V ab , E )whose projectiviaztion sits in the boundary of the incidence variety compactification P Ω g,n ( µ, E ). In this text we will only recall the definition of Ω(Γ , t, V ab , E ) for bi-colored graphs. Indeed, we will be interested in codimension 1 loci of P Ω g,n ( µ, E ),and we have the following lemma. Lemma 3.8.
If we denote by P e Ω g,n ( µ, E ) the union of the the projectivized bound-aries associated to • decorated graphs with level and or edge, • and bi-colored graphs,then P Ω g,n ( µ, E ) \ P e Ω g,n ( µ, E ) is of co-dimension in P Ω g,n ( µ, E ) .Proof. This lemma follows from the dimension computation of Section 6 of [BCGGM16].The co-dimension of the stratum associated to a decorated graph with N level is atleast N −
1. Besides, the horizontal nodes (nodes between two components of thesame level) can be smoothed independently from the other nodes. Thus, if a graph(Γ , t, V ab , ‘ ) has N horizontal edges, then it defines a stratum of co-dimension atleast N + N −
1. Therefore a graph defining a stratum of P Ω g,n ( µ ) of co-dimension1 has either one horizontal edge and 1 level, or 2 levels and no horizontal edges(bi-colored graph). (cid:3) Let (Γ , t, V ab ) be a bi-colored graph. For i = 0 or −
1, we denote by e Ω(Γ , t, V ab , E ) i = (cid:16) Y v ∈ ‘ − ( i ) v / ∈ V ab Ω kg ( v ) ,n ( v ) ( µ ( v ) , E ( v )) (cid:17) × (cid:16) Y v ∈ ‘ − ( i ) v ∈ V ab Ω k, ab g ( v ) ,n ( v ) ( µ ( v ) , E ( v )) (cid:17) , where for all v ∈ V (Γ): • µ ( 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 , t, V ab , E ) = e Ω(Γ , t, V ab , E ) .For the level −
1, we define Ω(Γ , t, V ab , E ) − as the substack of e Ω(Γ , t, V ab , E ) − of k -differentials satisfying the global resiude condition of ([BCGGM16], Defini-tion 1.4). We dot not state the precise definition of the global residue condition forgeneral bi-colored graphs, as we will only need to know that Ω(Γ , t, V ab , E ) − is asub-stack of e Ω(Γ , t, V ab , E ) − . However, at the end of the section we describe it for k -star graphs, as it will be required further in the text. Definition 3.9.
The stratum Ω(Γ , t, V ab , E ) is the productΩ(Γ , t, V ab , E ) × P Ω(Γ , t, V ab , E ) − . Moreover we denote by P Ω(Γ , t, V ab , E ) = P Ω(Γ , t, V ab , E ) × P Ω(Γ , t, V ab , E ) − and by P Ω(Γ , t, V ab , E ) its closure in Y i =0 , − P (cid:16) Y v ∈ ‘ − ( i ) V Ω kg ( v ) ,n ( v ) ( P ( µ ( v ))) (cid:17) . OLUMES OF MODULI SPACES OF FLAT SURFACES 13
Let ( C , ( x h ) , η ) × ( C − , ( x − h ) , η − ) ∈ P Ω(Γ , t, V ab , E ). We construct a nodalmarked curve ( C, x i , η ) by gluing markings of C and C − as prescribed by Γ. Wedefine a k -differential η on C by η | C = η and η | C − = 0. This constructiondefines a morphism ζ (Γ ,t,V ab ) : P Ω(Γ , t, V ab , E ) → P Ω kg,n ( µ, E ). The degree of anyirreducible component D of P Ω(Γ , t, V ab , E ) on its image is equal to:deg (cid:0) D/ζ (Γ ,t,V ab ) ( D ) (cid:1) = (cid:26) | Aut(Γ , t, V ab ) | if dim( D ) = dim( ζ (Γ ,t,V ab ) ( D )),0 otherwise . Global residue condition for k -star graphs. Let (Γ , t, V ab ) be a k -bi-coloredgraph. We denote by V ab ( E ) the set of vertices of Γ such that: • v ∈ V ab ∩ ‘ − (0); • E ( v ) = Pol k ( µ ( v )).The dimension count of Section 6 of [BCGGM16] gives the following inequalities: V ) − − V ab ( E )) ≤ dim( D ) − dim( ζ (Γ ,t,V ab ) ( D )) ≤ V ) − , t ) is a k -star graph, with central vertex v − . Then, we define the set E as the set of half-edges adjacent to v − and part of an edge to a vertex in V ab ( E ).Besides, we still denote by E ( v − ) the set of legs in E adjacent to v − . Finally wedenote by E − = E ∪ E ( v − ). With this notation, we have the isomorphism:Ω(Γ , t, V ab , E ) − ’ Ω kg ( v − ) ,n ( v − ) ( µ ( v − ) , E − ) , which is indeed a sub-stack of Ω kg ( v − ) ,n ( v − ) ( µ ( v − ) , E ( v − )).3.6. Relations in the cohomology of the P Ω kg,n ( µ ) . We will work with thefollowing cohomology classes in H ∗ ( P Ω kg,n ( µ ) , Q ). • For all 1 ≤ i ≤ n , we denote by ψ i = c ( L i ) ∈ H ( M g,n , Q ), where L i = σ ∗ i ω C g,n / M g,n is the cotangent line at the i th marked point. Wedenote by the same letter its pull-back in H ∗ ( P Ω kg,n ( µ ) , Q ). • ξ = c ( O (1)) ∈ H ∗ ( P Ω kg,n ( µ ) , Q ) . Notation 3.10.
For all 1 ≤ i ≤ n , we denote Bic kg,n ( µ, i ) and Star kg,n ( µ, i ) the setof k -star graphs such that the label i is adjacent to a vertex of level − E ⊂ Pol k ( µ ) and i ∈ Pol k ( µ ) \ E , then we denote by Bic kg,n ( µ, E, i ) andStar kg,n ( µ, E, i ) the set of graphs such that i is adjacent to either: • a vertex of level − • or a vertex v ∈ V ab such that for all i ∈ Pol k ( µ ), if i is adjacent to v then i ∈ E .The main purpose of the section is to prove the following proposition Theorem 3.11.
Let E ⊂ Pol k ( µ ) . For all (Γ , t, V ab ) ∈ Bic( µ ) , and all irreduciblecomponents D of P Ω(Γ , t, V ab , E ) , there exist an integer m D such that for all ≤ i ≤ n , we have: ξ + m i ψ i = X (Γ ,t,V ab ) ∈ Bic kg,n ( µ,i ) D ∈ Irr( P Ω(Γ ,t,V ab ,E )) m D | Aut(Γ , t, V ab ) | · ζ (Γ ,t,V ab ) ∗ ([ D ]);(5) and if i ∈ Pol k ( µ ) \ E , then we have: ξ = k [ P Ω kg,n ( µ, E ∪ { i } )] + X (Γ ,t,V ab ) ∈ Bic kg,n ( µ,E,i ) D ∈ Irr( P Ω(Γ ,t,V ab ,E )) m D | Aut(Γ , t, V ab ) | · ζ (Γ ,t,V ab ) ∗ ([ D ]) . (6) Moreover, if D is an irreducible component of P Ω(Γ , t, V ab , E ) for (Γ , t, V ab ) ∈ Star kg,n ( µ ) , and µ has only positive entries, then m D = m (Γ , t ) .Proof. Step 1: relation for fixed value of i . Let 1 ≤ i ≤ n . We consider the linebundle O (1) ⊗ L m i i → P Ω kg,n ( µ, E ). This line bundle has a natural section definedby s i : η m i th order of η at x i . This section does not vanish • on P Ω kg,n ( µ, E ); • on strata associated to decorated graphs with one level 0; • on strata associated to bi-colored graphs in Bic kg,n ( µ ) \ Bic kg,n ( µ, i ).Therefore, up to co-dimension 2 loci of P Ω kg,n ( µ ), the vanishing locus of s i is theunion of the irreducible component D ⊂ P Ω(Γ , t, V ab ) for (Γ , t, V ab ) in Bic kg,n ( µ, i ).Thus for each such D , there exists an integer m iD such that: ξ + m i ψ i = X (Γ ,t,V ab ) ∈ Bic kg,n ( µ,i ) D ∈ Irr( P Ω(Γ ,t,V ab ,E )) m iD | Aut(Γ , t, V ab ) | · ζ (Γ ,t,V ab ) ∗ ([ D ]) . (7)If i ∈ Pol( µ ) \ E , then we consider the line bundle O (1) and its section given by the i -th residue morphism. This section vanishes along P Ω kg,n ( µ, E ) with multiplicity k (Lemma 3.2). Besides, this section does not vanish identically on boundarycomponents associated to k -decorated graphs with one level neither on boundarycomponents associated to bi-colored graphs in Bic kg,n ( µ ) \ Bic kg,n ( µ, i ). Therefore,we have: ξ = k [ P Ω kg,n ( µ, E ∪ { i } )] + X (Γ ,t,V ab ) ∈ Bic kg,n ( µ,E,i ) D ∈ Irr( P Ω(Γ ,t,V ab ,E )) e m iD | Aut(Γ , t, V ab ) | · ζ (Γ ,t,V ab ) ∗ ([ D ]) , (8)where the e m iD are integers. Step 2: independence of ≤ i ≤ n . We will show that the numbers m iD can be cho-sen independently of i ∈ [[1 , n ]]. If D is of dimension smaller than dim( P Ω kg,n ( µ )) − m D = 0 thus, from now on we will only consider D of dimensiondim( P Ω kg,n ( µ )) − ≤ i = i ≤ n . Let D be an irreducible component of P Ω(Γ , t, V ab ) such that(Γ , t, V ab ) ∈ Bic kg,n ( µ, i ) ∩ Bic kg,n ( µ, i ). Let ∆ be an open disk of C parametrized by OLUMES OF MODULI SPACES OF FLAT SURFACES 15 (cid:15) . Let ∆ , → P Ω kg,n ( µ, E ) be a family of differentials such that the image of ∆ \ { } lies in P Ω kg,n ( µ, E ) while 0 is mapped to a generic point of D .Up to a choice of a smaller disk, there exists an integer ‘ and holomorphicfunctions f and f that do not vanish ∆ such that s i = (cid:15) ‘ f and s i = (cid:15) ‘ f (see the“necessary” part of Theorem 1.5 of [BCGGM16]). Thus s i and s i vanish with thesame multiplicity ‘ along (cid:15) = 0. Therefore the vanishing multiplicity of s i and s i along D are equal and the integers m iD can be chosen independently of 1 ≤ i ≤ n . Step 3: vanishing of residues.
Let 1 ≤ i ≤ n, and i ∈ Pol( µ ) \ E (not nec-essarily different). Let D be an irreducible component of P Ω(Γ , t, V ab ) such that(Γ , t, V ab ) ∈ Bic kg,n ( µ, i ) ∩ Bic kg,n ( µ, E, i ). We will show that m iD = e m i D .We chose a family ∆ , → P Ω g,n ( µ, E ) such that the image of (cid:15) = 0 is a genericpoint of D \ P Ω g,n ( µ, E ∪ { i } ) (this is a generic point of D ). Once again we canfind an integer ‘ and holomorphic functions f and f that do not vanish ∆ suchthat s i = (cid:15) ‘ f and res i = (cid:15) ‘ f . Thus the two functions vanish to the same order. Step 4: Computation of m D for k -star graphs. The fact that m D = m (Γ , t ) foran irreducible component of the stratum associated to a k -star graph is a directconsequence of the following Lemma. (cid:3) Lemma 3.12.
Let E ⊂ Pol k ( µ ) , and ≤ i ≤ n . Let (Γ , t ) ∈ Star kg,n ( i ) . If y is apoint of P Ω(Γ , t, E ) , then there exists an open neighborhood V in P Ω(Γ , t, E ) , a disk ∆ in C containing and a morphism ι : V × ∆ × G (Γ , t ) → P Ω g,n ( µ, E ) satisfying: • For all γ ∈ G (Γ , t ) , the morphism ι induces an isomorphism V × { } × g with V . • The image of V × (∆ \ { } ) × G (Γ , t ) lies in P Ω g,n ( µ, E ) . • The section s i vanishes with multiplicity lcm(Γ , t ) along V × { } × G (Γ , t ) . • The morpihsm ι is a degree parametrization of a neighborhood of y in P Ω g,n ( µ, E ) .Proof of Lemma 3.12. The proof is similar to the proof of Lemma 5.6 of [Sau19].In the case of k -star graph, the morphism p − : P Ω(Γ , t, E ) − → M g ( v ) ,n ( v ) is an embedding. In particular we can identify: P Ω(Γ , t, E ) = P Ω(Γ , t, E ) × P Ω(Γ , t, E ) − (see Section 3.5 for the notation). Thereore we can decompose thepoint y into y = y × y − = ( C , [ η ]) × ( C − , [ η − ]) , 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 P Ω(Γ , t, E ) i togetherwith a trivialization σ i of O ( − → P Ω(Γ , t, E ) i . We assume that U = U × U − has coordinate u = ( u , u − ) and that y = { u = (0 , } . We can rephrase thechoice 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 − Constructing a smoothing of η . Let e = ( h, h ) be an edge of Γ with twist t ( e ).Let σ : U → C and σ − : U → C − be the sections corresponding to the branchof 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 , andsuch that η i ( u i , z e,i ) = z ± t ( e ) e,i (cid:18) dz e,i z e,i (cid:19) k , where the sign is positive for i = 0 and negative for i = −
1. Note that no residueis involved because we assumed that (Γ , t ) is a k -star graph and that µ is positive.The coordinates z e,i are only defined up to a t ( e )-th root of unity. We fix such achoice for all edges e and i = 0 , − e ∈ E (Γ), we fix ζ e a t ( e )-th root of unity. This determine an element ζ ∈ (cid:0) Q e ∈ E (Γ) U t ( e ) (cid:1) . With this datum, we construct a family of curves C ζ → ∆ × U (where ∆ is a disk parametrized by (cid:15) ) as follows. Around a node corresponding to e ∈ E (Γ), we define C ζ ( (cid:15), u ) as the solution of z e, · z e, − = ζ e · (cid:15) lcm(Γ ,t ) /t ( e ) in ∆ e, × ∆ e, − . Outside a neighborhood of the nodes, we define C ζ ( u, (cid:15) ) ’ C ( u )or C − ( u ). On this family of curves, we can define a k -differential by η ζ = z t ( e ) e, (cid:18) dz e, z e, (cid:19) k = (cid:15) lcm(Γ ,t ) z t ( e ) e, − · (cid:18) dz e, − z e, − (cid:19) k in the chart z e, z e, − = ζ e · (cid:15) lcm(Γ ,t ) . Then this differential is extended by η or (cid:15) − lcm η − outside a neighborhood of the nodes. Neighborhood of the boundary.
Two deformations ( C ζ , η ζ ) and ( C ζ , η ζ ) are iso-morphic if and only if ζ = ρζ for some lcm(Γ , t )-th root of unity ρ . Therefore themorhpism: ι : P ( U ) × ∆ × G (Γ , t ) → P Ω kg,n ( µ )( u, (cid:15), γ ) ( C γ ( u, (cid:15) ) , η γ ( u, (cid:15) ))is of degree 1 on its image. To check that this morphism parametrizes a neigh-boorhod of y , we can show as in the case of abelian differentials that there existsa retraction η V : e V → V , where e V is a neighborhood of y in P Ω kg,n ( µ, E ). Besides,all points y in V lies in the image of { η ( y ) } × ∆ × G (Γ , t ) under ι (see “ Proof ofthe fourth point ” of Lemma 5.6 in [Sau19]). (cid:3)
Lemma 3.13.
We assume that µ has at least one negative entry. If D is anirreducible component of a component P Ω(Γ , t, V ab , E ) for a bi-colored graph withtwo vertices, then m D ≤ m (Γ , t ) .Proof. We refer to [CMZ19]. We define P Ω k, tot g,n ( µ, E ) = P Ω k, ab g,n ( µ, E ) S P Ω kg,n ( µ, E ),and by P Ω k, tot g,n ( µ, E ) its incidence variety compactification. There exists a smoothcompactification P Ξ k, tot g,n ( µ, E ) of P Ω k, tot g,n ( µ, E ) together with a forgetful morphism P Ξ kg,n ( µ, E ) → P Ω k, tot g,n ( µ )The functions s i can be defined on P Ξ kg,n ( µ, E ) and vanish with order lcm(Γ , t )along P Ξ(Γ , t ) if the marking i is adjacent to the vertex of level −
1. Therefore,the multiplicity m D of any irreducible component of P Ω(Γ , t, V ab , E ) in the divisordefined as the vanishing locus of s i is at most gcd(Γ , t ) × lcm(Γ , t ) = m (Γ , t ). (cid:3) OLUMES OF MODULI SPACES OF FLAT SURFACES 17 Flat recursion
Let g, and n be non-negative integers are such that 2 g − n >
0. We define: a g,n : D g,n → Q ( k, µ ) Z P Ω kg,n ( µ ) ξ g − n (the notation D g,n was given in 2.10). The purpose of the section is to prove thefollowing theorem. Theorem 4.1.
For all ( g, n ) , we have:(1) there exists a constant K > such that for all ( k, µ ) ∈ D g,n , we have (cid:12)(cid:12)(cid:12)(cid:12) a g,n ( k, µ ) k g − n − v g,n ( µ/k ) (cid:12)(cid:12)(cid:12)(cid:12) < K/k ; (2) the function v g,n is S n -invariant and vanishes at vectors α with at least oneintegral entry. Growth of sums on k -star graphs. Let (Γ , v ) be a star graph in Star g,n and α in b ∆ g,n . 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 h (Γ) (defined in theintroduction) by the action of Aut(Γ , v ). This action is free on an open densesubset of ∆(Γ , α ).If k ≥
2, then we denote by Twist k (Γ , α ) the set of k -twists on Γ compatiblewith kα (this set may be empty if kα is not integral). Lemma 4.2.
We assume that α is rational and that k is a positive integer such that k α is integral. Let f : Twist kk (Γ , α ) → R be a function and f ∞ : Twist(Γ , α ) → R a continuous function such that there exists a constant K > such that | f ( t ) − f ∞ ( t/kk ) | < K/ ( kk ) . Then we have lim k →∞ kk ) h (Γ) · X t ∈ Twist kk (Γ ,α ) f ( t ) | Aut(Γ , t ) | = 1 | Aut(Γ) | Z ∆(Γ ,α ) e f ∞ ( t ) , where e f ∞ is the composition ∆(Γ , α ) → Twist(Γ , α ) f ∞ → R .Proof. For all k ≥
2, we denote by ∆ k (Γ , α ) ⊂ Z E (Γ) > the set of vectors β such that β/k ∈ ∆(Γ , α ). Then Twist k (Γ , α ) is the quotient of ∆ k (Γ , α ) by Aut(Γ) and wecan rewrite X t ∈ Twist k (Γ ,α ) f ( t ) | Aut(Γ , t ) | = X β ∈ ∆ k (Γ ,α ) e f ( β ) | Aut(Γ) | where f is the composition ∆ kg,n → Twist k (Γ , α ) f → R . Then, the lemma followsfrom the the convergence of Riemann sums:lim k →∞ kk ) h (Γ) · X β ∈ ∆ kk (Γ ,α ) e f ( β ) = Z ∆(Γ ,α ) e f ∞ ( t ) . (cid:3) Recursion relations for fixed k . We begin by writing a recursion relationfor the a g,n ( k, µ ) with a fixed value of k >
1. In order to state it we will denote by a ab g = Z P Ω g, (2 g − ξ g − . These intersection numbers have been computed in [Sau18].
Lemma 4.3.
We assume that µ is non-negative. Let ≤ j ≤ g − n be aninteger. Let (Γ , t, V ab ) be a bi-colored graph in Bic kg,n ( µ, and D be an irreduciblecomponent of P Ω(Γ , t, V ab ) such that Z D ( − m ψ ) j ξ g − n − j = 0 , then:a) (Γ , t, V ab ) ∈ Star kg,n ( µ, ;b) all legs are adjacent to vertices of V \ V ab .c) the central vertex satisfies j = 2 g ( v ) − n ( v ) − Card( V ab ) . If (Γ , t ) is k -star graph satisfying these three conditions, then we have a (Γ , t ) def = Z P Ω(Γ ,t ) ( − m ψ ) j ξ g − n − j (9) = (cid:16) Y v | ‘ ( v )=0 ,v / ∈ V ab a g ( v ) ,n ( v ) ( k, µ ( v )) × Y v | ‘ ( v )=0 ,v ∈ V ab k g ( v ) − a ab g ( v ) (cid:17) × Z P Ω(Γ ,t ) − ) ( − m ψ ) j ! . We denote by Star kg,n ( µ, ∗ ⊂ Star kg,n ( µ,
1) the set of k -star graphs such that nolegs is adjacent to a vertex in V ab . Proof.
Let (Γ , t, V ab ) be a k bi-colored graph and D an irreducible component of P Ω(Γ , t, V ab ). Then we decompose: ξ g − n − j ψ j · [ D ] = (cid:0) ξ g − n − j · [ D ] (cid:1) × (cid:16) ψ j · [ D − ] (cid:17) , where D = D × D − and D i is an irreducible component of P Ω(Γ , t, V ab ) i for i = 0 , − j = dim P Ω(Γ , t, V ab ) − . We assume thatthis relation holds. Then we further decompose the first term as ξ g − n − j · [ D ] = (cid:16) Y v ∈ ‘ − ( i ) v / ∈ V ab ξ g ( v ) − n ( v ) [ D ( v )] (cid:17) × (cid:16) Y v ∈ ‘ − ( i ) v ∈ V ab ξ g ( v ) − n ( v ) [ D ( v )] (cid:17) , where D = P ( Q v ( O ( − | D v ) (the product of the total spaces of the line bun-dles O ( − → D v )) and D v is an irreducible component of P Ω kg ( v ) ,n ( v ) ( µ ( v )) or OLUMES OF MODULI SPACES OF FLAT SURFACES 19 P Ω k, ab g ( v ) ,n ( v ) ( µ ( v )). It was proved in [Sau18] (Proposition 3.3) that ξ g − n · [ P Ω g,n ( µ )] = (cid:26) a g if µ = (2 g − . Moreover, the argument used in [Sau18] implies that ξ g − n · [ D ] = 0 for anyirreducible component of [ P Ω g,n ( µ )] with ν = (2 g − k ≥
2, and D be an irreducible component of P Ω g,n ( µ ), where µ has at leastone entry divisible by k . Then the integral R D ξ g − n = 0. The argument is givenfor k = 2 in the proof of Theorem 1.6 of [CMS + P Ω kg,n ( µ ), seen as a subspace of P b Ω b g, b n ( b µ ) (cid:14) U k has directions in the strictly relativecohomology of the covering curve. However, the class ξ can be realized as a twoform involving only absolute periods of the covering curve (see Lemma 5.2 below).Therefore, the contribution of a bi-colored graph is trivial if the upper-verticescontain at least one vertex in V ab with more than two adjacent edges, or a vertexin 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 oflevel −
1. Indeed, if we assume that the a graph has at least two vertices of level − P Ω(Γ , t, V ab ) is of co-dimensionat least 2 in P Ω kg,n ( µ ) (see dimension computation of [BCGGM16]).All in all we have proved that ξ g − n − j ψ j · [ D ] = 0 for any irreducible component D of P Ω(Γ , t, V ab ) if (Γ , t, V ab ) is not in Star kg,n ( µ, ∗ . Besides, we have also provedthat if (Γ , t ) is in Star kg,n ( µ, ∗ then a (Γ , t ) is given by equation (9). (cid:3) An immediate corollary of Lemma 4.3 is the following lemma.
Lemma 4.4.
For all g, n and µ we have: a g,n ( k, µ ) = X (Γ ,t ) ∈ Star kg,n ( µ, ∗ m (Γ , t ) | Aut(Γ , t ) | · a (Γ , t ) . Proof.
We write ξ g − n = X j ≥ ξ g − n − j ( − m ψ ) j ( ξ + m ψ ) . Then we use formula (5) to express ( ξ + m ψ ) in terms of classes [ P Ω k (Γ , t )] for(Γ , t ) in Star kg,n ( µ, ∗ up to a term δ supported on the union of the P Ω k (Γ , t )for (Γ , t ) ∈ Bic g,n ( µ, \ Star kg,n ( µ, ∗ . The integral of ξ g − n − j ( − m ψ ) j on δ vanishes for all j by Lemma 4.3. Besides, the integral of ξ g − n − j ( − m ψ ) j onthe P Ω k (Γ , t ) for a k -star graph is also given by Lemma 4.3. (cid:3) Growth of intersection numbers on strata with residue conditions.
Let g, n, p be non-negative integers such that 2 g − n + p >
0. Let E ⊂ [[ n +1 , n + p ]]be a subset of cardinal r . Let B >
0. We define the set D Bg,n as the set of vectors( k, µ ) ∈ Z > × Z n + p satisfying:(1) | µ | = k (2 g − n + p − | ν | );(2) for all n ≤ i ≤ n + p , k | m i and m i < ≤ i ≤ n + p , we have m i k > − B ; (4) at most two entries of µ are positive, or k (cid:45) m i for all positive m i , with1 < i ≤ n .We consider the following function A g,n,E : D Bg,n → Q ( k, µ ) Z M kg,n + p ( µ,E ) ψ g − n + p − r The purpose of the section is to prove the following statement.
Lemma 4.5.
Let
B > . There exists a real constant K B > , such that for all ( k, µ ) ∈ D Bg,n we have: | A g,n,E ( k, µ ) | < K B · k g . If p = 0 is empty then there exists K B such that (cid:12)(cid:12)(cid:12)(cid:12) ( − m ) g − n k g − n · A g,n ( k, µ ) − A ∞ , g,n ( µ/k ) (cid:12)(cid:12)(cid:12)(cid:12) < K B /k. We begin by stating two lemmas.
Lemma 4.6.
A space P Ω k, • g,n + p ( µ, E ) with • ∈ {∅ , ab } , is of dimension 0 if andonly if one of the following situation holds:(1) • = ab , g = 0 , n = 1 , r = p − ;(2) • = ab , g = 0 , n = 2 , r = p − ;(3) • = ∅ , g = 0 , n = 2 , r = p − .(4) • = ∅ , g = 0 , n = 3 , r = p .In the third and fourth cases, k does not divide the entries of µ . In the second andfourth cases, the residue map is trivial on the total space. These four spaces areirreducible.Proof. We first assume that • = ∅ . The dimension of P Ω kg,n + p ( µ, E ) is equal to2 g − n + p − r . However, p − r ≥ g = 0 or 1. We can see that thecase g = 1 cannot occur as n = 1 would imply that k = 1. If g = 0 then 0 ≤ n ≤ p = r + n . The cases n = 0 is impossible from the condition | µ | = 2 g − p − r .The case n = 1 is not possible either as it would imply that µ 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 mapfor the second case follows from the fact that the residues of an holomorphic 1-formsum up to 0. (cid:3) Lemma 4.7.
Let i ∈ [[ n + 1 , n + p ]] \ E . Let D be an irreducible component of P Ω(Γ , t, V ab , E ) for some (Γ , t, V ab ) ∈ Bic g,n ( µ, g,n ( µ, i , E ) (where the notation ∆ is defined by A ∆ B = ( A ∪ B ) \ ( A ∩ B ) ). If (cid:18)Z D ψ g − n + p − r (cid:19) = 0 , then (Γ , t, V ab ) is a bi-colored with two vertices satisfying either: • V ⊂ V ab ; • or (Γ , t, V ab ) is a k -star graph. OLUMES OF MODULI SPACES OF FLAT SURFACES 21
Proof.
Let (Γ , t, V ab ) be a graph satisfying the hypothesis of the Lemma. Webegin by remarking that ψ is a pull-back from the moduli space of curves. There-fore this integral vanishes if the push-forward of [ D ] along the forgetful morphism P V Ω kg,n ( P ( µ ) → M g,n + p vanishes. This is the case if there are at least 2 verticesof 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. Then,as we require dim( D ) = dim ζ (Γ ,t,V ab ) ( D ), inequality (4) implies that V − ≤ V ab ) ≤ . We will finish the proof of the Lemma by studying separately all possibilities ofconfiguration: 1 is adjacent to a vertex of level 0 or −
1, and the same for i . If (Γ , t, V ab ) ∈ Bic g,n ( µ, \ Bic g,n ( µ, i , E ) . Then i is necessarily adjacent tothe vertex of level 0. We have Z D ψ g − n + p − r = (cid:0) Z D (cid:1) × (cid:0) Z D − ψ g − n + p − r (cid:1) . Therefore, this contribution vanishes if the space D is positive dimensional. Thisimposes that this vertex has to be of one of the types of Lemma 4.6. Besides, thisvertex can only be of type 1 or 3 as these are the only cases for which the residuemap is not trivial. • If D is of type 1, then n = 1 and there is at most one vertex of level − • If D 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 betweenthe two vertices are not divisible by k (and (Γ , t, V ab ) is a k -star graph). If (Γ , t ) ∈ Bic g,n ( µ, i , E ) \ Bic g,n ( µ, . Then we have: Z D ψ g − n + p − r = (cid:0) Z D ψ g − n + p − r (cid:1) × (cid:0) Z D − (cid:1) . The fact that Γ belongs to Bic g,n ( µ, i , E ) leads to two possibilities:(1) If i is adjacent to the vertex of level 0. Then this vertex is in V ab and allindices of [[ n + 1 , n + p ]] \ { i } adjacent to the upper vertex are in E (thecondition that residue vanishes at i follows from the fact that the sum ofresidues of a holomorphic 1-form vanishes). Then V ab ) = 0 and thusthere is one vertex of level −
1. This vertex has to be of type 1 or 3 inLemma 4.6 as the residue condition is empty. In the first case, the graphhas two vertices in V ab and in the second it is a k -star graph.(2) If i is adjacent to a vertex of level −
1, then the condition dim( D − ) = 0implies that all vertices of level − , D Bg,n : • if µ has at most two positive entries divisible by k ( m and anotherone), then there can be only one vertex of level − V ab ). • if µ has more entries, then all positive entries are not divisible by k ,thus all vertices of level − i to the level 0 has necessarily a vanishing residue.Thus the condition dim( D − ) = 0 imposes that there is only one vertexof level 0 (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 graphis a k -star graph. (cid:3) Proof of Lemma 4.5.
The proof will be done by induction on r and g . The base ofthe induction ( r = 0) is a direct consequence of Lemma 2.12.We assume that r >
0. Let
B >
0. Let E be a subset of [[ n + 1 , n + p ]] of cardinal r −
1, and i ∈ [[ n + 1 , n + p ]] \ E .We chose a µ ∈ D Bg,n . Taking the difference between the equation (5) for i = 1and the equation (6) for i = i , we get the following equation: m ψ = k [ P Ω g,n ( µ, E ∪ { i } )] + X (Γ ,t,V ab ) ∈ Bic g,n ( µ, g,n ( µ,i ,E ) D ∈ Irr( P Ω(Γ ,t,V ab ,E )) ± m D · ζ (Γ ,t,V ab ) ∗ ([ D ]) , where the ± depends on whether (Γ , t, V ab ) belongs to Bic g,n ( µ,
1) or Bic g,n ( µ, i , E ).If we multiply this expression by ψ g − n + p − r , we get: m A g,n,E ( k, µ ) − kA g,n,E ∪{ i } ( k, µ )= X (Γ ,t,V ab ) ∈ Bic g,n ( µ, g,n ( µ,i ,E ) D ∈ Irr( P Ω(Γ ,t,V ab ,E )) ± m D · Z D ψ g − n + p − r . Using both Lemma 3.13 and Lemma 4.7 we obtain the following inequality: | kA g,n,E ∪{ i } ( k, µ ) | ≤ | m A g,n,E ( k, µ ) | + X (Γ ,t,V ab ) ∈ Bic g,n ( µ, g,n ( µ,i ,E ) m (Γ , t, V ab ) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z P Ω(Γ ,t,V ab ,E ) k r − ψ g − n + p − r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . There are only a finite number of underlying star graphs in Bic g,n ( µ, i , E )∆Bic g,n ( µ, k, µ ) belong to the domain D Bg,n , imposes thatthe vectors ( k, µ ) and ( k, µ − ) belongs to domains of the form D B i g i ,n i for some B i > i = 0and − • If the vertex v i is not in V ab , then the integral is equal to A g i ,n i ,E i ( k, µ i )for i = 0 or − • As µ is bounded and the number of star graphs is finite, there are finitelymay values for the tuples ( g i , µ i /k, E i ). Besides the contribution of theintegral at a vertex in V ab depends only on these tuples. Thus the integralsat vertices in V ab are bounded by a common constant.Now using the induction hypothesis, there exists a constant K B such that: (cid:12)(cid:12) A g,n,E ∪{ i } ( k, µ ) (cid:12)(cid:12) ≤ K B · m k · k g + X (Γ ,t,V ab ) ∈ Bic g,n ( µ, g,n ( µ,i ,E ) m (Γ , t, V ab ) · k g − h (Γ) OLUMES OF MODULI SPACES OF FLAT SURFACES 23
The boundedness of the twists implies that m ≤ k · B , and m (Γ , t, V ab ) < k | E (Γ) | · B for some B >
0. Putting everything together, there exists a constant K B such that (cid:12)(cid:12) A g,n,E ∪{ i } ( k, µ ) (cid:12)(cid:12) ≤ K B · (cid:16) k g + X (Γ ,t,V ab ) ∈ Bic g,n ( µ, g,n ( µ,i ,E ) k g − h (Γ) (cid:17) . There are finitely many underlying star graphs in the last sum and for each suchstar graph the number of compatible k -twist is bounded by a constant times k h (Γ) .Therefore we obtain the desired estimate. (cid:3) Proof of Theorem 4.1.
We prove Theorem 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 ,n ( k, µ ) = 1.Now, let us fix some g, n ≥
0. We define the following set of vectors∆ g,n = (cid:26) ∆ g,n ∩ ( R × ( R \ Z ) n ) , if n ≥ g,n , otherwise.We also define D g,n ⊂ D g,n , the set of vectors ( k, µ ) such that µ/k ∈ ∆ g,n . Weproceed in two steps. First we prove that the first point of the theorem holds if wereplace D g,n by D g,n . Then in the second step, we prove the second point and thefact that the we can replace D g,n by D g,n . Step 1.
Let α ∈ ∆ g,n be a rational vector. Let Γ be star graph in Star g,n (1) andlet V ab ⊂ V be a subset of the outer vertices such that for all v ∈ V ab there is onlyone half-edge adjacent to v .For all k ≥
2, A twist t ∈ Twist k (Γ , α ) determines at a unique structure ofbi-colored graph. This is a k -star graph unless there exists an edge to a vertex in V \ V ab which is divisible by k . We define the following function: f Γ ,V ab : Twist k (Γ , α ) → R t m (Γ , t ) a (Γ , t ) k g − n (extended by 0 if t does not determine a k -star graph). There exists a constant K Γ ,V ab such that for all t ∈ Twist k (Γ , α ), we have a (Γ , t ) < K Γ ,V ab × Y v | ‘ ( v )=0 ,v / ∈ V ab k g ( v ) − n ( v ) × Y v | ‘ ( v )=0 ,v ∈ V ab k g ( v ) − × (cid:16) k g ( v ) − n ( v ) − V ab ) (cid:17) ≤ K Γ ,V ab · k g − n − h (Γ) − E (Γ)) − V ab ) . Here we have used the expression (9) to decompose a (Γ , t ) into a product of 3 terms.We bounded the first term by the induction hypothesis and the third by applyingLemma 4.5. In particular there exists a K Γ ,V ab such that X t ∈ Twist k (Γ ,α ) m (Γ , t ) a (Γ , t ) < K Γ ,V ab k g − n − V ab )4 ADRIEN SAUVAGET for all k ≥
2. Now, if V ab is empty, then we can show by the same arguments thatthere exist a constant K Γ such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v (Γ , α ) − X t ∈ Twist k (Γ ,α ) m (Γ , t ) a (Γ , t ) k g − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < K Γ /k. (here we have used the second part of Lemma 4.5 and Lemma 4.2). As the numberof star graphs appearing in the expression of the a g,n ( k, µ ) is finite, there exists aconstant K such that for all ( k, µ ) ∈ D g,n , we have: (cid:12)(cid:12)(cid:12)(cid:12) v g,n ( α ) − a g,n ( k, µ ) k g − n (cid:12)(cid:12)(cid:12)(cid:12) ≤ X Γ ∈ Star g,n V ab = ∅ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v (Γ , α ) − X t ∈ Twist k (Γ ,α ) m (Γ , t ) a (Γ , t ) k g − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X Γ ∈ Star g,n V ab = ∅ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t ∈ Twist k (Γ ,α ) m (Γ , t ) a (Γ , t ) k g − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < K/k. Step 2.
For all values of k , the function a g,n ( k, · ) is S n invariant by definition.Therefore, v g,n is S n invariant on ∆ g,n . As v g,n is continuous, it is S n -invariant on∆ g,n in general.If α ∈ Z > , then v g,n ( α ) = 0. Indeed, a g,n ( k, µ ) vanishes if one the entries of µ is divisible by k , and the first point of the theorem (restricted to D g,n ) impliesthat v g,n ( α ) is the limit of trivial sequence.Finally, the result of Step 1 is valid for all ( k, µ ) in D g,n as | a g,n ( k, µ ) − v g,n ( µ/k ) | vanishes if at least one entry of µ is divisible by k .4.5. Wall-crossing properties of the flat recursion.
By the flat recursion rela-tion (1), the function v g,n are continuous piece-wise polynomials on ∆ g,n of degreeat most 4 g − n . The chambers of polynomiality are delimited by walls of theform: P 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 func-tions v g,n along the walls. The results will we used further to prove Theorem 1.2using Theorem 1.1 Lemma 4.8.
For all g ≥ , we have lim α v g, ( α , g − α ) = 0 . Proof.
We use the fact that the only terms in the flat recursion formula (1) whichare not divisible by α are those for which the central component is a vertex ofgenus 0 with 3 half-edges. For small values of α , this condition is satisfied only bythe graph with the markings 1 , and 2, adjacent to a central vertex of genus 0 andwith one edge. Indeed, if α is smaller than 1 /
2, then the second markings belongsto the lower vertex as 2 g − − α > g −
1. Finally the contribution of this graphis equal to (2 g − v g, (2 g −
1) = 0 , as (2 g −
1) is integral. (cid:3)
OLUMES OF MODULI SPACES OF FLAT SURFACES 25
Proposition 4.9.
Let κ ∈ Z > . In the neighborhood of a generic point of the wall α i = κ , the function v g,n is of the form (cid:26) ( α i − κ ) e v g,n , if n ≥ α i − κ ) e v g,n , if n = 2 , where e v g,n is a continuous piece-wise polynomial.Proof. We prove the statement by induction on g and n . For ( g, n ) = (0 ,
3) thestatement is empty as ∆ , does not contain vectors with integral values.Let ( g, n ) = (0 , S n -invariance we can assume that i = 2. We begin bywriting the flat recursion formula (1): v g,n ( α ) = X Γ ∈ Star g,n v (Γ , α )Let Γ be a star graph in Star g,n . The function v Γ : α v (Γ , · ) is a piece-wisepolynomial on the domain ∆(Γ) bounded by the walls: X i v α i = 2 g v − n v for all vertices v of level 0. It is extended by 0 outside the domain ∆(Γ). In orderto understand the behavior of v Γ in the neighborhood of a generic point of the wall α = κ , we distinguish 3 cases: the label i = 2 is adjacent to the central vertex, ora vertex 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 − v Γ is polynomial on adomain containing a generic point of any wall of the form α = κ .If the label i = 2 is adjacent to a leg with at least one other marking, then ageneric point of the wall α i = κ is in the interior of ∆(Γ). Indeed, otherwise it iswould be at the intersection of two wall α i = κ and P α i = κ for all i adjacent tothe same vertex as i = 2 (non generic configuration). In the interior of ∆(Γ), thefunction v Γ is defined as the partial integration of a product of a polynomial andfunctions of the form v g v ,n v . Thus by induction hypothesis, v Γ = ( α − κ ) e v Γ forsome continuous piece-wise polynomial e v Γ .Finally, if i = 2 is the unique leg adjacent to its outer vertex v , then the wall α = 2 g v − n v is a boundary of the domain ∆(Γ). From the flat recursion thefunction v Γ is equal to Z ∆(Γ ,α ) v g v ,n v ( α , β , . . . , β n v − ) · ( β . . . β n v − ) · Q ( α, β ) . where Q is a continuous piecewise polynomial. Therefore, v Γ is of the form ( α − (2 g v − n v )) e v Γ for some continuous piece-wise polynomial e v Γ . Indeed for n v ≥ v Γ is the integral of a polynomial with valency atleast one in each β i for 1 ≤ i ≤ n v −
1. If n v = 2, it follows from the fact that v g v , ( α , (2 g v − n v ) − α ) tends to 0 as α goes to 2 g v − n v .Using these results we can write: v g,n = Q + ( α − κ ) v g,n + ( α − κ ) v g,n where Q is a polynomial (contribution of graphs with i = 2 adjacent to the centralvertex), and v g,n , v g,n are continuous piecewise polynomials (respectively contribu-tion of graphs with i = 2 adjacent to vertex with other legs or not). The polynomial Q vanishes along α = κ as v g,n does, thus if n ≥
2, we can indeed factorize v g,n by ( α − κ ).If n = 2, then term v g,n = 0 (as there are no graphs with at least two legson the outer vertices for n = 2). Thus we need to show that the derivative of α Q (2 g − α , α ) vanishes at κ . This follows from Theorem 1.1. Indeed, thefunction V g,n is non-negative for all rational entries and the sign of sin( πα )sin( πα )is constant when n = 2. Thus, by (2), the sign of v g, is constant on ∆ g, . Thisimplies that Q vanishes to the order at least 2. (cid:3) From intersection theory to volumes
In this section we recall the convention for the normalisation of volumes of modulispaces of flat surfaces and we complete the proof of Theorems 1.2 and 1.1.5.1. 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 ) > C p + q \ { } → P C h its projectivization. We can define twomeasures (in fact volume forms) on P C h . The first one is defined by ν ( U ) = Lebesgue measure (cid:0) proj − ( U ) ∩ { x | h ( x, x ) ≤ } (cid:1) . The second is defined by considering the line bundle O ( − → P C h . Indeed this linebundle is endowed with the hermitian metric equal h as we identify O ( − ∗ ’ C ∗ h .We denote by − ω h the curvature form of this metric h . Then we define the volumeform ν = ω p + q − h . Lemma 5.1.
We have ν = π p + q ( p + q )!det( h ) ν .Proof. The proof is similar to Lemma 2.1 of [Sau18] and Lemma 2.1 of [CMS + h is diagonal and given by h ( x, x ) = P ≤ i ≤ p + q h i | x i | with h i > ≤ i ≤ p . Using the action of the group U ( p + q ) ∩ U ( p, q ) it is sufficientto compare the form on the set of vectors of the form ( x , , . . . , , x p +1 , , . . . ).We consider the chart of P C h defined by x = 1. In this chart the measure ν isthe measure associated to the differential form:2 πh ( x, x ) p + q dim R ( C h ) · p + q Y i =2 ( i dx i ∧ dx i ) . In this same chart the form ω h is given by ω h = ( h + h p +1 | x p +1 | ) · ( P p + qi =2 h i dx i ∧ dx i ) − h p +1 | x p +1 | dx p +1 ∧ dx p +1 iπ ( h + h p +1 | x p +1 | ) . From this expression, we deduce the equality ω p + q − h = ( p + q − (cid:0) Q p + qi =1 h i (cid:1) (2 iπ ) p + q − h ( x, x ) p + q · p + q Y i =2 dx i ∧ dx i ! . = ( p + q )!det( h ) π p + q ν . (cid:3) OLUMES OF MODULI SPACES OF FLAT SURFACES 27
The holonomy map.
We fix a reference oriented marked surface (
S, s , . . . , s n )of genus g . Given α ∈ ∆ g,n , we denote by T ( α ) the moduli space of flat surfaces( C, x , . . . , x n , η ) with conical singularities prescribed by α together with an iso-morphism C → S preserving the markings. This is the Teichmüller moduli spaceof flat surfaces of type α .In [Vee93], Veech showed that there exists a map:hol : T ( α ) → U g , the holonomy character map. For all α , we denote by hol α : T ( α ) → K ( α ) ’ S g × { α } the restriction of this map. This map is a submersion for any value of α / ∈ N n , and the leaves are complex manifolds. For any value of λ ∈ K ( α ), wedenote by T λα = hol − α ( λ ) the leave assciated to λ .There exists a C ∞ -complex line bundle proj : L ( α ) → T ( α ) equipped with anhermitian metric h α . This line bundle is defined by fixing a choice of orientationand normalization of a flat surface. The restriction of this line bundle to any leaveof the holonomy foliation is holomorphic. The metric h α is the area of the flatsurface.For all λ ∈ K ( α ), the leave T λα has an atlas of charts of the form ϕ : U → P C h λ,U ⊂ P g − n for some hermitian form h λ,U depending on λ and U . Besides L ( α ) | U ’ ϕ ∗ O ( − h α is the pull-back of h λ,U (seen as a metric on O ( − U ( p, q ). Finally, the determinant andthe signature of h λ,U are independent of both λ and U .5.3. Measure on M g,n ( α ) . Let λ ∈ K ( α ). Using the U ( p ( α ) , q ( α )) structure on L ( α ) |T λα , we define a measure ν λα on T λα by ν λα ( U ) = Lebesgue measure (cid:0) proj − ( U ) ∩ { x | h α ( x, x ) ≤ } (cid:1) , (this is well-defined as U ( p, q ) transition maps are in U ( p, q )). As in the previoussection we can also consider the − ω λα the curvature form of the line bundle L ( α ) |T λα )for the hermitian metric h α . Lemma 5.2.
We have the equality: ν λα = 4 · ( − g + n − (2 π ) g − n (cid:0) Q ni =1 πα i ) (cid:1) · (2 g − n )! ( ω λα ) g − n . Proof.
Using Lemma 5.1 and the U ( p, q ) structure on L ( α ) |T λα ), we get the equality: ν λα = π g − n det( h α )(2 g − n )! ( ω λα ) g − n , where det( h α ) is the determinant of h λ,U for any chart U of T λα . This determinanthas been computed by Veech (see [Vee93], Lemmas 14.10, 14.17, and 14.32):det( h α ) = Q ( α )4 g − n , where the function Q ( α ) is defined by Q ( α ) = (2 i ) g n − Y i =1 (cid:12)(cid:12) − e iπα i (cid:12)(cid:12) ! · b n − c X a =0 ( − a S n − − a ((cotan( πα i ) ≤ i ≤ n − ) , and S ‘ is the ‘ th symmetric function. Then we use the following identity( − n − sin( πα n ) = sin( πα + . . . + πα n − )= X E ⊂ [[1 ,n − E ) odd i E ) − Y i ∈ E sin( πα i ) ! · Y i/ ∈ E cos( πα i ) ! = n − Y i =1 sin( πα i ) ! · b n − c X a =0 ( − a S n − − a ((cotan( πα i ) ≤ i ≤ n − ) . Combining this identity with the fact that (cid:12)(cid:12) − e iπα i (cid:12)(cid:12) = 4 sin( πα i ) , we deducethat Q ( α ) = (2 i ) g ( − n − · n Y i =1 sin( πα i ) . (cid:3) 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 λα depends continuously on the parameters λ . Thus, it defines a form in ^ g − n ) (cid:0) Ω( T ( α ) (cid:14) hol ∗ α Ω( U g ) (cid:1) . Therefore the form ν α = hol ∗ α ν U g ∧ ν λα (where ν U g is the Haar volume form) is a volume form on T ( α ). This form isinvariant under the action of the mapping class group on T ( α ) (see [Vee93], Theo-rem 13.14) and thus defines a volume form on the moduli space M g,n ( α ). Case of integral α . If α ∈ N n , then we denote by T ( α ) the pre-image of M g,n ( α )by the quotient morphism T ( α ) → M g,n ( α ). Veech showed that the holonomycharacter morphism hol α is a submersion outside T ( α ). Therefore the constructionof the volume form ν α for non-integral values of α also gives a continuous volumeform ν α on M g,n ( α ) \ M g,n ( α ).Therefore, we define the volume of M g,n ( α ) as the volume of M g,n ( α ) \M g,n ( α )for integral values of α .5.4. Reducing to moduli spaces of k -differentials. Let α ∈ ∆ g,n ∩ \ Z n . Tocompute the volume V g,n ( α ), we chose a sequence of sets ( E ‘ ) ‘ ∈ N ⊂ U g thatequidistributes (for the Haar measure of U g ) as ‘ goes to infinity. Then we havethe sequence of measures: 1Card( E ‘ ) X λ ∈ E ‘ ν λα , weakly converges to ν α as hol α is a submersion.Now, we assume that α is in ( Q \ Z ) n , and that k α is integral for some k > ‘ ≥
0, we define the following set: E ‘ = ( U k ‘ ) g . OLUMES OF MODULI SPACES OF FLAT SURFACES 29
Then for all ‘ , we have h − α ( E ‘ ) ’ P Ω k ‘g,n ( k ‘α ) , and the identification of line bun-dles: (cid:16) L ( α ) | P Ω k ‘g,n ( k ‘α ) (cid:17) ⊗ k ‘ (cid:40) (cid:40) ’ O ( − (cid:119) (cid:119) P Ω k ‘g,n ( k ‘α ) . By [CMZ19], the hermitian metric h k ‘α is good on P Ω k ‘g,n ( k ‘α ). Therefore we havethe equality: Z P Ω k ‘g,n ( k ‘α ) ( k ‘ω α ) g − n = Z P Ω k ‘g,n ( k ‘α ) ξ g − n where ω α is the curvature form of h α . Now using Lemma 5.2 and Theorem 4.1 weget the equality:det( h α )(2 g − n )! π g − n V g,n ( α ) = lim ‘ →∞ k ‘ ) g − n Z P Ω k ‘g,n ( k ‘ · α ) ξ g − n = v g,n ( α ) . Which finishes the proof of Theorem 1.1.5.5.
Finiteness of the volume function.
We finish here the proof of Theo-rem 1.2. Proposition 4.9 implies that the function b V g,n : ∆ g,n ∩ ( R \ Z ) n → R α · ( − g + n − (2 π ) g − n (cid:0) Q ni =1 πα i ) (cid:1) · (2 g − n )! v g,n ( α ) , admits a continuous extension to ∆ g,n (that we denote by the same letter). Lemma 5.3.
The function V g,n is lower semi-continuous, and V g,n ≤ b V g,n .Proof. Let α 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: ν α ( M g,n ) = sup compact K ⊂M g,n ( ν α ( K ))= sup compact K ⊂M g,n (cid:18) lim α α ν α ( K ) (cid:19) = sup compact K ⊂M g,n lim α α α ∈ ( Q \ Z ) n ν α ( K ) ≤ lim α α α ∈ ( Q \ Z ) n ν α ( M g,n ) = b V g,n ( α ) . (cid:3) End of the proof of Theorem 1.2.
We have seen that V g,n = b V g,n on a dense set ofof values and that b V g,n is continuous.Let (cid:15) >
0. We denote by U (cid:15) ⊂ ∆ g,n the set of vectors α such that V g,n ( α ) > b V g,n − (cid:15) . This set is open (as V g,n is lower semi-continuous) and dense (as it contains a dense subset). Now if we denote by U the set of vectors α such that V g,n ( α ) = b V g,n ( α ), then we have U = \ ‘ ≥ U /‘ which is a countable intersection of sets whose complement is of measure 0. There-fore the complement of U is of measure 0. (cid:3) References [BCGGM16] M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, and M. Moller. Compactifica-tion of strata of k-differentials. 2016, arXiv:1610.09238.[BHPSS20] Y. Bae, D. Holmes, R. Pandharipande, J. Schmitt, and R. Schwarz. Pix-ton’s formula and Abel-Jacobi theory on the Picard stack. 2020, available athttps://people.math.ethz.ch/~rahul/UniDR.pdf.[Chi08] Alessandro Chiodo. Towards an enumerative geometry of the moduli space of twistedcurves and r th roots. Compos. Math. , 144(6):1461–1496, 2008.[CMS +
19] D. Chen, M. Möller, A. Sauvaget, with an appendix of G. Borot, A. Giacchetto,and D. Lewanski. Masur-veech volumes and intersection theory on moduli spaces ofquadratic differentials: the principal strata. 2019, arXiv:1912.02267.[CMSZ19] D. Chen, M. Möller, A. Sauvaget, and D. Zagier. Masur-veech volumes and inter-section theory on moduli spaces of abelian differentials. 2019, arXiv:1901.01785.[CMZ19] Matteo Costantini, Martin Möller, and Jonathan Zachhuber. The area is a goodmetric. 2019, arXiv:1910.14151.[DM86] P. Deligne and G. D. Mostow. Monodromy of hypergeometric functions and nonlat-tice integral monodromy.
Inst. Hautes Études Sci. Publ. Math. , (63):5–89, 1986.[GP16] Selim Ghazouani and Luc Pirio. Moduli spaces of flat tori and elliptic hypergeometricfunctions. 2016.[HS19] David Holmes and Johannes Schmitt. Infinitesimal structure of the pluricanonicaldouble ramification locus. 2019, arXiv:1909.11981.[JPPZ17] F. Janda, R. Pandharipande, A. Pixton, and D. Zvonkine. Double ramification cycleson the moduli spaces of curves.
Publ. Math. Inst. Hautes Études Sci. , 125:221–266,2017.[KN18] Vincent Koziarz and Duc-Manh Nguyen. Complex hyperbolic volume and intersec-tion of boundary divisors in moduli spaces of pointed genus zero curves.
Ann. Sci.Éc. Norm. Supér. (4) , 51(6):1549–1597, 2018.[McM17] Curtis T. McMullen. The Gauss-Bonnet theorem for cone manifolds and volumes ofmoduli spaces.
Amer. J. Math. , 139(1):261–291, 2017.[Mir07] Maryam Mirzakhani. Simple geodesics and Weil-Petersson volumes of moduli spacesof bordered Riemann surfaces.
Invent. Math. , 167(1):179–222, 2007.[PZ] A. Pixton and D. Zagier. in preparation.[Sau18] Adrien Sauvaget. Volumes and Siegel-Veech constants of h(2g-2) and Hodge inte-grals.
Geom. Funct. Anal. , 28(6):1756–1779, 2018.[Sau19] Adrien Sauvaget. Cohomology classes of strata of differentials.
Geom. Topol. ,23(3):1085–1171, 2019.[Sch16] Johannes Schmitt. Dimension theory of the moduli space of twisted k-differentials.2016, arXiv:1607.08429.[Thu98] William P. Thurston. Shapes of polyhedra and triangulations of the sphere. In
TheEpstein birthday schrift , volume 1 of
Geom. Topol. Monogr. , pages 511–549. Geom.Topol. Publ., Coventry, 1998.[Tro86] Marc Troyanov. Les surfaces euclidiennes à singularités coniques.
Enseign. Math.(2) , 32(1-2):79–94, 1986.[Vee93] William A. Veech. Flat surfaces.
Amer. J. Math. , 115(3):589–689, 1993.
CNRS, Université de Cergy-Pontoise, Laboratoire de Mathématiques AGM, UMR8088, 2 av. Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France
E-mail address ::