PRINCIPAL BOUNDARY OF MODULI SPACES OF ABELIAN AND QUADRATIC DIFFERENTIALS

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ANNALES DE

L’INSTITUT FOURIER

LesAnnales de l’institut Fouriersont membres du Centre Mersenne pour l’édition scientifique ouverte

Dawei Chen & Qile Chen

Principal boundary of moduli spaces of abelian and quadratic differentials

Tome 69, n^o1 (2019), p. 81-118.

<http://aif.centre-mersenne.org/item/AIF_2019__69_1_81_0>

Cet article est mis à disposition selon les termes de la licence Creative Commons attribution – pas de modification 3.0 France.

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PRINCIPAL BOUNDARY OF MODULI SPACES OF ABELIAN AND QUADRATIC DIFFERENTIALS

by Dawei CHEN & Qile CHEN (*)

Abstract. — The seminal work of Eskin–Masur–Zorich described the principal boundary of moduli spaces of abelian differentials that parameterizes flat surfaces with a prescribed generic configuration of short parallel saddle connections. In this paper we describe the principal boundary for each configuration in terms of twisted differentials over Deligne–Mumford pointed stable curves. We also describe similarly the principal boundary of moduli spaces of quadratic differentials originally studied by Masur–Zorich. Our main technique is the flat geometric degeneration and smoothing developed by Bainbridge–Chen–Gendron–Grushevsky–

Möller.

Résumé. — Le travail fondateur d’Eskin–Masur–Zorich a décrit la limite principale des espaces de modules des différentielles abéliennes qui paramètre les surfaces plates possédant une configuration générique de petites connexions de selles parallèles prescrite. Dans cet article, nous décrivons la limite principale pour chaque configuration en terme de différentielles entrelacées sur les courbes stables pointées de Deligne–Mumford. Nous décrivons également la limite principale des espaces de modules des différentielles quadratiques étudiée à l’origine par Masur–Zorich. Nos principaux outils sont la dégénérescence géométrique plate et le lissage développés par Bainbridge–Chen–Gendron–Grushevsky–Möller.

1. Introduction

Many questions about Riemann surfaces are related to study their flat structures induced from abelian differentials, where the zeros of differentials correspond to the saddle points of flat surfaces. Loci of abelian differentials with prescribed type of zeros form a natural stratification of the moduli space of abelian differentials. These strata have fascinating geometry and can be applied to study dynamics on flat surfaces.

Keywords:Abelian differential, principal boundary, moduli space of stable curves, spin and hyperelliptic structures.

2010Mathematics Subject Classification:14H10, 14H15, 30F30, 32G15.

(*) D. Chen is partially supported by the NSF CAREER grant DMS-1350396 and Q. Chen is partially supported by the NSF grant DMS-1560830.

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Given a configuration of saddle connections for a stratum of flat surfaces, Veech and Eskin–Masur ([11, 23]) showed that the number of collections of saddle connections with bounded lengths has quadratic asymptotic growth, whose leading coefficient is called the Siegel–Veech constant for this configuration. Eskin–Masur–Zorich ([12]) gave a complete description of all possible configurations of parallel saddle connections on a generic flat surface.

They further provided a recursive method to calculate the corresponding Siegel–Veech constants. To perform this calculation, a key step is to describe theprincipal boundary whose tubular neighborhood parameterizes flat surfaces with short parallel saddle connections for a given configuration.

As remarked in [12], flat surfaces contained in the Eskin–Masur–Zorich principal boundary can be disconnected and have total genus smaller than that of the original stratum. Therefore, as the underlying complex curves degenerate by shrinking the short saddle connections, the Eskin–Masur–

Zorich principal boundary does not directly imply the limit objects from the viewpoint of algebraic geometry. In this paper we solve this problem by describing the principal boundary in the setting of the strata compactification [4] and consequently in the Deligne–Mumford compactification.

Main Result

For each configuration we give a complete description for the principal boundary in terms of twisted differentials over pointed stable curves.

This result is a combination of Theorems 2.1 and 3.4. Along the way we deduce some interesting consequences about meromorphic differentials on P¹ that admit the same configuration (see Propositions 2.3 and 3.8).

Moreover, when a stratum contains connected components due to spin or hyperelliptic structures ([19]), Eskin–Masur–Zorich ([12]) described how to distinguish these structures nearby the principal boundary via an analytic approach. Here we provide algebraic proofs for the distinction of spin and hyperelliptic structures in the principal boundary under our setting (see Sections 4.6 and 4.7 for related results).

Masur–Zorich ([20]) described similarly the principal boundary of strata of quadratic differentials. Our method can also give a description of the principal boundary in terms of twisted quadratic differentials in the sense of [3] (see Section 5 for details).

Twisted differentials play an important role in our description of the principal boundary, so we briefly recall their definition (see [4] for more details). Given a zero type µ = (m1, . . . , mn), a twisted differential η of

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typeµon ann-pointed stable curve (C, σ₁, . . . , σ_n) is a collection of (pos- sibly meromorphic) differentialsηi on each irreducible componentCiofC, satisfying the following conditions:

(0) ηhas no zeros or poles away from the nodes and markings ofCand ηhas the prescribed zero order mi at each markingσi.

(1) If a nodeqjoins two componentsC₁andC₂, then ord_qη₁+ord_qη₂=

−2.

(2) If ordqη₁= ordqη₂=−1, then Resqη₁+ Resqη₂= 0.

(3) IfC1andC2intersect atknodesq1, . . . , qk, then ordqiη1−ordqiη2

are either all positive, or all negative, or all equal to zero fori = 1, . . . , k.

Condition (3) provides a partial order between irreducible components that are not disjoint. If one expands it to a full order between all irreducible components of C, then there is an extra global residue condition which governs when such twisted differentials are limits of abelian differentials of typeµ. A construction of the moduli space of twisted differentials can be found in [2].

By using η on all maximum components and forgetting its scales on components of smaller order, [4] describes a strata compactification in the Hodge bundle over the Deligne–Mumford moduli spaceMg,n. As remarked in [4], if one forgetsηand only keeps track of the underlying pointed stable curve (C, σ1, . . . , σn), it thus gives the (projectivized) strata compactification inM_g,n. Hence our description of the principal boundary in terms of twisted differentials determines the corresponding boundary in the Deligne–

Mumford compactification. To illustrate our results, we will often draw such stable curves in the Deligne–Mumford boundary.

For an introduction to flat surfaces and related topics, we refer to the surveys [8, 24, 25]. Besides [4], there are several other strata compacti- fications, see [13] for an algebraic viewpoint, [9, 16] for a log geometric viewpoint and [21] for a flat geometric viewpoint. Algebraic distinctions of spin and hyperelliptic structures in the boundary of strata compactifica- tions are also discussed in [7, 9, 14].

This paper is organized as follows. In Sections 2 and 3 we describe the principal boundary of type I and of type II, respectively, following the roadmap of [12]. In Section 4 we provide algebraic arguments for distin- guishing spin and hyperelliptic structures in the principal boundary. Fi- nally in Section 5 we explain how one can describe the principal boundary of strata of quadratic differentials by using twisted quadratic differentials.

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Throughout the paper we also provide a number of examples and figures to help the reader quickly grasp the main ideas.

Notation

We denote byµthe singularity type of differentials, byH(µ) the stratum of abelian differentials of type µ and by Q(µ) the stratum of quadratic differentials of typeµ. An n-pointed stable curve is generally denoted by (C, σ1, . . . , σn). We use (C, η) to denote a twisted differential on C. The underlying divisor of a differential η is denoted by (η). Configurations of saddle connections are denoted byC and all configurations considered in this paper are admissible in the sense of [12].

Acknowledgements.We thank Matt Bainbridge, Alex Eskin, Quentin Gendron, Sam Grushevsky, Martin Möller and Anton Zorich for inspiring discussions on related topics. We also thank the referee for carefully reading the paper and many useful comments.

2. Principal boundary of type I

2.1. Configurations of type I: saddle connections joining distinct zeros

LetCbe a flat surface inH(µ) with two chosen zerosσ1andσ2of order m₁ and m₂, respectively. Suppose C has precisely p homologous saddle connectionsγ1, . . . , γp joiningσ1andσ2such that the following conditions hold:

• All saddle connectionsγi are oriented from σ1 toσ2with identical holonomy vectors.

• The cyclic order ofγ1, . . . , γp atσ1is clockwise.

• The angle betweenγ_i andγ_i+1 is 2π(a⁰_i+ 1) atσ₁ and 2π(a⁰⁰_i + 1) atσ2, wherea⁰_i, a⁰⁰_i >0.

Then we say thatChas aconfiguration of type C= (m₁, m₂,{a⁰_i, a⁰⁰_i}^p_i=1).

We emphasis here that this configurationC is defined with the two chosen zerosσ1andσ2. Ifp= 1, we also denote the configuration byC= (m1, m2) for simplicity. Since the cone angle at σ_i is 2π(m_i+ 1) for i = 1,2, we necessarily have

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p

X

i=1

(a⁰_i+ 1) =m₁+ 1 and

p

X

i=1

(a⁰⁰_i + 1) =m₂+ 1.

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2.2. Graphs of configurations

Given two fixed zeros σ₁ and σ₂ and a configuration C = (m₁, m₂, {a⁰_i, a⁰⁰_i}^p_i=1) as in the previous section, to describe the dual graphs of the underlying nodal curves in the principal boundary of twisted differentials, we introduce theconfiguration graphG(C) as follows:

(1) The set of vertices is{v_R, v₁,· · · , v_p}.

(2) The set of edges is{l1,· · ·, lp}, where eachli joinsvi andvR. (3) We associate to vR the subset of markings LR ={σ1, σ2} and to

eachv_ia subset of markingsL_i⊂ {σj}such thatL_RtL₁t · · · tL_p is a partition of{σ1, . . . , σn}.

(4) We associate to eachv_i a positive integerg(v_i) such that

p

X

i=1

g(vi) =g and X

σj∈Li

mj+ (a⁰_i+a⁰⁰_i) = 2g(vi)−2.

Figure 2.1 shows a pointed nodal curve whose dual graph is of typeG(C):

R

C1

Cp

σ1

σ2

L1

Lp

Figure 2.1. A curve with dual graph of typeC.

2.3. The principal boundary of type I

Denote by ∆(µ,C) the space of twisted differentialsη satisfying the following conditions:

• The underlying dual graph ofηis given byG(C), with nodesq_i and componentsCi corresponding to li andvi, respectively.

• The componentRcorresponding to the vertex v_R is isomorphic to P¹and contains only σ1 andσ2among all the markings.

• EachC_i has markings labeled byL_i and has genus equal tog(v_i).

• For eachi= 1, . . . , p, ordq_iηC_i =a⁰_i+a⁰⁰_i and ordq_iηR=−a⁰_i−a⁰⁰_i−2.

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• For eachi= 1, . . . , p, Res_q_iη_R= 0.

• ηRadmits the configurationC of saddle connections fromσ1toσ2. Recall that the twisted differentialη defines a flat structure onR(up to scale). Thus it makes sense to talk about the configurationConR. We say that ∆(µ,C) is theprincipal boundary associated to the configurationC.

SupposeC^ε∈ H(µ) has the configurationC= (m₁, m₂,{a⁰_i, a⁰⁰_i}^p_i=1) such that the phomologous saddle connections γ1, . . . , γp of C have length at mostε. We want to determine the limit twisted differential as the length of allγ_i shrinks to zero. To avoid further degeneration, suppose that C^ε does not have any other saddle connections shorter than 3ε (the locus of suchC^εis called thethick partof the configurationCin [12]). Take a small disk under the flat metric such that it containsσ1,σ2, allγi, and no other zeros (see [12, Figure 5]). Within this disk, shrinkγ_i to zero while keeping the configuration C, such that all other periods become arbitrarily large compared toγi.

Theorem 2.1. — The limit twisted differential of C^ε as γi → 0 is contained in ∆(µ,C). Conversely, twisted differentials in ∆(µ,C) can be smoothed to of typeC^ε.

Proof. — Sinceγiandγi+1are homologous and next to each other, they bound a surface C_i^ε with γ_i and γ_i+1 as boundary (see the lower right illustration of [12, Figure 5] whereC_i^ε is denoted by Si). The inner angle betweenγ_i andγ_i+1at σ₁ is 2π(a⁰_i+ 1) and atσ₂ is 2π(a⁰⁰_i + 1). Shrinking theγj to zero under the flat metric, the limit ofC_i^ε forms a flat surface Ci, and denote byqi the limit position ofσ1 andσ2in Ci. This shrinking operation is the inverse of breaking up a zero, see [12, Figure 3], which implies that the cone angle atqi is 2π(a⁰_i+a⁰⁰_i + 1), henceCi has a zero of ordera⁰_i+a⁰⁰_i at q_i.

On the other hand, instead of shrinking the γj, up to scale it amounts to expanding the other periods ofC_i^εarbitrarily long compared to the γ_j. Since a small neighborhoodNi enclosing both γi and γi+1 in C_i^ε consists of 2(a⁰_i +a⁰⁰_i + 1) metric half-disks, under the expanding operation they turn into 2(a⁰_i +a⁰⁰_i + 1) metric half-planes that form the basic domain decomposition for a pole of ordera⁰_i+a⁰⁰_i + 2 in the sense of [6]. Moreover, the boundary loop ofN_i corresponds to the vanishing cycle around q_i in the shrinking operation, which implies that the resulting pole will be glued toq_i as a node in the limit stable curve, hence we still useq_ito denote the pole. See Figure 2.2 for the casep= 2 and m1=m2= 0.

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γ1

γ1 γ2 γ2

L⁺₁ R₁⁺ L⁻₁ R⁻₁ L⁺₂ R₂⁺ L⁻₂ R⁻₂

Figure 2.2. The flat geometric neighborhood ofγ₁andγ₂for the case p = 2 and m1 = m2 = 0. Here we identify L⁻₁ = L⁺₂, L⁺₁ = L⁻₂, R⁺₁ =R⁻₂, and R⁻₁ =R⁺₂. As γ1, γ2 → 0, the middle two half-disks form a neighborhood of an ordinary point and the remaining two half- disks form a neighborhood of another ordinary point. Alternatively as L^±_i andR^±_j → ∞, the middle two half-planes form a neighborhood of a double pole and the remaining two half-planes form a neighborhood of another double pole. Both poles have zero residue.

Let (R, ηR) be the limit meromorphic differential out of the expanding operation. We thus conclude that

(ηR) =m1σ1+m2σ2−

p

X

i=1

(a⁰_i+a⁰⁰_i + 2)qi.

By the relation (2.1), the genus ofRis zero, henceR∼=P¹. Sinceq_i=C_i∩R is a separating node, it follows from the global residue condition of [4] that Resq_iηR = 0. Finally, in the expanding process the saddle connections γi

are all fixed, hence the configurationCis preserved in the limit meromorphic differential ηR. Summarizing the above discussion, we see that the limit twisted differential is parameterized by ∆(µ,C).

The other part of the claim follows from the flat geometric smoothing of [4], as twisted differentials in ∆(µ,C) satisfy the global residue condition and have the desired configuration of saddle connections.

Remark 2.2. — For the purpose of calculating Siegel–Veech constants, the Eskin–Masur–Zorich principal boundary only takes into account the non-degenerate componentsCi and discards the degenerate rational com- ponentR, though it is quite visible — for instance, R can be seen as the central sphere in [12, Figure 5].

2.4. Meromorphic differentials of type I onP¹

Recall that for a twisted differential η in ∆(µ,C), its restriction η_R on the component R ∼= P¹ has two zeros andp poles, where the residue at

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each pole is zero. Up to scale,η_Ris uniquely determined by the zeros and poles. In this section we study the locus of P¹ marked at such zeros and poles.

Given integersm1, m2>1 andn1, . . . , np>2 withm1+m2−Pp i=1ni=

−2, let Z ⊂ M0,p+2 be the locus of pointed rational curves (P¹, σ₁, σ₂, q₁, . . . , qp) such that there exists a differentialη0onP¹ satisfying that

(η₀) =m₁σ₁+m₂σ₂−

p

X

i=1

n_iq_i and Res_q_iη₀= 0 for eachi= 1, . . . , p.

For a given (admissible) configurationC= (m1, m2,{a⁰_i, a⁰⁰_i}^p_i=1), consider the subsetZ(C)⊂ Zparameterizing differentialsη₀onP¹(up to scale) that admit a configuration of typeC.

Proposition 2.3. — Z is a union ofZ(C)for all possible (admissible) configurationC and eachZ(C)consists of a single point.

Proof. — We provide a constructive proof using the flat geometry of meromorphic differentials. Let us make some observation first. Supposeη0

is a differential on P¹ whose underlying divisor corresponds to a point in Z. Sinceη0has zero residue at every pole, for any closed pathγ that does not contain a pole ofη0, the Residue Theorem says that

Z

γ

η₀= 0.

In particular, ifαandβ are two saddle connections joiningσ₁ toσ₂, then α−β represents a closed path onP¹, hence

Z

α

η0= Z

β

η0,

andαandβnecessarily have the same holonomy. It also implies thatη0has no self saddle connections. Collect the saddle connections from σ₁ to σ₂, list them clockwise atσ1, and count the angles between two nearby ones.

Since the saddle connections have the same holonomy, the angles between them are multiples of 2π, and hence they give rise to a configurationC. It implies that the underlying divisor of η0 corresponds to a point inZ(C).

Therefore,Z is a union ofZ(C).

Now supposeη0admits a configuration of typeC= (m1, m2,{a⁰_i, a⁰⁰_i}^p_i=1), i.e., up to scale it corresponds to a point inZ(C). Recall thatσ₁,σ₂, and qi are the zeros and poles of orderm1, m2, and a⁰_i+a⁰⁰_i + 2, respectively, wherei= 1, . . . , p, andγ₁, . . . , γ_p are the saddle connections joiningσ₁to σ2 such that the angle betweenγi andγi+1 in the clockwise orientation at

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σ₁is 2π(a⁰_i+ 1), and atσ₂ is 2π(a⁰⁰_i + 1). By the preceding paragraph, there are no other saddle connections betweenσ1 andσ2.

Rescale η₀ such that all the γ_i have holonomy equal to 1, that is, they are in horizontal, positive direction, and of length 1. Cut the flat surfaceη0

along all horizontal directions through σ₁ and σ₂, such that η₀ is decom- posed into a union of half-planes as basic domains in the sense of [6]. These basic domains are of two types according to their boundary half-lines and saddle connections. The boundary of the basic domains of the first type contains exactly one of σ1 and σ2 that emanates two half-lines to infinity on both sides. The boundary of the basic domains of the second type, from left to right, consists of a half-line ending atσ1, followed by a saddle connectionγi, and then a half-line emanating forσ₂.

Since the angle betweenγi andγi+1is given for eachi, the configuration Cdetermines uniquely how these basic domains are glued together to form η₀. More precisely, start from an upper half-plane S₁⁺ of the second type with two boundary half-linesL⁺₁ to the left andR⁺₁ to the right, joined by the saddle connectionγ₁. Turn aroundσ₁in the clockwise orientation. Then we will see a lower half-plane S₁⁻ of the second type with two boundary half-linesL⁻₁ andR⁻₁ joined byγ₁. Ifa⁰₁ = 0, i.e., if the angle betweenγ₁ andγ2 in the clockwise orientation is 2π, then next we will see an upper half-planeS₂⁺of the second type with two boundary half-linesL⁺₂ andR⁺₂ joined byγ₂, which is glued toS₁⁻by identifyingL⁺₂ withL⁻₁. See Figure 2.2 above for an illustration of this case.

On the other hand if a⁰₁ > 0, we will see a⁰₁ pairs of upper and lower half-planes of the first type containing onlyσ1in their boundary, and then followed by the upper half-plane of the second type containing γ₂ in the boundary. Repeat this process for each pairγi andγi+1 consecutively, and also use the angle betweenγiandγi+1atσ2to determine the identification of theR^±_i -edges emanated from σ₂. We conclude that the gluing pattern of these half-planes is uniquely determined by the configurationC.

Finally, since the angle between γ_i and γ_i+1 at σ₁ is 2π(a⁰_i+ 1) and at σ2 is 2π(a⁰⁰_i + 1), it determines precisely a⁰_i+a⁰⁰_i + 1 pairs of upper and lower half-planes that share the same point at infinity. In other words, they form a flat geometric neighborhood of a pole with ordera⁰_i+a⁰⁰_i + 2, which is the desired pole order ofqi fori = 1, . . . , p. We have thus verified that Z(C) is nonempty and all differentials up to scale parameterized byZ(C) have the same basic domain decomposition, henceZ(C) consists of a single

point.

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Example 2.4. — Consider the casem₁= 1,m₂= 1,n₁= 2 andn₂= 2.

The only admissible configuration is

a⁰₁=a⁰⁰₁ =a⁰₂=a⁰⁰₂ = 0,

henceZ consists of a single point. As a cross check, take σ₁ = 1, q₁= 0, andq2 =∞in P¹, and let z be the affine coordinate. Then up to scaleη0

can be written as

(z−1)(z−σ2)

z² dz.

It is easy to see that Res_q_iη₀= 0 if and only ifσ₂=−1.

Example 2.5. — Consider the casem₁= 1,m₂= 3 andn₁=n₂=n₃= 2. There do not exist nonnegative integersa⁰₁, a⁰₂, a⁰₃satisfying that

(a⁰₁+ 1) + (a⁰₂+ 1) + (a⁰₃+ 1) =m1+ 1 = 2,

because the left-hand side is at least 3. Since there is no admissible configuration, we conclude thatZ is empty. As a cross check, letq1= 0, q2= 1, andq3=∞. Up to scaleη0 can be written as

(z−σ₁)(z−σ₂)³ z²(z−1)² dz.

One can directly verify that there are noσ1, σ2∈P¹\ {0,1,∞}such that Res_q_iη₀= 0.

3. Principal boundary of type II

3.1. Configurations of type II: saddle connections joining a zero to itself

Let C be a flat surface in H(µ). Suppose C has precisely m homologous closed saddle connectionsγ₁, . . . , γ_m, each joining a zero to itself. Let L⊂ {1, . . . , m}be an index subset such that the curvesγlforl∈Lbound qcylinders. After removing the cylinders along with all theγ_k, the remaining part inCsplits intop=m−q disjoint surfacesC1, . . . , Cp, where the boundary of the closureCk of eachCkconsists of two closed saddle connec- tionsα_k andβ_k. These surfaces are glued together in a cyclic order to form C. More precisely, each Ck is connected to Ck+1 by either identifying αk

withβ_k+1 (as someγ_i in C) or inserting a metric cylinder with boundary αkandβk+1. The sum of genera of theCkisg−1, because the cyclic gluing procedure creates a central handle, hence it adds an extra one to the total genus (see [12, Figure 7]).

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There are two types of the surfacesC_k according to their boundary components. If the boundary saddle connectionsαi and βi of Ci are disjoint, we say that C_i has a pair of holes boundary. In this case α_i contains a single zeroziwith cone angle (2ai+ 3)πinsideCi, andβi contains a single zerowi with cone angle (2bi+ 3)πinsideCi, whereai, bi>0. We also take into account the special casem= 1, i.e., when we cutC along γ1, we get only one surfaceC1 with two disjoint boundary componentsα1andβ1. In this casez₁ is identified withw₁ in C, and we still say thatC₁ has a pair of holes boundary.

For the remaining case, ifα_j andβ_j form a connected component for the boundary ofCj, we say that Cj has afigure eight boundary. In this case αj andβj contain the same zero zj. Denote by 2(c⁰_j+ 1)πand 2(c⁰⁰_j + 1)π the two angles bounded byαj and βj inside Cj, wherec⁰_j, c⁰⁰_j >0, and let cj=c⁰_j+c⁰⁰_j.

In summary, the configuration considered above consists of the data (L,{ai, bi},{c⁰_j, c⁰⁰_j}).

Conversely, given the surfaces Ck along with some metric cylinders, lo- cal gluing patterns can create zeros of the following three types (see [12, Figure 12] and [5, Figures 6-8]):

(i) A cylinder, followed by k > 1 surfaces C₁, . . . , C_k, each of genus gi > 1 with a figure eight boundary, followed by a cylinder. The total angle at the newborn zero is

π+

k

X

i=1

(2c⁰_i+ 2c⁰⁰_i + 4)π+π, hence its zero order is

k

X

i=1

(c_i+ 2).

(ii) A cylinder, followed by k > 0 surfaces Ci, each of genus gi > 1 with a figure eight boundary, followed by a surfaceC_k+1 of genus gk+1 > 1 with a pair of holes boundary. The total angle at the newborn zero is

π+

k

X

i=1

(2c⁰_i+ 2c⁰⁰_i + 4)π+ (2bk+1+ 3)π, hence its zero order is

k

X

i=1

(c_i+ 2) + (b_k+1+ 1).

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(iii) A surfaceC₀of genusg₀>1 with a pair of holes boundary, followed by k > 0 surfaces Ci, each of genus gi > 1 with a figure eight boundary, followed by a surfaceC_k+1of genusg_k+1>1 with a pair of holes boundary. The total angle at the newborn zero is

(2a₀+ 3)π+

k

X

i=1

(2c⁰_i+ 2c⁰⁰_i + 4)π+ (2b_k+1+ 3)π, hence its zero order is

k

X

i=1

(ci+ 2) + (a0+ 1) + (bk+1+ 1).

For example, the flat surface in [12, Figure 7] is constructed as follows:S₁ with a pair of holes boundary, followed byS2with a pair of holes boundary, then a cylinder, followed byS₃ with a figure eight boundary, then another cylinder, followed byS4with a figure eight boundary, and finally back toS1.

3.2. The principal boundary of type II

SupposeC^ε∈ H(µ) has the configurationC= (L,{ai, bi},{c⁰_j, c⁰⁰_j}) with themhomologous saddle connectionsγ₁, . . . , γmof length at mostε. More- over, suppose thatC^εdoes not have any other saddle connections shorter than 3ε. As before, we degenerateC^εby shrinkingγito zero while keeping the configuration, such that the ratio of any other period toγ_i becomes arbitrarily large. Let ∆(µ,C) be the space of twisted differentials that arise as limits of such a degeneration process. Recall the three types of gluing patterns and newborn zeros in the preceding section. We will analyze the types of their degeneration as building blocks to describe twisted differentials in

∆(µ,C).

For the convenience of describing the degeneration, we view a cylinder as a union of two half-cylinders by truncating it in the middle. Then as its height tends to be arbitrarily large compared to the width, each half- cylinder becomes a half-infinite cylinder, which represents a flat geometric neighborhood of a simple pole. Moreover, the two newborn simple poles have opposite residues, because the two half-infinite cylinders have the same width with opposite orientations.

Proposition 3.1. — Consider a block of surfaces of type (i) in C^ε, that is, a half-cylinder, followed by k > 1 surfaces C₁^ε, . . . , C_k^ε, each of genusg_i>1with a figure eight boundary, followed by a half-cylinder. Let σbe the newborn zero of orderPk

i=1(ci+ 2). Asε→0, we have

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• The limit differential consists ofk disjoint surfaces C₁, . . . , C_k attached to a componentR∼=P¹at the nodesq1, . . . , qk, respectively.

• Rcontains onlyσamong all the markings.

• For eachi= 1, . . . , k,ordq_iηC_i =ci andordq_iηR=−ci−2.

• For eachi= 1, . . . , k,Resq_iηR= 0.

• ηR has two simple poles atq0 and qk+1 ∈ R\ {σ, q1, . . . , qk} with opposite residues±r.

• η_R admits a configuration of type (i), i.e., it has precisely k+ 1 homologous self saddle connections with angles 2(c⁰_i + 1)π and 2(c⁰⁰_i + 1)π in between consecutively fori= 1, . . . , k, and with holonomy equal torup to sign.

See Figure 3.1 for an illustration of the underlying curve of the limit differential.

R

C1 Ck

σ

q0 q1 qk qk+1

Figure 3.1. The underlying curve of the limit differential in Proposition 3.1.

Proof. — Asε→0, the limit of each C_i^ε is a flat surfaceCi, where the figure eight boundary of C_i^ε shrinks to a single zero qi with cone angle (2c_i + 2)π, i.e., q_i is a zero of order c_i. This shrinking operation is the inverse of the figure eight construction, see [12, Figure 10]. On the other hand, instead of shrinking the boundary saddle connectionsα_i, β_iof theC_i^ε, up to scale it amounts to expanding the other periods of theC_i^εarbitrarily long compared to theα_i, β_i. Since a small neighborhoodN_i enclosing both αi and βi in C_i^ε consists of 2(c⁰_i+c⁰⁰_i + 1) metric half-disks, under the expanding operation they turn into 2(c⁰_i+c⁰⁰_i + 1) = 2(ci+ 1) metric half- planes that form the basic domain decomposition for a pole of orderc_i+2 in the sense of [6]. The boundary loop ofNicorresponds to the vanishing cycle aroundq_iin the shrinking operation, which implies that the resulting pole will be glued toqias a node in the limit. In addition, the two half-cylinders expand to two half-infinite cylinders, which create two simple polesq₀ and qk+1with opposite residues±r, whererencodes the width of the cylinders.

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Let (R, η_R) be the limit meromorphic differential out of the expanding operation. We thus conclude that

(ηR) =

k

X

i=1

(ci+ 2)

! σ−

k

X

i=1

(ci+ 2)qi−q0−qk+1,

and hence the genus ofR is zero. Sinceq_i =C_i∩R is a separating node, it follows from the global residue condition of [4] that Resq_iηR = 0. As a cross check,

k+1

X

i=0

Res_q_iη_R= Res_q₀η_R+ 0 +· · ·+ 0 + Res_q_k+1η_R= 0,

hence ηR satisfies the Residue Theorem on R. Finally, the cylinders are glued to the figure eight boundary on both sides, hence thek+ 1 homologous self saddle connections have holonomy equal torup to sign. Their configuration (holonomy and angles in between) is preserved in the expanding process, hence the limit differentialη_Rpossesses the desired configura-

tion.

Proposition 3.2. — Consider a block of surfaces of type (ii) in C^ε, that is, a half-cylinder, followed by k > 0 surfaces C₁^ε, . . . , C_k^ε, each of genusg_i >1 with a figure eight boundary, followed by a surface C_k+1^ε of genusgk+1>1 with a pair of holes boundary. Let σbe the newborn zero of orderPk

i=1(ci+ 2) + (bk+1+ 1). Asε→0, we have

• The limit differential consists ofk+ 1disjoint surfacesC₁, . . . , C_k+1 attached to a componentR∼=P¹at the nodesq1, . . . , qk+1, respectively.

• ord_q_k+1η_C_k+1=b_k+1 andord_q_k+1η_R=−b_k+1−2.

• For eachi= 1, . . . , k,Resq_iηR= 0.

• η_R has a simple pole atq₀∈R\ {σ, q₁, . . . , q_k+1} withRes_q₀η_R=

−Resq_k+1ηR=±r.

• ηR admits a configuration of type (ii), i.e., it has precisely k+ 1 homologous self saddle connections with angles 2(c⁰_i + 1)π and 2(c⁰⁰_i + 1)π in between consecutively fori= 1, . . . , k, and with holonomy equal torup to sign.

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R

C1 Ck

Ck+1

σ

q0 q1 qk qk+1

Proof. — The proof is almost identical with the preceding one. The only difference occurs at the last surface. A small neighborhoodNk+1 enclosing β_k+1inC_k+1^ε consists of 2(b_k+1+ 1) half-disks, one of which is irregular as in [12, Figure 8], hence in the expanding process they turn into 2(bk+1+ 1) half-planes, giving a flat geometric neighborhood for a pole of orderbk+1+2.

Moreover,N_k+1 is homologous to the γ_i. The orientation of N_k+1 is the opposite to that of N0 enclosing the boundary α0 of the beginning half cylinder, hence their homology classes add up to zero. We thus conclude that Resq₀ηR = −Resq_k+1ηR. Alternatively, it follows from the Residue Theorem applied toR, since Resq_iηR= 0 for alli= 1, . . . , k. The holonomy of the saddle connections and the angles between them are preserved in the expanding process, henceηRhas the configuration as described.

Proposition 3.3. — Consider a block of surfaces of type (iii) in C^ε, that is, a surface C₀^ε of genus gk+1 > 1 with a pair of holes boundary, followed byk >0 surfacesC₁^ε, . . . , C_k^ε, each of genusgi >1 with a figure eight boundary, followed by a surfaceC_k+1^ε of genusg_k+1 >1 with a pair of holes boundary. Let σ be the newborn zero of order Pk

i=1(ci+ 2) + (a0+ 1) + (bk+1+ 1). Asε→0, we have

• The limit differential consists ofk+ 2disjoint surfacesC₀, . . . , C_k+1 attached to a componentR∼=P¹at the nodesq0, . . . , qk+1, respectively.

• ordq0η_C₀=a₀ andordq0η_R=−a0−2.

• ordq_k+1ηC_k+1=bk+1 andordq_k+1ηR=−bk+1−2.

• For eachi= 1, . . . , k,Res_q_iη_R= 0.

• Resq₀ηR=−Resq_k+1ηR=±r.

• η_R admits a configuration of type (iii), i.e., it has precisely k+ 1 homologous self saddle connections with angles 2(c⁰_i + 1)π and

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2(c⁰⁰_i + 1)π in between consecutively fori= 1, . . . , k, and with holonomy equal torup to sign.

R

C1 Ck

Ck+1

C0

σ

q0 q1 qk qk+1

Proof. — Since the beginning and ending surfaces both have a pair of holes boundary, the proof follows from the previous two.

Let us call the limit twisted differentials in Propositions 3.1, 3.2, and 3.3 surfaces of degenerate type(i), (ii), and (iii), respectively. In order to glue them to form a global twisted differential, the above proofs (and also the definition of twisted differentials) imply the following gluing pattern. The simple poleq0 (orqk+1) in a surface of degenerate type (i) has to be glued with a simple pole in another surface of type (i) or (ii), and the same description holds forq0 in a surface of type (ii). For a surface of type (ii), the component Ck+1 has to be contained in another surface of type (ii) or (iii). Namely, it has a zero of orderb⁰_k+1 that is glued with a pole q_k+1⁰ of orderb⁰_k+1+ 2 in the rational component R⁰ of the other surface. The same description holds forC₀ andC_k+1 in a surface of type (iii).

Theorem 3.4. — In the above setting, ∆(µ,C) parameterizes twisted differentials constructed by gluing surfaces of degenerate type (i), (ii), and(iii).

Proof. — SinceC^εadmits the configurationC= (L,{ai, bi},{c⁰_j, c⁰⁰_j}), it can be constructed by gluing blocks of surfaces of type (i), (ii), and (iii).

By applying Propositions 3.1, 3.2, and 3.3 simultaneously, we thus conclude that the limit twisted differential is formed by gluing surfaces of degenerate

type (i), (ii), and (iii) as above.

We summarize some useful observation out of the proofs.

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Remark 3.5. — If the homologous closed saddle connections in a configuration C of type II contains k distinct zeros, then a curve in ∆(µ,C) containskrational components. Moreover, if two rational components intersect, then each of them has a simple pole at the node, and the residues at the two branches of the node add up to zero. In general, at the polar nodes the residues are±rfor a fixed nonzero r∈C, such that their signs are alternating along the (unique) circle in the dual graph of the entire curve, and that the holonomy of the saddle connections is equal torup to sign.

Example 3.6. — The limit of the surface in [12, Figure 7] as theγishrink to zero is of the following type:S₁, followed by a markedP¹, followed byS₂, followed by a markedP¹, followed by a markedP¹with anS3tail, followed by a markedP¹with an S₄ tail, and back toS₁, see Figure 3.4, whereR₁ is of type (iii),R2is of type (ii),R3is of type (i) andR4 is of type (ii).

S1

R1

S2

R2

R3

S3

S4

R4

Figure 3.4. The underlying curve of the degeneration of[12, Figure 7].

Example 3.7. — The limit of the surface in [12, Figure 11] as the γ_i shrink to zero is of the following type: a flat torusE1, followed by a chain of two P¹, each with a marked simple zero, followed by a flat torus E2, followed by a chain of twoP¹, each with a marked simple zero, and back to E1, see Figure 3.5. Moreover, the differential on each P¹ has a double pole at the intersection with one of the tori and has a simple pole at the intersection with one of theP¹. Finally, the residues at the two poles of each P¹ are ±r for some fixed nonzero r ∈ C, such that their signs are alternating along the cyclic dual graph of the entire curve.

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E1 E2

Figure 3.5. The underlying curve of the degeneration of[12, Figure 11].

3.3. Meromorphic differentials of type II onP¹

Recall in Proposition 2.3 we showed that differentials on P¹ admitting a given configuration of type I are unique up to scale. The same result holds for differentials on P¹ admitting a given configuration of type (i), (ii), or (iii) as above.

Proposition 3.8. — Letη₀be a differential onP¹that admits a configuration of type either (i),(ii), or (iii)as described in Propositions 3.1, 3.2, and 3.3. Then up to scale suchη₀is unique.

Proof. — We provide a constructive proof for the case of type (i), which is analogous to the proof of Proposition 2.3. The other two types follow similarly.

Let us make some observation first. Suppose η0 is a differential on P¹ with a unique zeroσ and k+ 2 polesq₀, . . . , q_k+1 such that Res_q_iη₀ = 0 fori= 1, . . . , k, and that Resq₀η0=−Resq_k+1 =±r for a nonzeror. Let αand β be two self saddle connections ofη₀. Treat them as closed loops inC=P¹\ {qk+1}. Then the indices ofαandβ toq0 cannot be zero, for otherwise the integral ofη0 along them would be zero, contradicting that they are saddle connections of positive length. Therefore, both of them encloseq0 inC, hence by the Residue Theorem

Z

α

η₀= Z

β

η₀=±r.

We conclude that in this case all saddle connections ofη0are homologous with holonomy equal to±r.

Now suppose η₀ admits the configuration of type (i) (as the description forηRin Proposition 3.1). Rescaleη0such that the holonomy of the saddle connectionsγ₁, . . . , γ_k+1is 1. By the preceding paragraph,η₀has no other saddle connections. Cut the flat surfaceη0 along all horizontal directions through the unique zero σ. Since η₀ has two simple poles with opposite residues equal to±1, we see two half-infinite cylinders with boundary given

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by the first and the last saddle connectionsγ₁andγ_k+1, respectively. The rest part ofη0 splits into half-planes as basic domains in the sense of [6], which are of two types according to their boundary. The boundary of the half-planes of the first type contains σ that emanates two half-lines to infinity on both sides. The boundary of the half-planes of the second type, from left to right, consists of a half-line ending atσ, followed by a saddle connectionγi, and then a half-line emanated fromσ.

Since the angles between γ_i andγ_i+1 are given on both sides inside the open surface (after removing the two half-infinite cylinders), this configuration determines how these half-planes are glued together. More precisely, say in the counterclockwise direction the angle between γi and γi+1 is 2π(c⁰_i+ 1). Then starting from the upper half-planeS_i⁺ of the second type containingγiin the boundary and turning counterclockwise, we will seec⁰_i pairs of lower and upper half-planes of the first type, and then the lower half-planeS_i+1⁻ of the second type containingγ_i+1in the boundary. Repeat this process for eachion both sides. We conclude that the gluing pattern of these half-planes is uniquely determined by the configuration. After gluing, the resulting open surface has a single figure eight boundary formed byγ1

andγk+1at the beginning and at the end, which is then identified with the boundary of the two half-infinite cylinders to recoverη0. Finally, since the angles betweenγi andγi+1are 2π(c⁰_i+ 1) and 2π(c⁰⁰_i + 1) on both sides, it determines preciselyc⁰_i+c⁰⁰_i+ 1 =c_i+ 1 paris of upper and lower half-planes that share the same point at infinity. In other words, they give rise to a flat geometric representation of a pole of orderc_i+ 2, which is the desired

pole order fori= 1, . . . , k.

4. Spin and hyperelliptic structures

For special µ, the stratum H(µ) can be disconnected. Kontsevich and Zorich ([19]) classified connected components ofH(µ) for allµ. Their result says thatH(µ) can have up to three connected components, where the extra components are caused by spin and hyperelliptic structures.

4.1. Spin structures

We first recall the definition of spin structures. Supposeµ= (2k₁, . . . ,2k_n) is a partition of 2g−2 with even entries only. For an abelian differential (C, ω)∈ H(µ), let

(ω) = 2k1σ1+· · ·+ 2knσn