Measure bound for translation surfaces with short saddle connections
MMeasure bound for translation surfaces with shortsaddle connections
Benjamin Dozier ∗ February 25, 2020
Abstract
We prove that any ergodic SL ( R )-invariant probability measureon a stratum of translation surfaces satisfies strong regularity : the mea-sure of the set of surfaces with two non-parallel saddle connections oflength at most (cid:15) , (cid:15) is O ( (cid:15) · (cid:15) ). We prove a more general theoremwhich works for any number of short saddle connections. The proofuses the multi-scale compactification of strata recently introduced byBainbridge-Chen-Gendron-Grushevsky-M¨oller and the algebraicity re-sult of Filip. Contents V with M
355 Local volume bound 386 Proof of Theorem 1.3 51 ∗ Department of Mathematics, Stony Brook University, [email protected] . a r X i v : . [ m a t h . D S ] F e b Introduction A translation surface is a pair ( M, ω ), where M is a Riemann surface, and ω is a holomorphic 1-form on M . This data determines a flat metric withsingular points at the zeros of ω . The collection of all translation surfaceswith the same genus and combinatorics of zeros form a stratum H , whichadmits a natural SL ( R )-action.Questions about the dynamics of the straight-line flow on individualsurfaces are intimately connected to dynamics of this SL ( R )-action on thespace of surfaces. The study of these dynamical systems has produced aflourishing field of research, bringing together techniques from Teichm¨ullertheory, ergodic theory, homogeneous dynamics, and algebraic geometry.Measures µ on H that are invariant under the SL ( R )-action govern thedynamics. This paper concerns the volume of sets of degenerating surfaceswith respect to any such (ergodic) invariant measure.A saddle connection s is a geodesic segment on a translation surface thatconnects two singular points (with no singular points in the interior). Wedenote its length by | s | . Even the (complex) projectivized stratum P H isnon-compact; a sequence of translation surfaces escapes every compact setiff there are saddle connections whose lengths go to zero. A corollary of ourmain result is the following estimate on the volume of the set of surfaceswith multiple short saddle connections. Theorem 1.1 (Strong regularity) . Let H be any stratum of translationsurfaces, and let µ be any ergodic SL ( R ) -invariant probability measure onthe unit area locus H ⊂ H . Let L (cid:15) ,(cid:15) := { X ∈ H : X has non-parallel saddle connections s , s , with | s | < (cid:15) , | s | < (cid:15) } . Then µ ( L (cid:15) ,(cid:15) ) = O ( (cid:15) · (cid:15) ) , where the implicit constant depends on µ . Our proof establishes a new paradigm for exploring flat geometry us-ing the recently constructed multi-scale compactification of strata (due toBainbridge-Chen-Gendron-Grushevsky-M¨oller [BCG + We call the above property of the measure µ strong regularity . A substan-tially weaker version of regularity was needed by Eskin-Kontsevich-Zorich2EKZ14] as a crucial technical assumption in their proof of a striking for-mula relating sums of Lyapunov exponents of the Teichm¨uller geodesic flowto Siegel-Veech constants. This weaker form was proven by Avila-Matheus-Yoccoz [AMY13] using an intricate hands-on argument. But the abovestronger theorem is the natural bound that one expects based on indepen-dence heuristics.Strong regularity has been proved for the Masur-Veech measure µ MV (whose support is all of H ) by Masur-Smillie [MS91, proof of Theorem 10.3],and for measures coming from rank 1 affine invariant manifolds by Nguyen[Ngu12]. The natural analogue of the above for the locus of surfaces witha single (cid:15) -short saddle connection is a volume bound of O ( (cid:15) ). This boundcan be easily derived, for all measures µ , from the Siegel-Veech formula. What we will actually prove is a generalization of Theorem 1.1 (Strongregularity) that works for any number of short saddle connections.In order to state the generalization, we work in the context of an affineinvariant submanifold M of H . We will give a brief review of key con-cepts; for more background, the reader is encouraged to consult the survey[Wri15b].An affine invariant manifold is defined to be an immersed suborbifoldthat is locally cut out by homogeneous linear equations in period coordi-nates with real coefficients. An affine invariant manifold M comes with an affine measure µ M supported on M . Locally, this measure equals Lebesguemeasure on the linear subspace T X M of period coordinates correspondingto M . There is also a finite measure µ M supported on the unit area locus M ⊂ M , defined by “coning”: µ M ( S ) := µ M ( { sX : X ∈ S, ≤ s ≤ } ) , for any measurable subset S ⊂ H . We will take this measure to be normal-ized to have total mass 1.The landmark result of Eskin-Mirzakhani [EM18] states that any ergodic SL ( R )-invariant probability measure on H is equal to µ M for some affineinvariant manifold M ; these affine measures are thus of central importance.Our generalization will be stated in terms of affine invariant manifolds, andthen in Section 1.4 we will use [EM18] to deduce Theorem 1.1 (Strong reg-ularity) from the general theorem.The next definition gives the appropriate analogue of non-parallel forany number of saddle connections. Recall that the linear structure on H
3s given locally near X by the relative cohomology H ( X, Σ; C ), where Σdenotes the set of zeros of the differential. Definition 1.2.
Saddle connections s , . . . , s k on a surface X ∈ M aresaid to be M -independent if their relative homology classes define linearlyindependent functionals (over C ) on the linear subspace T X M ⊂ T X H ∼ = H ( X, Σ; C ). We can now state the main theorem that we will prove.
Theorem 1.3.
Let
M ⊂ H be an affine invariant manifold, and let L M (cid:15) ,...,(cid:15) k := { X ∈ M : X has M -independent saddle connections s , . . . , s k , with | s i | ≤ (cid:15) i } . Then µ M (cid:0) L M (cid:15) ,...,(cid:15) k (cid:1) = O ( (cid:15) · · · (cid:15) k ) , where the implicit constant depends on M .Remark . The only place in the proof where we use that M is cut outby linear equations with real coefficients (rather than arbitrary complexcoefficients) is when we apply Filip’s theorem [Fil16] to get that M is analgebraic subvariety. So the theorem above also holds for any subvariety ofa stratum locally cut out by linear equations in period coordinates, withany complex coefficients. In this setting, the existence of a globally definedlinear measure µ supported on the locus is not immediate due to monodromyissues, but provided that one exists we can apply the theorem to it. We willuse this more general result in the proof of Theorem 1.5 below.Part of the motivation for proving the above theorem was as a test ofour understanding of the structure of affine invariant manifolds near infin-ity. Certain structural results about affine invariant manifolds are known, forinstance Wright’s Cylinder Deformation Theorem [Wri15a]. But a full clas-sification of affine invariant manifolds in all strata is still beyond reach. Themethods developed here, in particular the use of the multi-scale compactifi-cation to understand degenerations in affine invariant manifolds, should behelpful in this quest. This definition makes sense at points X in the smooth locus M ∗ ⊂ M since at thesepoints T X M is a single linear subspace. At points of M − M ∗ , T X M is not a single linearsubspace, but rather a finite union of linear subspaces; however µ M ( M − M ∗ ) = 0, so forthe theorem below, it does not matter how we define M -independence at such X . .3 Applications of Theorem 1.1 (Strong regularity) Gadre has proved several results about excursions of Teichm¨uller geodesicsinto the thin part of the moduli space, for geodesics chosen randomly ac-cording to some invariant measure µ [Gad17, Gad19]. These results wereconditional on strong regularity of the measure, so they can now be givenunconditional proofs.There may be future applications of the theorem to counting problemsfor pairs of saddle connections. For instance, the L -norm of the Siegel-Veechtransform of a compactly supported indicator function is related to pairs ofsaddle connections. Athreya-Cheung-Masur [ACM19] recently proved thatthis L -norm is finite for the case of the Masur-Veech measure µ MV ; theirproof uses strong regularity of µ MV as well as some other special facts aboutthis measure. In the future we plan to investigate extending the result toany µ using Theorem 1.1 (Strong regularity). This deduction involves applying Eskin-Mirzakhani [EM18], and then re-lating M -independence of two saddle connections to the property of beingnon-parallel. Proof of Theorem 1.1 (Strong regularity) given Theorem 1.3.
By [EM18, The-orem 1.4], µ = µ M for some affine invariant manifold M . Let M ∗ ⊂ M bethe smooth locus (i.e. where the immersed suborbifold does not intersectitself). We claim that M ∗ ∩ L (cid:15) ,(cid:15) ⊂ L M (cid:15) ,(cid:15) . (1)Consider some X ∈ M ∗ ∩ L (cid:15) ,(cid:15) , and let s , s be non-parallel saddle con-nections with | s i | ≤ (cid:15) i . To prove the claim it suffices to show that s , s are M -independent. Otherwise, since s and s both have non-zero length, wewould have s − αs = 0 for some α ∈ C , where we think of both sides as lin-ear functionals on T X M ⊂ H ( X, Σ; C ). Now since M is locally cut out byequations with real coefficients, the annihilator Ann( T X M ) in H ( X, Σ; C )is spanned by real homology classes. Since s − αs is in Ann( T X M ), it thenfollows that s − Re( α ) s ∈ Ann( T X M ). Since neither saddle connectioncan have zero holonomy vector, Re( α ) (cid:54) = 0, and then we see that s and s must be parallel on X , contradiction. Hence s , s are M -independent, andso we get (1). 5sing that µ M is supported on M ∗ , the inclusion (1), and Theorem 1.3,we get µ ( L (cid:15) ,(cid:15) ) = µ M ( M ∗ ∩ L (cid:15) ,(cid:15) ) ≤ µ M ( L M (cid:15) ,(cid:15) ) = O ( (cid:15) (cid:15) ) . (cid:4) k -differentials A k -differential on a Riemann surface, for k a positive integer, is a formthat can locally be written as f ( z )( dz ) k , where f is holomorphic; i.e. it is asection of the k -th power of the canonical line bundle. Theorem 1.5 (Finiteness of volume) . Let µ be the canonical linear measurewhose support is a whole stratum K of k -differentials. Then µ (supportedon the unit area locus and defined via coning, as above) is finite: µ ( K ) < ∞ . This was proven recently by Nguyen [Ngu19], who also defines the mea-sures µ above using the symplectic form on the absolute homology. Thetheorem also follows easily from the results in this paper as we now show. Proof.
For any holomorphic k -differential ( M, q ), there is a canonical con-struction that produces a pair ( ˆ
M , ω ) with a finite cover f : ˆ M → M suchthat ω is a holomorphic 1-form on ˆ M satisfying ω k = f ∗ q . Applying thisconstruction to all elements of K embeds K as some locus ˆ K in some stratum H of holomorphic 1-forms. This locus is locally cut out by linear equationsin period coordinates, where the equations have complex coefficients (in fact,the equations can be chosen to lie in Q ( ζ k ), where ζ k is a primitive k -th rootof unity). The measure µ becomes a measure supported on ˆ K that is linearin period coordinates. And ˆ K is algebraic since it can be defined in termsof algebraic conditions. So according to Remark 1.4, we can apply Theorem1.3 to ˆ K , and in particular we get µ ( K ) < ∞ . (cid:4) Here is the reason that the bound in Theorem 1.3 is natural. The affinemeasure µ M is defined as Lebesgue measure on a linear subspace of periodcoordinates. Locally, we should be able to pick a basis for period coordinatesthat contains the short saddle connections. The period of a saddle connec-tion of length at most (cid:15) i lies in a ball of area (cid:15) i in C . So each independent (cid:15) i -short saddle connection should lead to an (cid:15) i factor in the volume.6hen we complete our set of short saddle connections to a full basis forfunctionals on the subspace given by M , each new period will be either (i)bounded, or (ii) a cross curve of a cylinder of small circumference. Dealingwith type (i) periods is easy, since they just increase the implicit constantin the O ( · ). Type (ii) periods are potentially unbounded, but they still liein a region of bounded area once the circumference curve of the cylinder hasbeen fixed.There are two main issues in making the above heuristic into a proof.1. We need to find a finite system of period coordinate charts to do the“local” computation; since H (and M ) are non-compact, some of thesecharts must necessarily also be non-compact. Our method is to use the Moduli space of multi-scale differentials , a nice compactification P H ofthe (complex) projectivization P H recently introduced by Bainbridge-Chen-Gendron-Grushevsky-M¨oller [BCG + • In Section 2, we give background on the multi-scale compactification P H , which is a compactification of the projectivized stratum P H ; itplays a key role in the proof. This space admits a nice system ofanalytic coordinates defined in terms of plumbing and scaling. • In Section 3, for each boundary point ¯ X ∈ P H , we find a finite systemof semianalytic period coordinate charts that cover a neighborhoodof ¯ X (minus the boundary). They are cut out by conditions on theanalytic coordinates. A major challenge is showing that the periodmap on each such set is injective (which is needed for it to be chart). • In Section 4, we use Filip’s theorem on algebraicity of affine invariantmanifolds to show that our M intersects each chart above in finitelymany components. This implies that in each chart, M is a union offinitely many linear subspaces. • In Section 5, we prove the desired volume bound for M restrictedto each chart. This involves several lemmas that give estimates onperiods of surfaces in each such chart.7 Finally, in Section 6, we deduce the global volume bound for M fromthe bound in each chart using the compactness of P H (and the factthat the volume bound is trivial in the compact part of H ). I am very grateful to Alex Wright for suggesting the problem and for helpfulconversations and guidance. I also would like to thank Matt Bainbridgeand Sam Grushevsky for patiently explaining their work to me. I thankFrederik Benirschke, Dawei Chen, Martin M¨oller, and Jenya Sapir for usefuldiscussions.Finally, I gratefully acknowledge the support of the Fields Institute dur-ing the Thematic Program on Teichm¨uller Theory and its Connections toGeometry, Topology and Dynamics, where some of this work was done.
We will use the multi-scale compactification P H introduced in [BCG +
19] todefine a finite system of period coordinate charts for H . Several other com-pactifications of strata have been considered, namely the “What You See IsWhat You Get” compactification studied by Mirzakhani-Wright [MW17],and the Incidence Variety Compactification [BCG + − P H We recall the definition and properties of the multi-scale compactification.The complete definition is quite complicated, so we refer the reader to[BCG +
19] for full details.Let H ( µ ), µ = ( m , . . . , m n ), be a stratum of holomorphic 1-forms with n zeroes with vanishing orders given by the m i . Since µ will be fixed,we will use the shorthand H = H ( µ ). The space P H will be formed byattaching to P H certain multi-scale differentials ¯ X which consist of stableRiemann surfaces with meromorphic differentials on the components, plussome combinatorial data, up to a certain equivalence relation.Our primary interest is in H , but the natural object to compactify in acomplex analytic way is the complex projectivization P H . So we work withthis space, and then translate statements back to H .Let M g,n denote the Deligne-Mumford compactification of the modulispace of genus g curves with n marked points. An element M ∈ M g,n is astable, nodal Riemann surface. Associated to M is the dual graph Γ M : thevertices v correspond to the components M v of M , and for each node of theRiemann surface there is an edge connecting the vertices corresponding tothe two components (possibly the same) joined by the node.A multi-scale differential will come with the following additional struc-ture on the dual graph; it is related to the differentials on the components:1. Half-edges , each having a single vertex as endpoint. There will beone of these for each of the n integers in the type µ that specifiesthe stratum H = H ( µ ). The purpose of these is to record on whichcomponents the zeros of surfaces in H end up in the limit. Each islabeled by a positive integer which is the order of the correspondingzero.2. A (surjective) level function (cid:96) : V (Γ) → N := { , − , . . . , − N } , which determines a weak order on the vertices. This ordering will de-termine which of two given components on smooth translation surfaces9 e v e l Figure 1: Left - A stable Riemann surface with a twisted differential. Thiscomes from a degeneration of surfaces in the stratum H (2 , H (2 ,
2) have. Right - Thecorresponding enhanced level graph.10ear the multi-scale differential is larger. Edges that join componentsat the same level will be called horizontal edges , while those that joincomponents at different levels will be called vertical edges .3. An assignment of a positive integer b e to each vertical edge. The coneangle around the node will be the same on either side of the node andwill equal 2 πb e .A dual graph equipped with all this structure will be called an enhancedlevel graph of type µ .A twisted differential on M compatible with the enhanced level graph Γis a collection of meromorphic differentials η = { η v } , one for each irreduciblecomponent M v of M , that is consistent with Γ in a sense suggested above(see [BCG +
19] for full definition). The term “twisted” is meant to remindus that these are meromorphic differentials, and so are related to sectionsof twists of the canonical bundle. Several consistency conditions ensurethat the limit objects can be smoothed to surfaces in P H . One conditionis that at a vertical node, the cone angle of the differential above equalsthe cone angle of the differential below. This is equivalent to the orders ofthe two differentials on either sides summing to −
2. A more subtle part ofthe definition of compatibility is the Global Residue Condition, which is notalways forced by the residue theorem.We will call the union of components at level i , i.e. those correspondingto the vertices of (cid:96) − ( i ), the level subsurface at level i and denote it ¯ X ( i ) .We will denote the restriction of η to ¯ X ( i ) by η ( i ) . At times, by ¯ X ( i ) we willactually mean the pair ( ¯ X ( i ) , η ( i ) ).The final piece of data is a prong-matching . This is only needed whenthere is at least one pole of order k ≤ −
3. At the other side of the corre-sponding node, there is a zero of order − − k ≥
1. The cone angle aroundeach is 2 π ( − k − Definition 2.1 (Multi-scale differential) . A multi-scale differential ¯ X oftype µ is the data ( M, Γ , η, σ ) where M is a stable Riemann surface, Γ is anenhanced level structure of type µ on the dual graph of M , η is a twisteddifferential on M compatible with Γ, and σ is a prong-matching.Two multi-scale differentials are considered equivalent if they differ bythe action of the level rotation torus . This action comes from simultaneously11escaling all the η v at the same level by some element of C ∗ . However, movingaround a circle in C ∗ may change the prong-matching at nodes at that level.This subtlety means that instead of directly taking an action of copies of C ∗ , we instead consider the universal cover C of each copy. Keeping trackof the prong-matching is needed to get a smooth orbifold, but its presencedoes not play a major role in our proof. (Note that we include rescaling ofthe top level in the level rotation torus, since we wish to directly define theprojectivized space; the definition in [BCG +
19] does not include rescalingof the top level.)
Definition 2.2.
The (projectivized)
Moduli space of multi-scale differentials P H ( µ ) is the set of multi-scale differentials of type µ , modulo the equivalencerelation described above.The following is part of [BCG +
19, Theorem 1.2].
Theorem 2.3.
The set P H has a natural structure of a compact smoothcomplex orbifold. P H From the way it is constructed in [BCG + P H admits a nice system ofcomplex-analytic coordinates (in the orbifold sense). We now describe thesenear a boundary point ¯ X . There are two types of coordinates:(i) Moduli parameters s i . Changing these moves the surface parallel tothe boundary stratum.(ii) Smoothing parameters . Deforming a smoothing parameter away from0 will smooth certain nodes of ¯ X . These parameters fall into twocategories:(a) a scaling parameter t i for each level i below the top level (whichis level 0), and(b) a horizontal node parameter t for each horizontal node.We will describe how to construct a surface X specified by moduli andsmoothing parameters. There are several choices that need to be madeto define the coordinates; we will discuss these along the way, but theyshould really be made at the beginning of the construction. We will describeplumbing applied to single surface, but to get a complex orbifold, it is crucialthat this construction can be done in a holomorphically varying way forholomorphic families. This is subtle; it is dealt with carefully in [BCG + distinguished surface X ∈ H , not just a projective equivalenceclass in P H . In some of the lemmas we will use this as a reference surfaceto discuss sizes. We first describe the role of the moduli parameters s i . Note that ¯ X liesin a boundary stratum consisting of twisted differentials in P H that sharethe same combinatorial data Γ , σ (the enhanced level graph and prong-matching). This boundary stratum can be identified with a suborbifold ofa certain stratum of meromorphic differentials cut out by certain conditionson residues coming from the Global Residue Condition and the requirementthat residues match at pairs of simple poles. This space naturally has acomplex structure, and thus we can take a system of complex-analytic co-ordinates for (the projectivization of) the boundary stratum near ¯ X .A good choice is to use period coordinates. Strata of meromorphic differ-entials admit period coordinates given by the cohomology group H ( Y \ P, Z ; C )of the surface Y minus the poles P , relative to the set Z of zeros and markedpoints. Since the residues are given by periods of cycles that encircle punc-tures, all the residue conditions are linear in period coordinates. Hence weget coordinates for our boundary stratum by taking the projectivization ofthe relevant subspace of period coordinates for the meromorphic stratum.These will be the moduli parameters s i . We will sometimes translate toarrange that all s i are 0 at ¯ X . We now move onto the smoothing parameters, which are more complicated.To start off, we must pick a particular multi-scale differential ( M, Γ , η, σ )in the equivalence class of ¯ X (recall that P H is the set of equivalence classesof multi-scale differentials). In fact, we must pick such a representative forevery surface in a small neighborhood of the boundary stratum of ¯ X , andfurther, we do this in such a way that the differentials η v that compose η vary holomorphically. In other words, we are choosing a distinguishedholomorphic section of the projectivization map locally for each level. Infact, we will use a particular local section defined as follows. Stability impliesthe existence of a relative cycle γ at this level with non-zero period at ¯ X (otherwise, integrating the 1-form would give a branched covering to ˆ C , butthe number of preimages of ∞ , counted with multiplicity, would be too13mall). We then choose the section such that the period of γ is 1 for allnearby surfaces in the same boundary stratum. To perform plumbing,we find standard coordinates on ¯ X near its nodes. For each node we canfind complex-analytic coordinates on the Riemann surface near that nodesuch that the differential has the form z k dz if k ≥ rz dz if k = − (cid:0) z k + rz (cid:1) dz if k ≤ − , (2)where k is the order of vanishing of the differential at the node. The exis-tence of these coordinates is guaranteed by [BCG +
19, Theorem 4.1], wherethey further prove that standard coordinates are essentially unique and varyholomorphically as we vary ¯ X . When k ≥
0, the coordinates are unique upto multiplication by a ( k + 1)-st root unity. When k = −
1, we can rescaleby any complex number. When k ≤ −
2, the coordinates are unique afterspecifying a fixed point near the node to have some fixed coordinate.
We will first discuss the role of the scalingparameters t i . Such a parameter determines both the size of the level i subsurface relative to the level i + 1 subsurface and the way that the level i subsurface is plumbed into the higher levels. Poles of order at most , no residues. We will first describe thesimpler case in which all poles of ¯ X are of order 1 or 2, and all residuesat poles are zero (at ¯ X as well as at all nearby surfaces). Consider twolevel subsurfaces ¯ X ( i ) and ¯ X ( j ) of ¯ X , with i > j , and suppose that u and v are standard coordinates near two points of ¯ X ( i ) , ¯ X ( j ) , respectively, thatare joined together in a node (we will assume that the discs { u : | u | ≤ } and { v : | v | ≤ } are contained in the coordinate charts; otherwise weshould remove a somewhat larger disc in the next paragraph before doingthe plumbing). The orders of vanishing at the two points that get joinedwill sum to −
2. Let T = t i − t i − · · · t j . To smooth the node, at the level of Riemann surfaces, we perform classicalplumbing , as in Figure 2. That is, we first remove the small discs { u : | u | ≤ | T |} , { v : | v | ≤ | T |} from ¯ X ( i ) , ¯ X ( j ) , respectively. For remaining points inthe sets parameterized by u, v , we glue u to v whenever uv = T. For the differential on ¯ X ( i ) , we take t − t − · · · t i · η ( i ) , and on ¯ X ( j ) , we take t − t − · · · t j · η ( j ) . Next we choose standard coordinates (2) near the nodes (actually, it iscrucial that this choice be made at the very beginning of the construction,and that it can be made in a holomorphically varying way for families; seediscussion at beginning of Section 2.2). To get uniqueness, we must choose adistinguished point near the node, which we do holomorphically over surfacesnear ¯ X . Near the nodes, the differentials then look like t − t − · · · t i · du, and t − t − · · · t j · dvv . t − t − · · · t i · du = t − t − · · · t i · d ( T /v )= − t − t − · · · t i · t i − t i − · · · t j · dvv . (Because of the factor of −
1, on the lower component we modify the coordi-nates so that the differential is the negative of the standard form (2) above).Hence in this case we get a well-defined differential on the plumbed surface.
Poles of any order, no residues.
We now generalize to the case inwhich all pole orders are allowed, but we still assume that all residues arezero (at ¯ X as well as at all nearby surfaces). For each level i , we assign aninteger a i , taken to be the least common multiple of the b k associated to allnodes that join a component at level greater than i to a component at level i or smaller, where 2 πb k is the cone angle around the node.For the differential on ¯ X ( i ) , we take t a − − · · · t a i i · η ( i ) , and on ¯ X ( j ) , we take t a − − · · · t a j j · η ( j ) . At a node with cone angle 2 πb , we choose standard coordinates (2) (thischoice is actually made at the very beginning of the construction). Thecoordinates are not quite unique, and the prong-matching σ gives a conditionon how the coordinates at the two sides of the nodes should relate.So in these standard coordinates near the node, the differential on ¯ X ( j ) has the form t a − − · · · t a i i · u b − du, and near the node on ¯ X ( j ) the differential has the form t a − − · · · t a j j · dvv b +1 . We perform plumbing as in the previous case, gluing together points u, v whenever uv = T , but now we use T = t a i − /bi − · · · t a j /bj . −
1) at points that are glued together: t a − − · · · t a i i · u b − du = t a − − · · · t a i i · ( T /v ) b − d ( T /v )= − t a − − · · · t a i i · ( T /v ) b − · T · dvv = − t a − − · · · t a i i · t a i − i − · · · t a j j · dvv b +1 . Modification differentials to account for residues.
In the casewhen the residue r (at a node on the lower level subsurface ¯ X ( j ) ) is non-zero, we have to work to create a residue on the upper level subsurface ¯ X ( i ) so that the differentials will match when we do the plumbing. The solutionis to create a modification differential ξ on the underlying Riemann surfaceof ¯ X ( i ) that has a simple pole at the node with residue r . The existence ofholomorphically varying modification differentials with the required proper-ties is furnished by [BCG +
19, Proposition 11.3] (or [BCG +
18, Lemma 4.6]).However, the modification differential will not be uniquely specified, so wemust make a choice of one at the beginning of the construction of coordi-nates. In our discussion here, we will focus on the modification differentialthat comes from residues on ¯ X ( j ) . In general there will be several modifi-cation differentials coming from residues at different levels and these shouldall be added to the higher level differential.So we consider the differential u b − du + t a i − i − · · · t a j j ξ on ¯ X ( i ) . Now[BCG +
19, Theorem 4.2 and Theorem 4.3] (or [BCG +
18, Theorem 4.3]) givesthat by performing a change of coordinates (depending holomorphically onthe t k , as well as on the moduli parameters) on a fixed annulus near 0, wecan assume that this differential has the form (cid:16) u b − + t a i − i − · · · t a j j ru (cid:17) du. On the disk bounded by the inner circle of this annulus, we change thedifferential to have the form above.Then upon rescaling, the differential on the upper component ¯ X ( i ) , nearthe node, has the form t a − − · · · t a i i (cid:16) u b − + t a i − i − · · · t a j j ru (cid:17) du, and near the node on ¯ X ( j ) , in standard coordinates (2), and after rescaling,the differential has the form t a − − · · · t a j j (cid:18) v b +1 + rv (cid:19) dv.
17e glue together points with uv = T = t a i − /bi − · · · t a j /bj , as in the casewithout residues. A short calculation similar to the one in the previous caseshows that the two differentials above match up at the points glued together. Merging zeros.
The modification differential solves the residue match-ing issue but introduces a new problem: it may split up some of the higherorder zeros of the original differential. So afterwards, we must merge thezeros that have been separated. We can do this in a local manner using[BCG +
19, Theorem 4.2] (or [BCG +
18, Lemma 4.7]).
Thecase when we have a node joining two points that lie on the same level sub-surface ¯ X ( i ) is similar to the higher order pole case, but we don’t have toworry about modification differentials. On either side of the node are simplepoles. On nearby surfaces, this node will become a degenerating cylinder.We find standard coordinates u, v given by (2) on each side of the node. Theresidues at the two simple poles are forced to be negatives of one another, bythe definition of multi-scale differential. So in these coordinates, on eitherside of the two nodes the differential has the form t a − − · · · t a i i · ru du, and t a − − · · · t a i i · − rv dv, respectively. On the level of Riemann surfaces, we do classical plumbingas above, gluing u to v whenever uv = t , where t is the horizontal nodeparameter. A short calculation very similar to the ones done in the case ofhigher order poles gives that the two differentials match up at points thatare glued together. Hence we get a well-defined differential on the plumbedsurface. In this section we will work with a fixed boundary point ¯ X ∈ P H .We let p : H → P H be the natural projectivization map (we intentionally take the target to bethe compactification so that we can pull back neighborhoods of a boundarypoint). For a subset S ⊂ H , we define P S := p ( S ).18ur goal is to define a finite system of semianalytic period coordinatecharts { V k } in H , each invariant under C ∗ -scaling, and such that { P V k } cover U ∩ P H for some neighborhood U of ¯ X in P H . In each such periodcoordinate chart, an affine invariant manifold M will be given by a finiteunion of linear subspaces. Hence, equipped with this system of finitely manyperiod coordinate charts, we can understand M in terms of finitely manylinear spaces.Each P V k will be “multi-sector” i.e. a product of small sectors withrespect to analytic coordinates near ¯ X . V k Choose a system of analytic coordinates near the boundary point ¯ X , as inSection 2.2, where t are the smoothing parameters, and s are the moduliparameters (translated so that the value is 0 at the ¯ X ). Each P V k (whichdepend on this choice of coordinates) will then be defined by conditions asfollows:(i) Restrict 0 < | s | < (cid:15) for each moduli parameter s , for (cid:15) > (cid:15) will need to be chosen yet smaller.)(ii) Restrict 0 < | t | < (cid:15) for each smoothing parameter t .(iii) If t is a horizontal node parameter, then we consider a restriction ofthe form arg t ∈ ( α, α + π/ . We choose finitely many α so that the union of the intervals above forthese α cover the full circle of directions. Each P V k will correspondto an interval given by one choice of α . (We need such conditions toensure that V k admits an injective period map; see Example 3.5.)(iv) Suppose t i is the scaling parameter for level i . We have defined anassociated integer a i in Section 2.2.2.2 related to higher order poles.We can find connected interval conditions on the arg t i that imply thatarg t a i i satisfies a condition of the form in (iii) above. With finitelymany such interval conditions, we cover all possibilites for t i . Each19 V k will correspond to one of these interval conditions for each such t i .We then take V k := p − ( P V k ). Remark . In the above we were assuming that ¯ X was not an orbifoldpoint. If it is an orbifold point, we define sets in the local manifold cover asabove, and then get the P V k by pushing down to P H . Example . Consider the example of a point ¯ X in P H (3 ,
1) which has twolevels connected by a single node, where the lower level piece is genus 1,with one pole of order 3 and one zero of order 3, and the upper level piece isgenus 2 with two zeroes of order 1, one of which lies at the node. See Figure3. There are moduli parameters s , . . . , s n that vary the bottom the topand bottom pieces within their strata. There is one scaling parameter t − .We have that a − = 2, since there is only one pole, and its order is 3 (recallthat a − is the least common multiple over all the poles of the associatedinteger b k , which is defined so that the cone angle at that pole is 2 πb k ). For k = 0 , . . . ,
15, define intervals: S k +1 = (cid:16) k π , ( k + 1) π (cid:17) , (3) S k +2 = (cid:16) . k π , . k + 1) π (cid:17) . (4)The sets V k in the definition can then be taken to be V k = { X : | s | , . . . , | s j | < (cid:15), | t − | < (cid:15), arg t − ∈ S k } , for k = 1 , . . . , V In a real-analytic manifold, a semianalytic set is a subset that is locally cutout by real-analytic equalities and inequalities. In a real-analytic orbifold, asemianalytic set is a subset such that its preimage under some real-analyticorbifold chart is a semianalytic subset of R n . See [Kan11] for a completedefinition. Lemma 3.3.
Each P V k is simply connected and semianalytic, and (cid:91) k P V k = U ∩ P H , where U is the neighborhood of ¯ X given by restricting | s | < (cid:15) for each moduliparameter, and | t | < (cid:15) for each smoothing parameter. P H (3 , Proof.
Because of the conditions on angles, each P V k is a product of convexsubsets of C , and is hence itself convex, hence contractible, and in particularsimply connected.The P V k are semianalytic because each of the defining conditions can beexpressed as an inequality on the real or imaginary part of a locally definedanalytic coordinate function (the arg expression in (iii) and (iv) is not awell-defined analytic function, but equivalent conditions can be expressed interms of ratios of real and imaginary parts).The last claim about the union of the P V k follows immediately from theirdefinition. (cid:4) Definition 3.4 (Period coordinate charts) . A connected, open subset Q ⊂H is said to be a period coordinate chart if it admits an injective map to C n that is locally linear (the stratum locally has a linear structure comingfrom local period coordinates; near orbifold points one should work in anappropriate cover).In particular, any sufficiently small neighborhood of a point X ∈ H willbe a period coordinate chart. However, it is not necessarily true that for asmall neighborhood U of a boundary point ¯ X in P H , the set p − ( U ) ∩ H is a period coordinate chart. One issue is that the set might not be simplyconnected, in which case it is not possible to consistently choose a basis ofrelative homology to take periods of. Example . Even on a simply connected subset of p − ( U ) ∩H ,the period coordinate map might not be injective. To see this, consider a21oundary point with two levels, connected by a single node, with a pole oforder 3 below (e.g. the surface in Figure 3). Let X (cid:48) , X (cid:48)(cid:48) be two surfacesnear this boundary point whose coordinates are identical, except that forthe scaling parameters t (cid:48)− , t (cid:48)(cid:48)− , we have t (cid:48)− = − t (cid:48)(cid:48)− . All periods of relativecycles coming from the top surface are the same for X (cid:48) , X (cid:48)(cid:48) . Since the bottomdifferentials are multiplied by ( t (cid:48)− ) = ( t (cid:48)(cid:48)− ) , the periods of cycles from thebottom surface are also all equal. Since we can form a basis out of suchcycles that do not cross the nodes, we have that the all periods for X (cid:48) , X (cid:48)(cid:48) are equal. On the other hand, the two surfaces must in fact be different,since they have different analytic coordinates (this also can be seen directlyfrom the flat pictures).Instead we use the sets V defined in Section 3.1. Lemma 3.6.
Each set V = V k ⊂ H from Section 3.1 is a period coordi-nate chart in the above sense, provided that the (cid:15) in the defintion of V issufficiently small (see Section 3.1, item (i)). The proof of this lemma is somewhat involved; the necessary tools aredeveloped in Section 3.3, and then proof is completed in Section 3.4.
Our first goal is estimate periods of various cycles on surfaces in V . Since P V is simply connected (Lemma 3.3), if we pick a relative cycle α representingan element of H ( X, Σ; Z ) on some surface X ∈ V , we can consistentlytransport it to a cycle on all surfaces in V . We will study how the period ofthis cycle depends on the analytic coordinates describing X .To study the period of the cycle α on smooth surfaces, we will firstintroduce the perturbed period , which comes from taking the period of thepart of α that comes from the highest level subsurface that it interacts with.This will be easier to understand than the full period, since it will give awell-defined holomorphic function in a full neighborhood of ¯ X , including atthe boundary. Our perturbed periods are a special case of the constructionin [BCG +
19, Section 11].
Definition 3.7.
Given a relative cycle γ on the level subsurface ¯ X ( (cid:96) ) , theperturbed period is the holomorphic function γ pert : U → C , from a small neighborhood U of ¯ X in P H , defined as follows. In the plumbingconstruction of Section 2.2, after the various choices of coordinates, but22igure 4: Construction of the perturbed period corresponding to a relativecycle γ on a piece of the limit surface.before doing the plumbing, we truncate γ by taking each representativecurve that ends at a node and replacing it by a curve that ends at the pointwith coordinate p (some fixed small complex number) in the coordinateschosen about that node. Then γ pert ( X ) is defined as the period over thistruncated curve, before doing any rescaling. See Figure 4.Suppose γ is a relative cycle on some level subsurface ¯ X ( (cid:96) ) of ¯ X . If wewish to consider this as a class on smooth surfaces in V , we can extend γ “downwards” as follows. Represent γ by curves, and extend each curvegoing toward a vertical node downwards until a locally minimal component(in the level graph) is reached. Such a component must have a zero of thedifferential, so a relative homology class can be produced. See, for example,Figure 5. This produces cycles ˆ γ on surfaces in X . Note that ˆ γ will notcross any degenerating cylinders (coming from horizontal nodes).In the next lemma, we estimate the period of γ in terms of the perturbedperiod part, the part that crosses the plumbing region, and the lower levelpart. We will focus on how this period depends on moduli parameters for¯ X ( (cid:96) ) and the scaling parameter t (cid:96) − . The expression in the lemma hassome non-explicit terms, but each of these either extends to a holomorphicfunction on a full neighborhood of ¯ X (and so the function, as well as itsderivative, will enjoy good boundedness properties), or only depends onlower level moduli or scaling parameters. Lemma 3.8.
Let ˆ γ be a cycle on smooth surfaces in V obtained by starting ith a relative homology class γ on ¯ X ( (cid:96) ) and extending downwards (withoutcrossing degenerating cylinders). Let t := t a (cid:96) − (cid:96) − . Then ˆ γ ( X ) = t a − − · · · t a (cid:96) (cid:96) (cid:0) γ pert ( X ) + c + tf ( X ) + ( t log t ) g ( X ) + th ( X ) (cid:1) , for X ∈ V , where: • c is a constant (which we can take small, if we choose the p in thedefinition of perturbed periods to be small), • γ pert is the perturbed period coordinate, • f, g are functions that extend to holomorphic functions in a full neigh-borhood of ¯ X , • h is a function of X , but it only depends on X through moduli param-eters at level lower than (cid:96) , and scaling parameters t k with k < (cid:96) − .Also, h ( X ) is bounded above in absolute value as X ranges over V (wedo not require that h gives a holomorphic function on a full neighbor-hood of ¯ X ).Proof. We begin by expressing ˆ γ as the union of three disjoint parts:(i) the part that lies on ¯ X ( (cid:96) ) and away from the plumbing regions(ii) the part that lies in the plumbing regions(iii) the part that lies on level subsurfaces at level j < (cid:96) and away from theplumbing regions.For (i), by the definition of the perturbed period γ pert above, the contri-bution is equal to t a − − · · · t a (cid:96) (cid:96) γ pert ( X ).For (ii), we will analyze each such plumbing region which ˆ γ crosses sep-arately, and then we can add all the expressions together. Each such regioncorresponds to a node joining level i to level j , where (cid:96) ≥ i > j . Thecontribution to ˆ γ ( X ) is given by t a − − · · · t a i i (cid:90) pT (cid:16) u b − + T b ru (cid:17) du, where T = t a i − /bi − · · · t a j /bj , and r (which comes from the modification differ-ential) extends to a holomorphic function in a full neighborhood of ¯ X . Weclaim that such an integral can be written in the form c + tf ( X ) + ( t log t ) g ( X ) + th ( X ) .
24o prove this, first note that u b − term in the integrand gives a c + tf ( X )contribution, where f is just some constant multiple of a power of t .Now we split up into cases based on i, j to analyze the contribution ofthe T b ru term in the integrand. • When i = (cid:96), j = (cid:96) −
1, we have T = t a (cid:96) − /b(cid:96) − , and the contribution is (cid:90) pT T b ru du = T b r (log p − log T ) = tr log p − ( r/b ) t log t, which becomes part of the tf ( X ) + ( t log t ) g ( X ) term. • When i < (cid:96) , we do a computation similar to the one immediatelyabove, and we ultimately get a contribution of the form tf ( X )+ th ( X ).Here h is a product of powers of various t k and a term of the form τ log τ , where τ is also a product of powers of t k . Since τ log τ → τ →
0, this h varies in a bounded away over V . • When i = (cid:96) , j < (cid:96) −
1, we get a contribution of the form tf ( X ) + ( t log t ) g ( X ) + th ( X ) . To see this, we split up the log( t a (cid:96) − /b(cid:96) − · · · t a j /bj ) in the expression weget by evaluating the integral intolog( t a (cid:96) − /b(cid:96) − ) + log( t a (cid:96) − /b(cid:96) − · · · t a j /bj ) . The first term behaves as in the i = (cid:96), j = (cid:96) − i < (cid:96) case.For pieces of type (iii), the period has the form t a − − · · · t a (cid:96) (cid:96) t a (cid:96) − (cid:96) − h ( X ),where h depends only parameters at level lower than (cid:96) . This h varies in abounded way as X varies over V , since the moduli and scaling parameterson which it depends are bounded.Combining the estimates for the pieces in (i), (ii), (iii) gives the desiredestimate. (cid:4) Proof of Lemma 3.6.
Since P V is simply connected (Lemma 3.3), we canchoose a basis for relative homology at some surface in V and then consis-tently transport it to a basis at all other surfaces in V . Taking the periodsof these classes gives the map φ : V → C n , which is clearly locally linear.25t remains to show that φ is injective, which is rather involved, sinceit involves carefully estimating periods of cycles, including those that crossbetween different levels. The reader is encouraged to first consider the proofin the case of a boundary point ¯ X of the form given by Figure 5, wherethere are just two levels.Suppose for the sake of contradiction that φ ( X (cid:48) ) = φ ( X (cid:48)(cid:48) ), with X (cid:48) , X (cid:48)(cid:48) ∈ V and X (cid:48) (cid:54) = X (cid:48)(cid:48) . First suppose that the projective classes [ X (cid:48) ] , [ X (cid:48)(cid:48) ] areequal. We have that X (cid:48) = c (cid:48) X and X (cid:48)(cid:48) = c (cid:48)(cid:48) X , where X is the distinguishedrepresentative of the projective class produced by the plumbing construction(as discussed in Section 2.2), and c (cid:48) (cid:54) = c (cid:48)(cid:48) . Thus φ ( X (cid:48) ) = c (cid:48) φ ( X ) (cid:54) = c (cid:48)(cid:48) φ ( X ) = φ ( X (cid:48)(cid:48) ), contradiction.So we assume for the rest of the argument that [ X (cid:48) ] (cid:54) = [ X (cid:48)(cid:48) ], and we willprove that [ φ ( X (cid:48) )] (cid:54) = [ φ ( X (cid:48)(cid:48) )]. Since every relative homology class is a linearcombination of the classes defining φ , it is sufficient to find classes γ , γ suchthat γ ( X (cid:48) ) /γ ( X (cid:48) ) (cid:54) = γ ( X (cid:48)(cid:48) ) /γ ( X (cid:48)(cid:48) ). To prove this statement about ratios,it clearly suffices to assume that X (cid:48) , X (cid:48)(cid:48) are the distinguished representativesproduced in Section 2. Since the projective classes are different, the analyticcoordinates for X (cid:48) , X (cid:48)(cid:48) must be different.Our first task is to determine at which levels to look for these classes γ , γ , in terms of the analytic coordinates for X (cid:48) , X (cid:48)(cid:48) . To this end wewill define the effective level (cid:96) of each moduli, horizontal node, and scalingparameter (WARNING: this is somewhat different than the notion of levelthat will be used for subsurfaces and curves in Section 5). The motivationfor the definition is that if two surfaces agree for all analytic coordinates ateffective levels (cid:96) and lower, then ratios of periods that interact only withsubsurfaces at level (cid:96) or lower should also be equal. Here is the definition:(i) For a moduli parameter s that deforms the level subsurface ¯ X ( i ) , theeffective level is i .(ii) For a scaling parameter t i , the effective level is i + 1. (The reason wetake the effective level higher than i is that t i does not affect ratios ofperiods at levels i and below.)(iii) For a horizontal node parameter t , the effective level is i + , where i is the level subsurface on which the pair of simple poles correspondingto the horizontal node lie.Consider all parameters that take different values for X (cid:48) , X (cid:48)(cid:48) ; among theeffective levels of these, let (cid:96) be the lowest.We now split into three cases based on the value of (cid:96) . In the first twocases (in which (cid:96) is an integer), we begin by choosing cycles γ , . . . , γ n in26he relative homology of ¯ X ( (cid:96) ) that give a basis for the space of functionalson the subspace of period coordinates cut out by the residue conditions,as in Section 2.2.1. See Figure 5. The periods of these give the moduliparameters corresponding to ¯ X ( (cid:96) ) . We take γ to be the cycle that hasperiod 1 on all surfaces in the boundary stratum (recall from Section 2.2.2that we are choosing the holomorphic section of the projectivization map bynormalizing a particular cycle to have period 1).Case 1: (cid:96) = − N (i.e. the surfaces differ at the lowest effective level) This case is straightforward because periods on the bottom level subsur-face have a simple expression in terms of the analytic coordinates.By the above definition of effective level, there must be some moduliparameter s for the lowest level subsurface ¯ X ( − N ) that has two differentvalues s (cid:48) (cid:54) = s (cid:48)(cid:48) for X (cid:48) , X (cid:48)(cid:48) , respectively. The parameter s is a period of somerelative homology class on ¯ X ( − N ) , which we can assume is γ . Now since γ is on the lowest level, it extends uniquely on all smooth surfaces near ¯ X to a relative homology class which does not cross any of the curves that arepinched at ¯ X .Now, the period of γ differs between X (cid:48) , X (cid:48)(cid:48) , while the period of γ doesnot; hence the ratio of the two periods differs. In fact we have the followingexplicit expressions for the periods on a smooth surface X with parameter s : γ ( X ) = t a − − · · · t a (cid:96) (cid:96) · s,γ ( X ) = t a − − · · · t a (cid:96) (cid:96) . Hence γ ( X (cid:48) ) γ ( X (cid:48) ) = ( t (cid:48)− ) a − · · · ( t (cid:48) (cid:96) ) a (cid:96) · s (cid:48) ( t (cid:48)− ) a − · · · ( t (cid:48) (cid:96) ) a (cid:96) = s (cid:48) (cid:54) = s (cid:48)(cid:48) = ( t (cid:48)(cid:48)− ) a − · · · ( t (cid:48)(cid:48) (cid:96) ) a (cid:96) · s (cid:48)(cid:48) ( t (cid:48)(cid:48)− ) a − · · · ( t (cid:48)(cid:48) (cid:96) ) a (cid:96) = γ ( X (cid:48)(cid:48) ) γ ( X (cid:48)(cid:48) ) , and we are done since we have found a pair of periods with different ratiosfor X (cid:48) , X (cid:48)(cid:48) .Case 2: (cid:96) > − N , and (cid:96) is an integer (i.e. the surfaces are the same at thebottom level, and the lowest effective level where they differ does not comefrom a degenerating cylinder) This case is rather involved since we have to understand how periodscrossing between levels depend on the analytic coordinates (logarithmicterms appear) and how ratios of such periods behave.Recall that we have chosen γ , . . . , γ n in the relative homology of ¯ X ( (cid:96) ) that give a basis for the space of functionals on the subspace of period27igure 5: Extension of cycles on limit surface ¯ X to a smooth surface X inthe multi-sector V . Case 2 applies if the moduli parameters for the bottomcomponents of X (cid:48) , X (cid:48)(cid:48) are all the same. We are then forced to consider thecycles ˆ γ , ˆ γ crossing between levels.coordinates cut out by the residue conditions; γ is the cycle whose periodis 1 on all surfaces in the boundary stratum.We let β be the relative homology class on ¯ X ( (cid:96) − whose period is exactly1 at ¯ X and on nearby surfaces in the boundary stratum.Now the γ i classes, as well as the β class, can be extended downwardson the smooth surfaces in V to give relative homology classes ˆ γ i and ˆ β . Werequire that these extended cycles do not cross any degenerating cylinders(coming from horizontal nodes). This is the same procedure described inthe setup before Lemma 3.8.We will show that either ˆ γ i ( X (cid:48) ) / ˆ γ ( X (cid:48) ) (cid:54) = ˆ γ i ( X (cid:48)(cid:48) ) / ˆ γ ( X (cid:48)(cid:48) ), for some i ,or ˆ β ( X (cid:48) ) / ˆ γ ( X (cid:48) ) (cid:54) = ˆ β ( X (cid:48)(cid:48) ) / ˆ γ ( X (cid:48)(cid:48) ). To do this, we need to compute theperiods of ˆ γ i and ˆ β with respect to the analytic parameters s i for i > t := t a (cid:96) − (cid:96) − . Here s i is the moduli parameter corresponding to the period of γ i at the boundary.By our assumption on effective levels, we know that( s (cid:48) , . . . , s (cid:48) n , t (cid:48) ) (cid:54) = ( s (cid:48)(cid:48) , . . . , s (cid:48)(cid:48) n , t (cid:48)(cid:48) ) , (note that if t (cid:48) = t (cid:48)(cid:48) , then t (cid:48) (cid:96) − = t (cid:48)(cid:48) (cid:96) − , since our surfaces are in the multi-sector V and so satisfy conditions given by (iv) in Section 3.1).28e will define a map P on the subset of P V given by S := { X ∈ P V : for every analytic coordinate a with effective level < (cid:96), the value of a at X agrees with the common value for [ X (cid:48) ] , [ X (cid:48)(cid:48) ] } . Define P by P = ( P , . . . , P n ) : S → C n ( s , . . . , s n , t ) (cid:55)→ (cid:32) ˆ γ ( X )ˆ γ ( X ) , . . . , ˆ γ n ( X )ˆ γ ( X ) , ˆ β ( X )ˆ γ ( X ) (cid:33) , where X is the distinguished representative produced by the plumbing con-struction for the projective equivalence class with coordinates ( s , . . . , s n , t ).We use the common value for coordinates at lower levels; coordinates athigher levels will only affect the scaling of periods on the right, so the ratiosare still well-defined. Since each P k is ratio of periods, to finish the proof inthis case, it suffices to find some k such that P k ([ X (cid:48) ]) (cid:54) = P k ([ X (cid:48)(cid:48) ]). So it issufficient to show that P is injective. In our proof we will need to make theset V sufficiently small, by choosing the (cid:15) in Section 3.1 sufficiently small.It is clearly crucial that this choice depends only on ¯ X , not on X (cid:48) , X (cid:48)(cid:48) (or S ). Our strategy is to get injectivity of P by estimating the derivative andthen integrating. We will work with the distinguished representative X ofthe projective equivalence class throughout.First we write explicit expressions for ˆ γ i ( X ) , ˆ β ( X ), which are crucial forthe calculations needed for the rest of the proof.By Lemma 3.8,ˆ γ ( X ) = t a − − · · · t a (cid:96) (cid:96) (cid:0) γ pert ( X ) + c + tf ( X ) + ( t log t ) g ( X ) + th ( X ) (cid:1) . The expression for the period of ˆ β is simpler, since this class only inter-acts with subsurfaces at levels (cid:96) − β ( X ) = t a − − · · · t a (cid:96) (cid:96) · ( tb ) , where b is some constant (for fixed S ) that tends to 1 as [ X ] → ¯ X (i.e. as S gets closer to ¯ X ).We now aim to estimate the partial derivatives of the components of P . We will use the usual o ( · ) and O ( · ) notations to denote constants thatare asymptotically smaller than, respectively, smaller than or equal to, theargument (in the limit as all parameters tend towards their values at the29oundary point ¯ X ). The implied constants will depend only on ¯ X (not on S ). First we need estimates on how the perturbed periods vary. At theboundary, the perturbed period is just a fixed translate of the correspondingrelative period, which implies that ∂γ pert i ∂s i (cid:12)(cid:12)(cid:12) ¯ X = 1 , for i (cid:54) = 1, and ∂γ pert i ∂s j (cid:12)(cid:12)(cid:12) ¯ X = 0 , for i (cid:54) = j . Since γ pert i is a holomorphic function, it follows that at any X forwhich [ X ] ∈ S . ∂γ pert i ∂s i = 1 + o (1) , for i (cid:54) = 1 ,∂γ pert i ∂s j = o (1) , for i (cid:54) = j. Also, since γ pert i is holomorphic, we have ∂γ pert i ∂t = O (1) . Now we proceed to estimate the partial derivatives of components of P ,evaluated at a point X for which [ X ] ∈ S : • For i (cid:54) = 1: ∂∂s i ˆ γ i ˆ γ = ∂∂s i γ pert i + c i + tf i + ( t log t ) g i + th i γ pert1 + c + tf + ( t log t ) g + th = (1 + o (1)) (cid:16) γ pert1 + c + tf + ( t log t ) g + th (cid:17) ( γ pert1 + c + tf + ( t log t ) g + th ) − o (1) (cid:16) γ pert i + c i + tf i + ( t log t ) g i + th i (cid:17) ( γ pert1 + c + tf + ( t log t ) g + th ) = d + o (1) , where d = γ pert1 ( ¯ X )+ c . Note that we can take γ pert1 ( ¯ X ) + c to be non-zero, since the disc removed for perturbed period coordinates is small,so γ pert1 ( ¯ X ) is close to γ ( ¯ X ) = 1, and c can be taken to be small.30 For i (cid:54) = j : ∂∂s i ˆ γ j ˆ γ = ∂∂s i γ pert j + c j + tf j + ( t log t ) g j + th j γ pert1 + c + tf + ( t log t ) g + th = o (1) (cid:16) γ pert1 + c + tf + ( t log t ) g + th (cid:17)(cid:16) γ pert1 + c + tf + ( t log t ) g + th (cid:17) − o (1) (cid:16) γ pert j + c j + tf j + ( t log t ) g j + th j (cid:17)(cid:16) γ pert1 + c + tf + ( t log t ) g + th (cid:17) = o (1) . • For any i : ∂∂t ˆ γ i ˆ γ = ∂∂t γ pert i + c i + tf i + ( t log t ) g i + th i γ pert1 + c + tf + ( t log t ) g + th = ( O (1) + O (log t )) (cid:16) γ pert1 + c + tf + ( t log t ) g + th (cid:17) ( γ pert1 + c + tf + ( t log t ) g + th ) − (( O (1) + O (log t ))) (cid:16) γ pert i + c i + tf i + ( t log t ) g i + th i (cid:17) ( γ pert1 + c + tf + ( t log t ) g + th ) = O (log t ) . • For i (cid:54) = 1: ∂∂s i ˆ β ˆ γ = ∂∂s i btγ pert1 + c + tf + ( t log t ) g + th = − o (1) bt (cid:16) γ pert1 + c + tf + ( t log t ) g + th (cid:17) = o ( t ) . Finally: ∂∂t ˆ β ˆ γ = ∂∂t btγ pert1 + c + tf + ( t log t ) g + th = b ( γ pert1 + c + tf + ( t log t ) g + th ) − ( O (1) + O (log t )) bt (cid:16) γ pert1 + c + tf + ( t log t ) g + th (cid:17) = d + o (1) , where d = γ pert1 ( ¯ X )+ c , as in the first bullet above (recall also that b → P : DP = d + o (1) O (log t ). . . ... d + o (1) O (log t ) o ( t ) · · · o ( t ) d + o (1) , where all blank entries are o (1).The idea for the remainder of the proof is as follows. Notice that for t small, the above matrix has non-zero determinant (all products, except theproduct of diagonal entries, are O ( t log t ), which goes to 0 as t → DP were close to a fixed invertible matrix, then we can get injectivity byintegrating the derivative and using the Fundamental Theorem of Calculus.However, the O (log t ) entries in the above can blow up as t →
0. Thesolution is to divide into two cases. In (I) we assume that the values t (cid:48) , t (cid:48)(cid:48) for X (cid:48) , X (cid:48)(cid:48) have quite different magnitudes and then directly show that P n takes different values. In (II) we assume that t (cid:48) , t (cid:48)(cid:48) have somewhat similarmagnitudes, in which case the relevant values of DP are in fact close to afixed invertible matrix and so we can use the integration argument.So we consider two cases:(I) Suppose | t (cid:48)(cid:48) | / | t (cid:48) | / ∈ [1 / , | t (cid:48) | > | t (cid:48)(cid:48) | . In this case, we don’t need the derivative estimate.Note that | P n ( X (cid:48) ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) bt (cid:48) γ pert1 ( X (cid:48) ) + r t (cid:48) log t (cid:48) + t (cid:48) c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > . | t (cid:48) | | b || γ pert1 ( ¯ X ) | > . | t (cid:48)(cid:48) | | b || γ pert1 ( ¯ X ) | , V (hence t (cid:48) ) is small. Similar estimates give | P n ( X (cid:48)(cid:48) ) | < . | t (cid:48)(cid:48) | | b || γ pert1 ( ¯ X ) | for V small, so P n ( X (cid:48) ) (cid:54) = P n ( X (cid:48)(cid:48) ), and we are done.(II) Suppose | t (cid:48)(cid:48) | / | t (cid:48) | ∈ [1 / , v = ( v , . . . , v n − , τ ) T . Multiplying by the matrixabove gives the following expressions for the directional derivatives ofthe component functions of P : ∇ v ( P i ) = dv i + o (1) max i | v i | + τ · O (log t ) , for i = 1 , . . . , n − , ∇ v ( P n ) = o ( t ) max i | v i | + τ ( d + o (1)) . Now P V is a convex region (with respect to the analytic coordinates),and hence S is also convex. So we can connect X (cid:48) , X (cid:48)(cid:48) by a straightline segment L lying in S (and L has non-zero length, since X (cid:48) , X (cid:48)(cid:48) have different analytic coordinates). Let v be the direction of thissegment.Note that for every surface X on the segment L , the value of the t parameter is within a factor of 2 in absolute value from | t (cid:48) | , since t liesin a sector of angle at most π/
4, by conditions (iii) and (iv) in Section3.1. (The distance from 0 to t is a convex function along the lineconnecting t (cid:48) , t (cid:48)(cid:48) , so | t | ≤ max( | t (cid:48) | , | t (cid:48)(cid:48) | ) ≤ | t (cid:48) | . If the triangle formedby 0 , t (cid:48) , t (cid:48)(cid:48) is obtuse, then | t | is monotone along the segment connecting t (cid:48) , t (cid:48)(cid:48) , and hence | t | ≥ min( | t (cid:48) | , | t (cid:48)(cid:48) | ) ≥ | t (cid:48) | /
2. If the triangle is acute,then | t | ≤ | t (cid:48) | cos θ for some θ ∈ [0 , π/ | t | ≥ | t (cid:48) |√ / ≥ | t (cid:48) | / i | v i | , which we take to be | v j | :(a) | v j | (cid:28) | τ || t (cid:48) | . We then get ∇ v ( P n ) | X = o ( t ) | v j | + τ ( d + o (1)) = o (2 t (cid:48) ) | v j | + τ ( d + o (1))= τ ( d + o (1)) . More precisely, the above means that for each κ >
0, there issome constant c > | v j | < c τ | t (cid:48) | , then (cid:12)(cid:12) ∇ v ( P n ) | X − dτ (cid:12)(cid:12) < κ | dτ | . (5)In particular, we can take the constant c for κ = 1 /
2. This c depends on the implied constant in o ( · ) above (which depends on¯ X , but not on S ). 33b) | v j | (cid:29) | τ | · | log t (cid:48) | . Then ∇ v ( P j ) | X = dv j + o (1) | v j | + τ · O (log 2 t (cid:48) ) = v j ( d + o (1)) . More precisely, the above means that for each κ >
0, there issome constant
C > | v j | > Cτ | log t (cid:48) | , then (cid:12)(cid:12) ∇ v ( P j ) | X − dv j (cid:12)(cid:12) < κ | dv j | . (6)In particular, we can take the constant C for κ = 1 /
2. This C depends on the implied constant in O ( · ) above (which dependson ¯ X , but not on S ).We now choose (cid:15) in the definition of V in Section 3.1 small dependingon the constants c, C above. Specifically, we take (cid:15) such that | (cid:15) · log (cid:15) | 0) for the pair ( c, C ) thatwe get in the above for κ = 1 / 2. Note that c, C depend only on ¯ X ,not S . Since X (cid:48) ∈ V , we have that | t (cid:48) | < (cid:15) , hence | t (cid:48) | · | log t (cid:48) | < c/C ,which means Cτ | log t (cid:48) | < c τ | t (cid:48) | , and so the subcases (a) and (b) coverall possibilites for | v j | .Now define z, k as follows. In case (a), we take z = dτ and k = n .In case (b), take z = dv j and k = j (if | v j | falls into both cases,choose one arbitrarily). Note that z (cid:54) = 0, since some v i or τ must benon-zero (and d (cid:54) = 0), and then the inequality for (a) or (b) impliesthat z is also non-zero. Consider the open half-plane H in C whoseboundary is the line through the origin perpendicular to z . From theinequalities (5) or (6) above, we get that ∇ v ( P k ) | X lies in H for all X ∈ L . Then by the Fundamental Theorem of Calculus, the difference P k ( X (cid:48)(cid:48) ) − P k ( X (cid:48) ) can be expressed as an integral of ∇ v ( P k ) | X overthe segment S . Since the integrand lies in H , and H is closed underaddition, we get that P k ( X (cid:48)(cid:48) ) − P k ( X (cid:48) ) ∈ H . In particular it is non-zero, hence P ( X (cid:48) ) (cid:54) = P ( X (cid:48)(cid:48) ), and we are done.Case 3: (cid:96) is not an integer (i.e. the lowest effective level where the twosurfaces differ comes from a degenerating cylinder). Here we must understand the period of a curve crossing the degeneratingcylinder that witnesses the effective level.Let j = (cid:96) − / 2. All the moduli parameters for the level subsurfaces ¯ X ( j ) and below are the same for X (cid:48) , X (cid:48)(cid:48) . For the scaling parameters, we have t (cid:48) i = t (cid:48)(cid:48) i for i ≤ j − 1. By assumption, there is some degenerating cylinder34t level j for which the parameter values t (cid:48) , t (cid:48)(cid:48) for X (cid:48) , X (cid:48)(cid:48) , are different. Onsurfaces obtained from ¯ X ( j ) by smoothing out the horizontal node, we candefine a relative homology class α that crosses this cylinder once. We extend α downwards to a relative homology class ˆ α on smooth surfaces in V , andwe can do this in such a way that ˆ α does not cross any other degeneratingcylinders (as in Case 2). We take γ to be the class on ¯ X ( j ) that has period1 on all surfaces near ¯ X in the boundary stratum. This can be extendeddownwards to a relative homology class ˆ γ on smooth surfaces in V .In the plumbing construction to smooth the horizontal node, plumbingis performed on the discs of radius 1 in standard coordinates centered atthe nodes. The class ˆ α can be decomposed into a piece that lies outsidethis disc; the period of this piece does not depend on t . There is alsoa portion that crosses the plumbing region. The period here is equal to (cid:82) √ t ru du = ( r/ 2) log t , where r is the residue at the simple pole. Since weare working in a sector, log can be consistently defined.We then getˆ α ( X (cid:48) )ˆ γ ( X (cid:48) ) = ( t (cid:48)− ) a − · · · ( t (cid:48) j ) a j ( c + ( r/ 2) log t (cid:48) )( t (cid:48)− ) a − · · · ( t (cid:48) j ) a j d = c + ( r/ 2) log t (cid:48) d , ˆ α ( X (cid:48)(cid:48) )ˆ γ ( X (cid:48)(cid:48) ) = ( t (cid:48)(cid:48)− ) a − · · · ( t (cid:48)(cid:48) j ) a j ( c + ( r/ 2) log t (cid:48)(cid:48) )( t (cid:48)(cid:48)− ) a − · · · ( t (cid:48)(cid:48) j ) a j d = c + ( r/ 2) log t (cid:48)(cid:48) d . It is crucial in the above that c, r, d are the same for X (cid:48) , X (cid:48)(cid:48) ; this is becausethese depend only on moduli parameters at level j and below, and levelscaling parameters for level j − X (cid:48) , X (cid:48)(cid:48) .So, since we are assuming that t (cid:48) (cid:54) = t (cid:48)(cid:48) , we get ˆ α ( X (cid:48) )ˆ γ ( X (cid:48) ) (cid:54) = ˆ α ( X (cid:48)(cid:48) )ˆ γ ( X (cid:48)(cid:48) ) , and weare done. (cid:4) V with M We now introduce the affine invariant manifold M into the picture. Our goalis to show that M intersects each period coordinate chart V constructed inSection 3.1 nicely. We will use the following deep result of Filip. Theorem 4.1 ([Fil16]) . Any affine invariant manifold M is a quasi-projectivesubvariety of H . orollary 4.2. P M is a semianalytic subset of P H , thought of as a real-analytic orbifold.Proof. This would be easier if P H was a projective algebraic variety; howeverthis is currently unknown. Instead, we will use(i) the fact that we have available the Incidence Variety Compactification P H IVC of P H [BCG + π : P H → P H IVC that is a complex-analytic(hence real-analytic) map of orbifolds. See [BCG + 19, Theorem 1.2 (6)and Theorem 1.6].By Theorem 4.1, P M is a quasi-projective subvariety of P H IVC . Infact, P M = W − Z , where W is the Zariski closure of P M in P H IVC , and Z = P H IVC − P H , which is also an algebraic variety. Then, thought of as asubset of P H , we have P M = π − ( P M ) = π − ( W ) − π − ( Z ) . Since W, Z are both algebraic varieties in the IVC, their preimages underthe real-analytic map π are real-analytic varieties, and hence their differenceis semianalytic. (cid:4) The following basic facts give finiteness of components of semianalyticsets. Fact 4.3. Let S be a semianalytic subset of a real-analytic orbifold M . Thenfor any p ∈ M , and W (cid:48) a neighborhood of p in M , we can pass to a smallerneighborhood W ⊂ W (cid:48) , p ∈ W , such that W ∩ S has finitely many connectedcomponents.Proof. This is proved in [BM88, proof of Corollary 2.7] for the case when M is a real-analytic manifold. The generalization to orbifolds is straightfor-ward. See also [Kan11, Theorem 6.4, part (7)] for a related statement. (cid:4) Fact 4.4. Let M be a compact real-analytic orbifold, and let S ⊂ M be asemianalytic subset. Then S has finitely many components.Proof. By Fact 4.3, for each p ∈ M , we can find an open set W p ⊂ M , p ∈ W p , such that W p ∩ S is a union of finitely many connected sets. By com-pactness of M , we can cover M by finitely many of these, say W p , . . . , W p k .So S is a union of finitely many connected sets, and it follows that it hasfinitely many connected components. (cid:4) 36e now apply the above facts to V ∩ M . Lemma 4.5 (Finite intersections) . The intersection of the smooth locus M ∗ ⊂ M with each set V from Section 3.1 has finitely many connectedcomponents.Proof. First we show that P V ∩ P M ∗ has finitely many components. ByLemma 3.3, P V is a semianalytic subset of P H . By Corollary 4.2, P M is asemianalytic subset of P H , and hence P M ∗ also is (since the singular locusof variety is a subvariety). Thus the intersection P V ∩ P M ∗ is semianalytic.Finiteness of components then follows by applying Fact 4.4.The connected components of V ∩ M ∗ are in bijection with those of P V ∩ P M ∗ , hence also finite. (cid:4) We now show that, in our volume estimation problem, µ M can be re-placed by a finite sum of linear measures on subspaces W , and M -independenceof saddle connections can be replaced by W -independence, defined below.Fix a period coordinate chart Q ⊂ H , and W a subset of Q given bypulling back a single linear subspace from C n . Let W ⊂ W be the locus ofunit area surfaces. We define L W(cid:15) ,...,(cid:15) k := { X ∈ W : X has W -independent saddle connections s , . . . , s k with | s i | ≤ (cid:15) i } . Here a collection of saddle connections on X ∈ W is said to be W -independent if their relative homology classes define linearly independentfunctionals on W .We define a measure µ W on Q to be the natural Lebesgue measure onthe linear subspace W (we can pick an arbitrary normalization). Let µ W be the corresponding measure supported on the unit area locus W , i.e. forany measurable subset S ⊂ Q , µ W ( S ) := µ W ( { sX : X ∈ S, ≤ s ≤ } ) . Lemma 4.6 (Finitely many subspaces) . Fix a V from Section 3.1 satisfyingLemma 3.6, and an affine invariant submanifold M . There exist finitelymany linear subspaces W , . . . , W j ⊂ V , and a constant C , such that forany (cid:15) , . . . , (cid:15) k > , µ M (cid:0) L M (cid:15) ,...,(cid:15) k ∩ V (cid:1) ≤ C · (cid:88) (cid:96) µ W (cid:96) (cid:0) L W (cid:96) (cid:15) ,...,(cid:15) k (cid:1) . Proof. By Lemma 4.5 (Finite intersections), the smooth locus M ∗ intersects V in finitely many components N , . . . , N j . Now, by the definition of an37ffine invariant manifold, locally near a point of N (cid:96) , we have that M ∗ agreeswith a linear subspace W (cid:96) ⊂ V since V is a period coordinate chart, byLemma 3.6. Furthermore, the measures µ M and µ W (cid:96) agree locally, up to aconstant scaling factor C (cid:96) . The subset of N (cid:96) where N (cid:96) agrees locally with W (cid:96) is open and closed in N (cid:96) . Since N (cid:96) is connected, this means that thisset of local agreement is all of N (cid:96) . Hence N (cid:96) ⊂ W (cid:96) . And near points in N (cid:96) , the measures µ M and µ W (cid:96) agree up to the factor C (cid:96) , and furthermorethe notion of saddle connections being M -independent coincides with thenotion of W (cid:96) -independence at such points. Hence µ M (cid:0) L M (cid:15) ,...,(cid:15) k ∩ N (cid:96) (cid:1) ≤ C (cid:96) · µ W (cid:96) (cid:0) L W (cid:96) (cid:15) ,...,(cid:15) k (cid:1) . Using these observations, and the fact that µ M ( M − M ∗ ) = 0, we get that µ M (cid:0) L M (cid:15) ,...,(cid:15) k ∩ V (cid:1) ≤ (cid:88) (cid:96) µ M (cid:0) L M (cid:15) ,...,(cid:15) k ∩ N (cid:96) (cid:1) ≤ (cid:88) (cid:96) C (cid:96) · µ W (cid:96) (cid:0) L W (cid:96) (cid:15) ,...,(cid:15) k (cid:1) , and we get the desired result by taking C = C + · · · + C (cid:96) . (cid:4) Below is the key local (near a boundary point) result needed to prove The-orem 1.3 Proposition 5.1 (Local volume bound) . Let ¯ X ∈ P H . Then there exists asmall neighborhood U ⊂ P H containing ¯ X and a constant C such that µ M ( L M (cid:15) ,...,(cid:15) k ∩ p − ( U )) ≤ C(cid:15) · · · (cid:15) k , for any (cid:15) , . . . , (cid:15) k > . First we will develop the necessary tools to understand the relationshipbetween short saddle connections and the structure of the boundary point¯ X . This culminates in Lemma 5.7, which is the above estimate but for asingle linear subspace W in the period coordinate chart V . Then in Section5.5 we will combine this estimate with Lemma 4.6 (Finitely many subspaces)to prove the above proposition.We will work with surfaces X in a period coordinate chart V from Section3.1 that covers part of a neighborhood of a boundary point ¯ X . We need tounderstand where the short saddle connections on X are in terms of datafrom ¯ X . One challenge is understanding the interaction of degeneratingcylinders with small subsurfaces. 38 .1 Sizes of subsurfaces and orderings Recall that the boundary point ¯ X has level subsurfaces ¯ X ( i ) consisting ofall those components at level i . Each of these corresponds to a subsurface ofeach X ∈ V , defined up to isotopy. For each horizontal node (correspondingto a pair of simple poles) there is a degenerating cylinder on X , which isalso a subsurface defined up to isotopy.Let S be the union of the set of the level subsurfaces and the set of de-generating cylinders. Each element of S is a topological subsurface, definedup to homotopy. Once we have chosen a particular X ∈ V , we can talkabout a definite size of each element of S : Definition 5.2 (Size) . Given a surface X ∈ V , we define size X ( Y ) of anelement Y ∈ S to be(i) (cid:12)(cid:12) t a − − · · · t a i i (cid:12)(cid:12) if Y is the level i subsurface ¯ X ( i ) ; this is the magnitude ofthe scaling parameter in the plumbing construction for the projectiveclass of X (see Section 2.2.2.2)(ii) (cid:12)(cid:12) log | t | · t a − − · · · t a i i (cid:12)(cid:12) if Y is a degenerating cylinder with horizontal nodeparameter t whose circumference lies at level i (this is approximatelythe width of the cylinder, since the 1 /u pole of the differential con-tributes (log u ) | √ t to the integral of the curve crossing the annuluswhere the plumbing happens). Ordering of subsurfaces. We wish to use the structure of the boundarypoint ¯ X to understand something about the relative size of saddle connec-tions on surfaces X ∈ V . There is a natural ordering on the set of levelsubsurfaces, just given by the level. If there are no degenerating cylinders,the ordering induced on saddle connections from their level roughly agreeswith the orderings of their lengths on surfaces in V . However, for saddleconnections that cross degenerating cylinders at lower levels the situation ismore complicated.We wish to extend the ordering on level subsurfaces to an order on all of S , including the degenerating cylinders. Since the expression in (ii) abovecontains both large terms ( | log | t || ), and small terms (the others), the limitpoint ¯ X does not by itself tell us about the relative magnitudes of degener-ating cylinder cross curves compared to other subsurfaces of X . Thus, wehave several possible orderings compatible with a given ¯ X ; however, sincethere are only finitely many, our strategy will be to do a separate volumecomputation for each ordering, and then take a sum.39et O denote the set of total orderings (cid:31) on S that restrict to the naturalordering on the level subsurfaces and have the additional property that if C is a degenerating cylinder and ¯ X ( i ) is the level subsurface at which thecircumference of C lies, then C (cid:23) ¯ X ( i ) .We say that an ordering (cid:31)∈ O is consistent with X if for any Y , Y ∈ S with Y (cid:31) Y we have that size X ( Y ) ≥ size X ( Y ). For each X , there is atleast one (cid:31)∈ O that is consistent with X .Given a relative integral homology class γ on surfaces in V , we definethe level of γ with respect to (cid:31)∈ O to be the minimal Y ∈ S with respect to (cid:31) such that γ has a representative that does not intersect any subsurfaces Y (cid:48) with Y (cid:48) (cid:31) Y . Here we consider those representatives of γ that are unionsof oriented arcs/curves (such a representative always exists).We will call a degenerating cylinder (cid:31) - wide if it is (cid:31) -greater than anylevel subsurface. In this section we prove bounds on the periods of various homology classesin terms of the sizes of the subsurfaces that they intersect. Here we areonly interested in coarse bounds (constant factors do not matter). Parts ofthese lemmas could be proven using the more delicate estimates in Section3.3 (though the estimates there do not handle homology classes crossing de-generating cylinders). Since this precision is not necessary for our purposeshere, we do not make use of those more precise estimates in this section.Our first lemma states that the period of a relative homology class iscoarsely bounded above by the size of its level. A similar estimate has beenused recently by Chen-Wright [CW19, Theorem 8.1]. Lemma 5.3 (Period bounds) . For γ a relative (integral) homology classdefined on the surfaces in V , there exists a constant C > with the followingproperty. Fix X ∈ V of area . Let (cid:31)∈ O be consistent with X , and let Y be the level of γ with respect to (cid:31) . Then | γ ( X ) | ≤ C · size X ( Y ) . Proof. Since X is in V , there is some C -rescaling X (cid:48) = αX = ( M, ω ) thatis the distinguished surface coming from the plumbing construction. Wewill work with this X (cid:48) and then compare to X at the end. By definition,size X ( Y ) = size X (cid:48) ( Y ) for any subsurface Y ∈ S .The underlying Riemann surface M can be decomposed into pieces corre-sponding to elements of S . For a level subsurface ¯ X ( i ) in S , the correspond-ing piece of M is very close to the complement of a fixed small neighborhood40igure 6: Neighborhoods of nodes of ¯ X are replaced by plumbing annuliand degenerating cylinders to get X .of the nodes in ¯ X ( i ) . For a degenerating cylinder C in S whose circumfer-ence lies at level i , the corresponding piece of M is an annulus. These piecesdo not quite cover all of M ; there are also plumbing annuli for the nodesconnecting level subsurfaces. See Figure 6.Now for γ , choose representative curves/arcs (which we will also call γ ) on the surfaces in V that exhibit the level Y of γ , as defined above inSection 5.1. The intersection of γ with a piece of M that corresponds toa level subsurface can be taken to be a fixed arc. Because of the way V was defined in terms of restrictions on the plumbing coordinates, γ windsaround each plumbing annulus a bounded number of times (with the boundonly depending on the boundary point ¯ X , not X ). See Figure 7.Now we consider the restriction of the differential ω to the pieces of M .Level subsurface:On a piece ˆ Z of X (cid:48) corresponding to the level subsurface Z = ¯ X ( i ) , therestriction of ω gives a translation surface that is close to a piece of ¯ X ( i ) scaled by t a − − · · · t a i i . The (small) discrepancy between the period of γ on ˆ Z versus on Z (suitably rescaled) is caused by:(i) Moduli parameters. This error has order at most (cid:12)(cid:12) t a − − · · · t a i i (cid:12)(cid:12) .(ii) Modification differential(s) coming from residues of poles on lower com-ponents, and the remerging of zeros. The order of the contributionfrom a modification differential added to account for residue on ¯ X ( j ) 41s at most (cid:12)(cid:12)(cid:12) t a − − · · · t a i i · t a i − i − · · · t a j j (cid:12)(cid:12)(cid:12) = o (cid:0)(cid:12)(cid:12) t a − − · · · t a i i (cid:12)(cid:12)(cid:1) . It follows that the restriction of γ to such a piece has period bounded aboveby a constant multiple of (cid:12)(cid:12) t a − − · · · t a i i (cid:12)(cid:12) , which equals size X ( Z ).Degenerating cylinder:On a piece ˆ C of X (cid:48) corresponding to a degenerating cylinder C ∈ S withparameter t whose circumference lies at level i , the restriction of ω gives acylinder of width on the order of (cid:12)(cid:12) log | t | · t a − − · · · t a i i (cid:12)(cid:12) and circumference onthe order of (cid:12)(cid:12) t a − − · · · t a i i (cid:12)(cid:12) . We saw above that γ winds arounds the cylindera bounded number of times, so the restriction of γ to the cylinder has pe-riod bounded above by a constant multiple of (cid:12)(cid:12) log | t | · t a − − · · · t a i i (cid:12)(cid:12) , which issize X ( C ).Plumbing annulus:Finally, consider a plumbing annulus A such that the higher adjacentpiece ˆ Z corresponds to the level subsurface Z = ¯ X ( i ) , while the lower adja-cent piece is the level subsurface ¯ X ( j ) . There are two contributions to theperiod of γ restricted to A :(i) Winding around the annulus. As discussed above, the winding happensonly a bounded number of times. The contribution to the period is oforder at most | t a − − · · · t a i i | .(ii) Crossing the annulus. In the general case where there are non-zeroresidues (and hence modification differentials), the total contributionis at most O (cid:0)(cid:12)(cid:12) t a − − · · · t a i i (cid:12)(cid:12)(cid:1) . Included in the above is the contribution from the modification differ-ential itself, which is order at most (cid:12)(cid:12)(cid:12) t a − − · · · t a i i · (cid:16) t a i − i − · · · t a j j log (cid:12)(cid:12)(cid:12) t a i − /bi − · · · t a j /bj (cid:12)(cid:12)(cid:12)(cid:17)(cid:12)(cid:12)(cid:12) = o (cid:0)(cid:12)(cid:12) t a − − · · · t a i i (cid:12)(cid:12)(cid:1) . Hence the total contribution to γ from A is of order at most | t a − − · · · t a i i | =size X ( Z ).We have now seen that for all the pieces of X (cid:48) , if γ intersects that piece,the period over the intersection is bounded above by a constant multiple ofsize X ( Z ) for some Z ∈ S that γ also intersects. Since γ is a curve/arc that42igure 7: The two annuli are glued together, with the circles of radius (cid:112) | t | being glued to one another. The two points labeled x get identified. The redsegments form the part of a curve that crosses the degenerating cylinder.exhibits that its relative homology class is at level Y , any such Z satisfies Z ≺ Y . And since (cid:31) is consistent with X (and X (cid:48) ), it follows that theperiod of each such intersection is bounded above by a constant multiple ofsize X ( Y ). Putting all the parts of γ together gives | γ ( X (cid:48) ) | ≤ C · size X ( Y ) , (7)For some constant C depending only on ¯ X . Now we claim that the α fromthe beginning of the proof is bounded below in magnitude, with the bounddepending only on ¯ X . This follows from the fact that the area of X (cid:48) = αX is at least (approximately) the area of the differential η on the top levelsubsurface ¯ X (0) , which is bounded below, while X is assumed to have area1. We then get from (7) that | γ ( X ) | ≤ | α | | γ ( X (cid:48) ) | ≤ C · size X ( Y ) , for some new choice of C , completing the proof. (cid:4) Our next lemma allows us to control the period of a curve that crosses asingle degenerating cylinder. This period is potentially large. But we showthat if we fix the period of the circumference curve of the cylinder, then theperiod of the crossing curve lies in a rectangle in C of bounded area.43igure 8: The result of plumbing together the annuli in Figure 7. Lemma 5.4 (Cylinder bounds) . Let β be a relative (integral) homology classdefined on the surfaces in V with the following properties:(i) β has a representative that crosses exactly one closed curve γ that is thecore curve of a degenerating cylinder. We let Y be the level subsurfacecontaining γ .(ii) β has a representative such that the level subsurfaces which the repre-sentative intersects all lie at or below the level of Y .Then there exists a family of rectangles R ( w ) ⊂ C , w ∈ C , of area boundedabove by some R (depending on ¯ X and β , but not on X ), with the followingproperty. For any X ∈ V of area , we have β ( X ) ∈ R ( γ ( X )) . Furthermore, the rectangles have the property that s R ( w ) ⊂ R ( sw ) for any w and < s ≤ .Proof. The set R ( w ) will be a bounded neighborhood of a rectangle of fixedarea that gets longer and thinner as w gets smaller.As in the proof of Lemma 5.3 (Period bounds), we divide up X intopieces corresponding to elements of S , and then take the intersection of β with these pieces. Let β (cid:48) be the piece that crosses the cylinder withcircumference γ , and let β (cid:48)(cid:48) be the remaining part of β . See Figure 7 andFigure 8.To control β (cid:48) , we can assume that the situation is as in Figure 7, wherethe differential on the left is given by γ ( X )2 πi duu , and on the right by − γ ( X )2 πi dvv (since γ is the cylinder circumference curve, the residues of the simple polesare ± γ ( X )). We assume for notational simplicity that γ ( X ) is purely imag-inary. In the figure, β (cid:48) consists of the red arcs that intersect the shaded44nnuli. There are two contributions to β (cid:48) . The first, which we call β iscomposed of the two straight segments connecting the circles of radii 1 and (cid:112) | t | , on either side of the nodes. The second, β , is an arc of the circle ofradius √ t . We see that Re( β (cid:48) ( X )) = β ( X ) and i · Im( β (cid:48) ( X )) = β ( X ). Theheight of the cylinder is approximately | β ( X ) | , so its area is approximately | γ ( X ) | · | β ( X ) | . Since this must be less than 1, we have | β ( X ) | ≤ / | γ ( X ) | . To bound β ( X ), we note that β winds around the circle at most once(because of the way V was defined in terms of restrictions of plumbingcoordinates). Suppose its endpoint is at angle ρ ∈ [0 , π ]. Then β ( X ) = (cid:90) β γ ( X )2 πi duu = γ ( X )2 π (cid:90) ρ dθ = γ ( X )2 π ρ. Thus we have that Re( β (cid:48) ( X )) = β ( X ) is a real number of magnitudeat most 1 / | γ ( X ) | , while Im( β (cid:48) ( X )) = β ( X ) is a purely imaginary numberof magnitude at most | γ ( X ) | . Thus β (cid:48) ( X ) lies in a rectangle one of whosesides has length 2 | γ ( X ) | , and the other length 2 / | γ ( X ) | .Now we bound the remaining piece β (cid:48)(cid:48) . Let X (cid:48) = αX be the surface inthe projective class of X that is produced in the plumbing construction. Byassumption β (cid:48)(cid:48) lies on level subsurfaces at or below the level Y . Arguingas in Lemma 5.3 (Period bounds), we see that | β (cid:48)(cid:48) ( X ) | ≤ C size X ( Y ). Nowsize X ( Y ) is very close to a constant multiple of γ ( X (cid:48) ), since γ lies at level Y . Hence | β (cid:48)(cid:48) ( X (cid:48) ) | ≤ C (cid:48) | γ ( X (cid:48) ) | , where C (cid:48) is a constant depending on β (and¯ X ). Since X is just a rescaling of X (cid:48) , we get | β (cid:48)(cid:48) ( X ) | ≤ C (cid:48) | γ ( X ) | . Finally, by combing the estimates for β (cid:48) ( X ) and β (cid:48)(cid:48) ( X ), we see that theirsum β ( X ) must lie in a modest enlargement of the rectangle discussed abovefor β (cid:48) . The sides have length (2 + 2 C (cid:48) ) | γ ( X ) | and 2 / | γ ( X ) | + 2 C (cid:48) | γ ( X ) | . Wetake R ( γ ( X )) to be this rectangle. For γ ( X ) small, which is the relevantregime, the area of this rectangle is bounded by some R depending only on C (cid:48) . The property s R ( w ) ⊂ R ( sw ) holds for any w and 0 < s ≤ (cid:4) The next lemma bounds the length of a saddle connection from below interms of the period of any fixed relative homology class that lies at or belowthe level of the saddle connection. 45 emma 5.5 (Saddle connection bounds) . For γ a relative homology classdefined on the surfaces in V , there exists a constant c > with the followingproperty. Fix X ∈ V . Let (cid:31)∈ O be consistent with X , and let Y be the levelof γ with respect to (cid:31) . Let s be a saddle connection on X whose relativehomology class has level at least Y . Then | s ( X ) | ≥ c · | γ ( X ) | . Proof. Since the desired inequality is invariant under scaling X , we canassume that X is the distinguished surface in the projective equivalenceproduced by the plumbing construction.Define the injectivity radius inj( Y ) of a subsurface Y (with boundary)of a translation surface to be the infimum of flat length over all curves/arcs(arcs are allowed to have endpoints at singular points or on the boundary)that cannot be homotoped (rel endpoints) to lie in the boundary.As in the proof of Lemma 5.3 (Period bounds), we divide up X intopieces corresponding to elements of S , together with plumbing annuli.We claim that | s ( X ) | ≥ inj( ˆ Z ) , (8)for some subsurface ˆ Z ⊂ X corresponding to an element Z ∈ S with Z (cid:23) Y . In fact, by the definition of level of a relative homology class, s mustintersect a subsurface Z ∈ S of level at least Y in a curve/arc that cannotbe homotoped rel endpoints to lie in subsurfaces at smaller levels. Hencethe length of s on X must be at least the injectivity radius of ˆ Z .Now we claim that inj( ˆ Z ) ≥ c (cid:48) · size X ( Z ) , (9)where c (cid:48) is a constant depending only on ¯ X . In fact, if we remove fixedsize neighborhoods from the nodes of Z as in the plumbing construction,the resulting surface with boundary has some non-zero injectivity radius r .We then obtain ˆ Z from this by scaling by the parameter t a − − · · · t a i i , addinga small modification differential, and then merging zeros. The latter twooperations only change the injectivity radius by a small amount, so theinjectivity radius of ˆ Z is close to r · | t a − − · · · t a i i | , and | t a − − · · · t a i i | is exactlysize X ( Z ), so (9) follows.Now Z was chosen such that Z (cid:23) Y , and since X is assumed to be (cid:31) -consistent, we have size X ( Z ) ≥ size X ( Y ) . (10)46inally, a direct application of Lemma 5.3 (Period bounds) gives thatsize X ( Y ) ≥ C · | γ ( X ) | . (11)Stringing together the inequalities (8), (9), (10), (11) gives the desiredinequality, for some c depending only on ¯ X . (cid:4) In this section we describe how to choose a basis of H ( X, Σ; Z ), for surfaces X ∈ V , that is adapted to the surfaces in V in the sense that homologyclasses living on smaller subsurfaces will generally come earlier in the basis.The basis will depend on an ordering (cid:31)∈ O of subsurfaces S .Given Z ∈ S , we define H (cid:22) Z ⊂ H ( X, Σ; C ) to be the span of all elements γ ∈ H ( X, Σ; Z ) that lie at level (cid:22) Z . Lemma 5.6 (Basis) . Fix an ordering (cid:31)∈ O . There is a basis α , . . . , α m (cid:48) for H ( X, Σ; C ) consisting of integral classes with the following properties.(i) For each Z a basis for the vector space H (cid:22) Z is given by α , . . . , α j forsome j .(ii) The number of crossings of each α i with the set of (cid:31) -wide cylindercircumference curves is at most .(iii) If α i intersects a (cid:31) -wide cylinder circumference γ , then α i has a rep-resentative such that the level subsurfaces which the representative in-tersects lie at or below the level of γ (of course α i also crosses thecylinder, which may be at a higher level, but the cylinder is not a levelsubsurface).Proof. Begin by choosing an integral basis for H (cid:22) ¯ X ( − N ) (note that for any (cid:31)∈ O , ¯ X ( − N ) is always the ≺ -smallest element of S ). Then continue byappending elements that together with the previously added elements givea basis for H (cid:22) Z , where Z is the second ≺ -smallest element of S . Continuein this manner through Z = ¯ X (0) , the largest level subsurface. None of the α i added up to this point will cross (cid:31) -wide cylinder circumference curves,so (ii) and (iii) are satisfied for these α i .By definition of (cid:31) -wide, every Z with Z (cid:31) ¯ X (0) is a (cid:31) -wide cylinder.Each (cid:31) -wide cylinder joins two components, possibly equal. We can findzeros of the differential by moving to either side of the cylinder and then47oving down level subsurfaces until we come to a locally minimal compo-nent in the level graph (such a component must have a zero). We connectthese zeros (which might coincide), giving a relative homology class α i thatsatisfies (ii) and (iii). We add this α i to the basis. We do this one byone for each new (cid:31) -wide cylinder cross curve whose circumference curve isindependent of the current basis elements. (cid:4) Using the lemmas established in the previous two sections, we can now provethe volume estimate for a single linear subspace. Lemma 5.7 (Volume for single linear subspace) . Let W be a linear subspaceof a V from Section 3.1. Then µ W (cid:0) L W(cid:15) ,...,(cid:15) k (cid:1) = O ( (cid:15) · · · (cid:15) k ) . Proof. Our first goal is to choose a basis for linear functionals on W whichwe will use to do the volume estimation. Fix an ordering (cid:31) , and use Lemma5.6 (Basis) to choose a basis α , . . . , α m (cid:48) for H ( X, Σ; C ) satisfying the threeconditions given by that lemma.The choice of { α i } basis above was made independently of the subspace W . Now extract a basis { β , . . . , β m } ⊂ { α , . . . , α m (cid:48) } for W ∗ as follows. Let β = α , and suppose we have chosen β , . . . , β j . Take β j +1 to be α i , where i is the smallest index such that α i , β , . . . , β j give linearly independentfunctionals on the subspace W . Continue doing this until no additional α i can be added satisfying the conditions. Note that by construction, for each Z ∈ S the subspace of W ∗ given by elements of H (cid:22) Z has basis β , . . . , β j for some j .Our next goal is to show that certain sets defined by inequalities on the β i functionals contain all the surfaces in L W(cid:15) ,...,(cid:15) k that are (cid:31) -consistent.We claim that if X is (cid:31) -consistent, and X ∈ L W(cid:15) ,...,(cid:15) k , then there existdistinct indices φ (1) , . . . , φ ( k ) ∈ { , . . . , k } such that X is in the set B φ, (cid:31) (cid:15) ,...,(cid:15) k := (cid:8) X (cid:48) ∈ W : X (cid:48) is (cid:31) -consistent , | β φ (1) ( X (cid:48) ) | ≤ (cid:15) /c, . . . , | β φ ( k ) ( X (cid:48) ) | ≤ (cid:15) k /c (cid:9) , where c is the constant from Lemma 5.5 (Saddle connection bounds). In fact,by definition of L W(cid:15) ,...,(cid:15) k , such an X must have saddle connections s , . . . , s k that are W -independent and such that | s i ( X ) | ≤ (cid:15) i . Because of the way the β i were chosen, for each s i we can find an element of β φ ( i ) such that48a) the level of s i (with respect to (cid:31) ) is at least that of β φ ( i ) , and(b) all the φ ( i ) are distinct.(Concretely, if the s i are in increasing order in terms of (cid:31) level, we can simplytake φ ( i ) = i ; otherwise, the situation is just a permutation of this). Now byLemma 5.5 (Saddle connection bounds), we have (cid:15) i ≥ | s i ( X ) | ≥ c | β φ ( i ) ( X ) | ,so X is in the set B φ, (cid:31) (cid:15) ,...,(cid:15) k .Our goal now is to establish the inclusion (15) below, which we willthen use to estimate the volume of B φ, (cid:31) (cid:15) ,...,(cid:15) k . The definition of B φ, (cid:31) (cid:15) ,...,(cid:15) k givesbounds on β , . . . , β k ; we now control the rest of the β i for X (cid:48) that are (cid:31) -consistent. Roughly, some of these β i do not cross any (cid:31) -wide cylinders;for these β i , the absolute value is bounded. The remaining β i can be longbecause they cross (cid:31) -wide cylinders, but nevertheless, after fixing the otherperiods, such a β i lies in a set of bounded area.To formalize the above, we find the index (cid:96) such that all β i with i ≤ (cid:96) lieat or below level ¯ X (0) and all β i with i > (cid:96) lie above level ¯ X (0) (and hencecross (cid:31) -wide cylinders).First consider the β i for i ≤ (cid:96) . Using Lemma 5.3 (Period bounds) and thefact that, over surfaces in V , the size of any level subsurface is bounded fromabove, we get that there exists a K such that for any X (cid:48) ∈ W consistentwith (cid:31) , | β i ( X (cid:48) ) | < K, (12)when i ≤ (cid:96) .Next we consider β i for i > (cid:96) . Such a β i crosses exactly one (cid:31) -wide cylin-der circumference curve γ i , and since (iii) of Lemma 5.6 (Basis) also holds,all hypotheses of Lemma 5.4 (Cylinder bounds) are satisfied. Hence by thatlemma, β i ( X (cid:48) ) ∈ R ( γ i ( X (cid:48) )), where R is a rectangle of area at most R . Nowthere is some linear function f i such that γ i ( X (cid:48) ) = f i ( β ( X (cid:48) ) , . . . , β (cid:96) ( X (cid:48) ))for all X (cid:48) ∈ W . Let R i ( z , . . . , z (cid:96) ) := R ( f i ( z , . . . , z (cid:96) )) . So β i ( X (cid:48) ) ∈ R i ( β ( X (cid:48) ) , . . . , β (cid:96) ( X (cid:48) )) , (13)when i > (cid:96) . Furthermore, by the inclusion property given by Lemma 5.4(Cylinder bounds), for any s ≤ s R i ( z , . . . , z (cid:96) ) ⊂ R i ( sz , . . . , sz (cid:96) )) . (14)49ow putting the cases (12) and (13) together yields the desired inclusion: B φ, (cid:31) (cid:15) ,...,(cid:15) k ⊂{ X (cid:48) ∈ W : | β φ (1) ( X (cid:48) ) | ≤ (cid:15) /c, . . . , | β φ ( k ) ( X (cid:48) ) | ≤ (cid:15) k /c, | β k +1 ( X (cid:48) ) | ≤ K, . . . , | β (cid:96) ( X (cid:48) ) | ≤ K,β (cid:96) +1 ( X (cid:48) ) ∈ R (cid:96) +1 ( β ( X (cid:48) ) , . . . , β (cid:96) ( X (cid:48) )) , . . . , β j ( X (cid:48) ) ∈ R j ( β ( X (cid:48) ) , . . . , β (cid:96) ( X (cid:48) )) } . (15)Now we can compute µ W volume, up to a constant factor, by takingstandard Lebesgue volume λ on C j with respect to any basis. In particular,we can work with the basis z , . . . , z j such that the β , . . . , β j are the co-ordinate functions with respect to this basis. We then compute using (15)and (14): µ W ( B φ, (cid:31) (cid:15) ,...,(cid:15) k ) = µ W (cid:16)(cid:110) sX (cid:48) : 0 ≤ s ≤ , X (cid:48) ∈ B φ, (cid:31) (cid:15) ,...,(cid:15) k (cid:111)(cid:17) ≤ λ (cid:0)(cid:8) s ( z , . . . , z j ) : 0 ≤ s ≤ , | z φ (1) | ≤ (cid:15) /c, . . . , | z φ ( k ) | ≤ (cid:15) k /c, | z k +1 | , . . . , | z (cid:96) | ≤ K,z (cid:96) +1 ∈ R (cid:96) +1 ( z , . . . , z (cid:96) ) , . . . , z j ∈ R j ( z , . . . , z (cid:96) ) (cid:9)(cid:1) = λ (cid:0)(cid:8) s ( z , . . . , z j ) : 0 ≤ s ≤ , | z φ (1) | ≤ (cid:15) /c, . . . , | z φ ( k ) | ≤ (cid:15) k /c, | z k +1 | , . . . , | z (cid:96) | ≤ K,sz (cid:96) +1 ∈ s R (cid:96) +1 ( z , . . . , z (cid:96) ) , . . . , sz j ∈ s R j ( z , . . . , z (cid:96) ) } ) ≤ λ (cid:0)(cid:8) s ( z , . . . , z j ) : 0 ≤ s ≤ , | sz φ (1) | ≤ (cid:15) /c, . . . , | sz φ ( k ) | ≤ (cid:15) k /c, | sz k +1 | , . . . , | sz (cid:96) | ≤ K,sz (cid:96) +1 ∈ R (cid:96) +1 ( sz , . . . , sz (cid:96) ) , . . . , sz j ∈ R j ( sz , . . . , sz (cid:96) ) (cid:9)(cid:1) = λ (cid:0)(cid:8) ( z , . . . , z j ) : | z φ (1) | ≤ (cid:15) /c, . . . , | z φ ( k ) | ≤ (cid:15) k /c, | z k +1 | , . . . , | z (cid:96) | ≤ K,z (cid:96) +1 ∈ R (cid:96) +1 ( z , . . . , z (cid:96) ) , . . . , z j ∈ R j ( z , . . . , z (cid:96) ) (cid:9)(cid:1) = (cid:90) B ( (cid:15) /c ) · · · (cid:90) B ( (cid:15) k /c ) (cid:90) B ( K ) · · · (cid:90) B ( K ) (cid:90) R (cid:96) +1 ( z ,...,z (cid:96) ) · · · (cid:90) R j ( z ,...,z (cid:96) ) dz j ∧ d ¯ z j · · · dz ∧ d ¯ z = O (cid:16) (cid:15) · · · (cid:15) k · K (cid:96) − k · R j − (cid:96) (cid:17) = O (cid:0) (cid:15) · · · (cid:15) k (cid:1) . Since every X ∈ L W(cid:15) ,...,(cid:15) k is in some B φ, (cid:31) (cid:15) ,...,(cid:15) k for one of the finitely manychoices of ( φ, (cid:31) ), adding up the estimates above over all the ( φ, (cid:31) ) gives thedesired inequality. (cid:4) .5 Proof of local volume bound In the previous section we established the volume bound for a single linearsubspace. We now combine this with the facts established in Section 3 toprove Proposition 5.1 (Local volume bound). Proof of Proposition 5.1 (Local volume bound). By Lemma 3.3, we get a fi-nite collection of semianalytic sets P V i that cover U ∩ P H , where U is someneighborhood of ¯ X in P H . Hence p − ( U ) ⊂ (cid:83) V i , and so µ M (cid:0) L M (cid:15) ,...,(cid:15) k ∩ p − ( U ) (cid:1) ≤ (cid:88) i µ M (cid:0) L M (cid:15) ,...,(cid:15) k ∩ V i (cid:1) . Let { W (cid:96),i } be the finite collection of all the linear subspaces W (cid:96) that weget from applying Lemma 4.6 to each of the sets V i . After summing, thatlemma gives (cid:88) i µ M (cid:0) L M (cid:15) ,...,(cid:15) k ∩ V i (cid:1) ≤ C · (cid:88) (cid:96),i µ W (cid:96),i (cid:16) L W (cid:96),i (cid:15) ,...,(cid:15) k (cid:17) , for some constant C .Then applying Lemma 5.7 to each subspace W (cid:96),i , we get (cid:88) (cid:96),i µ W (cid:96),i (cid:16) L W (cid:96),i (cid:15) ,...,(cid:15) k (cid:17) = O ( (cid:15) · · · (cid:15) k ) . Combining the three inequalities above gives µ M (cid:0) L M (cid:15) ,...,(cid:15) k ∩ p − ( U ) (cid:1) = O ( (cid:15) · · · (cid:15) k ) , as desired. (cid:4) To prove the main theorem, all that remains is to combine the local volumeestimate with compactness of P H . Proof of Theorem 1.3 given Proposition 5.1 (Local volume bound). We definean open cover { U X } X ∈ P H of P H as follows. For a boundary point X ∈ P H − P H , take U X to be the open set given by Proposition 5.1 (Local vol-ume bound). For a point X ∈ P H , we take U X to be any open neighborhoodof X contained in some compact subset of P H .51e have an open cover of the set P H , which is a compact space (Theorem2.3). Thus there is a finite subcover U X , , . . . , U X n . For the X i in this listthat are in the boundary, we use the volume estimate given in Proposition5.1 (Local volume bound). We take C (cid:48) to be the maximum of the C ’s givenby that proposition over the different X i in the boundary. So for such i , µ M ( p − ( U X i ) ∩ L M (cid:15) ,...,(cid:15) k ) ≤ C (cid:48) (cid:15) · · · (cid:15) k . For the X i not in the boundary, U X i is contained in some compact set K , andall surfaces in K have a uniform lower bound on saddle connection length.Thus if any of the (cid:15) i are small, the above estimate automatically holds, sincethe left hand side is zero. 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