HHIGH RANK INVARIANT SUBVARIETIES
PAUL APISA AND ALEX WRIGHT
Dedicated to Maryam Mirzakhani,who contributed to key ideas in this project,and whose vision continues to inspire us
Contents
1. Introduction 12. Structure of the proof 63. The boundary of an invariant subvariety 94. Cylinder degenerations 145. A cylinder degeneration dichotomy (with Mirzakhani) 286. Double degenerations (with Mirzakhani) 337. Classification using a nested free cylinder 398. Finding useful cylinders 489. Primality of the boundary 5210. Proof of Proposition 2.6 5411. Typical rank-preserving degenerations 5812. Proof of Proposition 2.7 62References 661.
Introduction
Main result.
Eskin, Mirzakhani, and Mohammadi showed that GL + (2 , R ) orbit closures of translation surfaces are properly immersedsmooth sub-orbifolds, and Filip showed they are moreover algebraicvarieties [EM18, EMM15, Fil16] . Conversely, all irreducible closed GL + (2 , R )-invariant subvarieties of strata of translation surfaces, orinvariant subvarieties for short, are GL + (2 , R ) orbit closures.Despite these strong structure theorems, as well as a great deal ofolder work preceding them and newer work building upon them, itremains a major open problem to classify invariant subvarieties. Inthis paper we give a significant portion of such a classification. Theorem 1.1.
Let M be an invariant subvariety of genus g transla-tion surfaces with rank( M ) ≥ g + 1 . Then M is either a connected a r X i v : . [ m a t h . D S ] F e b APISA AND WRIGHT component of a stratum, or the locus of all holonomy double covers ofsurfaces in a stratum of quadratic differentials.
Recall that the tangent space T ( X,ω ) ( M ) to M at a point ( X, ω ) is asubspace of the relative cohomology group H ( X, Σ , C ), where Σ is theset of zeros of ω . There is a natural map p : H ( X, Σ , C ) → H ( X, C )from relative to absolute cohomology, and rank is defined byrank( M ) = 12 dim p ( T ( X,ω ) ( M )) . This is a positive integer that is at most the genus g . We say that M has high rank if rank( M ) ≥ g + 1.Rank is the most important notion of the size of an invariant sub-variety. Even when an orbit closure is not explicitly known, its rankcan often be bounded below either by finding cylinders [Wri15a , The-orem 1.10 ] or using analytic techniques [MW18 , Section 7 ] . Rank alsoplays a central role in algebro-geometric descriptions of orbit closures [Fil16] .In Theorem 1.1 it is implicit that the surfaces do not have markedpoints, but a version with marked points follows immediately fromknown results, as in the proof of Theorem 2.8 below.In Corollary 7.3, we will see that the conclusion of Theorem 1.1 alsoholds if M ⊂ H (2 g −
2) and rank( M ) ≥ g + , which is a slightimprovement in odd genus.1.2. Context.
Prior to the work of Eskin, Mirzakhani, and Moham-madi, McMullen classified orbit closures in genus 2 [McM07] ; see also [McM03,Cal04] . More recently, orbit closures of full rank (rank( M ) = g ) were classified by Mirzakhani and Wright [MW18] ; and orbit clo-sures of rank at least 2 were classified in genus 3 primarily by Aulicinoand Nguyen [NW14, ANW16, AN16, AN] , and in hyperelliptic con-nected components of strata by Apisa [Api18] .New orbit closures of rank 2 were recently constructed by Mukamel,McMullen and Wright in [MMW17] , and, with Eskin, in [EMMW20] .See also [Api19, Ygo] for recent progress in rank 1 and [EFW18,BHM16, LNW17] for recent finiteness results.1.3. Hopes for a magic table.
To solve many dynamical and geo-metric problems about translations surfaces, including questions aris-ing in the study of rational billiards, interval exchange transformations,and other applications, it is necessary to first know the orbit closure ofthe translation surface. Indeed, the orbit closure can be viewed as thearena for re-normalization, and the answers to many questions dependquantitatively on its geometry.
IGH RANK INVARIANT SUBVARIETIES 3
Zorich wrote that his most “optimistic hopes” for the study of trans-lation surfaces were that a “magic table” could be created, containinga list of orbit closures and their numerical invariants [Zor06 , Section9.3 ] . Classification results like Theorem 1.1 bring us closer to the com-plete realization of these hopes.1.4. Mirzakhani’s conjecture.
Mirzakhani conjectured a classifica-tion of orbit closures of rank at least 2; see [AWc , Remark 1.3 ] fora more detailed discussion. The new orbit closures constructed in [MMW17, EMMW20] disprove her conjecture, but we view The-orem 1.1 as substantiating the vision behind it.Mirzakhani participated in our early efforts to prove Theorem 1.1,and we are very grateful for the insights she contributed, especially inSections 5 and 6, which we have marked as containing joint work withher.1.5. Three key difficulties.
Our approach to Theorem 1.1 is induc-tive, with genus 2 as the base case. Using cylinder deformations, wecan pass from a surface in M to a surface in a component M (cid:48) in theboundary of M . The first key difficulty is that it is not always possibleto arrange for such codimension 1 degenerations M (cid:48) to have high rank.For this reason, we develop a notion of “double degenerations”.Given a collection of cylinders on a surface in M , in many cases thisgives a degeneration of the surface in a codimension 2 component ofthe boundary of M . See Figure 1.1 for one of the most concrete ex-amples when M is a stratum. The second key difficulty is arranging Figure 1.1.
From left to right: A surface in H (4), witha vertical cylinder highlighted; the single degenerationof this cylinder is a surface in H (1 , H (0) × H (0).for the component of the boundary containing a double degenerationto be high rank. APISA AND WRIGHT
Throughout, we must deal with the possibility that degenerationscan disconnect the surface and create marked points. Using a struc-ture theorem from [CW19] , we are able to easily show that the singledegenerations we require are connected, but avoiding the possibilitythat double degenerations are disconnected is the third key difficultyin the paper.1.6.
Certificates.
Our proof that double degenerations exist is notexplicit, and these degenerations themselves are surprisingly subtle anddifficult to work with. So we were not able to rule out in general thepossibility that the component of the boundary containing a doubledegeneration satisfies the conclusion of Theorem 1.1 even if M doesnot.For this reason, we rely on the main result of [AWa] , which showsthat the failure of an invariant subvariety to satisfy the conclusion ofTheorem 1.1 must be witnessed by cylinders. Proceeding by contradic-tion, we consider a minimal dimensional counterexample M to Theo-rem 1.1. The main result of [AWa] produces a collection of cylinderson a surface in M which could not exist if M satisfied the conclusionof Theorem 1.1. We think of these cylinders as being a certificate that M does not satisfy the conclusion of Theorem 1.1. We are able toarrange for this certificate to persist on the double degeneration, thusshowing that the associated codimension 2 boundary component of M also does not satisfy the conclusion of Theorem 1.1. We arrange for thisto contradict the fact that M was chosen to be a minimal dimensionalcounterexample to Theorem 1.1.1.7. The Cylinder Degeneration Dichotomy.
Most of the argu-ments in this paper are novel and broadly applicable to the classifi-cation problem, and we hope they will shed significant light on thestructure of invariant subvarieties.We wish to highlight especially a general dichotomy for cylinder de-generations, proven in Section 5, which arose from joint work withMirzakhani. This result shows in particular that when cylinders de-generate to a collection of saddle connections, this collection is eitheracyclic or strongly connected when viewed as a directed graph. In Sec-tion 6, we show that in the acyclic case this graph can be contractedin a natural way, thus producing the double degeneration.1.8.
Organization.
In Section 2, we reduce the proof of Theorem 1.1to two propositions. We then turn in Section 3 to the boundary ofinvariant subvarieties, observing the “symplectic compatibility” of theboundary and recalling the notion of primality from [CW19] . IGH RANK INVARIANT SUBVARIETIES 5
Section 4 develops the general theory of cylinder degenerations. Thisis followed in Sections 5 and 6 with the proof of the Cylinder Degener-ation Dichotomy and the construction of double degenerations.In Section 7, we prove that if a certain simple configuration of twocylinders appears on a surface in an invariant subvariety, then thatsubvariety satisfies the conclusion of Theorem 1.1. This allows us toavoid such simple configurations, which give rise to components of theboundary containing double degenerations that may not be high rank.In Sections 8 and 10, we produce the cylinders that will be doubledegenerated and then perform the double degeneration. These resultsare all phrased for invariant subvarieties of rank at least 2 or 3, withthe high rank assumption only used to ensure prime degenerations areconnected, via a short argument in Section 9.Sections 11 and 12 conclude the paper by handling the cases whereonly a single degeneration is required.1.9.
Prerequisites.
We assume the reader is familiar with cylinderdeformations and the What You See Is What You Get (WYSIWYG)partial compactification, as in [Wri15a, MW17, CW19] , but we donot assume familiarity with any other recent results.This paper is the culmination of a large project, which motivated andrelies on [AWc] and [AWa] . In spite of this, here we have isolated thedependence on those two papers to a single statement, which appearsas Theorem 2.2 below. So, it is not necessary or even helpful to haveread those two papers.1.10.
Next steps.
The next step in this line of investigation would bethe following.
Conjecture 1.2 (Almost High Rank Conjecture) . If M is an orbitclosure of surfaces of genus g andrank( M ) = g , then M is a locus of double covers of a component of a stratum ofAbelian differentials, a locus of double covers of a hyperelliptic locus ina stratum of Abelian differentials, or the locus of all holonomy doublecovers of surfaces in a stratum of quadratic differentials.Although there are many directions in which the classification prob-lem can be explored, we think the following deserves special attention. Question 1.3.
Is every invariant subvariety of rank at least 3 trivial?More precisely, is every such subvariety a full locus of covers accordingto the definition given in [AWc] ? APISA AND WRIGHT
Although we feel there is still insufficient evidence to upgrade thisquestion to a conjecture, there are a number of reasons to hope for apositive answer. It would be interesting to investigate the special caseof rank 3 orbit closures of dimension 6 in genus 6.
Acknowledgements.
We thank Maryam Mirzakhani for generouslysharing her ideas, vision, and enthusiasm with us. We also thank BarakWeiss for helpful conversations, and Francisco Arana-Herrera and ChrisZhang for helpful comments on an earlier draft.During the preparation of this paper, the first author was partiallysupported by NSF Postdoctoral Fellowship DMS 1803625, and the sec-ond author was partially supported by a Clay Research Fellowship,NSF Grant DMS 1856155, and a Sloan Research Fellowship.2.
Structure of the proof
Here we show, using previous results, that Theorem 1.1 follows fromtwo technical results, namely Propositions 2.6 and 2.7 below. Thisgives the high level structure of our approach. The remainder of thepaper will then develop the novel and broadly applicable results thatwe use to derive Propositions 2.6 and 2.7.We begin with the following observation.
Lemma 2.1.
In an invariant subvariety M of high rank, there existAbelian differentials which are not translation covers of smaller genusAbelian differentials.Proof. Suppose otherwise, and consider a surface with dense orbit in M that is a translation cover. Let g denote the genus of surfaces in M , and h denote the genus of the codomain of the covering map.So M is contained in an invariant subvariety of covers of surfaces ina genus h stratum of Abelian differentials, and we see that M has rankat most h .The Riemann-Hurwitz formula gives that 2 − g ≤ − h , which isequivalent to h ≤ g + . This contradicts the assumption that M hashigh rank. (cid:3) In light of Lemma 2.1, we can state the following special case of [AWa , Theorem 1.1 ] . Recall that an orbit closure M is said to be geminal if for any cylinder C on any ( X, ω ) ∈ M , either • any cylinder deformation of C remains in M , or • there is a cylinder C (cid:48) such that C and C (cid:48) are parallel and havethe same height and circumference on ( X, ω ) as well as on all
IGH RANK INVARIANT SUBVARIETIES 7 small deformations of (
X, ω ) in M , and any cylinder deforma-tion that deforms C and C (cid:48) equally remains in M .In the first case we say that C is free, and in the second case we saythat C and C (cid:48) are twins. Theorem 2.2.
Any geminal invariant subvariety of high rank is a com-ponent of a stratum of Abelian differentials or the set of all holonomydouble covers of surfaces in a stratum of quadratic differentials.Proof.
Without the high rank assumption, [AWa , Theorem 1.1 ] allowsfor two additional possibilities where M consists of covers of lowergenus Abelian differentials; but these are ruled out by Lemma 2.1. (cid:3) Our approach to Theorem 1.1 will be to prove the following:
Theorem 2.3.
Every high rank invariant subvariety consisting of sur-faces without marked points is geminal.
This immediately implies Theorem 1.1.
Proof of Theorem 1.1 assuming Theorem 2.3.
Let M be high rank. The-orem 2.3 gives that M is geminal, and hence Theorem 2.2 implies that M is a component of a stratum or the set of all holonomy double coversof surfaces in a stratum of quadratic differentials. (cid:3) Theorem 2.3 in turn will follow from the next two results. We beginwith the following definitions.
Definition 2.4.
Let p be a marked point on a translation surface( X, ω ) in an invariant subvariety M . The point p is said to be free if M contains all surfaces obtained by moving p while fixing the rest ofthe surface. Definition 2.5. A rel vector is defined to be an element of the sub-space ker( p ) ∩ T ( X,ω ) ( M ), where p is the map from relative to absolutecohomology. The dimension of this subspace is called the rel of M .We say M has no rel if the rel of M is zero, and that it has rel if therel is positive. Proposition 2.6.
Suppose that M is an invariant subvariety of genus g surfaces with no marked points, and assume that M has high rank,is not geminal, and has no rel. Additionally assume rank( M ) ≥ .Then the boundary of M contains an invariant subvariety M (cid:48) that(1) consists of connected surfaces,(2) has rank( M (cid:48) ) = rank( M ) − ,(3) is not geminal, APISA AND WRIGHT (4) does not have free marked points, and(5) consists of surfaces of genus at most g − . Proposition 2.7.
Suppose M is an invariant subvariety of genus g surfaces without marked points, and assume M has high rank, is notgeminal, and has rel.Then the boundary of M contains an invariant subvariety M (cid:48) that(1) consists of connected surfaces,(2) has rank( M (cid:48) ) = rank( M ) ,(3) is not geminal, and(4) does not have free marked points. We will also need the following special case of results in [Api20,AWb] . If M is an invariant subvariety of surfaces with marked points,we let F ( M ) denote the corresponding invariant subvariety with markedpoints forgotten. Theorem 2.8. If M (cid:48) is an invariant subvariety of high rank, possiblywith marked points but without free marked points, and F ( M (cid:48) ) is acomponent of a stratum of Abelian differentials or the locus of holonomydouble covers of surfaces in a stratum of quadratic differentials, then M (cid:48) is geminal.Moreover, either M (cid:48) = F ( M (cid:48) ) and M (cid:48) is a component of a stratumof Abelian differentials, or M (cid:48) is a quadratic double. Following [AWc] and [AWa] , we say M (cid:48) is a quadratic double if F ( M (cid:48) ) is the locus of holonomy double covers of surfaces in a stra-tum of quadratic differentials and the marked points in M (cid:48) occur inpairs exchanged by the holonomy involution or at fixed points of theholonomy involution, with no further constraints. We will sometimesuse this terminology even when there are no marked points, since it ismore concise. Proof.
First assume F ( M (cid:48) ) is a non-hyperelliptic component of a stra-tum of Abelian differentials. In this case, [Api20] implies, with no ad-ditional assumptions, that all marked points are free; since here thereare no free marked points, we get that M (cid:48) = F ( M (cid:48) ) is a componentof a stratum and hence geminal.Next assume F ( M (cid:48) ) is a hyperelliptic component of a stratum ofAbelian differentials of genus at least two. In this case, [Api20] implies,with no additional assumptions, that all marked points are free, oroccur in pairs exchanged by the hyperelliptic involution, or are fixedby the hyperelliptic involution. Since here there are no free marked IGH RANK INVARIANT SUBVARIETIES 9 points, M is a quadratic double. We also see that every cylinder C (cid:48) ona surface in M (cid:48) is either fixed by the hyperelliptic involution, in whichcase it is free, or its image is another cylinder C (cid:48) , in which case C and C (cid:48) are a pair of twins. In particular, M (cid:48) is geminal.Finally assume that F ( M (cid:48) ) is a quadratic double. We first claim thatthe associated stratum of quadratic differentials is not hyperelliptic.Indeed, if ( X, q ) is hyperelliptic with hyperelliptic involution τ , thenits holonomy double cover is a translation cover of the holonomy doublecover of ( X, q ) /τ ; so the claim follows from Lemma 2.1.Because the associated stratum of quadratic differentials is not hy-perelliptic, [AWb] gives that, without any additional assumptions, allmarked points are free, or occur in pairs exchanged by the holonomyinvolution, or are fixed points for the holonomy involution. We see that M (cid:48) is geminal and a quadratic double as in the previous case. (cid:3) Proof of Theorem 2.3 assuming Propositions 2.6 and 2.7.
Suppose, inorder to find a contradiction, that Theorem 2.3 is not true, and let M be a counterexample of minimal dimension.First we claim that M cannot have rank 2. Indeed, in that casethe high rank assumption implies that the genus is 2, and the onlyrank 2 invariant subvarieties in genus 2 are strata: this follows fromMcMullen’s classification in genus 2 [McM07] , and very short proofsusing newer technology have also been given in [MW18 , Lemma 3.2 ] and [Wri15b , Lemma 5.14, Proposition 5.16 ] . So assume M has rankat least 3.If M has no rel, let M (cid:48) be the invariant subvariety given by Propo-sition 2.6. Because the genus has decreased by at least two, and therank has decreased by one, M (cid:48) is high rank.If M has rel, let M (cid:48) be the invariant subvariety given by Proposition2.7. Because the rank has not changed and the genus has not increased, M (cid:48) is high rank.Since M (cid:48) is in the boundary of M , it has smaller dimension than M .Hence F ( M (cid:48) ) also has smaller dimension than M . By our minimalityassumption, F ( M (cid:48) ) must be geminal, and hence Theorem 2.2 givesthat F ( M (cid:48) ) is a component of a stratum of Abelian differentials or aquadratic double.Thus Theorem 2.8 gives that in fact M (cid:48) is geminal, contradictingthe statement of Proposition 2.6 or Proposition 2.7. (cid:3) The boundary of an invariant subvariety
This section recalls and establishes foundational results on the bound-ary of an invariant subvariety. We begin in Section 3.1 by recalling the structure of invariant subvarieties of multi-component surfaces from [CW19] ; in Sections 3.2 and 3.3 we prove new results showing thatthe symplectic form on first cohomology is as compatible as could beimagined with invariant subvarieties and their boundaries; and we endin Section 3.4 with some preliminary applications.3.1.
Prime boundary components.
In this subsection, we will sup-pose that M is an invariant subvariety in a product of strata of Abeliandifferentials H × · · · × H k . Thus, we are assuming that the k compo-nents of surfaces in M are ordered or labelled, but one can apply thesame discussion when the components are not ordered or labelled bylifting to a product of strata, as in [CW19] . We start by recalling aconcept introduced in [CW19] . Definition 3.1. M is called prime unless, possibly after reorderingthe components, there is some 1 ≤ s < k , and invariant subvarieties M (cid:48) ⊆ H ×· · ·×H s and M (cid:48)(cid:48) ⊆ H s +1 ×· · ·×H k such that M = M (cid:48) ×M (cid:48)(cid:48) .The study of invariant subvarieties of multi-component surfaces re-duces to the prime case, due to the following observation from [CW19 ,Corollary 7.10 ] . Theorem 3.2 (Chen-Wright) . M can be written uniquely as a productof prime subvarieties. The following result from [CW19 , Theorem 1.3 ] strongly restrictsthe structure of prime invariant subvarieties, and will play a prominentrole in the proof of Proposition 2.6. Here π i denotes the projectiononto the i -th factor. Theorem 3.3 (Chen-Wright) . If M is prime, then the following hold:(1) There is an invariant subvariety M i and a finite union M (cid:48) i ofproper invariant subvarieties of M i such that M i − M (cid:48) i ⊆ π i ( M ) ⊆ M i . (2) Locally in M , the absolute periods on any component of a multi-component surface in M determines the absolute periods on allother components.(3) The rank of M i is independent of i . The main statement in Theorem 3.3 is the second statement; thethird statement follows from the second, and the first statement is justa preliminary observation.
IGH RANK INVARIANT SUBVARIETIES 11
Clarifying the definition of rank.
In the multi-componentcase, rank is defined as in the single component case: given a point(
X, ω ) in M with singularities Σ, it is still the case that T ( X,ω ) ( M )is a subset of H ( X, Σ; C ). If p denotes the projection to absolutecohomology, then we definerank( M ) := 12 dim p ( T ( X,ω ) ( M )) . The absolute cohomology group H ( X, R ) has a natural symplecticform. Indeed, if the components of X are X , . . . , X k , then H ( X, R ) = (cid:76) ki =1 H ( X i , R ), and the sum of the symplectic forms on the H ( X i , R )is a symplectic form on H ( X ). We denote the symplectic form by (cid:104)· , ·(cid:105) .Our first goal in this section is to show p ( T ( X,ω ) ( M )) is symplec-tic, generalizing a result of Avila-Eskin-M¨oller [AEM17] to the multi-component case. Lemma 3.4. If ( X, ω ) ∈ M and M is prime, then p ( T ( X,ω ) ( M )) issymplectic and its dimension is twice the rank of M i for any i .Proof. Consider the flat subbundle of L of T M whose fiber over a point( X, ω ) in M consists of elements v such that (cid:104) p ( v ) , p ( w ) (cid:105) = 0 for all w ∈ T ( X,ω ) ( M ). Clearly, L contains ker( p ) as a sub-bundle.The symplectic pairing of the real and imaginary parts of ω is thearea of ( X, ω ), and so in particular cannot be zero. Hence, the real andimaginary parts of ω are not in L , and hence L is not all of T M .The statement of [CW19 , Theorem 7.13 ] , which generalizes that of [Wri15a , Theorem 5.1 ] , gives precisely that any proper flat sub-bundleof T M must be contained in ker( p ). Hence L = ker( p ).In other words, for every element v of T ( X,ω ) ( M ) that is not in thekernel of p , there is an element w of T ( X,ω ) ( M ) so that (cid:104) p ( v ) , p ( w ) (cid:105) (cid:54) = 0.This shows that (cid:104)· , ·(cid:105) induces a nondegenerate skew-symmetric bilinearform on p ( T ( X,ω ) ( M )) and hence that p ( T ( X,ω ) ( M )) is symplectic asdesired.The claim about dimension follows since, letting ( X i , ω i ) denote the i th component of ( X, ω ), ( π i ) ∗ : p ( T ( X,ω ) ( M )) → p ( T ( X i ,ω i ) ( M i )) is anisomorphism by Theorem 3.3 (2). (cid:3) It is now easy to generalize to the case when M is not prime. Corollary 3.5. If ( X, ω ) ∈ M and M is any invariant subvariety ofmulti-component surfaces, then p ( T ( X,ω ) ( M )) is symplectic.Moreover, the rank of M is half the dimension of a maximal di-mensional symplectic subspace of T ( X,ω ) ( M ) with respect to the form (cid:104) p ( · ) , p ( · ) (cid:105) . Proof.
By Theorem 3.2, there are a collection of prime invariant subva-rieties M , . . . , M d such that M = M × · · · × M d . This correspondsto a partition of ( X, ω ) as ( X , ω ) (cid:116) · · · (cid:116) ( X d , ω d ) where each ( X i , ω i )is a union of connected components of ( X, ω ). We can write p (cid:0) T ( X,ω ) ( M ) (cid:1) = p (cid:0) T ( X ,ω ) ( M ) (cid:1) ⊕ · · · ⊕ p (cid:0) T ( X d ,ω d ) ( M d ) (cid:1) . Lemma 3.4 gives that each summand is symplectic, so the first claimholds. The second claim follows immediately from the first. (cid:3)
Symplectic compatibility of the boundary.
Following [MW17] ,if (
X, ω ) is “close” to a point (
Y, ω Y ) in the boundary of the WYSI-WYG compactification, we can view the tangent space H ( Y, Σ Y ) ofthe stratum of ( Y, ω Y ) as a subspace of the tangent space H ( X, Σ X )of the stratum of ( X, ω ). We now explain how the symplectic formon H ( X ) is related to the symplectic form on H ( Y ), showing thatthe induced bilinear form on H ( Y, Σ Y ) is the restriction of the corre-sponding bilinear form on H ( X, Σ X ).Let X be an oriented surface. Let f : X → Y (cid:48) be a map obtained asfollows: First collapse a simple multi-curve γ on X to obtain a nodalsurface. Then collapse a subset of the components of this nodal surface,so each component in this subset gets collapsed to a point. The result, Y (cid:48) , is a collection of surfaces glued together at a collection of points.Thus we assume that each component of X \ γ maps either to a point,or maps homeomorphically to a subset of Y (cid:48) . Figure 3.1.
An example of the collapse map f and thegluing map g .There is a possibly disconnected surface Y equipped with a surjectivegluing map g : Y → Y (cid:48) that glues together finitely many points, as inFigure 3.1. The components of Y correspond to the components of X \ γ that don’t map to a point under f . IGH RANK INVARIANT SUBVARIETIES 13
Now, let Σ X ⊂ X, Σ Y (cid:48) ⊂ Y (cid:48) , and Σ Y ⊂ Y be finite sets such that f (Σ X ) = Σ Y (cid:48) and Σ Y = g − (Σ Y (cid:48) ), and suchthat Σ Y (cid:48) contains f ( γ ). The map g ∗ : H ( Y (cid:48) , Σ Y (cid:48) ) → H ( Y, Σ Y )is an isomorphism, and the map f ∗ ◦ ( g ∗ ) − : H ( Y, Σ Y ) → H ( X, Σ X )is an inclusion. So we can view H ( Y, Σ Y ) as a subspace of H ( X, Σ X ).If ( X, ω ) is a surface near a surface (
Y, ω Y ) in the WYSIWYG bound-ary, this inclusion of H ( Y, Σ Y ) into H ( X, Σ X ) is the same one used [MW17 , Lemma 9.2 ] , which underlies the main results of [MW17] , [CW19] . In this context we call f a collapse map; it is produced in [MW17 , Proposition 2.4 ] , and can be seen to have the form above byfactoring it through a map to a nearby point in the Deligne-Mumfordcompactification.Each component of Y \ Σ Y is homeomorphic to a subset of X − ( γ ∪ Σ X ), and we pick the orientation on Y corresponding to the orienta-tion on X . Both H ( X, Σ X ) and H ( Y, Σ Y ) thus have natural bilinearforms, obtained by first mapping from relative cohomology to absolutecohomology, and then taking the symplectic pairing of the absolutecohomology classes. Working with relative rather than absolute coho-mology means the bilinear form may be degenerate, but allows us tostate the following. Lemma 3.6.
The bilinear form on H ( Y, Σ Y ) is the restriction of thebilinear form on H ( X, Σ X ) .Proof. There is a Poincare duality isomorphism from H ( Y, Σ Y ) to H ( Y − Σ Y ), obtained by viewing H ( Y, Σ Y ) as the dual of H ( Y, Σ Y ),and defining an intersection number pairing between H ( Y, Σ Y ) and H ( Y − Σ Y ). This isomorphism sends the bilinear form on H ( Y, Σ Y )defined above to the bilinear form on H ( Y − Σ Y ) defined by algebraicintersection number.Similarly, H ( X, Σ X ) is isomorphic to H ( X − Σ X ). The inclusion of H ( Y − Σ Y ) into H ( X, Σ X ) is induced by the homomorphism of Y − Σ Y to a subset of X , which does not affect the number of intersectionsbetween curves or their signs. (cid:3) Preliminary applications.
In this subsection we will let M (cid:48) bea component of a boundary of an invariant subvariety M . Corollary 3.7. If M (cid:48) is a codimension d boundary of M then the rankof M (cid:48) is at least rk( M ) − d .Proof. By Corollary 3.5, rank is half the dimension of a maximal sym-plectic subspace of the tangent space.Pick a point (
X, ω ) ∈ M close to a point ( Y, ω Y ) ∈ M (cid:48) . The mainresults of [MW17, CW19] allow us to view T ( Y,ω Y ) ( M (cid:48) ) as a subspaceof T ( X,ω ) ( M ). Lemma 3.6 gives that the bilinear form on T ( Y,ω Y ) ( M (cid:48) )is the restriction of the bilinear form on T ( X,ω ) ( M ).There is a symplectic subspace of S of T ( X,ω ) ( M ) of dimension 2 rank( M ).Basic linear algebra gives that S ∩ T ( Y,ω Y ) ( M (cid:48) ) has dimension at least2 rank( M ) − d . Any codimension d subspace of a symplectic vectorspace contains a symplectic subspace of codimension 2 d . Hence M (cid:48) has rank at least rank( M ) − d . (cid:3) If (
X, ω ) ∈ M is close to a point ( Y, ω Y ) ∈ M (cid:48) , then we can definethe vanishing cycles , using the notation of the previous subsection, as V = ker( f ∗ : H ( X, Σ) → H ( Y (cid:48) , Σ Y (cid:48) )) . In this case, the annihilator Ann( V ) ⊂ H ( X, Σ) can be identifiedwith H ( Y, Σ Y ), and the main results of [MW17, CW19] identify T ( Y,ω Y ) ( M (cid:48) ) with Ann( V ) ∩ T ( X,ω ) ( M ). Lemma 3.8. M (cid:48) has rank less than that of M if and only if there isa vanishing cycle v ∈ V that is nonzero as a functional on T ( X,ω ) ( M ) ,but which is zero on ker( p ) ∩ T ( X,ω ) ( M ) .Proof. The characterization of rank in Corollary 3.5 together with Lemma3.6 give that rank( M ) = rank( M (cid:48) ) if and only if p (Ann( V ) ∩ T ( X,ω ) ( M )) = p ( T ( X,ω ) ( M )) . Recall the following statement from linear algebra.
Sublemma 3.9.
Let T : W → W be a linear map between finitedimensional vector spaces, and let U be a subspace of W ∗ . Then T (Ann( U )) = T ( W ) if and only if every non-zero element of U isnon-zero on ker( T ) . The lemma follows by setting W = T ( X,ω ) ( M ), T = p , and U to bethe image of V in W ∗ . (cid:3) Cylinder degenerations
The twist space.
In this section we will suppose that C is anequivalence class of generic cylinders on a surface ( X, ω ) in an invariantsubvariety M . Recall that a cylinder is said to be M -generic , or just IGH RANK INVARIANT SUBVARIETIES 15 generic if M is clear from context, if its boundary saddle connectionsremain parallel to the core curve of the cylinder in a neighborhood of( X, ω ) in M .Following [AWc , Definition 3.10 ] , we define cylinders to be opensubsets, so that a cylinder does not include its boundary saddle con-nections. We can view C as a set of cylinders, but we also frequentlyview it as a subset of the surface.For each cylinder C ∈ C , let γ C denote its core curve, and let h C denote its height. Orient the γ C , C ∈ C , consistently, which meansthat their holonomies are all positive multiples of each other. By theCylinder Deformation Theorem, the standard deformation σ C = (cid:88) C ∈ C h C γ ∗ C belongs to T ( X,ω ) ( M ); see [Wri15a , Theorem 1.1 ] for the original state-ment, [MW17 , Section 4.1 ] for the reformulation we are using here,and [BDG] for a novel proof and generalization. Here γ ∗ C denotes thecohomology class dual to γ C , using the same duality used in Section3.3 and [MW17 , Section 4.1 ] .Sometimes the tangent space contains other elements of the sameform. Definition 4.1.
The twist space Twist( C , M ) is the complex vectorsubspace of T ( X,ω ) ( M ) consisting of vectors that can be represented as (cid:80) C ∈ C a C γ ∗ C with a C ∈ C .This space is so-named because Twist( C , M ) is the complexificationof the real subspace of T ( X,ω ) ( M ) tangent to deformations of ( X, ω ) in M that twist or shear the cylinders of C while leaving the complementof C unchanged.For any v ∈ T ( X,ω ) ( M ) small enough, there is a surface in M whoseperiods coordinates differ from those of ( X, ω ) by v . We denote thatsurface by ( X, ω ) + v .If v ∈ Twist( C , M ) is small enough, then ( X, ω ) + v can be obtainedfrom ( X, ω ) by deforming the cylinders in C . In this case there isnatural piecewise linear map T v : ( X, ω ) → ( X, ω ) + v, whose derivative is the identity off of C , and which shears and dilatesthe cylinders in C . Different cylinders in C may be sheared and dilateddifferent amounts.Recall the following result from [MW17 , Theorem 1.5 ] , which is alsodiscussed in [AWc , Lemma 6.10 ] . Theorem 4.2 (Mirzakhani-Wright) . p (Twist( C , M )) = C · p ( σ C ) . This theorem says that, up to purely relative cohomology classes(the kernel of p ), the only cylinder deformations of C that remain in M are multiples of the standard deformation. It can be viewed as apartial converse to the Cylinder Deformation Theorem. The followingconsequence, recorded in [MW17 , Corollary 1.6 ] , can be viewed as apartial generalization of the Veech dichotomy. Corollary 4.3 (Mirzakhani-Wright) . If M has no rel, then M -parallelcylinders have rational ratios of moduli. In particular, this implies that the ratio of moduli of M -parallelcylinders is locally constant in M .4.2. The definition of cylinder degenerations.
Thus, for any v ∈ Twist( C , M ), the surface ( X, ω ) + tv is well defined and containedin M for all t > t v when some cylinder in C reaches zero height.We assume from now on that the height of some cylinder in C decreasesalong this path, so t v < ∞ , and we denote by C v the subset of cylinderswhose heights go to zero as t approaches t v . Example 4.4. If C is horizontal and v = (cid:80) C ∈ C a C γ ∗ C , then t v = min( − h C / Im( a C ) : Im( a C ) < , and C ∈ C v if and only if it realizes this minimum. Definition 4.5. If C is an equivalence class of generic cylinders, and v ∈ Twist( C , M ), then the path ( X, ω ) + tv ∈ M , t ∈ [0 , t v ) is calleda collapse path if it diverges in the stratum and if C v is not the wholesurface.The second assumption prevents the area from going to zero, andthe first ensures that this path does in fact degenerate the surface. Example 4.6. If C is horizontal and v = − iσ C = − i (cid:80) C ∈ C h C γ ∗ C ,then deforming in the v direction vertically collapses all the cylindersin C , so C v = C . In this case t v = 1, and the path diverges in thestratum if and only if C contains a vertical saddle connection.For each t < t v , the map T tv : ( X, ω ) → ( X, ω ) + tv has constant derivative on each cylinder in C , and this derivative hasa 1 eigenvector in the direction of the cylinders. For each cylinder in C v , the other eigenvalue of the derivative must be in (0 , ⊆ R , since IGH RANK INVARIANT SUBVARIETIES 17 the area of these cylinders decreases, and we will call the associatedeigendirection the maximally contracted direction . Example 4.7. If C is horizontal and v = (cid:80) C ∈ C a C γ ∗ C , the derivativeof T tv on C is (cid:32) t Re( a C ) h C t Im( a C ) h C (cid:33) . Example 4.8.
Figures 4.1 and 4.2 both illustrate collapse paths where C = { C } is a single horizontal cylinder, and v = − ih C γ ∗ C , so thedegeneration path vertically collapses C and converges to a surfaceCol v ( X, ω ) where C is completely collapsed. In both examples, if weglue together the two points z , z on the limit, it is possible to definea collapse map Col v . We can choose not to glue these points together,at the expense of allowing the map Col v to be multivalued. Figure 4.1.
For any point p on the vertical segment α ,one may define Col v ( p ) = { z , z } . Figure 4.2.
For any point p in the top left square, onemay define Col v ( p ) = { z , z } .Note that in Figure 4.1, Col v is only multivalued on single saddleconnection, whereas in Figure 4.2, Col v is multivalued on an open set. Recall from Section 3.3 or [CW19] that a point in the WYSIWYGpartial compactification is obtained in two steps, first via a collapseand then ungluing a finite set of points. Because of the ungluing,multivalued collapse maps are typical in this context.In the next subsection, we verify the following.
Lemma 4.9.
The collapse path converges as t → t v . The limit Col v ( X, ω ) is the image of PL map Col v : ( X, ω ) → Col v ( X, ω ) . The derivative of
Col v is the identity off C ; is constant and invertibleon each cylinder of C \ C v ; and has kernel equal to the maximallycontracted direction on each cylinder of C v .At a point p where Col v is multivalued, Col v ( p ) is a finite subset ofthe singularities and marked points of Col v ( X ) . Col v is single-valuedexcept possibly on a finite union of line segments. All but the final claim in this lemma should be viewed as intuitive,and all of the claims may even be clear in any given example. Howeverthere is non-trivial content to the final claim that Col v is only as badlymultivalued as the example in Figure 4.1; we will give a proof that thebehaviour in Figure 4.2 is ruled out by the requirement that C consistsof generic cylinders. Definition 4.10.
The limit Col v ( X, ω ) will be called the cylinder de-generation corresponding to the collapse path (
X, ω ) + tv . The com-ponent of the boundary of M containing Col v ( X, ω ) will be denoted M v .4.3. The proof of Lemma 4.9.
The discussion in this section, whichis somewhat technical, is not used the remainder of the paper, exceptfor one appeal to Lemma 4.12 in the proof of Lemma 11.6. This sectioncan thus be skipped on a first reading of this paper.Throughout this subsection we will assume that C is an M -equivalenceclass of cylinders on a surface ( X, ω ) in an invariant subvariety M andthat v ∈ Twist( C , M ). We will assume without loss of generality thatthe cylinders in C are horizontal.We begin with two general lemmas, which may be of some indepen-dent interest. The first does not require our assumption that C consistsof generic cylinders. Lemma 4.11. If C v (cid:54) = C , then the imaginary part of the holonomy ofany closed loop in C v is zero.Proof. Let α be a closed loop in C v . Recall that σ C = (cid:80) h C γ ∗ C denotesthe standard deformation. IGH RANK INVARIANT SUBVARIETIES 19
Without loss of generality, assume that v is purely imaginary. ByTheorem 4.2, we can write v as v = ciσ C + r , where c ∈ R , and r ∈ Twist( C , M ) is purely relative.If we write r = (cid:80) ir C γ ∗ C , then the computation in Example 4.4 showsthat C ∈ C v if and only if t v = − h C / ( r C + ch C ) . We now claim that r C is not zero when C ∈ C v . Indeed, the factthat C v is neither empty nor all of C implies that r is non-zero. Since r is purely relative, this means that some r C are positive and somenegative. On ( X, ω ) + tv , the height of C is (1 + tc + tr C h C ) h C so wenotice that the heights of cylinders with r C < r C >
0. Sincethe cylinders in C v reach height zero first, we have that r C < C ∈ C v .For all C ∈ C v , we compute that r C = − ct v + 1 t v h C . It is important that r C is a constant non-zero multiple of h C , and thatthe constant r C /h C does not depend on C .Hence, since α is contained in C v , r ( α ) = − i ct v + 1 t v (cid:88) h C γ ∗ C ( α ) . Again since α is contained in C v , the expression (cid:80) h C γ ∗ C ( α ) computesthe imaginary part of the holonomy of α , and we get r ( α ) = − i ct v + 1 t v Im (cid:18)(cid:90) α ω (cid:19) . Since r is purely relative and α is a closed loop, by definition r ( α ) =0. Hence the above equality gives Im( (cid:82) α ω ) = 0, as desired. (cid:3) Lemma 4.12.
Assume that C consists of generic cylinders and that C (cid:54) = ( X, ω ) . Then the imaginary part of the holonomy of any closedloop in C that is disjoint from the singularities of ω is zero. Compared to the previous lemma, the additional assumption thatthe cylinders in C are generic and that the loop avoids singularitieswill ensure that the loop continues to be a closed loop and continuesto be contained in C even after degenerations and perturbations. Sothe assumptions in Lemma 4.12 should be considered as a more robustversion of the assumptions in Lemma 4.11. Proof.
Suppose not in order to deduce a contradiction. Let M be acounterexample with smallest possible dimension.Since C (cid:54) = ( X, ω ), Smillie-Weiss [SW04 , Corollary 6 ] gives the exis-tence of a cylinder D disjoint from C . Perturbing if necessary, we canassume that D is not parallel to C .Let D be the equivalence class of D . A short argument using theCylinder Deformation Theorem gives that D is disjoint from C ; com-pare for example to [NW14 , Propostion 3.2 ] .We can now degenerate D using a standard deformation to obtaina surface in a smaller dimensional orbit closure M (cid:48) . Since C and D are not parallel and hence not adjacent, each saddle connection on theboundary of C persists under this degeneration, and C continues toconsist of generic cylinders. Similarly the assumption that C does notcover the whole surface continues to hold.Since we considered a minimal counterexample, it must be that C is not an M (cid:48) equivalence class; the degeneration must have causedcylinders to become generically parallel to those in C . So C is actu-ally a strict subset of an M (cid:48) -equivalence class E . In this case, taking v = − iσ C to be the standard deformation, which decreases heights ofcylinders in C , we have that v ∈ Twist( E , M (cid:48) ) and E v = C . ThusLemma 4.11 gives that the imaginary part of the holonomy of any rele-vant closed loop is in fact zero, which contradicts the assumption that M is a counterexample to the claim. (cid:3) For any C ∈ C v , define a collapsed segment of C to be a maximal linesegment contained in C in the direction of the maximally contracteddirection, oriented in the upwards direction. We do not require theendpoints to be singularities or marked points. Lemma 4.13.
Suppose that ( X, ω ) (cid:54) = C v and that C consists of genericcylinders if C v = C . Any path in C v that is a concatenation of col-lapsed segments and that is disjoint from the singularities can only entereach cylinder in C v at most once.Proof. Otherwise, we can form a closed loop α by starting in the middleof a cylinder C in C v , traveling upwards along the path until we returnto C , and then travelling along the core curve of C . The imaginarypart of the holonomy of α is positive, contradicting Lemma 4.11 (when C v (cid:54) = C ) or Lemma 4.12 (when C v = C ). (cid:3) The following immediate corollary of Lemma 4.13 is the key to avoid-ing the behavior illustrated in Figure 4.2.
Corollary 4.14.
Under the assumptions of Lemma 4.13, starting atany point in C v , it is possible to travel both upwards and downwards IGH RANK INVARIANT SUBVARIETIES 21 along a concatenation of collapsed segments and reach either a singu-larity or the boundary of C v . It is possible to define the surface Col v ( X, ω ) and the map Col v without Corollary 4.14, but we will make use of it to simplify thediscussion. Proof of Lemma 4.9:
Assuming v defines a cylinder degeneration, wewill define Col v ( X, ω ) in several steps, starting from (
X, ω ). First, weperform the part of the cylinder degeneration that does not in factdegenerate any cylinders. Namely, we consider (
X, ω ) + t v v C \ C v , wherewe define v C \ C v = (cid:88) C ∈ C \ C v a C γ ∗ C to be the part of v = (cid:80) a C γ ∗ C corresponding to the cylinders in C \ C v .Then, we cut out the interior of C v , to obtain the translation surfacewith boundary (( X, ω ) + t v v C \ C (cid:48) v ) \ int( C v ) . We then identify pairs of points on the boundary if they are joinedby a concatenation of collapsed segments that are disjoint from thesingularities. Since we have avoided singularities, the result is a punc-tured surface, and we fill in the punctures to obtain Col v ( X, ω ). Anynon-singular filled-in points are declared to be marked points.We can now define the map Col v as follows. Off C v , it is induced bythe cylinder deformation T t v v C \ C v : ( X, ω ) → ( X, ω ) + t v v C \ C v . For each point p in C v not reachable from a singularity via a concate-nation of collapsed segments, Col v ( p ) is equal to the single point ofCol v ( X, ω ) obtained by identifying the two points on the boundary of C v reachable from p via a concatenation of collapsed segments.All other points p are in the set where Col v is potentially multival-ued, and we can define Col v ( p ) to be the set of limits of Col v ( p n ), forsequences p n of points where Col v has previously been defined with p n → p . The construction gives that Col v ( p ) is a subset of the sin-gularities and marked points of Col v ( X, ω ). By construction Col v ismultivalued off the set reachable from singularities by travelling alongcollapsed segments, and Corollary 4.14 gives that that set is a finiteunion of line segments.Since the derivative of Col v is as described in Lemma 4.9, it remainsonly to show that the collapse path converges to Col v ( X, ω ). We sketch this now, explaining how to extend the strategy of [MW17 , Lemma3.1 ] to this slightly more general situation.We will use the criterion for convergence from [MW17 , Definition2.2 ] , which requires that we build maps g t : Col v ( X, ω ) \ U → ( X, ω ) + tv that distort the flat metric very little, where U is a small neighbourhoodof the zeros and marked points of Col v ( X, ω ) and where t is close to t v . The criterion also requires that the complement of the image of g t be small in a precise sense, which, in particular, is satisfied when thiscomplement is contained in a small neighborhood of the zeros.To this end, let U be the union of the L ∞ balls centered at the zerosand marked points on Col v ( X, ω ) of radius ε > t , the discussion abovegives PL collapse maps f t : ( X, ω ) + tv → Col v ( X, ω )for all t ∈ [0 , t v ). Here we avoid our previous notation for collapsemaps, since it does not specify the domain surface.The map f t is an isometry when restricted to the complement of C .By Example 4.7, the derivative of the restriction of f t to a cylinder C ∈ C − C v is (cid:32) ( t v − t ) Re( a C ) h C ( t v − t ) Im( a C ) h C (cid:33) . Notice that this matrix is arbitrarily close to the identity when t issufficiently close to t v .Let U (cid:48) be an (cid:15) neighborhood of Col v ( C v ) in the L ∞ metric. We beginby defining g t to be f − t on the complement of U (cid:48) . Ultimately, we wantto extend this function over points in U (cid:48) \ U .The cylinders in C v give rise to a finite collection Col v ( C v ) of horizon-tal saddle connections on Col v ( X, ω ). For each such saddle connection s , consider a rectangle R s centered on s of width (cid:96) s − (cid:15) and height2 (cid:15) , where (cid:96) s is the length of s . Let γ tops and γ bots denote the top andbottom line segments of this rectangle. See Figure 4.3.Given a vertical line η in R s joining a point in γ tops to γ bots there is acorresponding sequence of lines f − t ( η ), which can be pulled tight, whilefixing endpoints, to a geodesic representative η t . Let P s,t be the unionof all line segments formed this way. Since the height of the cylindersin C v is proportional to t − t v , which can be taken to be much smallerthan 2 ε (which is the height of R s ), η t is indeed a line segment and P s,t is a parallelogram. IGH RANK INVARIANT SUBVARIETIES 23
Figure 4.3. f − t ( η ) is a sequence of line segments(shown on the left in green), which can be tightened toa straight line η t . The red region on the left is a subset, f − t ( s ) ⊂ C v .For each s ∈ Col v ( C v ) extend g t to R s by defining g t | R s to be thelinear map taking R s to P s,t that agrees with g t on γ tops and γ bots . It canbe shown that the derivative of this linear map may be made arbitrarilyclose to the identity if t is sufficiently close to t v .We have now extended g t to a PL homeomorphism from Col v ( X, ω ) \ U to ( X, ω ) + tv whose derivatives, where defined, are arbitrarily closeto the identity. It is also clear that, when t is sufficiently close to t v ,the image of g t contains the complement of a 2 (cid:15) L ∞ neighborhood ofthe zeros of ( X, ω ) + tv .Since, where defined, the derivatives of g t were made arbitrarily closeto the identity, it is possible to approximate g t by a smooth diffeo-morphism whose derivative is also arbitrarily close to the identity andwhose image also contains the complement of a 2 (cid:15) L ∞ neighborhoodof the zeros of ( X, ω ) + tv . (cid:3) A slightly stronger genericity assumption.
Up until now, wehave considered equivalence classes all of whose cylinders are generic.In this subsection, we introduce and discuss a slightly stronger notionof genericity for an equivalence class of cylinders.
Definition 4.15.
An equivalence class C of M -parallel cylinders on( X, ω ) ∈ M is called M -generic , or just generic when M is clearfrom context, if each cylinder in C is M -generic, and there does notexist a perturbation of ( X, ω ) in M that creates new cylinders that are M -parallel to those in C . Example 4.16.
Consider a surface ( X (cid:48) , ω (cid:48) ) with a pair of simple ho-mologous cylinders C, C (cid:48) . Let M be the stratum containing ( X, ω ).Assume there are no other cylinders homologous to C and C (cid:48) . Collapse C (cid:48) without degenerating the surface, to obtain a surface ( X, ω ) ∈ M .The cylinder C on ( X, ω ) is simple and hence M -generic. Since ( X, ω )has no cylinders homologous to C , it is its own equivalence class. Thus { C } is an equivalence class of generic cylinders. However, we can de-form ( X, ω ) to make C (cid:48) reappear, so { C } is not a generic equivalenceclass; see Figure 4.4. Figure 4.4.
Left: (
X, ω ). Right: ( X (cid:48) , ω (cid:48) ).Despite the example, there are mild conditions which guarantee thatan equivalence class of generic cylinders is a generic equivalence class. Lemma 4.17.
Let C be an equivalence class of generic cylinders on ( X, ω ) ∈ M . Suppose either(1) M has no rel, or(2) every saddle connection parallel to C is generically parallel to C .Then C is a generic equivalence class. In particular, the second criterion of Lemma 4.17 implies that theset of surfaces in M where all equivalence classes are generic is densein M . The proof will use the following definition. Definition 4.18.
Recall that a cross curve of a cylinder D is a saddleconnection contained in D that crosses the core curve exactly once. Proof.
First suppose M has no rel. Then Corollary 4.3 implies that C is generic.Next suppose that every saddle connection parallel to C is genericallyparallel to C . Suppose, in order to find a contradiction, that there is asequence of surfaces ( X n , ω n ) ∈ M that converge to ( X, ω ), and suchthat the cylinders of C persist on ( X n , ω n ), but where each ( X n , ω n )has a cylinder D n that is generically parallel to C but not contained in C .By the Cylinder Finiteness Theorem of [MW17 , Theorem 5.1 ] or [CW19 , Theorem 5.3 ] , the circumference of the D n is uniformly boundedin n . After passing to a subsequence, D n converges to a collection ofsaddle connections on ( X, ω ) parallel to the core curves of C . IGH RANK INVARIANT SUBVARIETIES 25 If γ n are bounded length cross curves of D n , then similarly we mayassume that γ n converges to a collection of saddle connections paral-lel to C . Since γ n is not parallel to C , at least one of these saddleconnections is not generically parallel to C .This contradicts the assumption, so the sequence must not exist, andwe conclude that C is generic. (cid:3) The twist space decomposition.Lemma 4.19.
In relative homology, the difference between any twocross curves of a cylinder C that cross the cylinder in the same di-rection can be written as a linear combination of the boundary saddleconnections of C .Let C be any collection of parallel cylinders on a translation surface.In relative homology, every saddle connection in C can be written as alinear combination of boundary saddle connections and cross curves.Proof. The first statement follows from the definition of relative ho-mology.Every saddle connection in C is homotopic to a concatenation ofcross curves of cylinders in C and boundary saddle connections of cylin-ders in C . (cid:3) Proposition 4.20.
Let C be a generic equivalence class of cylinderson ( X, ω ) ∈ M .Fix an element w ∈ T ( X,ω ) ( M ) that pairs non-trivially with the corecurve of a cylinder in C . Then any η ∈ T ( X,ω ) ( M ) admits a uniquedecomposition η = aw + η C + η ( X,ω ) \ C where a ∈ C , η C ∈ Twist( C , M ) , and η ( X,ω ) \ C ∈ T ( X,ω ) ( M ) evaluatesto zero on every saddle connection in C . For any C ∈ C , we have a = η ( γ C ) w ( γ C ) . Recall that γ C denotes the core curve of C . Proof.
With the specified value of a , the class η − aw ∈ T ( X,ω ) ( M )evaluates to zero on γ C . Since every saddle connection in the boundaryof C is generically parallel to γ C , η − aw evaluates to zero on the corecurves of C . For simplicity of notation, we will replace η with η − aw so that we may suppose that η evaluates to zero on the core curves of C .For each C ∈ C , fix a cross curve s C of C , oriented so γ ∗ C ( s C ) = 1.Define η C = (cid:88) C ∈ C η ( s C ) γ ∗ C . Lemma 4.21. η − η C is zero on each saddle connection in C .Proof. We have assumed that η evaluates to zero on every saddle con-nection on the boundary of a cylinder of C . By construction, the sameholds for η C . Hence, η − η C is zero on every such boundary saddle con-nection of a cylinder in C as well as on every cross curve in { s C } C ∈ C .The result now follows by Lemma 4.19. (cid:3) Lemma 4.22. η C ∈ T ( X,ω ) ( M ) .Proof. Assume that the cylinders in C are horizontal. We will showthat both the real and imaginary parts of η C are in the tangent space.Since η ∈ T ( X,ω ) ( M ) and since M is defined by real linear equations,Re( η ) and Im( η ) both belong to T ( X,ω ) ( M ). Therefore, for sufficientlysmall (cid:15) , ( X (cid:48) , ω (cid:48) ) = ( X, ω ) + i(cid:15)
Re( η )belongs to M . The standard deformation on ( X (cid:48) , ω (cid:48) ) is σ (cid:48) C = (cid:88) C ∈ C ( h C + ε Re( η )( s C )) γ ∗ C , where h C continues to denote the height of C on ( X, ω ). Here, cru-cially, we use that C is a generic equivalence class, rather than justan equivalence class of generic cylinders, to ensure that C is still anequivalence class on the deformation ( X (cid:48) , ω (cid:48) ).By the Cylinder Deformation Theorem, the standard deformationof C on ( X (cid:48) , ω (cid:48) ) belongs to T ( X (cid:48) ,ω (cid:48) ) ( M ). By parallel transport, wecan also consider this vector as an element of T ( X,ω ) ( M ). The stan-dard deformation at ( X, ω ), namely σ C = (cid:80) C ∈ C h C γ ∗ C , also belongs to T ( X,ω ) ( M ). Hence σ (cid:48) C − σ C = ε (cid:88) C ∈ C Re( η )( s C ) γ ∗ C ∈ T ( X,ω ) ( M ) . This shows that Re( η C ) ∈ T ( X,ω ) ( M ). A similar argument shows thatIm( η C ) ∈ T ( X,ω ) ( M ) and hence that η C ∈ T ( X,ω ) ( M ). (cid:3) Proposition 4.20 now follows from Lemmas 4.21 and 4.22; uniquenessis left as an exercise. (cid:3)
We now establish the following version of the twist space decomposi-tion using p ( T ( X,ω ) ( M )) instead of T ( X,ω ) ( M ). Recall that T Col v ( X,ω ) ( M v )can be naturally identified with Ann( V ) ∩ T ( X,ω ) ( M ), where V is thespace of vanishing cycles. It will also be convenient to use the followingdefinition. IGH RANK INVARIANT SUBVARIETIES 27
Definition 4.23.
A cylinder degeneration specified by v ∈ Twist( C , M )will be called rank-reducing if rank( M v ) < rank( M ), and rank-preservingif rank( M v ) = rank( M ). Proposition 4.24.
Let C be a generic equivalence class of cylinders on ( X, ω ) ∈ M and let v ∈ Twist( C , M ) define a rank-reducing cylinderdegeneration. Then p ( T ( X,ω ) ( M )) = p (Ann( V ) ∩ T ( X,ω ) ( M )) ⊕ p ( C · σ C ) . Before we give the proof, we note the following.
Lemma 4.25.
Let C be an equivalence class on a surface in an in-variant subvariety M . For any cylinder degeneration determined by v ∈ Twist( C , M ) , there is a basis of V consisting of -chains sup-ported in C v .Proof. Letting Col v be the PL collapse map defined in Lemma 4.9, V isthe kernel of (Col v ) ∗ on H ( X, Σ; C ) where Σ denotes the singularitiesof ω . The claim now follows from the fact that for any simplicial mapthat is surjective on 2-chains, the kernel of first homology is generatedby 1-chains that map to zero. (cid:3) Proof of Proposition 4.24.
Consider u ∈ T ( X,ω ) ( M ).Let η denote the cohomology class of the Abelian differential onCol v ( X, ω ), viewed as an element of Ann( V ) ∩ T ( X,ω ) ( M ) via the iden-tification of this space and T Col v ( X,ω ) ( M v ).We see that η is nonzero on core curves of C , since the Abelian differ-ential on Col v ( X, ω ) is nonzero on nonzero, non-negative linear combi-nations of consistently oriented parallel saddle connections in Col v ( C v ),and every core curve of a cylinder in C v degenerates to such a sum.Using Proposition 4.20, write u = aη + u C + u ( X,ω ) \ C . Since u ( X,ω ) \ C is zero on all saddle connections in C , it is contained inAnn( V ) by Lemma 4.25.By Theorem 4.2, there is a constant c such that p ( u C ) = cp ( σ C ), sowe can write p ( u ) = p ( aη + u ( X,ω ) \ C ) + cp ( σ C ) . Since both η and u ( X,ω ) \ C are in Ann( V ) ∩ T ( X,ω ) ( M ), the first summandis in p (Ann( V ) ∩ T ( X,ω ) ( M )).Since this holds for all u , this shows that p ( T ( X,ω ) ( M )) = p (Ann( V ) ∩ T ( X,ω ) ( M )) + p ( C · σ C ) . Since p ( C · σ C ) is one dimensional, to show that this is a direct sumdecomposition it suffices to show that p ( T ( X,ω ) ( M )) (cid:54) = p (Ann( V ) ∩ T ( X,ω ) ( M )). That follows from Corollary 3.5 and Lemma 3.6, since thedegeneration is rank-reducing. (cid:3) Corollary 4.26.
Under the same assumptions as Proposition 4.24, if rank( M v ) < rank( M ) , then rank( M v ) = rank( M ) − .Proof. This follows from Proposition 4.24, Lemma 3.4, and Corollary3.5. (cid:3) A cylinder degeneration dichotomy (with Mirzakhani)
Any collection of parallel saddle connections on a translation surfacecan be considered as a directed graph in two ways, by directing each ofthe saddle connections so they all point in the same direction. The twodirected graphs obtained in this way differ by changing the direction ofeach edge. It will not matter in the subsequent analysis which of thetwo directed graphs are chosen. We will consider these graphs both asabstract graphs and as subsets of the surface.In this section, we will consider degenerations in which some cylin-ders become a collection of parallel saddle connections. We will showthat the resulting directed graph of saddle connections is either stronglyconnected or satisfies a very restrictive property that, in particular,implies that it is acyclic. Recall that a directed graph is acyclic if itcontains no directed cycles, and it is strongly connected if every edge iscontained in a directed cycle, so this is a dichotomy between oppositeextremes.5.1.
Statement.
We will call a collection of parallel saddle connec-tions Γ on a translation surface ( X (cid:48) , ω (cid:48) ) in an invariant subvariety M (cid:48) M (cid:48) -rel-scalable (or simply rel-scalable when M (cid:48) is clear from context)if there is purely relative cohomology class s ∈ T ( X (cid:48) ,ω (cid:48) ) ( M (cid:48) ) which eval-uates on each edge of Γ to give the holonomy of the edge. In that casewe say that s certifies that Γ is rel-scalable .By definition, a cohomology class is purely relative if it evaluates tozero on any absolute homology class. Hence, if Γ is rel-scalable, thenthe integral of ω (cid:48) along every loop in Γ is zero. This implies that Γ isacyclic as a directed graph.Fix an equivalence class of cylinders C on a translation surface ( X, ω )in an invariant subvariety M . Pick a direction v ∈ Twist( C , M ) inwhich to deform the cylinders in C in order to degenerate the surface.We will continue to use the notation introduced in the previous section: C v ⊂ C denotes the subset of the cylinders that reach zero height first IGH RANK INVARIANT SUBVARIETIES 29 along the degeneration path; Col v ( X, ω ) denotes the limit surface; and M v is the boundary invariant subvariety that contains it. Theorem 5.1 (The Cylinder Degeneration Dichotomy) . Let C be ageneric equivalence class of cylinders on a translation surface ( X, ω ) in an invariant subvariety M . Consider a cylinder degeneration givenby v ∈ Twist( C , M ) . • If rank( M v ) < rank( M ) , then C v = C and Col v ( C ) is M v -rel-scalable. • If rank( M v ) = rank( M ) , then Col v ( C v ) is strongly connected. The following special case of Theorem 5.1 was initially discoveredand proved jointly with Mirzakhani.
Corollary 5.2.
Let C be an equivalence class of M -generic cylinderson a translation surface ( X, ω ) in an invariant subvariety M . Considera cylinder degeneration given by v ∈ Twist( C , M ) . If M has no rel,then C v = C and Col v ( C ) is M v -rel-scalable.Proof. If M has no rel, then rank( M v ) < rank( M ), and Lemma 4.17gives that every equivalence class of generic cylinders is a generic equiv-alence class. (cid:3) Theorem 5.1 evolved from the “no rel” special case (Corollary 5.2)discovered with Mirzakhani, so we feel it is appropriate to attributeTheorem 5.1 jointly to her. In the course of this evolution, our per-spective has changed significantly, so the arguments we present willhave a different flavor than those we discussed with Mirzakhani.5.2.
Proof of Theorem 5.1.
Let V denote the collection of vanishingcycles for the collapse from ( X, ω ) to Col v ( X, ω ). The first statement(on rel-scalability) of Theorem 5.1 will be derived from the following.The reader should keep in mind that relative cohomology is the dualof relative homology.
Proposition 5.3. If rank( M v ) < rank( M ) , then the restriction ofthe rel subspace of T Col v ( X,ω ) ( M v ) to the subspace of relative homologygenerated by saddle connections in Col v ( C v ) is equal to the restrictionof T Col v ( X,ω ) ( M v ) to this subspace. Moreover, C = C v . Thus, any change to Col v ( C v ) that can be accomplished locally in M v can also be obtained with a rel deformation in M v .Before we give the proof, we explain how to apply Proposition 5.3. Proof of Theorem 5.1 in the rank-reducing case.
Let ω denote the holo-morphic one-form that induces the flat structure on Col v ( X, ω ). By Proposition 5.3, there is a purely relative class r ∈ T Col v ( X,ω ) ( M v )whose restriction to the subspace of relative homology generated bysaddle connections in Col v ( C v ) is the same as the restriction of [ ω ] ∈ T Col v ( X,ω ) ( M v ). Since ω evaluated on any saddle connection gives itsholonomy, this means that r evaluated on any saddle connection inΓ gives its holonomy. The existence of such an r is the definition ofrel-scalability, so this, together with the fact that C = C v proves theresult. (cid:3) Proof of Proposition 5.3.
Let w ∈ T Col v ( X,ω ) ( M v ). We wish to find apurely relative class r ∈ T Col v ( X,ω ) ( M v ) which is equal to w on saddleconnections in Col v ( C v ). We start with finding some relative class r ,which can be thought of as an initial guess for r , and then we will showhow to modify r to get a class r with the desired restriction. Lemma 5.4.
There is a purely relative class r ∈ T Col v ( X,ω ) ( M v ) that,when viewed using the inclusion T Col v ( X,ω ) ( M v ) ⊂ T ( X,ω ) ( M ) , is non-zero on core curves of cylinders in C . Moreover, C v = C . Since core curves are absolute cohomology classes, r is not purelyrelative in T ( X,ω ) ( M ). So one might say that r becomes purely relativein the passage from M to M v . Proof.
Recall that σ C denotes the standard deformation in C . Sublemma 5.5.
For any w ∈ T ( X,ω ) ( M ) , if (cid:104) σ C , w (cid:105) (cid:54) = 0 , then w ( γ ) (cid:54) =0 for each core curve γ of C .Proof of Sublemma. Given any vector space V with a non-degeneratebilinear form (cid:104)· , ·(cid:105) , the dual vector space V ∗ is endowed with a dualbilinear form. The fundamental property is that v (cid:55)→ φ v = (cid:104)· , v (cid:105) sends the bilinear form to the dual bilinear form. We have that (cid:104) φ v , ψ w (cid:105) = (cid:104) v, w (cid:105) = ψ w ( v ).In our situation, we apply this to V = p ( T ( X,ω ) ( M )) with its symplec-tic form. The standard deformation σ C is dual to a linear combination (cid:80) h i γ i of the core curves of C . From (cid:104) w, σ C (cid:105) (cid:54) = 0 we then get that w is non-zero on (cid:80) h i γ i . Since all the γ i give the collinear functionals on T ( X,ω ) ( M ), it is non-zero on all of them. (cid:3) By Proposition 4.24, since rank( M v ) < rank( M ), p ( T ( X,ω ) ( M )) = p (Ann( V ) ∩ T ( X,ω ) ( M )) ⊕ p ( C · σ C ) . In particular, there is an element r of Ann( V ) ∩ T ( X,ω ) ( M ) so that(1) (cid:104) r , σ C (cid:105) (cid:54) = 0, and IGH RANK INVARIANT SUBVARIETIES 31 (2) r pairs trivially with every element of Ann( V ) ∩ T ( X,ω ) ( M ).Using the identification of T Col v ( X,ω ) ( M v ) with Ann( V ) ∩ T ( X,ω ) ( M ), thesecond property shows that r is rel in T Col v ( X,ω ) ( M v ). Together withSublemma 5.5, the first shows that r , when viewed as an element of T ( X,ω ) ( M ), pairs with the core curve of every cylinder in C nontrivially.Finally, if C (cid:54) = C v , then there is a cylinder C in C that remainsa cylinder on Col v ( X, ω ). Since r pairs nontrivially with the corecurve of every cylinder in C , the same is true for Col v ( C ). But thiscontradicts the fact that r is rel and hence must evaluate to zero onany closed curve. This concludes the proof of Lemma 5.4. (cid:3) Since the r supplied by Lemma 5.4 is non-zero on core curves of C ,there is constant c such that w − cr is zero on core curves of C . (Allthe core curves are generically parallel, so being zero on one core curveimplies being zero on all of them.)For notational convenience, define η = w − cr . By Proposition 4.20,we can decompose η as η = η C + η ( X,ω ) \ C , where η C ∈ Twist( C , M ) and η ( X,ω ) \ C pairs trivially with any saddleconnection contained in C .By definition, η is in T Col v ( X,ω ) ( M v ), which we identify with Ann( V ) ∩ T ( X,ω ) ( M ). By Lemma 4.25, η ( X,ω ) \ C is also in Ann( V ), and hence sois η C = η − η ( X,ω ) \ C . Lemma 5.6. η C is purely relative in T ( X,ω ) ( M ) .Proof. Since η C ∈ Twist( C , M ), we use Theorem 4.2 to write it as η C = aσ C + τ, where a ∈ C and τ is a purely relative element of Twist( C , M ). Wemust show that a = 0.The result now follows from Proposition 4.24, since p ( η C ) = ap ( σ C )and η C ∈ Ann( V ) ∩ T ( X,ω ) ( M ). (cid:3) We can now conclude. We have that r is purely relative, and w = ( cr + η C ) + η ( X,ω ) \ C , where the first summand is purely relative and the second is zero onall saddle connection in C , and both summands are in Ann( V ). Sinceall saddle connections in Col v ( C ) arise from saddle connections in C ,and since η ( X,ω ) \ C is zero on saddle connections in C , we get that w and cr + η C are equal on Col v ( C ), concluding the proof of Proposition5.3. (cid:3) It remains only to consider the rank-preserving case.
Proof of Theorem 5.1 in the rank-preserving case.
For concreteness, as-sume that C is horizontal, and direct Col v ( C v ) so the edges point inthe positive real direction.We will verify that Col v ( C v ) is strongly connected using the followingcriterion: A directed graph is strongly connected if and only if the edgescan be assigned positive weights such that for each vertex the sum ofthe incoming edge weights is equal to the sum of the outgoing edgeweights.To each edge of Col v ( C v ), we assign a positive weight as follows. Pickany point p on the edge, and define the weight to be the imaginary partof the holonomy of Col − v ( p ). More informally, this can be describedas the sum of the heights of the cylinders degenerating to this edge. Itdoes not depend on the choice of the point p .Let q be any singularity in Col v ( C v ) and let α q denote the clockwiseoriented boundary of any small embedded flat disk centered at q thatcontains no other singularities or marked points of Col v ( X, ω ), as inFigure 5.1. The following two lemmas complete the proof.
Figure 5.1.
An example Col − v ( α q ) (left) and α q (right). Lemma 5.7.
The imaginary part of the holonomy of the loop
Col − v ( α q ) is equal to sum of the weights of the edges leaving q minus the sum ofthe weights of the edges entering q .Proof. Let P q (resp. N q ) denote the set of points of positive (resp.negative) intersection of α q with the saddle connections in Col v ( C v ).The imaginary part of the holonomy of Col − v ( α q ) isIm (cid:88) p ∈ P q (cid:90) Col − v ( p ) ω − (cid:88) p ∈ N q (cid:90) Col − v ( p ) ω , IGH RANK INVARIANT SUBVARIETIES 33 as can be seen by using smaller and smaller embedded discs to define α q .This quantity is precisely the sum of the incoming edge weights minusthe sum of the outgoing edge weights for q . (cid:3) Lemma 5.8.
The holonomy of the loop
Col − v ( α q ) is zero.Proof. Notice that Col − v ( α q ) is a vanishing cycle. Since Col − v ( α q ) is aclosed loop, any class in ker( p ) ∩ T ( X,ω ) ( M ) evaluates to zero on it.Since rank( M ) = rank( M v ), Lemma 3.6 and Corollary 3.5 implythat p (cid:0) T ( X,ω ) ( M ) (cid:1) = p (cid:0) Ann( V ) ∩ T ( X,ω ) ( M ) (cid:1) , where V continues to denote the vanishing cycles. This shows thatevery element of T ( X,ω ) ( M ) evaluates to zero on Col − v ( α q ). Since thecohomology class of ω is in T ( X,ω ) ( M ), we get the result. (cid:3) At each vertex q of the graph Col v ( C v ), Lemmas 5.7 and 5.8 togethershow that the incoming weights equal the outgoing weights, establish-ing the criterion for being strongly connected. (cid:3) Remark . In the rank-preserving case, the assumption that C isgeneric is only used to ensure that the PL map Col v is defined (byLemma 4.9). In fact that only requires that C consists of genericcylinders, and likely even that could be weakened.6. Double degenerations (with Mirzakhani)
Using Theorem 5.1, when rank( M v ) < rank( M ) we will prove theexistence of a degeneration of Col v ( X, ω ) that contracts the graphCol v ( C ). When M has no rel, this degeneration will be canonicallydefined, but even in this case we remark in Warning 6.11 that it ismore subtle than may be expected.The degeneration will involve deforming in the direction of a vectorthat verifies rel-scalability. The end result, given in Definition 6.10, willbe called a double degeneration , keeping in mind that it results fromthe two step process of first degenerating ( X, ω ) to obtain Col v ( X, ω )and then degenerating Col v ( X, ω ) to contract the graph Col v ( C ).The main ideas in this section were initially discovered in the specialcase when M has no rel jointly with Mirzakhani.6.1. Rel.
In this subsection and the next, we start by recalling somewell-known material; compare to [BSW , Section 6 ] , [KZ03 , Section4.2 ] , [EMZ03 , Section 8.1 ] and [McM13, McM14, MW14, Wol18] .Let ( X, ω ) be a translation surface and let Σ = { z , . . . , z s } be theset of zeros of ω . Consider a vector λ = ( λ , . . . , λ s ) ∈ C s . For each i , we can consider the set of directions at z i that point inthe same direction as λ i ; if the cone angle is 2 πk i then there are k i suchdirections. For example, if λ i is positive and real, these are the eastpointing directions at z i .Define Star i ( λ ) ⊂ ( X, ω ) to be the union of the k i line segments oflength | λ i | leaving z i in the direction of λ i . We will assume that allthese line segments are embedded. We will moreover assume that thedifferent Star i ( λ ) are disjoint, and defineStar( λ ) = ∪ i Star i ( λ ) . Under these assumptions, we now define a surface Sch λ ( X, ω ), whichlies in the same stratum as (
X, ω ). The notation is chosen to emphasizethe similarity to classical Schiffer variations. The surface Sch λ ( X, ω ) isconstructed from (
X, ω ) via the following two step process illustratedin Figure 6.1.(1) First cut each Star i ( λ ). For each i , we thus obtain a degenerate2 k i -gon, all of whose sides have length | λ i | .(2) Then, for each i , re-glue the collection edges thus created sothat each is glued to the adjacent edge that it wasn’t previouslyglued to. Figure 6.1.
A local surgery at a cone point of angle6 π .The cut and re-gluing surgeries corresponding to different i are inde-pendent, and can be done simultaneously or sequentially.The relevance of this construction is given by the following observa-tion. Lemma 6.1.
Along the path
Sch τλ ( X, ω ) , τ ∈ [0 , , the absolute pe-riods are constant, and the derivative of a cycle joining z i to z j isconstant and equal to λ j − λ i . IGH RANK INVARIANT SUBVARIETIES 35
Corollary 6.2.
For ξ ∈ ker( p ) , there is a unique λ ∈ C s with (cid:80) λ i = 0 such that for τ sufficiently small, the surface ( X, ω ) + τ ξ is equal to
Sch τλ ( X, ω ) .If ξ is real, then so is λ . Real rel.
When ξ ∈ ker( p ) ∩ H ( X, Σ , R ) is real, the construc-tion above is somewhat nicer, because all the segments in Star( λ ) arehorizontal and hence parallel to each other.We consider the path ( X, ω ) − τ ξ, τ ≥
0. (The minus sign will proveconvenient in our application.) Either (
X, ω ) − τ ξ is defined for all τ >
0, or it is defined for τ less than some constant τ ξ such that ahorizontal saddle connection on ( X, ω ) reaches zero length as τ → τ ξ .To this standard discussion, we add the following observation. Lemma 6.3. If ( X, ω ) − τ ξ is only defined until time τ ξ < ∞ , thenthe path ( X, ω ) − τ ξ converges as τ → τ ξ in the WYSIWYG partialcompactification.The limit can be obtained from ( X, ω ) by a cutting and regluingsurgery. Conversely, ( X, ω ) − τ ξ can be obtained from the limit bymaking cuts at the zeros of size proportional to τ − τ ξ , and regluingthese cuts in some pattern.The space V of vanishing cycles is spanned by the horizontal saddleconnections that reach zero length as τ → τ ξ , which are exactly thesaddle connections s where ω ( s ) = τ ξ · ξ ( s ) .Proof. ( X, ω ) − τ ξ is obtained by cutting a collection of line segmentsemanating from the zeros, as described above. The length of the seg-ments is proportional to τ .Define a translation surface ( X (cid:48) , ω (cid:48) ) by the same process, with cutsof lengths proportional to τ = τ ξ . The τ = τ ξ case differs from the τ < τ ξ case only in that the endpoints of two different line segmentsbeing cut may coincide, or an endpoint of one of these line segmentsmay now be a zero.This ( X (cid:48) , ω (cid:48) ) is the candidate limit for the path. By definition, itis obtained by local surgeries, and conversely ( X, ω ) can be recoveredfrom ( X (cid:48) , ω (cid:48) ) by reversing these surgeries, cutting all the line segmentsproduced in the re-gluing and then identifying them in their originalgluing to obtain ( X, ω ).Similarly, ( X (cid:48) , ω (cid:48) ) can be obtained from ( X, ω ) − τ ξ via cutting andre-gluing, where the cuts have size proportional to τ − τ ξ . Conversely,we can obtain ( X, ω ) − τ ξ from ( X (cid:48) , ω (cid:48) ) by cutting and re-gluing acollection of segments of size proportional to τ − τ ξ , all of which startat the zeros and marked points of ( X (cid:48) , ω (cid:48) ). We thus see that there is an isometric map from the complement ofa small neighborhood of the zeros on ( X (cid:48) , ω (cid:48) ) to ( X, ω ) − τ ξ , verifyingthe criterion for convergence from [MW17 , Definition 2.2 ] .For any relative homology class s , its holonomy on ( X, ω ) − τ ξ is bydefinition ω ( s ) − τ ξ ( s ). Thus, the horizontal saddle connections thatreach zero length are exactly those where ω ( s ) = τ ξ · ξ ( s ). That thereare no other vanishing cycles follows as in the proof of Lemma 4.25. (cid:3) Assumptions.
We will require some assumptions to define thedouble degeneration, which are motivated by the following two lemmas.
Lemma 6.4.
If the assumptions of Theorem 5.1 hold and rank( M v ) < rank( M ) , then(1) if M has no rel, then the rel vector in T Col v ( X,ω ) ( M v ) that cer-tifies that Col v ( C ) is rel-scalable is unique, and(2) regardless of whether M has rel, this vector can be chosen to bea complex multiple of a real vector.Proof. We begin by proving the first statement. Corollary 4.26 impliesthat rank( M v ) = rank( M ) −
1. Since Col v ( C ) is rel-scalable, M v hasat least one dimension of rel. This implies that, in the case that M hasno rel, M v has exactly one dimension of rel, since dim( M v ) < dim( M ).In particular, the vector certifying rel-scalability must be unique up toscaling.To prove the second statement, it suffices to show that the vectorcan be chosen to be real when C is horizontal. When C is horizontal,the real part of a vector certifying rel-scalability again certifies rel-scalability. (cid:3) We will now establish the following, which is a slight variant of [AWc , Corollary 3.23 ] . Lemma 6.5.
Suppose that C is an equivalence class of generic cylin-ders on a surface ( X, ω ) in an invariant subvariety M . Let v ∈ Twist( C , M ) define a cylinder degeneration.Suppose that the ratio of heights of any two cylinders in C v is con-stant on a neighborhood of ( X, ω ) in M . Then all the saddle connec-tions in Col v ( C v ) are generically parallel on M v . Moreover, Ann( V ) ∩ T ( X,ω ) ( M ) is codimension one in T ( X,ω ) ( M ) .Remark . By Corollary 4.3, if M has no rel then it is automaticallythe case that the ratio of heights of any two cylinders in C is constantin any neighborhood where those cylinders persist. IGH RANK INVARIANT SUBVARIETIES 37
Proof.
Pick an arbitrary cylinder in C v . Let γ denote its core curve,and let s denote one of its cross curves.By Lemmas 4.19 and 4.25, every vanishing cycle can be written as alinear combination of cross curves and core curves of cylinders in C v .Since the heights of any two cylinders in C v have a constant ratio in aneighborhood of ( X, ω ), it follows that the holonomy of any such chainis a linear combination of the holonomies of γ and s . Since γ is not avanishing cycle, there is some constant c ∈ R so that s (cid:48) := cγ + s hasgenerically the same holonomy as a vanishing cycle.For any 1-chain, α in C v , the holonomy of α can be written as aγ + bs (cid:48) for some real constants a and b . Since s (cid:48) is a vanishing cycle, the ho-lonomy of Col v ( α ) can be written as a times the holonomy of Col v ( γ ).This shows that every saddle connection contained in Col v ( C v ) is gener-ically parallel to Col v ( γ ).If α is a vanishing cycle, Col v ( α ) has zero holonomy, and so a = 0.This shows that all vanishing cycles induce collinear functionals on T ( X,ω ) ( M ), giving the codimension one statement. (cid:3) The definition.
Motivated by the previous two lemmas, we makethe following assumption for the rest of this section.
Standing Assumption 6.7.
Suppose that
Col v ( C v ) is rel-scalable andthat all the saddle connections parallel to Col v ( C v ) are generically par-allel to each other. Let η denote a rel vector in T Col v ( X,ω ) ( M v ) thatcertifies that Col v ( C v ) is rel-scalable and that is a complex multiple ofa real vector.Remark . When M has no rel and all saddle connections parallelto C are generically parallel, Assumption 6.7 holds. The “genericallyparallel” condition can always be obtained by an arbitrarily small per-turbation. Lemma 6.9.
The path
Col v ( X, ω ) − tη is defined and remains in M v for t ∈ [0 , . It converges as t → , and the space V of vanishingcycles is spanned by the saddle connections parallel to Col v ( C ) .Proof. Rotating the surface if required, we can assume that C is hori-zontal and η is real. The result then follows from Lemma 6.3, notingthat t η = 1 since η ( s ) = ω ( s ) for all horizontal saddle connections. (cid:3) Definition 6.10.
Let Col doubv ( X, ω ) denote the limit as t approaches t η = 1 of Col v ( X, ω ) − tη . Let M doubv denote the component of theboundary containing Col doubv ( X, ω ).Note that these definitions depend on the choice of η . In this paper,they will be exclusively used in the case when M has no rel, where Lemma 6.4 gives that η is unique, so the dependence on η is not re-flected in the notation. Warning 6.11.
It is dangerously incomplete to describe the doubledegeneration simply by saying it contracts the graph Col v ( C ). Firstof all, there might be saddle connections generically parallel to thisgraph but not contained in this graph, and they get contracted as well.But more importantly, our proof of the rel-scalability of Col v ( C ) doesnot give an explicit description of a cohomology class η certifying therel-scalability, even when this η is unique. So our understanding of thedouble degeneration is not very explicit. We have not ruled out thepossibility that the local surgeries required to obtain ( X, ω ) − tη mightrequire surgeries at zeros not on saddle connections generically parallelto Col v ( C ), so singularities not on the graph Col v ( C v ) might a priori“move around” along the path Col v ( X, ω ) − tη .6.5. Basic results.
Keeping in mind that double degenerations areonly defined under Assumption 6.7, we compare M doubv to M v and M . Lemma 6.12. dim M doubv = dim M v − and rank M doubv = rank M v .Proof. By Lemma 6.3, the vanishing cycles are spanned by saddle con-nections parallel to Col v ( C ), which are all generically parallel by As-sumption 6.7. Hence dim M doubv = dim M v −
1. The fact that Col v ( C )is rel-scalable, together with Lemma 3.8, gives that rank M doubv =rank M v . (cid:3) Corollary 6.13. rank( M doubv ) = rank( M ) − . If rel( M ) = 0 then rel( M doubv ) = 0 .Proof. The first statement follows from Lemma 6.12 and Corollary 4.26.The second statement follows from the first, since M doubv has dimensionat least two less than M , and dimension is twice rank plus rel. (cid:3) Lemma 6.14.
Let L C ⊂ H ( X, Σ) be the span of all saddle connectionscontained in C as well as all saddle connections parallel to C . Then T Col doubv ( X,ω ) ( M doubv ) = T ( X,ω ) ( M ) ∩ Ann( L C ) . Moreover, the genus of
Col doubv ( X, ω ) is at least d less than that of ( X, ω ) , where d is the dimension of the span of the core curves of C inhomology.Proof. The first statement follows because L C is the preimage under(Col v ) ∗ of the span (in relative homology) of the saddle connectionsparallel to Col v ( C ), which span the vanishing cycles for the degenera-tion from Col v ( X, ω ) to Col doubv ( X, ω ). IGH RANK INVARIANT SUBVARIETIES 39
The second statement follows because all the core curves of cylindersare in the kernel of the induced map on homology for the compositionof the collapse maps that map (
X, ω ) to Col doubv ( X, ω ). Alternatively,the second statement can be derived from the fact that Ann( L C ) is thetangent space to the stratum of Col doubv ( X, ω ), the fact that genus isthe rank of the stratum, Lemma 3.4, and Corollary 3.5. (cid:3) Classification using a nested free cylinder
In this section we develop some results of independent interest.
Definition 7.1.
Suppose that C is a cylinder on a surface ( X, ω ) in aninvariant subvariety M . We say that C is nested in another cylinder H if C ⊂ H and C crosses H exactly once.Nested cylinders are automatically simple. If C is nested in H , thenthere is a saddle connection which appears in both the top and bottomboundaries of H and is a cross curve for C . The property of beingnested can typically be destroyed by perturbation, so when we discussnested cylinders we will typically be considering “non-generic” surfaces. Theorem 7.2.
Let M be an invariant subvariety with no rel. If asurface ( X, ω ) ∈ M has a nested free cylinder then M is a stratum ofAbelian differentials or a quadratic double. Recall that “free” is defined before Theorem 2.2. After proving The-orem 7.2, we will derive from it Proposition 7.13. These results areimportant ingredients in the next section.Proposition 7.13 also has the following surprisingly strong conse-quence.
Corollary 7.3.
Suppose that M has no rel and is contained in a genus g stratum with s zeros. Then if rank( M ) > g + s − then M is either a component of a stratum or a quadratic double. The rank bound in Corollary 7.3 is strictly weaker than that in The-orem 1.1 when s >
2, equivalent when s = 2, and slightly strongerwhen s = 1. The g = 3, s = 1 case recovers the classification of ranktwo invariant subvarieties of H (4), established in [NW14, ANW16] .We prove Corollary 7.3 at the end of this section. The proof.
The arguments in this section use only the CylinderDeformation Theorem and its partial converse Theorem 4.3.
Proof of Theorem 7.2:
Assume M has a surface with a nested freecylinder. Lemma 7.4. M contains a horizontally periodic surface ( X, ω ) withonly a single horizontal cylinder H , such that H contains a vertical freecylinder V , and such that any pair of horizontal saddle connections on ( X, ω ) that have the same length continue to have the same length in aneighborhood of ( X, ω ) in M .Proof. Start with any surface with a cylinder V nested in a cylinder H . Without loss of generality, assume H is horizontal and the surfacedoesn’t contain any vertical saddle connections. Note that H must befree, since deformations of V can be used to change the circumferenceof H without changing the circumference of any parallel cylinders.We now describe how to “collapse the complement of H onto H ”,via a procedure illustrated in Figure 7.1. Formally, this is a limit as Figure 7.1.
The two step process illustrated with (cid:15) = . (cid:15) → (cid:15) using the GL (2 , R ) action, and then vertical stretching H by (cid:15) − using a cylinder deformation, so that the height of H staysconstant. Since there are no vertical saddle connections, Masur’s Com-pactness Criterion implies this process does not degenerate the surface.(In fact, the limit can be described concretely: it is obtained by gluingeach point p on the top of H to the point on the bottom of H wherethe straight line which leaves p in the north direction first returns to H ; this gluing is well defined on a co-finite set.)To conclude, perform a small generic real deformation so that anytwo horizontal saddle connections that are not M -parallel have differ-ent lengths, and then shear the surface so V is vertical. (cid:3) The rest of the proof takes place on a surface produced by Lemma7.4. For each horizontal saddle connection a not contained in V , define IGH RANK INVARIANT SUBVARIETIES 41 W a to be the cylinder that crosses H once, has a as a cross curve, anddoes not intersect V . See Figure 7.2. Figure 7.2
Lemma 7.5.
For each W a , there is at most one cylinder that is M -parallel to W a . If such a cylinder exists, it has the same height as W a .Proof. This follows by “over-collapsing V to attack W a ”, a procedureillustrated in Figure 7.3. The same idea was used previously in [AWc] and [AWa] . Figure 7.3
To accomplish this, start by shearing V slightly so that it no longercontains a horizontal saddle connection. Then use the standard cylin-der deformation of V to reduce the height of V without changing thehorizontal foliation of the surface. Continuing in this direction, eventu-ally V reaches zero height, but because it does not contain horizontalsaddle connections this does not degenerate the surface. Hence, thislinear path in period coordinates can be continued past the point where V reaches zero height, thus “over-collapsing V ”.In this case the overcollapse can be understood concretely. We canthink of V has being composed of two triangles, and the deformationabove simply changes these triangles as in Figure 7.3. The over-collapse can be obtained by removing two triangles from H − V , and makingappropriate edge identifications.When these triangles first hit a cylinder M -parallel to W a , it can hitat most two such cylinders. As the attack continues, these two cylin-ders change height (and indeed they lose an equal amount of height).Corollary 4.3 thus gives that there are at most two cylinders in theequivalence class of W a .The same argument shows that if there are two cylinders in theequivalence class then their heights must be equal, since the modulimust stay rationally related throughout the attack. (cid:3) If there is a cylinder M -parallel to W a , we will denote it W (cid:48) a . If not,we will say that W (cid:48) a does not exist. Lemma 7.6. If W (cid:48) a exists, then every horizontal saddle connection thatit passes through has the same length as a .Proof. Otherwise, since W a and W (cid:48) a have the same height, W (cid:48) a passesthrough a horizontal saddle connection c that is longer than a .Consider the cylinder deformation that horizontally compresses W c and, if it exists, W (cid:48) c ; this deformation does not change the horizontalfoliation. This deformation eventually reduces the length of c to zero,and also changes the lengths of the horizontal saddle connections that W (cid:48) c passes through. But, since W (cid:48) c has the same height as W c , and a is smaller than c , W (cid:48) c does not pass through a . So, the length of a isunchanged.Our explicit definition of W a shows that W a persists along this de-formation, and its height does not go zero. But, the height of W (cid:48) a doesgo to zero, because W (cid:48) a passes through c , and the length of c goes tozero. This contradicts the fact that W a and W (cid:48) a must always have thesame height. (cid:3) Lemma 7.7. If W (cid:48) a exists, it is equal to W b for some horizontal saddleconnection b . This is equivalent to the statement that W (cid:48) a passes through only onehorizontal saddle connection, rather than more. If that saddle connec-tion is b , then W (cid:48) a = W b . Equivalently, W (cid:48) a has the same circumferenceas W a . Equivalently, the core curve of W (cid:48) a intersects the core curve of H only once. Proof.
By Lemma 7.6, every horizontal saddle connection that W (cid:48) a passes through is a core curve of W (cid:48) a . In particular, all these sad-dle connections are of the same length. Let b , . . . , b k be the horizontal IGH RANK INVARIANT SUBVARIETIES 43 saddle connections that W (cid:48) a passes through. We want to show k = 1,so suppose to the contrary that k > b is the closest of a, b , . . . , b k to V on the topboundary of H in the left direction, i.e. the left-to-right distance fromthe right endpoint of b to the left side of V is minimal. Let s be theclosest of a, b , . . . , b k to V on the bottom boundary of H in the leftdirection.Because W (cid:48) a does not cross V , it must pass up through s and thenthrough b , so H ∩ W (cid:48) a contains a parallelogram with s on the bottomand b on the top. Hence, because k >
1, we conclude that s (cid:54) = b . Onecan also check that s (cid:54) = a , since W a and W (cid:48) a are parallel and disjointfrom V . So, without loss of generality, assume s = b , as in Figure 7.4. Figure 7.4.
The proof of Lemma 7.7.Lemma 7.5 gives that W (cid:48) b has the same height as W b . Our assump-tion on ( X, ω ) implies that the only horizontal saddle connections ofthe same length as a are b , . . . , b k . Hence Lemma 7.6 implies that W (cid:48) b , if it exists, cannot pass through any horizontal saddle connectionsother than a, b , . . . , b k . However, W (cid:48) b cannot pass through b , because W (cid:48) b is disjoint from V , and by our choice of b and b .Hence, considering the cylinder deformation in W b , and W (cid:48) b if itexists, we see that in fact b and b are not generically the same length.This is a contradiction, so we get the desired result. (cid:3) Lemma 7.8.
If, for every horizontal saddle connection a , W (cid:48) a does notexist, then M is a component of a stratum of Abelian differentials.Proof. We will show that every deformation of (
X, ω ) in its stratumcan be obtained by deforming the cylinders H , V , and the various W a .Indeed, these deformation are sufficient to arbitrarily deform each edgein the rectangular presentation of ( X, ω ) illustrated above. (cid:3)
Therefore we assume that there is at least one horizontal saddleconnection s such that W (cid:48) s exists. Lemma 7.9.
Under this assumption, the involution of the interior of H that maps V to itself extends to an involution of the surface.Proof. Let a be any horizontal saddle connection for which W (cid:48) a exists,and let W (cid:48) a = W b . As in the proof of Lemma 7.5, by attacking from V we see that W a and W b are interchanged by the involution. So inparticular, a and b are interchanged by the involution.It now suffices to show that any saddle connection c for which W (cid:48) c does not exist is fixed by the involution. This follows since, like V , W c is a nested free cylinder. So, letting a and b be horizontal saddleconnections so that W (cid:48) a = W b , the previous paragraph implies that theinvolution fixing W c must exchange a and b and hence coincide withthe involution fixing V . (cid:3) We now claim that M is equal to a quadratic double, having alreadyshown in the previous lemma that it is contained in one. To do this,we will show that every deformation of ( X, ω ) on which the holonomyinvolution persists can be obtained using cylinder deformations of thepairs { W a , W (cid:48) a } when W (cid:48) a exists, of individual W a when W (cid:48) a does notexist, of V , and of H . Indeed, these deformation are sufficient toarbitrarily deform each fixed edge or exchanged pair of edges in therectangular presentation of ( X, ω ) illustrated above. (cid:3)
Background on cylindrical stability.
Given a horizontally pe-riodic surface (
X, ω ) ∈ M , its twist space is defined as the subspace of T ( X,ω ) ( M ) of vectors that can be written as linear combinations of du-als of core curves of horizontal cylinders. Its cylinder preserving space is defined to be the subspace of T ( X,ω ) ( M ) of cohomology classes eval-uating to zero on all the core curves of horizontal cylinders on ( X, ω ).It is immediate from the definition that the twist space is contained inthe cylinder preserving space.Following the terminology in [AN , Definition 2.4 ] , we say a surfacein M is M -cylindrically stable , or just cylindrically stable when M isclear from context, if it is horizontally periodic and its twist space isequal to its cylinder preserving space.We summarize a number of results that we will need on cylindricallystable surfaces. Lemma 7.10.
Let M be an invariant subvariety.(1) A surface is cylindrically stable if and only if for every hori-zontal saddle connection there is a linear combination of corecurves of horizontal cylinders that has the same period locallyin M . IGH RANK INVARIANT SUBVARIETIES 45 (2) A surface is cylindrically stable if and only if its twist space hasdimension rank( M ) + rel( M ) . No surface has twist space oflarger dimension.(3) If M has no rel, then a surface is cylindrically stable if and onlyif it has rank( M ) horizontal equivalence classes. No surface hasmore horizontal equivalence classes.(4) If a surface is cylindrically stable, then the horizontal core curvesspan a subset of T ( X,ω ) ( M ) ∗ of dimension rank( M ) .(5) If M has no rel and a surface is cylindrically stable, then takingone core curve of each equivalence class gives a linearly inde-pendent subset of T ( X,ω ) ( M ) ∗ .(6) If k ( M ) = Q , every surface in M can be perturbed to becomecylindrically stable and square-tiled in such a way that all hor-izontal cylinders on the original surface stay horizontal in theperturbation. Here k ( M ) denotes the smallest field which can be used to definethe linear equations locally defining M in period coordinates, as in [Wri14] . Proof.
This result can be thought of as a black box coming from [Wri15a ,Section 8 ] , but nonetheless we give more specific references.The Cylinder Deformation Theorem and Theorem 4.2 imply thatwhen M has no rel, the dimension of the twist space of a horizontallyperiodic surface is equal to the number of horizontal equivalence classes.This observation shows that Claim (3) follows from Claim (2). Noticetoo that Claim (5) follows from Claims (3) and (4). So we will provethe remaining claims.Claim (1) is immediate from [Wri15a , Definition 8.1, Corollary 8.3 ] ,which gives that, the twist space can also be defined as the subspace of T ( X,ω ) ( M ) of cohomology classes that are zero on all horizontal saddleconnections.We will now prove Claim (4). Let T be the span of the core curves ofthe horizontal cylinders in ( T ( X,ω ) ( M )) ∗ . By [Wri15a , Lemma 8.8 andCorollary 8.11 ] , dim T ≥ rank( M ). Since the core curves of parallelcylinders span an isotropic subspace of p ( T ( X,ω ) ( M )), and any isotropicsubspace has dimension at most rank( M ), dim p ( T ) ≤ rank( M ). Sincethe core curves of the cylinders are absolute homology classes, theirspan in ( p ( T ( X,ω ) ( M )) ∗ is isomorphic to their span in ( T ( X,ω ) ( M )) ∗ .Therefore, dim( T ) = rank( M ).For the first part of Claim (2), note that if a surface is cylindricallystable, Claim (4) and [Wri15a , Lemma 8.8 ] imply that the cylinder preserving space has codimension rank( M ) in T ( X,ω ) ( M ). Hence it hasdimension rank( M ) + rel( M ). By definition of cylindrically stable, thetwist space is equal to the cylinder-preserving space, and hence mustalso have dimension rank( M ) + rel( M ). Conversely, if the twist spacehas this dimension, then [Wri15a , Lemma 8.10 ] implies that the corecurves span a subspace of ( T ( X,ω ) ( M )) ∗ of dimension rank( M ), andhence that the cylinder preserving space has codimension rank( M ).The second part of Claim (2) is [Wri15a , Corollary 8.11 ] For Claim (6), recall that since k ( M ) = Q , square-tiled surfacesare dense in M ; see for example [AWc , Lemma 3.3 ] . Given this,the claim follows from [Wri15a , Sublemma 8.7 ] , keeping in mind thatperturbations can be arranged to be square-tiled and hence horizontallyperiodic. The same argument is used in [AWc , Lemma 6.9 ] . (cid:3) An application.
We start with a somewhat cumbersome defini-tion.
Definition 7.11.
Suppose that C is a collection of parallel cylinders ona surface ( X, ω ) in an invariant subvariety M . A C -path is a continuousmap γ : [0 , → M such that γ (0) = ( X, ω ), and C persists at all pointsalong this path, and the ratio of moduli and circumferences of any twocylinders in C is constant along the path. We will say that γ (1) is C -related to ( X, ω ). Remark . If M has no rel, Corollary 4.3 gives that the conditionthat “the ratio of moduli and circumferences of any two cylinders in C is constant along the path” is automatic whenever the cylinders in C are all pairwise M -equivalent. Proposition 7.13.
Suppose that k ( M ) = Q and M has no rel. Let C be a horizontal equivalence class on a surface ( X, ω ) in M . Then thereis a C -related cylindrically stable surface ( X (cid:48) , ω (cid:48) ) on which the cylindersin C remain horizontal and where one of the following occurs:(1) There is a free horizontal cylinder not in C that contains anested free cylinder.(2) None of the horizontal cylinders not in C are free.Remark . Throughout this work, when we have considered collec-tions C of cylinders, there has been a tacit understanding that C isnonempty. In this lemma, however, it is permissible to take C to beempty. We tacitly use this observation in the proof of Corollary 7.3. Proof.
Assume that every cylindrically stable C -related surface ( X (cid:48) , ω (cid:48) )on which C remains horizontal contains a free horizontal cylinder not IGH RANK INVARIANT SUBVARIETIES 47 in C . Assume without loss of generality, perhaps after replacing ( X, ω )with a C -related surface, that ( X, ω ) is square-tiled and that it containsas few horizontal free cylinders as possible, subject to the constraintsthat the cylinders in C remain horizontal and that the surface is cylin-drically stable (such a surface exists by Lemma 7.10 (6)).By assumption there is a free horizontal cylinder F on ( X, ω ) thatdoes not belong to C . Let γ F denote its core curve. Let A denote thecollection of all horizontal cylinders on ( X, ω ).By Lemma 7.10 (5), there is a nonzero tangent vector v ∈ T ( X,ω ) ( M )that evaluates to zero on the core curves of every cylinder in A − { F } but so that v ( γ F ) is a nonzero purely imaginary number. Since k ( M ) = Q , we may also assume that Re( v ) = 0 and that Im( v ) is rational.By assuming that v is sufficiently small, we can assume that thecylinders in A − { F } persist (and necessarily remain horizontal) on( X, ω ) + v . Since v is rational, ( X, ω ) + v is square-tiled and hencehorizontally periodic. Since ( X, ω ) had the fewest number of free hor-izontal cylinders, there must be a new free horizontal cylinder F (cid:48) on( X, ω ) + v .By Corollary 4.3, { F (cid:48) } is an equivalence class, and the perturbationdoes not create new cylinders generically parallel to those in A − { F } .By Lemma 7.10 (3), since ( X, ω ) + v has at least rank( M ) many equiv-alence classes, it is cylindrically stable. Since it cannot have morehorizontal equivalence classes, horizontal cylinders on ( X, ω ) + v areexactly the ones in ( A − { F } ) ∪ { F (cid:48) } .Assuming v is sufficiently small, we can assume that F has verysmall non-zero slope on ( X, ω ) + v . Let s be one of its boundary saddleconnections. Since F must be contained in F (cid:48) , we see that F (cid:48) mustcross s , and hence that the height of F (cid:48) must be small. Since the areaof F (cid:48) is at least that of F on ( X, ω ) + v , we get that F (cid:48) must have longcircumference. In particular, we can assume that F (cid:48) has circumferencelonger than the sum of the circumferences of the cylinders in A − { F } .That ensures that F (cid:48) contains a saddle connection on its top boundarythat also appears on its bottom boundary.Therefore, the new horizontal cylinder F (cid:48) contains a nested cylinder,call it T . Let T be the equivalence class of T . Lemma 7.15. T = { T } .Proof. The proof will use a variant of the technique used in the proofof Lemma 7.4 to obtain a surface where the cylinders in A − { F } stayhorizontal, and where T becomes horizontal. Our assumption that( X, ω ) had the minimum possible number of free horizontal cylinderswill then give that T consists of a single free cylinder. By replacing (
X, ω ) with the result of slightly shearing F (cid:48) , we canassume that F (cid:48) does not contain any vertical saddle connections. Since T is nested in F (cid:48) this implies in particular that the real part of theholonomy of the core curve of T is nonzero.We will now build a sequence of surfaces depending on (cid:15) that havethe same vertical foliation as ( X, ω ), via a two step process illustratedin Figure 7.5. First deform in the direction of − iσ F (cid:48) until F (cid:48) has height (cid:15) . Then deform in the iσ T direction until the area of T is one. Callthis surface ( X (cid:15) , ω (cid:15) ). Figure 7.5.
The two step process for creating ( X (cid:15) , ω (cid:15) ).As (cid:15) tends to zero, the holonomy of the core curve of T on ( X (cid:15) , ω (cid:15) )tends to the real part of the holonomy of the core curve of T on ( X, ω ),and the area of T is constant and equal to one. This shows that T converges to a horizontal cylinder. Moreover, since we assumed that F (cid:48) had no vertical saddle connections it follows the limit of ( X (cid:15) , ω (cid:15) )exists and is contained in M . Since the original surface was chosen tominimize the number of horizontal free cylinders, T must be a singlefree cylinder. (cid:3) Since T is a free cylinder nested in a free cylinder F (cid:48) , this concludesthe proof. (cid:3) Proof of Corollary 7.3.
A horizontally periodic surface in a stratum hasat most g + s − (cid:3) Finding useful cylinders
The goal of the section is the following result, which we will useto prove Proposition 2.6. Given an equivalence class D of cylinderson ( X, ω ) ∈ M , we will let (cid:98) D denote the union of D and all saddleconnections parallel to D , and we will let (cid:98) D c = ( X, ω ) \ (cid:98) D denote itscomplement. Theorem 8.1.
Assume that M is an invariant subvariety with rank atleast 3, no rel, and k ( M ) = Q , and suppose that M isn’t a componentof a stratum or a quadratic double. Suppose that C is an equivalence IGH RANK INVARIANT SUBVARIETIES 49 class of cylinders on ( X, ω ) . Then, there exists a C -related surface withgeneric equivalence classes D and D (cid:48) such that(1) C , D and D (cid:48) are disjoint,(2) D has at least two cylinders,(3) every saddle connection parallel to D is generically parallel to D , and(4) every component of (cid:98) D c contains a cylinder from C and onefrom D (cid:48) . The main purpose of Theorem 8.1 is to produce D with a favourablerelationship with C ; the existence D (cid:48) is of secondary importance.We will derive Theorem 8.1 from the following. Theorem 8.2.
Assume M is an invariant subvariety with rank atleast 2 and no rel, and suppose that all parallel saddle connections on ( X, ω ) ∈ M are generically parallel. Suppose that C , D , . . . , D rank( M ) − are disjoint equivalence classes on ( X, ω ) . Then, for some ≤ i ≤ rank( M ) − , every component of (cid:98) D ci contains a cylinder in C and acylinder in each D j , j (cid:54) = i .Proof of Theorem 8.1 assuming Theorem 8.2. Since M is neither a stra-tum nor a quadratic double, Theorem 7.2 gives that no surface in M has a nested free cylinder. By Proposition 7.13, we can therefore find acylindrically stable C -related surface where the cylinders in C are hori-zontal and the only free horizontal cylinders (if there are any) belong to C . Since an equivalence class with only one cylinder is a free cylinder,this implies that every equivalence class of horizontal cylinders asidefrom C contains at least two cylinders.Let C , D , D , . . . , D rank( M ) − denote the equivalence classes of hor-izontal cylinders on the newly constructed surface and perturb so thatthese equivalence classes persist and become generic and so that anytwo parallel saddle connections on the perturbed surfaces are actuallygenerically parallel. By Theorem 8.2, there is some i such that everycomponent of (cid:98) D ci contains a cylinder from C and from every D j , j (cid:54) = i .Setting D to be D i , and D (cid:48) to be any D j , j (cid:54) = i , gives the result. (cid:3) Basic lemmas.
The following result will be a convenient tool forour analysis in this subsection.
Lemma 8.3.
Let ( X, ω ) be a translation surface, and let M be a mini-mal component for the horizontal straight line flow on ( X, ω ) . Then, forall ε > , there is a cylinder C contained in M whose ε -neighborhoodcontains M . Sketch of proof.
A result of Smillie-Weiss implies that M contains cylin-ders such that the imaginary part of the holonomy of the core curveis arbitrarily small [SW04 , Corollary 6 ] ; such cylinders are necessarilyalmost horizontal. Because M is a minimal component, every sequenceof cylinders in M that become more and more horizontal must becomemore and more dense. (cid:3) We also use the following direct corollary of the Cylinder Deforma-tion Theorem, which is a variant of [NW14 , Proposition 3.2 ] . Corollary 8.4.
Let M be an invariant subvariety. Let A and B beequivalence classes of cylinders. Then either A is disjoint from B andthe saddle connections generically parallel to B , or A intersects everycylinder in B and every saddle connection generically parallel to B .Proof. The result follows from deforming A using the Cylinder Defor-mation Theorem. (cid:3) The next two lemmas are morally related to Theorem 3.2, but theproofs we give will not make use of that result.
Lemma 8.5.
Suppose D and D (cid:48) are disjoint equivalence classes ofcylinders on a surface ( X, ω ) in an invariant subvariety M . Supposeall parallel saddle connections on ( X, ω ) are generically parallel.Then every component of (cid:98) D c that contains a saddle connection par-allel to D (cid:48) also contains a cylinder in D (cid:48) .Proof. In the direction of D , ( X, ω ) is partitioned into cylinders (whichnecessarily belong to D ) and minimal components for straight line flow.Lemma 8.3 implies there is a cylinder E which intersects D (cid:48) but not D . Let E be the equivalence class of E . Corollary 8.4 gives that everycylinder of E must intersect a cylinder of D (cid:48) , and that E is disjointfrom (cid:98) D .Deforming E thus only affects the components of (cid:98) D c containing cylin-ders of D (cid:48) . Since deforming E changes D (cid:48) , it must also change all saddleconnections generically parallel to D (cid:48) , giving the result. (cid:3) Lemma 8.6.
Suppose D , D (cid:48) and D (cid:48)(cid:48) are disjoint equivalence classes ofcylinders on a surface ( X, ω ) in an invariant subvariety M . Supposeall parallel saddle connections on ( X, ω ) are generically parallel.Then either every component of (cid:98) D c that contains a cylinder in D (cid:48) contains a cylinder in D (cid:48)(cid:48) , or no component of (cid:98) D c contains both acylinder in D (cid:48) and a cylinder in D (cid:48)(cid:48) . IGH RANK INVARIANT SUBVARIETIES 51
Proof.
Suppose there is a component of (cid:98) D c that contains both a cylin-der in D (cid:48) and a cylinder in D (cid:48)(cid:48) . Lemma 8.3 implies there is a cylinder E which intersects D (cid:48) and D (cid:48)(cid:48) but not D .Let E be the equivalence class of E . Corollary 8.4 gives that everycylinder of E must intersect both a cylinder in D (cid:48) and one in D (cid:48)(cid:48) , thatevery cylinder of D (cid:48) ∪ D intersects a cylinder of E , and that E is disjointfrom (cid:98) D , giving the result. (cid:3) Proof of Theorem 8.2.
For each 1 ≤ i ≤ rank( M ) −
1, define G i to be the union of the components of (cid:98) D ci that contain a cylinder from C . Note that because G i is bounded by saddle connections parallel to D i , we have G i (cid:54) = G j if i (cid:54) = j . Lemma 8.7.
For any i (cid:54) = j , if D j is not contained in G i , then G i ⊂ G j .Proof. If D j is not contained in G i , it follows from Lemma 8.6 thatall cylinders of D j are contained in components of (cid:98) D ci not containingcylinders of C . Lemma 8.5 thus gives that all the saddle connectionsparallel to D j are contained in components of (cid:98) D ci that do not containcylinders of C .Thus, each component of (cid:98) D ci that contains a cylinder of C is con-tained in such a component of (cid:98) D cj . (cid:3) For the remainder of the proof, fix i such that G i is not contained in G j for any j (cid:54) = i . For this i , Lemma 8.7 gives the following immediateconsequence. Corollary 8.8. G i contains D j for all j (cid:54) = i . Lemma 8.6 thus implies that every component of (cid:98) D ci that containsa cylinder from C also contains a cylinder from each D j , j (cid:54) = i . So thefollowing lemma completes the proof of Theorem 8.2. Lemma 8.9. ( X, ω ) = (cid:98) D i ∪ G i .Proof. If not, there is a component B of (cid:98) D ci that doesn’t contain anycylinders of any of the D j or of C .Lemma 8.3 gives that there is a cylinder E contained in S , whichmust be disjoint from all the cylinders in D j , j = 1 , . . . , rank( M ) − C . Corollary 8.4 thus gives that the equivalence class E of E isalso disjoint from all these cylinders. Thus, we have found rank( M )+1disjoint equivalence classes. This contradicts the fact that M has norel, because p ( T ( X,ω ) ( M )) is symplectic and of dimension 2 rank( M ),and the standard shears in these equivalence classes span an isotropicsubspace of dimension rank( M ) + 1. (cid:3) Primality of the boundary
The goal of this short section is to find a simple criterion for whensurfaces in a boundary component of a high rank invariant subvari-ety are connected. Throughout this section we will use the notationintroduced in Section 4.9.1.
Cylinder degenerations are prime.
We begin with a generalargument that shows that any cylinder degeneration produces a primecomponent of the boundary of M . This argument does not assumethat M has high rank. Lemma 9.1.
Let C be a collection of generic cylinders on a surface ( X, ω ) in an invariant subvariety M . Let v ∈ Twist( C , M ) specify acylinder degeneration. Then M v is prime.Proof. We begin with the following observation.
Sublemma 9.2.
Each component of
Col v ( X, ω ) contains a finite set ofparallel saddle connections, such that the sum of the holonomies of thisfinite set on one component is M v -generically parallel to the analogoussum on any other component.Proof. Consider any saddle connection s on the boundary of C v . This s remains a saddle connection along the collapse path, but on Col v ( X, ω )may split into a finite set s v of saddle connections.Along the collapse path ( X, ω ) − C is unchanged. Moreover, thecylinders in C − C v may have their heights change along the collapsepath, but they persist on the boundary. In particular, this meansthat each component of ( X, ω ) − C v is associated to a component ofCol v ( X, ω ). Two components of (
X, ω ) − C v could be associated to thesame component of Col v ( X, ω ), and all components of Col v ( X, ω ) areassociated to at least one component of (
X, ω ) − C v .If s is contained in the boundary of a component of ( X, ω ) − C v ,then s v is contained in the associated component of Col v ( X, ω ).Since the cylinders in C are generic, all s arising this way are M -generically parallel, and hence the sums associated to each s v are M v -generically parallel. (cid:3) By Theorem 3.2, if M v is not prime, then it is possible to produce anew surface in M v by applying an arbitrary element of GL(2 , R ) to thecomponents of Col v ( X, ω ) in one prime factor while fixing the compo-nents of Col v ( X, ω ) in the other prime factors, contradicting Sublemma9.2. (cid:3)
IGH RANK INVARIANT SUBVARIETIES 53
Prime boundary in high rank.
We now turn to a somewhatmiraculous property that distinguishes high rank invariant subvarieties.
Lemma 9.3.
Suppose that M (cid:48) is a prime boundary component of ahigh rank invariant subvariety M and that either • rank( M (cid:48) ) = rank( M ) , or • rank( M (cid:48) ) = rank( M ) − and the surfaces in M (cid:48) have genusat least one less than those in M .Then the surfaces in M (cid:48) are connected.Proof. Let (
Y, η ) be a surface in M (cid:48) and let( Y, η ) = ( Y , η ) ∪ · · · ∪ ( Y k , η k )be its decomposition into connected components. Suppose that thegenus of Y i is g i and that the genus of a surface in M is g . Our goal isto show that k = 1.Since M is high rank, rank( M ) ≥ g . Since g ≥ (cid:80) g i , we haverank( M ) ≥ k (cid:88) i =1 g i . Since M (cid:48) is prime, the rank of any component of M (cid:48) is the sameas the rank of M (cid:48) (by Lemma 3.4). The rank of the component of M (cid:48) containing ( Y i , η i ) is bounded above by g i by definition of rank.Therefore, min( g i ) ki =1 ≥ rank( M (cid:48) ) and so,rank( M ) ≥ k M (cid:48) ) . Set r = rank( M ). If rank( M (cid:48) ) = r , this equation becomes − ≥ (cid:18) k − (cid:19) r, which implies k = 1.If rank( M (cid:48) ) = r −
1, then, by assumption, the surfaces in M (cid:48) havegenus at least one less than those in M and so g ≥ (cid:80) g i . This isan improvement over the estimate g ≥ (cid:80) g i used above. By the samereasoning as in the preceding paragraphs, we have r ≥
32 + k r − , which implies r − > k r − , which again implies k = 1. (cid:3) Proof of Proposition 2.6
In this section we will use Theorem 8.1 to derive the following.
Theorem 10.1.
Suppose M that has no rel, rank at least 3, and k ( M ) = Q . Assume that C is an equivalence class of cylinders on ( X, ω ) ∈ M and that ( X, ω ) has no marked points. Then there is a C -related ( X (cid:48) , ω (cid:48) ) ∈ M and an equivalence class D of generic cylin-ders on ( X (cid:48) , ω (cid:48) ) that is disjoint from C such that either a single ordouble degeneration of D has genus at least two less than ( X (cid:48) , ω (cid:48) ) andis contained in a prime component of the boundary.Moreover, on this single or double degeneration, C persists and thereare no free marked points.Remark . In the double degeneration case, it is implicit that allsaddle connections on ( X (cid:48) , ω (cid:48) ) that are parallel to D are genericallyparallel, so that Assumption 6.7 is satisfied and the double degenerationis defined.Theorem 10.1 in turn easily implies Proposition 2.6. Proof of Proposition 2.6 given Theorem 10.1:
In general, [Wri14 , The-orem 1.5 ] gives rank( M ) · deg( k ( M )) ≤ g. Hence the high rank assumption implies that k ( M ) = Q . So theassumptions in Proposition 2.6 imply those in Theorem 10.1.Since M is not geminal, it contains a surface ( X, ω ) and an equiva-lence class C of cylinders that cannot be partitioned into free cylindersand pairs of twins (defined before Theorem 2.2). Let M (cid:48) be the bound-ary of M constructed in Theorem 10.1 and let ( Y, η ) ∈ M (cid:48) be thesingle or double degeneration produced in Theorem 10.1, so C persistson ( Y, η ).Note that rank( M (cid:48) ) = rank( M ) − D is generic, which is required to apply Corol-lary 4.26). The fact that the surfaces in M (cid:48) are connected is immediatefrom Lemma 9.3.Finally, we will show that M (cid:48) is not geminal. This is expectedbecause C certifies that M is not geminal, and C persists on ( Y, η ) ∈M (cid:48) . We will give one of several possible technical justifications.We begin by claiming that the standard deformation of C is con-tained in the tangent space of M (cid:48) , using the identification of the tan-gent space of M (cid:48) with a subspace of the tangent space of M . Lemma 10.3. σ C ∈ T ( Y,η ) ( M (cid:48) ) . IGH RANK INVARIANT SUBVARIETIES 55
Proof.
In the single degeneration case, this is immediate since C and D are disjoint. (More formally, Lemma 4.25 gives that C does notintersect the vanishing cycles.)So suppose that we are in the case where we must double-degenerate D . Lemma 6.14 states that the tangent space to the component ofthe boundary containing the double degeneration is identified with T ( X,ω ) ( M ) ∩ Ann( L D ), where L D ⊂ H ( X, Σ) is the span of all saddleconnections contained in D as well as all saddle connections parallel to D .By assumption, C is disjoint from D and hence, by Corollary 8.4and Remark 10.2, from any saddle connection parallel to D . Hence σ C ∈ Ann( L D ) as desired. (cid:3) Suppose to a contradiction that M (cid:48) is geminal. Since σ C remainsan element of T ( Y,η ) ( M (cid:48) ), no cylinder in Col( C ) can be a twin of acylinder not in Col( C ), where Col( C ) denotes the cylinders on ( Y, η )that persist from C .Moreover, since Twist(Col( C ) , M (cid:48) ) is isomorphic to Twist( C , M ), itfollows that Twist(Col( C ) , M (cid:48) ) is one-dimensional and hence Col( C )consists of either one free cylinder or two twins. This implies in par-ticular that σ C is a multiple of (cid:80) i γ ∗ i where { γ i } i is the set of corecurves of cylinders in Col( C ). However, this cannot be the case since C contains two cylinders with distinct heights by assumption. This isa contradiction. (cid:3) Lemmas for the single degeneration case.
In general, it ischallenging to prove that genus decreases by more than one in (single)cylinder degenerations, and indeed there are surprisingly complicatedsituations where this does not occur. We can however verify a genusreduction of at least two in the following narrow circumstance.
Lemma 10.4.
Let D be a generic equivalence class of cylinders on asurface ( X, ω ) in an invariant subvariety M . Let v ∈ Twist( D , M ) define a rank-reducing cylinder degeneration.Suppose additionally that there are homologous cylinders H , H / ∈ D such both components of the complement of the core curves of the H i contain cylinders of D .Then the genus of Col v ( X, ω ) is at least two smaller than that of ( X, ω ) .Proof. By Theorem 5.1, Col v ( D ) is an acyclic graph. This cylinderdegeneration can be achieved in two steps, by first collapsing the cylin-ders in D on one side of H , H , and then collapsing those on the other side. If either step didn’t reduce the genus, then the portion of Col v ( D )on the corresponding side of H , H could not be acyclic. (cid:3) Lemma 10.5.
Let D be a generic equivalence class of cylinders ona surface ( X, ω ) with no marked points in an invariant subvariety M with no rel. Let v ∈ Twist( D , M ) define a rank-reducing cylinderdegeneration. If Col v ( C ) contains at least two saddle connections, then Col v ( X, ω ) does not have any free marked points.Proof. Theorem 5.1 gives that D v = D , and hence Lemma 6.5 impliesthat all saddle connections in Col v ( D v ) are generically parallel.Since ( X, ω ) does not have marked points, any marked points ofCol v ( X, ω ) must lie on Col v ( D ). We first note that no saddle con-nection in Col v ( D ) can join a marked point to itself. If this were thecase, this saddle connection would border a cylinder, which it wouldbe generically parallel to, contradicting the fact that D v = D .Hence, if any of the marked points were free, moving its positionwould contradict the fact that all saddle connections in Col v ( D v ) aregenerically parallel. (cid:3) Proof of Theorem 10.1.
Replacing (
X, ω ) with a C -relatedsurface, let D and D (cid:48) be the equivalence classes produced by Theorem8.1. The single degeneration case:
Suppose that D contains a pair ofhomologous cylinders H , H such that both components of the com-plement of the core curves of the H i intersect (cid:98) D c . (This fails if onecomponent is covered by cylinders in D and their boundary saddleconnections.) In this case, the statement of Theorem 8.1 gives thatboth components contain cylinders from D (cid:48) .Lemma 10.4 gives that a single degeneration of D (cid:48) loses two genus,and Lemma 10.5 gives that this single degeneration does not containfree marked points. This proves Theorem 10.1 in this case. The double degeneration case:
Suppose that D does not containsuch a pair of a pair of homologous cylinders H , H . In this case wewill not make use of D (cid:48) . This case will however require an in-depthunderstanding of Section 6.Recall that Corollary 8.4 gives that (cid:98) D is disjoint from C . Fix v ∈ Twist( D , M ) that specifies a cylinder degeneration of M , and recallfrom Section 6 that since M has no rel this also uniquely specifies adouble degeneration. Lemma 10.6.
Possibly after replacing ( X, ω ) with the result of a cylin-der deformation in C , we can assume that C persists on the double IGH RANK INVARIANT SUBVARIETIES 57 degeneration of D , and each component of the double degeneration con-tains a cylinder of C .Proof. We first arrange for C to survive the double degeneration, usingthe following outline: the single degeneration of D doesn’t affect C ,and, using a cylinder deformation of C , we can assume that the heightsof the cylinders in C are so large that they survive the surgeries thatproduce the double degeneration. We now give the details.Consider the one parameter path ( X t , ω t ) of translation surfaceswhere C is deformed by adding the multiple of the standard defor-mation σ C that preserves and stretches out the direction parallel to D .(For example, if D is vertical, this is accomplished by deforming in thedirection of iσ C .) No new saddle connections parallel to D are createdalong the path, and the height of all cylinders in C can be assumed tobe very large at the endpoint.The single degeneration Col v ( X t , ω t ) also varies continuously, and theunique vector certifying rel-scalability of Col v ( D ) is locally constant.Thus the surgeries required to get from Col v ( D ) to the double degen-eration, which are described in detail in Section 6, are locally constant.These surgeries involve making cuts of certain sizes at zeros and thenre-gluing the boundary saddle connections of the cut surface in a pre-scribed pattern.We can assume at the end of the path that the height of each cylinderin C is much greater than the size of the surgeries required to passfrom Col v ( D ) to the double degeneration. Thus, at the endpoint of thepath, we can assume that each cylinder of C persists on the doubledegeneration.By the construction of the double degeneration described in Section6, the components of the double degeneration correspond bijectively tothe components of (cid:98) D c . The statement of Theorem 8.1 gives that allcomponents of (cid:98) D c contain cylinders of C , and hence all components ofthe double degeneration contain cylinders of C . (cid:3) Corollary 10.7.
The invariant subvariety M doubv containing the doubledegeneration of D is prime.Proof. This follows from Theorem 3.2, since if the component of theboundary containing the double degeneration were not prime, it wouldbe a product, and that would contradict the fact that the cylinders in C are generically parallel to each other and appear on each component. (cid:3) Lemma 10.8.
The double degeneration of D loses at least two genus. Proof.
If the core curves of D span a k -dimensional subspace in ho-mology, then the double degeneration loses at least k genus by Lemma6.14. So if k >
1, then the double degeneration of D loses at least twogenus.Hence we suppose k = 1, which means exactly that all cylinders in D are homologous. Recalling that D contains at least two cylinders,pick two distinct cylinders D , D ∈ D . Let the core curve of D i be γ i .By assumption, one of the two components of ( γ ∪ γ ) c does notintersect (cid:98) D c , since otherwise we would be in the single degenerationcase. Thus, there is a component that is covered by cylinders in D andtheir boundary saddle connections. Since all the cylinders in D arehomologous, this means that there are cylinders D, D (cid:48) ∈ D such thatthe top of D is glued entirely to the bottom of D (cid:48) .Since there are no marked points, D ∪ D (cid:48) contains a subsurface ofgenus at least 2. Since double-degenerating D collapses this subsurfacethe result follows. (cid:3) Lemma 10.6 gives that C persists on the double degeneration. Corol-lary 10.7 gives that M doubv is prime. Corollary 6.13 gives that M doubv doesn’t have any rel, and hence cannot have free marked points. Thisconcludes the proof of Theorem 10.1.11. Typical rank-preserving degenerations
Throughout this section (
X, ω ) will be a surface in an invariant sub-variety M and we will use the notation introduced in Section 4. Inparticular, if C is an equivalence class on ( X, ω ) and v ∈ Twist( C , M ),we will use the notation C v , t v , and Col v introduced in Section 4.It will be necessary to place a mild genericity assumption on ourdegenerations, using the following definition. Definition 11.1.
A cylinder degeneration of (
X, ω ) ∈ M defined by v ∈ Twist( C , M ) will be called typical if C is generic and all cylindersin C v have constant ratio of heights on all perturbations of ( X, ω ) in M .This definition is motivated by Lemma 6.5, which states that, fortypical degenerations, M v is codimension one and that all the saddleconnections in Col v ( C v ) are generically parallel to each other.The purpose of this section is to prove three results. The firstproduces typical degenerations, the second shows that such degen-erations are often rank-preserving, and the third shows that typicalrank-preserving degenerations do not create free marked points. IGH RANK INVARIANT SUBVARIETIES 59
Lemma 11.2.
For any generic equivalence class C on a surface ( X, ω ) in an invariant subvariety M , there exists v ∈ Twist( C , M ) whichdefines a typical cylinder degeneration.Proof. Assume without loss of generality that C is horizontal. For eachpair of cylinders in C , the subset of v ∈ Twist( C , M ) where these twocylinders have constant ratio of heights along the path ( X, ω ) + tv isdefined by a single linear equation, which is vacuous if the two cylindershave generically a constant ratio of heights and otherwise defines ahyperplane.Consider v ∈ Twist( C , M ) not contained in any of the hyperplanesjust described. By construction, for any deformation in a direction inTwist( C , M ), the cylinders in C v have constant ratio of height, sinceotherwise v would lie in one of the hyperplanes above.Proposition 4.20 with w = ω now implies that all the cylinders in C v have constant ratios of heights for any deformation in M .A final issue is that v might not define a cylinder degeneration, ifthe path ( X, ω ) + tv, ≤ t < t v does not diverge in the stratum.This however can be easily corrected as follows. First replace v by itsimaginary part. Then add to v a multiple cσ C of the standard shearin C , with c ∈ R chosen so that ( X, ω ) + t v cσ C has a vertical saddleconnection in some cylinder of C v . Both changes only affect the realpart of v , and since C is horizontal C v only depends on the imaginarypart, so C v remains the same. Because of the choice of c , now thereis a saddle connection in C v that has length going to zero along thecylinder degeneration path. (cid:3) Definition 11.3.
We will say that C is involved with rel if some vec-tor in ker( p ) ∩ T ( X,ω ) ( M ) evaluates non-trivially on a cross curve of acylinder in C . Lemma 11.4. If C is generic and involved with rel, then every typicalcylinder degeneration of C is rank-preserving.Proof. First suppose that not all pairs of cylinders in C have genericallyconstant ratio of height. In this case the definition of typical gives that C v (cid:54) = C , and so Theorem 5.1 gives that the degeneration is rank-preserving.Next suppose that all pairs of cylinders in C have generically con-stant ratio of height. In this case C v = C . Also v must be a multiple ofthe standard deformation of C , and hence there is a saddle connection s in C whose length goes to zero along the cylinder degeneration path.Let w ∈ ker( p ) ∩ T ( X,ω ) ( M ) evaluate non-trivially on a cross curve ofa cylinder in C . Because the cylinders in C have generically the same height, possibly after re-scaling we can assume that w evaluated on anycross curve of any cylinder in C gives the height of that cylinder. Since s must cross some of the curves in C , we can conclude that w ( s ) (cid:54) = 0.Hence Lemma 3.8 gives that the degeneration is rank-preserving. (cid:3) For the third main result of this section, it will be helpful to firstnote the following consequence of Theorem 4.2.
Corollary 11.5.
Let C be an equivalence class on a surface ( X, ω ) inan invariant subvariety M . If v ∈ Twist( C , M ) is such that deforming ( X, ω ) in the direction of any complex multiple of v does not changethe area, then p ( v ) = 0 .In particular, if γ and γ are the core curves of cylinders in C with equal circumference and γ ∗ − γ ∗ ∈ T ( X,ω ) ( M ) , then γ and γ arehomologous.Proof. Using Theorem 4.2, we can write v = cσ C + r , where c ∈ C and r is purely relative. Deforming in the direction of purely relativeclasses doesn’t change the area of a translation surface, but there is amultiple of σ C which corresponds to dilating (rather than shearing) C ,and this does change the area. This gives c = 0, establishing the mainclaim.The second claim follows immediately from the first. (cid:3) Lemma 11.6. If v ∈ Twist( C , M ) defines a typical rank-preservingdegeneration, and ( X, ω ) does not have marked points, then Col v ( X, ω ) does not have any free marked points.Proof. Suppose, in order to find a contradiction, that Col v ( X, ω ) doeshave a free marked point p .Without loss of generality, we assume that C is horizontal and thatthe only horizontal cylinders on ( X, ω ) are contained in C . Since ( X, ω )contains no marked points, the free point p on Col v ( X, ω ) must be theendpoint of a saddle connection s in Col v ( C v ).We will show now that s joins p to itself. If s joins p to anotherzero or marked point, then s cannot be generically parallel to anothersaddle connection, by definition of free marked point. Lemma 6.5 givesthat the saddle connections in Col v ( C v ) must be generically parallel toeach other, so we conclude Col v ( C v ) = { s } . Since M and M v have thesame rank, Theorem 5.1 gives that Col v ( C v ) is a strongly connecteddirected graph. But a single saddle connection joining two distinctpoints is not strongly connected. This proves our claim that s joins p to itself.Any saddle connection joining a marked point to itself is a corecurve of a cylinder on the corresponding surface without marked points. IGH RANK INVARIANT SUBVARIETIES 61
Figure 11.1.
The saddle connection s .Thus, as in Figure 11.1, there are two cylinders D and D on Col v ( X, ω ),both of which have an entire boundary component consisting of s ; theseare the cylinders above and below s . Suppose without loss of generalitythat D lies below s .Let C and C be the cylinders on ( X, ω ) with Col v ( C i ) = D i ; theseare simply the same cylinders viewed before the degeneration. Sublemma 11.7. C and C are homologous.Proof. Let γ i denote the core curve of D i . Deforming in the directionof γ ∗ − γ ∗ moves p while keeping the rest of the surface unchanged. Sosince p is a free marked point, γ ∗ − γ ∗ belongs to T Col v ( X,ω ) ( M v ).Notice that since the D i are horizontal, the C i are horizontal andhence contained in C . (In particular, this shows that C (cid:54) = C v .) Wewill continue to use γ i to denote the core curve of C i .Since γ ∗ − γ ∗ belongs to Twist( C , M ), Corollary 11.5 gives that γ and γ are homologous. (cid:3) Keeping in mind that C and C are homologous, let C be set ofcylinders in C v that lie above C and below C , so that the closure C covers the region above C and below C .Let D be the cylinder that lies directly above C . If D and C havethe same circumference, then there must be a marked point separatingthem, which gives a contradiction. If not, we have the following. Sublemma 11.8. If D and C do not have the same circumference,then there is oriented closed loop in C v consisting of segments thattravel along core curves of cylinders in C v , and vertical downwardsoriented segments.Proof. The assumption gives that there is a saddle connection on thebottom of D that does not border C . On the other side of this saddleconnection lies a cylinder D (cid:48) ∈ C . See Figure 11.2 for an example Figure 11.2.
The proof of Sublemma 11.8where D (cid:48) = D . Because C and C are homologous, the bottom of D (cid:48) is glued entirely to cylinders in C ∪ { C } .We can form a path by following the core curves of D , then goingvertically down into D (cid:48) , and then if necessary following the core curveof D (cid:48) , and then going into a cylinder D (cid:48)(cid:48) below that. Eventually, thispath must visit the same cylinder twice, and in this way we can obtaina closed loop as desired. See Figure 11.2 for an example where D (cid:48) = D . (cid:3) The imaginary part of the holonomy of such a loop is non-zero, con-tradicting Lemma 4.12. (cid:3)
Proof of Proposition 2.7
Suppose that M is an invariant subvariety of genus g surfaces with-out marked points, and assume that M has high rank, is not geminal,and has rel. Since M is not geminal, there is a generic equivalenceclass C of cylinders on a surface ( X, ω ) in M so that C cannot bepartitioned into free cylinders and pairs of twins.We will give the proof of Proposition 2.7 in two cases. In bothcases, we will use without further comment that a typical cylinderdegeneration of an equivalence class involved in rel • is rank-preserving by Lemma 11.4, • is connected by Lemmas 9.1 and 9.3, and • does not have free marked points by Lemma 11.6.Hence, it suffices to find such a degeneration that is not geminal.The following lemma addresses the easier case, when the twist spaceof C doesn’t contain the rel of M . IGH RANK INVARIANT SUBVARIETIES 63
Lemma 12.1. If Twist( C , M ) does not contain T ( X,ω ) ( M ) ∩ ker( p ) ,then Proposition 2.7 holds.Proof. By Lemma 7.10 (6), we can perturb (
X, ω ) in such a way that C remains horizontal and the surface becomes cylindrically stable. Itfollows from the definition of cylindrically stable that the twist spaceof ( X, ω ) then contains T ( X,ω ) ( M ) ∩ ker( p ), since the cylinder pre-serving space always contains T ( X,ω ) ( M ) ∩ ker( p ) by definition. SinceTwist( C , M ) does not contain all the rel there is an equivalence class D of horizontal cylinders that is disjoint from C and involved in rel.After perturbing again, we may assume that D is generic. UsingLemma 11.2, pick v ∈ Twist( D , M ) that defines a typical cylinderdegeneration.It suffices to show that M v is not geminal. Indeed, no cylinder inCol v ( C ) can be the twin of a cylinder not in Col v ( C ), since the standardtwist in C can also be performed in M v , as in Lemma 10.3. So, because C cannot be partitioned into free cylinders and pairs of twins, we getthe same statement for the equivalence class of Col v ( C ). (cid:3) In light of Lemma 12.1, we will suppose that Twist( C , M ) containsall the rel in T ( X,ω ) ( M ). So in particular, C is involved in rel. The restof this section gives the proof of Proposition 2.7 in this case, by assum-ing that that all typical degenerations of C are geminal and deriving acontradiction.Using Lemma 11.2, pick v ∈ Twist( C , M ) defining a typical cylinderdegeneration. By assumption, M v is geminal, so since C v is not all of C we can find a set C of either one or two cylinders in C \ C v suchthat Col v ( C ) is either a single free cylinder or a pair of twins.In particular, Twist( C , M ) is one-dimensional and, using the iden-tification of T Col v ( X,ω ) ( M v ) with a subspace of T ( X,ω ) ( M ), spanned bythe standard shear σ of Col v ( C ). Lemma 12.2.
There are two cylinders in C and they do not havegenerically equal heights.Proof. Suppose in order to derive a contradiction that the cylinders in C all have generically identical heights; this holds vacuously if C hasonly a single cylinder. For simplicity of notation assume additionallythat C contains a vertical saddle connection; this can be arranged byshearing the surface.Thus − iσ defines a typical cylinder degeneration, and hence M − iσ isgeminal. Hence Col − iσ ( C ) is partitioned into twins and free cylinders.Since Col − iσ ( C − C ) consists of cylinders isometric to those in C − C , it follows that C has a decomposition into twins and free cylinders,which is a contradiction. (cid:3) By replacing (
X, ω ) with a perturbation and using Lemma 12.2, wemay assume that the two cylinders in C have different heights. Wewill denote the two cylinders C and C , where C is shorter than C .Letting γ and γ denote their (consistently oriented) core curves, upto scaling, σ = γ ∗ + γ ∗ .Now that we have perturbed, C − iσ = { C } and so − iσ is typi-cal. Since C is not free, there is another cylinder C ∈ C suchthat { Col − iσ ( C j ) } j =2 is a pair of twins. Letting γ denote the (con-sistently oriented) core curve of C , we have that γ ∗ + γ ∗ also belongsto T ( X,ω ) ( M ). Lemma 12.3. C and C are homologous to each other.Proof. Note that γ ∗ − γ ∗ = ( γ ∗ + γ ∗ ) − ( γ ∗ + γ ∗ ) ∈ Twist( C , M ) . Since C and C become twins in a degeneration, they have the samecircumference. Since C and C become twins in a degeneration, theyalso have the same circumference. So Corollary 11.5 gives the result. (cid:3) For convenience, we will now assume that C and C contain verticalcross curves. This can be arranged for example by shearing the wholesurface so a cross curve of C becomes vertical, and then using a realmultiple of the shear γ ∗ + γ ∗ so that a cross curve of C becomesvertical. Hence, if w = i ( γ ∗ − γ ∗ ), then both w and − w define cylinderdegenerations. Since C w = { C } and C − w = { C } both contain only asingle cylinder, these cylinder degenerations are typical.We can thus observe that Theorems 2.8 and 2.2 imply that both M ± w are quadratic doubles. The idea of the remainder of the proofis to show that the holonomy involutions in these two degenerationsare in some sense incompatible. This incompatibility is expected since(the images of) C and C are exchanged in one of these degenerations,while (the images of) C and C are exchanged in the other.Recall that a cylinder is said to be half-simple if one of its bound-aries is a single saddle connection and the other consists of two saddleconnections of equal length. We will refer to these two boundaries asthe simple and half-simple boundaries of the cylinder respectively. Lemma 12.4.
In a high rank quadratic double, all generic cylindersare simple or half-simple.
IGH RANK INVARIANT SUBVARIETIES 65
Proof.
The proof follows from the following two straightforward conse-quences of foundational results.
Sublemma 12.5.
A quadratic double is high rank if and only if thecorresponding stratum of quadratic differentials has at least 6 odd orderzeros.Proof.
This follows from Riemann-Hurwitz formula together with theformula for the rank of a stratum of quadratic differentials recorded in [AWb , Lemma 4.2 ] . (cid:3) In [MZ08] , Masur and Zorich showed that the generic cylinders instrata of quadratic differentials other than Q ( − ) are one of five types.These cylinder types are reviewed in [AWb , Section 4.1 ] , where theywere named simple cylinders, simple envelopes, half-simple cylinders,complex envelopes, and complex cylinders. Sublemma 12.6.
Any stratum of quadratic differentials with at least6 odd order zeros does not have generic cylinders that are complexcylinders or complex envelopes.Proof.
This requires the work of Masur-Zorich [MZ08 , Theorems 1 and2 ] ; the exact consequence of their work we need is recalled in [AWc ,Theorem 4.8 (2) ] .By this result, the complement of a complex envelope is a connectedtranslation surface with boundary, and the complement of a complexcylinder is the union of two connected translation surfaces with bound-ary. Therefore, any odd order zeros must be on the boundary of thecomplex cylinder or complex envelope. This shows that there are atmost four odd order zeros in strata with these types of cylinders. (cid:3) A simple cylinder on a quadratic differential gives rise to a twin pairof simple cylinders on the holonomy double cover; a half-simple cylindergives rise to a twin pair of half simple cylinders; and a simple envelopegives rise to either a simple cylinder, or possibly a twin pair of simpleor half-simple cylinders if there are marked points. These possibilitiesare illustrated in [AWc , Figure 4.1 ] . Thus the lemma follows from thetwo sublemmas. (cid:3) Corollary 12.7. C i is either simple or half-simple.Proof. We show the statement for i = 1; the i = 2 , w ( C ) is a generic cylinder. Hence Lemma 12.4gives that Col w ( C ) is simple or half-simple. It follows that C is simpleor half simple, since Col w ( C ) has at least as many saddle connectionson each boundary component as C . (cid:3) Perturb once more so that (
X, ω ) is horizontally periodic, while pre-serving the property that C is generic. Assume too after perturbationthat any two horizontal cylinders that have identical heights in facthave generically identical heights.Assume without loss of generality that the bottom boundary of C is simple. Let D be the horizontal cylinder below C , so the singleboundary saddle connection that is the bottom of C appears in thetop of D .Notice that since ( X, ω ) has no marked points, D cannot have thesame length as C and hence D / ∈ { C , C } . In particular, Col ± w ( D )remains a cylinder, and the single saddle connection that is the bottomof Col ± w ( C ) appears in the top of Col ± w ( D ).Let J ± w denote the holonomy involution on Col ± w ( X, ω ). Since thedeformations specified by ± w fix the complement of { C , C } there arecylinders D ± such that Col ± w ( D ± ) = J ± w ( D ). Lemma 12.8. D + = D − and this cylinder is glued to the top boundaryof C and the top boundary of C .Proof. D + = D − since otherwise Col w ( D + ), Col w ( D − ), and Col w ( D )would be generically isometric, contradicting the fact that in a qua-dratic double it is possible to deform a cylinder and its image underthe holonomy involution, without changing the rest of the surface.Recall that C w = { C } . Since the single saddle connection whichis the bottom boundary of C is glued to the top boundary of D , thecorresponding statement is true for Col w ( C ) and Col w ( D ). Hence thetop of J w (Col w ( C )) = Col w ( C ) consists of a single saddle connection,which is glued to the bottom of J w (Col w ( D )) = Col w ( D + ).The top boundary of C cannot border C or C , since it consistsof a single saddle connection, and this would result in a marked point.Hence, keeping in mind that C w = { C } , the fact that the top ofCol w ( C ) is contained in the bottom of Col w ( D + ) implies that the topof C is contained in the bottom of D + .The same argument with − w instead of w shows that the top bound-ary of C is contained in the bottom boundary of D − . (cid:3) We now have a contradiction since D + = D − appears on the topboundary of both C and C , and yet these cylinders have homologouscore curves by Lemma 12.3. References [AEM17] Artur Avila, Alex Eskin, and Martin M¨oller,
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