An Analytic Proof of the Stable Reduction Theorem
aa r X i v : . [ m a t h . DG ] S e p AN ANALYTIC PROOF OF THE STABLE REDUCTION THEOREM
JIAN SONG ∗ , JACOB STURM ∗∗ , XIAOWEI WANG † ∗ A BSTRACT . The stable reduction theorem says that a family of curves of genus g ≥ C , using the K¨ahler-Einstein metricson the fibers to obtain the limiting stable curves at the punctures. Introduction
Let X , X , ... be a sequence of compact Riemann surfaces of genus g ≥
2. A con-sequence of the Deligne-Mumford construction of moduli space is the following. Thereexists
N > T i : X i ֒ → P N such that after passing to a subsequence, T i ( X i ) = W i ⊆ P N converges to a stable algebraic curve, i.e. a curve W ∞ ⊆ P N whose sin-gular locus is either empty or consists of nodes, and whose smooth locus carries a metricof constant negative curvature. The stable reduction theorem [DM] (stated below) is theanalogue of this result with { X i : i ∈ N } replaced by an algebraic family { X t : t ∈ ∆ ∗ } where ∆ ∗ ⊆ C is the punctured unit disk.The imbeddings T i are determined by a canonical (up to a uniformly bounded auto-morphism) basis of H ( X i , mK X i ) (here m ≥ T i explicitly? In Theorem 1.1 we give anaffirmative answer to this question.The main goal of this paper is to give an independent analytic proof of these algebraiccompactness results, which is the content of Theorem 1.2. We start with the Bers com-pactness theorem, which says that after passing to a subsequence, the X i converge to anodal curve in the Cheeger-Colding topology. We then use the technique of Donaldson-Sun [DS] which uses the K¨ahler-Einstein metric to build a bridge between analytic con-vergence (in Teichmuller space) to algebraic convergence (in projective space). The maindifficulty is that unlike the [DS] setting, the diameters of the X i are unbounded and as aconsequence, some of the pluri-canonical sections on X ∞ are not members of L ( X ∞ , ω KE ),so one can’t apply the L -Bergman imbedding/peak section method directly. In orderto solve this problem, we introduce the “ ǫ -Bergman inner product” on the vector space H ( X i , mK X i ), which is defined by the L norm on the thick part of the X i (unlike thestandard Bergman inner product which is the L norm defined by integration on all of X i ) and we show that for fixed m ≥ T i is an an orthnormalbasis for this new inner product. This establishes Theorem 1.1 which we then use toprove Theorem 1.2 (the stable reduction theorem).We start by reviewing the corresponding compactness results for Fano maniolds estab-lished by Donaldson-Sun in [DS]. Let ( X i , ω i ) be a sequence of K¨ahler-Einstein manifoldsof dimension n with c >
0, volume at least V and diameter at most D , normalized so thatRic( ω i ) = ω i . The first step in the proof of the Donaldson-Sun theorem is the application ∗ Research supported in part by National Science Foundation grant DMS-1711439, DMS-1609335 andSimons Foundation Mathematics and Physical Sciences-Collaboration Grants, Award Number: 631318. of Gromov’s compactness theorem which implies that after passing to a subsequence, X i converges to a compact metric space X ∞ of dimension n in the metric sense, i.e. theCheeger-Colding (CC) sense. This first step is not not available in the c < g ≥ T i : X i ֒ → P N withthe following properties. Let X i → X ∞ in the Cheeger-Colding sense as above. Thenthere is a K-stable algebraic variety W ∞ ⊆ P N such that if W i = T i ( X i ) then W i → W ∞ in the algebraic sense (i.e. as points in the Hilbert scheme). Moreover, T ∞ : X ∞ → W ∞ is a homeomorphism, biholomorphic on the smooth loci, where(1.1) T ∞ ( x ∞ ) = lim i →∞ T i ( x i ) whenever x i → x ∞ .We summarize this result with the following diagram:(1.2) X i W i P N X ∞ W ∞ P NT i CC Hilb T ∞ Here the vertical arrows represent convergence in the metric (Cheeger-Colding) senseand the the algebraic (Hilbert scheme) sense respectively. The horizontal arrows iso-morphisms: T i is an algebraic isomorphism, and T ∞ is a holomorphic isomorphism. For1 ≤ i ≤ ∞ , the maps W i ֒ → P N are inclusions.The imbeddings T i : X i → P N are the so called “Bergman imbeddings”. This means T i = ( s , ..., s N ) where the s α form an orthonormal basis of H ( X i , − mK X i ) with respectto the Bergman inner product:(1.3) Z X i ( s α , s β ) ω ni = δ α,β Here m is a fixed integer which is independent of i and the pointwise inner productis defined by ( s α , s β ) = s α ¯ s β ω mi . Since the definition of T i depends on the choice oforthonormal basis s = ( s , ..., s N ), we shall sometimes write T i = T i,s when we want tostress the dependence on s .Thus we assume Ric( ω i ) = − ω i and we wish to construct imbeddings T i : X i → P N such that the sequence W i = T i ( X i ) ⊆ P N converges to a singular K¨ahler-Einstein variety W ∞ with K W ∞ > W ∞ is a “singular K¨ahler-Einstein variety” can be made preciseas follows. Let W ⊆ P N be a projective variety with K W ample. The work of Berman-Guenancia [BG] combined with the results of Odaka [O] tell us that the following condi-tions are equivalent. (1) There is a K¨ahler metric ω on W reg such that Ric( ω ) = − ω satisfying the volumecondition R W reg ω n = c ( K W ) n .(2) W has at worst semi-log canonical singularities.(3) W is K-stableWe wish to construct T i in such a way that W ∞ = lim i →∞ T i ( X i ) has at worst semi-logcanonical singularities. In this paper we restrict our attention to the case n = 1.Our long-term goal is to generalize the above theorem of [DS] to the case where the( X i , ω i ) are smooth canonical models, of dimension n , i.e. X i is smooth and c ( X i ) < Remark 1.1.
One might guess, in parallel with the Fano setting, that the T i : X i → P N should be the pluricanonical Bergman imbeddings, that is T i = T i,s where s = ( s , ..., s N ) and the s α form an orthonormal basis of H ( X i , mK X i ) with respect to the inner product(1.3). But as we shall see, this doesn’t produce the correct limit, i.e. W ∞ , the limitingvariety, is not stable. In order to get the right imbedding into projective space, we needto replace T i,s with T ǫi,s , the so called ǫ -Bergman imbedding, defined below. We first need to establish some notation. Fix g ≥ ǫ >
0. If X is a compactRiemann surface of genus g , or more generally a stable analytic curve (i.e. a Riemannsurface with nodes whose universal cover is the Poincar´e disk) of genus g , we define the ǫ -thick part of X to be X ǫ := { x ∈ X : inj x ≥ ǫ } Here inj x is the injectivity radius at x and the metric ω on X is the unique hyperbolicmetric satisfying Ric( ω ) = − ω . It is well known that there exists ǫ ( g ) > X of genus g , and for all 0 < ǫ < ǫ ( g ), that X \ X ǫ is a finite disjoint union ofholomorphic annuli.Next we define the “ ǫ -Bergman imbedding” T ǫs : X → P N . Fix 0 < ǫ < ǫ ( g ) and fix m ≥
3. For each stable analytic curve of genus g , we choose a basis s = { s , ..., s N m } of H ( X, mK X ) such that Z X ǫ ( s α , s β ) ω = δ α,β Here ( s α , s β ) = s α ¯ s β ω − mi is the usual pointwise inner product. Such a basis is uniquelydetermined up to the action of U ( N +1). Let T ǫs : X ֒ → P N m be the map T ǫs = ( s , ..., s N m ).Let W = T ǫs ( X ). One easily checks that W is a stable algebraic curve and T ǫs : X → W is a biholomorphic map. In particular, we have the following simple lemma. Lemma 1.1. If X and X ′ are stable analytic curves, and s, s ′ are orthonormal bases for H ( X , mK X ) and H ( X , mK X ′ ) respectively, then the following conditions are equiva-lent(1) X ≈ X ′ (i.e. X and X ′ are biholomorphic).(2) [ T ǫs ′ ( X ′ )] ∈ U ( N + 1) · [ T ǫs ( X )] (3) [ T ǫs ′ ( X ′ )] ∈ SL ( N + 1 , C ) · [ T ǫs ( X )] Here [ T ǫs X ] ∈ Hilb is the point representing T ǫs X ⊆ P N in Hilb , the Hilbert scheme.
Now let X i be a sequence of stable analytic curves of genus g (e.g Riemann surfaces ofgenus g ). Then a basic theorem of Bers [B] (we shall outline the proof below) says thereexists a stable analytic curve X ∞ (for a precise definition see Definition 2.1) such thatafter passing to a subsequence, X i → X ∞ . By this we mean X reg i → X reg ∞ in the pointedCheeger-Colding topology (see Definition 2.2). Here, for 1 ≤ i ≤ ∞ , X reg i ⊆ X i is thesmooth locus. This provides the analogue of the left vertical arrow in (1.2). Theorem 1.1.
Let X i be a sequence of stable analytic curves of genus g . After passingto a subsequence we have X i → X ∞ in the Cheeger-Colding sense as above. Then thereis a stable algebraic curve W ∞ and orthonormal bases s i of H ( X i , mK X i ) , such that if W i = T ǫi ( X i ) then W i → W ∞ in the algebraic sense, i.e. as points in the Hilbert scheme.Moreover, T ∞ | X reg i satisfies property (1.1). The idea of using Teichmuller theory to understand moduli space was advocated by Bers[B, B1, B2, B3] in a of project he initiated, and which was later completed by Hubbard-Koch [HK]. They define an analytic quotient of “Augmented Teichmuller Space” whosequotient by the mapping class group is isomorphic to compactified moduli space as an-alytic spaces. Our approach is different and is concerned with the imbedding of theuniversal curve into projective space.
Remark 1.2. . As we vary ǫ , the maps T ǫi differ by uniformly bounded transformations.We shall see that if < ǫ , ǫ < ǫ ( g ) then T ǫ i = g i ◦ T ǫ i where the change of basis matrices g i ∈ GL ( N + 1 , C ) converge: g i → g ∞ ∈ GL ( N + 1 , C ) . In particular, lim i T ǫ i ( X i ) and lim i T ǫ i ( X i ) are isomorphic. As a corollary of our theorem we shall give a “metric” proof of the stable reductiontheorem due to Deligne-Mumford [DM]:
Theorem 1.2.
Let C be a smooth curve and f : X → C be a flat family of stableanalytic curves over a Zariski open subset C ⊆ C . Then there exist a branched cover ˜ C → C and a flat family ˜ f : ˜ X → ˜ C of stable analytic curves extending X × ˜ C C .Moreover, the extension is unique up to finite base change. In addition we show that the central fiber can be characterized as the Cheeger-Coldinglimit of the general fibers. More precisely:
Proposition 1.1.
Endow X t with its unique K¨ahler-Einstein metric normalized so that Ric( ω t ) = − ω t . Then for every t ∈ C there exist points p t , ...., p µt ∈ X t := f − ( t ) suchthat the pointed Cheeger-Colding limits Y j = lim t → ( X t , p jt ) are the connected componentsof ˜ X \ Σ where ˜ X := ˜ f − (0) and Σ ⊆ ˜ X is the set of nodes of ˜ X . Moreover the limitingmetric on X ∞ is its unique K¨ahler-Einstein metric. Remark 1.3.
A slightly modified proof also gives the log version of stable reduction, i.efor families ( X t , D t ) where D t is an effective divisor supported on n points and K X t + D t is ample. We indicate which modifications are necessary at the end of section 3. Remark 1.4. In [S] and [SSW] , Theorems 1.1 and Corollary 2.1 are shown to hold forsmooth canonical models of dimension n > . But these papers assume the general versionof Theorem 1.2, i.e. of stable reduction. In this paper we do not make these assumptions.In fact, our main purpose here is to prove these algebraic geometry results using analyticmethods. We shall first prove Theorem 1.1 under the assumption that the X i are smooth, andTheorem 1.2 under the assumption that the generic fiber of f smooth. Afterwards wewill treat the general case. 2. Background
Let X be a compact connected Hausdorff space, let r ≥ { z , ..., z r } ⊆ X . Wesay that X is a nodal analytic curve if X \ Σ is a disjoint union Y ∪ · · · ∪ Y µ of puncturedcompact Riemann surfaces and if for every z ∈ Σ, there is a small open set z ∈ U ⊆ X and a continuous function f : U → { ( x, y ) ∈ C : xy = 0 } with the properties:(1) f ( z ) = (0 , f is a homeomorphism onto its image(3) f | U \{ z } is holomorphicIf r = 0 then X is a compact Riemann surface. Definition 2.1.
We say that a nodal analytic curve X is a stable analytic curve if eachof the Y j is covered by the Poincar´e disk. In other words, each of the Y j carries a uniquehyperbolic metric (i.e. a metric whose curvature is − ) with finite volume. If X is a stable analytic curve we let K X be its canonical bundle. Thus the restrictionof K X to X \ Σ is the usual canonical bundle. Moreover, in the neighborhood of a point z ∈ Σ, that is in a neighborhood of of { xy = 0 } ⊆ C , a section of K X consists of a pairof meromorphic differential forms η and η defined on x = 0 and y = 0 respectively, withthe following properties: both are holomorphic away from the origin, both have at worstsimple poles at the origin, and res ( η ) + res ( η ) = 0.We briefly recall the proof of the above characterization of K X for nodal curves. Anodal singularity is Spec( B ) where B = C [ X, Y ] / ( Y − X ). Then C [ X ] → C [ X, Y ] isgenerated by Y which satisfies the monic equation Y − X = 0. According the Lipman’scharacterization of the canonical sheaf [Lip] if B = C [ Y ] / ( f ) where C = C [ X , ..., X n ]and f is a monic polynomial in Y with coefficients in C , and if X = Spec( B ), then K X is the sheaf of holomorphic ( n, n ) forms on X reg which can be written as F · π ∗ ( dx ∧··· dx n ) f ′ ( Y ) where π : X → Spec( C ) and F is a regular function on X . In our case, f ( Y ) = Y − X so f ′ ( Y ) = 2 Y which means that K X is free of rank one, generated by dx y or equivalently dxy . If we consider the map C → X given by t ( t, t ) then dxy pulls back to dtt . On theother hand, if we consider t ( t, − t ) then dxy pulls back to − dtt . If X is a compact Riemann surface of genus g ≥
2, then vol( X ) = 2 g −
2. If X is a stable analytic curve, we say that X has genus g if P j vol( Y j ) = 2 g −
2. Here thevolumes are measured with respect to the hyperbolic metric and the Y j are the irreduciblecomponents of X reg .Let X be a stable analytic curve. The following properties of K X are proved in Harris-Morrison [HM]:(1) h ( X, mK X ) = (2 m − g −
1) := N m − m ≥ mK X is very ample if m ≥ m ≥ m -pluricanonical imbedding of X is a stable algebraic curve in P N m Next we recall some basic results from Teichmuller theory. Fix g > S ,a smooth surface of genus g . Teichmuller space T g is the set of equivalence classes ofpairs ( X, f ) where X is a compact Riemann surface of genus g and f : S → X is adiffeomorphism. Two pairs ( X , f ) and ( X , f ) are equivalent if there is a bi-holomorphicmap h : X → X such that f − ◦ h ◦ f : S → S is in Diff ( S ), diffeomorphisms isotopicto the identity. The pair ( X, f ) is called a “marked Riemann surface”. The space T g hasa natural topology: A sequence τ n ∈ T g converges to τ ∞ if we can find representatives f n : S → X n , 1 ≤ n ≤ ∞ such that the sequence of diffeomorphisms f − ∞ ◦ h ◦ f n convergesto the identity.The space T g has a manifold structure given by Fenchel-Nielsen Coordinates whoseconstruction we now recall. Choose a graph Γ with the following properties: Γ has g vertices, each vertex is connected to three edges (which are not necessarily distinct sincewe allow an edge to connect a vertex to itself). For example, if g = 2, then there are twosuch graphs: Either v and v are connected by three edges, or they are connected by oneedge, and each connected to itself by one edge.Fix such a graph Γ. It has 3 g − e , ..., e g − on the edges.Once we fix Γ and we fix an edge ordering, we can define a map ( R + × R ) n → T g asfollows. Given ( l , θ , ..., l n , θ n ) ∈ R n we associate to each vertex v ∈ Γ the pair of pantswhose geodesic boundary circles have lengths ( l i , l j , l k ) where e i , e j , e k are the three edgesemanating from v . Each of those circles contains two canonically defined points, whichare the endpoints of the unique geodesic segment joining it to the other geodesic boundarycircles.If all the θ j = 0, then we join the pants together, using the rules imposed by the graphΓ, in such a way that canonical points are identified. If some of the θ j are non-zero, thenwe rotate an angle of l j θ j before joining the boundary curves together.Thus we see that T g is a manifold which is covered by a finite number of coordinatecharts corresponding to different graphs Γ (each diffeomorphic to ( R + × R ) n ) If we allowsome of the l j to equal zero, then we can still glue the pants together as above, but thistime we get a nodal curve. In this way, ( R ≥ × R ) n parametrizes all stable analytic curves.Teichmuller proved that the manifold T g has a natural complex structure, and thatthere exists a universal curve C g → T g , which is a map between complex manifolds, suchthat the fiber above ( X, f ) ∈ T g is isomorphic to X . Moreover, if X → B is any family ofmarked Riemann surfaces, then there exists a unique holomorphic map B → T g such that X is the pullback of C g . Fenchel-Nielsen coordinates are compatible with the complexstructure, i.e. they are smooth, but not holomorphic (although they are real-analytic). Remark 2.1.
One consequence of Teichmuller’s theorem is the following. Let
X → B bea holomorphic family of marked Riemann surfaces and let F : B → ( R + × R ) n be the mapthat sends t to the Fenchel-Nielsen coordinates of X t . Then F is a smooth function. Inparticular, X t → X . This shows that in the stable reduction theorem, if a smooth fill-inexists then it is unique. Now let X be a compact Riemann surface. A theorem of Bers [B], Theorem 15 (a sharpversion appears in Parlier [P], Theorem 1.1) says that for g ≥ C ( g ), now known as the Bers constant, with the following property. For every Riemannsurface X of genus g there exists a representative τ = ( X, f ) ∈ T g and a graph Γ (i.e. acoordinate chart) such that the Fenchel-Nielsen coordinates of τ are all bounded aboveby C ( g ). This is analogous to the fact that P N is covered by N + 1 coordinate charts,each biholomorphic to C N , and that give a point x ∈ P N we can choose a coordinatechart so that x ∈ C N has the property | x j | ≤ j . In particular, this proves P N issequentially compact.Bers [B] uses the existence of the Bers constant to show that the space of stable analyticcurves is compact with respect to a natural topology (equivalent to the Cheeger-Coldingtopology). For the convenience of the reader, we recall the short argument. Let X j bea sequence of Riemann surfaces. Then after passing to a subsequence, there is a graphΓ and representatives τ j = ( X j , f j ) ∈ T g such that the Fenchel-Nielsen coordinates of τ j with respect to Γ are all bounded above by C ( g ) (this is due to the fact that thereare only finite many allowable graphs). After passing to a further subsequence, we see τ j → τ ∞ ∈ ( R ≥ × R ) n . If τ ∞ ∈ ( R + × R ) n then the limit is a smooth Riemann surface.Otherwise, it is a stable analytic curve X ∞ . Thus(2.4) X ∞ = ∪ µi =1 X α , and X reg ∞ = ⊔ µα =1 Y α where the second union is disjoint, and Y α = X α \ F α where X α is a compact Riemannsurface and F α ⊆ X α a finite set, consisting of the cusps. Corollary 2.1.
Let p α ∞ ∈ Y α . Then there exist p i , ...., p µi ∈ X i such that in the pointedCheeger-Colding topology, ( Y α , p α ∞ ) = lim j →∞ ( X j , p αj ) . Moreover, for every open set p α ∞ ∈ U α ∞ ⊆⊆ Y α there exist open sets p αi ⊆ U αi ⊆ X i and diffeomorphisms f αj : U α ∞ → U αj so that ( f αj ) ∗ ω αj → ω α ∞ and ( f αj ) ∗ J αj → J α ∞ where ω αj and ω α ∞ are the hyperbolic metricson U αj and U α ∞ , and J αj and J α ∞ are the complex structures on U αj and U α ∞ Definition 2.2.
In the notation of Corollary 2.1, we shall say ω j → ω ∞ in the pointedCheeger-Colding sense and we shall write X i → X ∞ . Remark: Odaka [O2] uses pants decompositions to construct a “tropical compactification”of moduli space which attaches metrized graphs (of one real dimension) to the boundaryof moduli space. These interesting compactifications are compact Hausdorff topologicalspaces but are no longer algebraic varieties. Limits of Bergman imbeddings
Now let X be as in the theorem, and let t i ∈ C with t i →
0. Let X i = X t i andfix a pants decomposition of X i . Then Bers’ theorem implies that after passing to asubsequence we can find a nodal curve X ∞ as above so that X j → X ∞ .In order to prove the theorem, we must show:(1) X ∞ is independent of the choice of subsequence.(2) After making a finite base change, we can insert X ∞ as the central fiber in sucha way that the completed family is algebraic.We begin with (2). Let X be a hyperbolic Riemann surface with finite area (i.e. possiblynot compact, but only cusps). The Margulis “thin-thick decomposition” says that thereexists ǫ ( g ) > g − ǫ ( g ). Moreover, for every ǫ ≤ ǫ ( g ) the set X \ X ǫ := { x ∈ X : inj x < ǫ } is a finite union of of holomorphic annuli (which are open neighborhoods of short geodesics)if X is compact, and a finite union of annuli as well as punctured disks, which correspondto cusp neighborhoods if X is has singularities. We call these annuli “Margulis annuli”.Moreover, V ( ǫ ), the volume of X \ X ǫ , has the property lim ǫ → V ( ǫ ) = 0. An elementaryproof is given in Proposition 52, Chapter 14 of Donaldson [D].Now we define a modified Bergman kernel as follows: For convenience we write ǫ = ǫ ( g ).This is a positive constant, depending only on the genus g . Let X be a stable analyticcurve. For η , η ∈ H ( X, mK X ) let(3.5) h η , η i ǫ = Z X ǫ η ¯ η h mKE ω KE and k η k ǫ = h η, η i ǫ . If we replace X ǫ by X , we get the standard Bergman inner product.Now fix m ≥
3. Choosing orthonormal bases with respect to the inner product (3.5)defines imbeddings T ǫi : X i → P N m and T ǫ ∞ : X ∞ → P N m , which we call ǫ -Bergmanimbeddings. Our goal is to show Theorem 3.1.
Let X , X , ... be a sequence of compact Riemann surfaces of genus g .Then there exists a stable analytic curve X ∞ such that after passing to a subsequence, X i → X ∞ in the Cheeger-Colding topology. For ≤ i < ∞ , we fix an orthonormal basis s i of H ( X i , mK X ) . Then there exists a choice of orthonormal basis s ∞ for X ∞ such thatafter passing to a subsequence, (3.6) lim i →∞ T ǫi,s i = T ǫ ∞ ,s ∞ In other words, if x i ∈ X i and x ∞ ∈ X ∞ with x i → x ∞ , then T ǫi ( x i ) → T ǫ ∞ ( x ∞ )The proof of Theorem 3.1 rests upon the following. Theorem 3.2.
Fix g ≥ and m, ǫ > . Then there exist C ( g, m, ǫ ) with the followingproperty. k s k ǫ ≤ k s k ǫ/ ≤ C ( g, m, ǫ ) k s k ǫ for all Riemann surfaces X of genus g and all s ∈ H ( X, mK X ) . To prove the theorem, we need the following adapted version of a result of Donaldson-Sun. We omit the proof which is very similar to [DS] (actually easier since the onlysingularities of X ∞ are nodes so the pointed limit of the X i in the Cheeger-Coldingtopology is smooth). Proposition 3.1.
Let X i → X ∞ be a sequence of Riemann surfaces of genus g con-verging in the pointed Cheeger-Colding sense to a stable curve X ∞ . Fix { s ∞ , ..., s ∞ M } ⊆ H ( X ∞ , mK X ∞ ) an ǫ -orthonormal basis of the bounded sections (i.e. the L ( X ∞ ) sec-tions, i.e. the sections which vanish at all nodes). Then there exists an ǫ -orthonormalsubset { s i , ..., s iM } ⊆ H ( X i , mK X i ) such that for ≤ α ≤ M , we have s iα → s ∞ α in L and uniformly on compact subsets of X reg ∞ . In particular, if x i ∈ X reg i (3.7) x i → x ∞ ⇐⇒ s iα ( x i ) → s ∞ α ( x ∞ ) for all 0 ≤ α ≤ M ⇐⇒ T ν,ǫi ( x i ) → T ν,ǫ ∞ ( x ∞ ) .where T ν,ǫi : X reg i ֒ → P M is the map x i ( s i , ..., s iM )( x i ) for ≤ i ≤ ∞ .Proof of theorem 3.2. Let X i → X ∞ as in Proposition 3.1. Choose ( s ∞ , ..., s ∞ M ) and( t ∞ , ..., t ∞ M ) which are ǫ and ǫ/ H ( X ∞ , mK X ∞ ) in such a way that t ∞ α = λ ∞ α s ∞ α for real numbers 0 < λ ∞ α <
1. Choose s iα → s ∞ α and t iα → t ∞ α as in Proposition 3.1 in such a way that t iα = λ iα s iα with 0 < λ iα < λ iα → λ ∞ α > ≤ α ≤ M Choose additional sections s iα and t iα for M +1 ≤ α ≤ N so that { s i , ..., s iN } and { t i , ..., t iN } are ǫ and ǫ/ H ( X i , mK X i ) and t iα = λ iα s iα with 0 < λ iα < ≤ α ≤ N .Now assume the theorem is false. Then there exists X i → X ∞ as above such that λ iα → α . We must have α ≥ M + 1 by (3.8). Choose M + 1 ≤ A < N suchthat λ iα → A ≤ α ≤ N .Since k s iα k L ( X ǫ ) = 1 we may choose s ∞ α ( ǫ ) ∈ H ( X ǫi , K X ∞ | X ǫ ∞ ) such that(3.9) s iα | X ǫi → s ∞ α ( ǫ ) for M + 1 ≤ α ≤ N uniformly on compact subsets of X ǫ Let T ǫi : X i → W ǫi ⊆ P N be the Kodaira map given by the sections s i , ..., s iN and let W ǫ ∞ = lim i →∞ W ǫi . Let T ǫ ∞ : X ǫ ∞ ֒ → W ǫ ∞ and T ν,ǫ ∞ : X reg ∞ ֒ → P M be the Kodaira maps given by ( s ∞ , ..., s ∞ M , s ∞ M +1 ( ǫ ) , ..., s ∞ N ( ǫ )) and ( s ∞ , ..., s ∞ M ). Thus(3.10) π ◦ T ǫ ∞ = T ν,ǫ ∞ | X ǫ ∞ where π : P NM := P N \{ z = · · · = z M = 0 } → P M is defined by ( z , ..., z N ) ( z , ..., z M ).Moreover π ( W ǫ ∞ ∩ P NM ) ⊆ T ν,ǫ ∞ ( X reg ∞ )Now the definition of A implies T ǫ/ ∞ ( X ǫ ∞ ) ⊆ Z ǫ/ ∞ := { z ∈ W ǫ/ ∞ : z A = z A +1 = · · · = z N = 0 } Thus (3.10) implies T ν,ǫ/ ∞ ( X reg ∞ ) ⊃ π ( Z ǫ/ ∞ ∩ P NM ) ⊃ π ( T ǫ/ ∞ ( X ǫ ∞ )) = T ν,ǫ/ ∞ ( X ǫ ∞ )Since the second set is constructible, π ( Z ǫ/ ∞ ∩ P NM ) = T ν,ǫ/ ∞ ( X reg ∞ \ Σ ǫ )where Σ ǫ ⊆ X reg ∞ \ X ǫ ∞ is a finite set. Moreover, Σ ǫ is monotone in ǫ .Let x ∞ ∈ X reg ∞ \ Σ ǫ . Then T ν,ǫ/ ∞ ( x ∞ ) = π ( w ∞ ) for some w ∞ ∈ Z ǫ/ ∞ ∩ P NM . Choose w i ∈ W ǫ/ i such that w i → w ∞ and choose x i ∈ X i such that T ǫ/ i ( x i ) = w i . Then (3.7)implies T ǫ/ i ( x i ) → w ∞ = ⇒ π ( T ǫ/ i ( x i )) → π ( w ∞ ) = ⇒ T ν,ǫ/ i ( x i ) → T ν,ǫ/ ∞ ( x ∞ ) = ⇒ x i → x ∞ Thus we see that if x ∞ ∈ X reg ∞ \ Σ ǫ there exists x i → x ∞ such thatlim i →∞ T ǫ/ i ( x i ) ∈ Z ǫ/ ∞ Lemma 3.1. If ǫ < ǫ then Σ ǫ ⊆ Σ ǫ ∩ X ∞ \ X ǫ/ ∞ .Proof. Assume x ∞ / ∈ Σ ǫ . Then there exists x i → x ∞ such that lim i →∞ s iα ( x i ) = 0 forall A ≤ α ≤ N . Then lim i →∞ t iα ( x i ) = lim i →∞ λ iα s iα ( x i ) = 0 since 0 < λ iα < (cid:3) We may there assume, after possibly decreasing ǫ , that Σ ǫ = ∅ . This means that forevery x ∞ ∈ X reg ∞ there exists x i → x ∞ such that lim i →∞ s iα ( x i ) = 0 for all A ≤ α ≤ N .Let x ∞ ∈ X reg ∞ . We say that x ∞ is an ǫ -good point if for every x i → x ∞ , lim i →∞ s iα ( x i ) =0 for all A ≤ α ≤ N . The set of ǫ -bad points is finite (otherwise W ǫ ∞ would have infinitelymany components by the intermediate value theorem). Also, every point in X ǫ ∞ is ǫ -good. Lemma 3.2. If x ∞ is ǫ -good then it is ǫ -good for all ǫ < ǫ .Proof. Assume x ∞ is ǫ good, let x i → x ∞ and let ǫ < ǫ . Then lim i →∞ t iα ( x i ) =lim i →∞ λ iα s iα ( x i ) = 0 since 0 < λ iα < (cid:3) Lemma 3.2 implies that by decreasing ǫ if necessary, that all points x ∞ ∈ X reg ∞ are ǫ -good. But k s iA k ǫ = 1 so there exists x i ∈ X ǫi such that | s iA ( x i ) | = 1. After passing to asubsequence, x i → x ∞ ∈ X ǫ ∞ but lim i | s iA ( x i ) | = 1 = 0 a contradiction.We conclude that if η j ∈ H ( X j , mK X j ) is a sequence such that the norms k η j k ǫ = h η j , η j i ǫ = 1, then after passing to a subsequence, we have ( f αj ) ∗ η j → η ∞ for some η ∞ ∈ H ( X reg ∞ , mK X ∞ | X reg ∞ ) with k η k ǫ = 1. Here the f αj : U αj → U α are as in thestatement of Corollary 1 and this is true for all U α and all α . Moreover, an orthornormalbasis of H ( X j , mK X j ), which is a vector space of dimension (2 m − g − to an orthonormal set of (2 m − g −
1) elements in H ( X reg ∞ , mK X ∞ ). The main problemis to now show that these (2 m − g −
1) elements extend to elements of H ( X ∞ , mK X ∞ ).If they extend, then they automatically form a basis since H ( X ∞ , mK X ∞ ) has dimension(2 m − g −
1) and this would prove Theorem 3.1.To proceed, we make use of the discussion of the Margulis collar in section 14.4.1 of[D]. Let λ > C a collapsing geodesic in X j which forms a node in thelimit in X ∞ . We fix j and we write X = X j . Let A λ = { z ∈ C : 1 ≤ | z | ≤ e πλ , λ ≤ arg( z ) ≤ π − λ } / ∼ where the equivalence relation identifies the circles | z | = 1 and | z | = e πλ . Then [D]shows A injects holomorphically into X in such a way that 1 ≤ y ≤ e πλ maps to C . Thepoint is that the segment 1 ≤ y ≤ e πλ is very short - it has size λ . But the segments A ∩ { arg( z ) = λ } and A ∩ { arg( z ) = π − λ } have size 1. So for λ small, A is a topologicallya cylinder, but metrically very long and narrow in the middle but not narrow at the ends.In other words, the middle of A is in the thin part, but the boundary curves are in thethick part.The transformation τ = exp (cid:0) i ln zλ (cid:1) maps A λ to the annulus A ′ λ = { exp( − ( π − λ ) /λ ) ≤ | τ | ≤ exp( − } To summarize: We are given a sequence X j , and a geodesic C j in X j that collapses to anode ν in Y α for some α . We are also given a sequence of orthonormal bases { η j, , ..., η j,N } of H ( X j , kK X j ) where N = (2 k − g −
1) and η j,µ → η ∞ ,µ . Here η ∞ ,µ is a section of kK X ∞ on X reg ∞ . Fix µ and write η j = η j,µ and η ∞ = η ∞ ,µ . We need to show that η ∞ extends to all of X ∞ .We may view η j as a k form on A λ j or on A ′ λ j and η ∞ as a k form on the punctureddisk A ′ . Write η j = f j ( z ) dz k = h j ( τ ) dτ k and η ∞ = h ∞ ( τ ) dτ k . The discussion in [D]shows that if we fix a relatively compact open subset U ⊆ A ′ , then h j → h ∞ uniformlyon U (this makes sense since U ⊆ A ′ λ j for j sufficiently large).Since k η j k L = 1 we have uniform sup norm bounds on the thick part of X j . Thus(3.11) k η j k L ∞ (( X i ) ǫ ≤ C ( ǫ )We want to use (3.11) to get a bound on the thin part. In z coordinates, (3.11) implies(3.12) | η | ω = | Im( z ) | k · | f ( z ) | ≤ C ( ǫ ) if arg( z ) = λ or arg( z ) = 2 π − λ since the boundary curves arg( z ) = λ and arg( z ) = 2 π − λ are in the thick part. Here wewrite η for η j and f for f j to lighten the notation. Now(3.13) Im( z ) = − exp( λ arg τ )(sin( λ ln | τ | )if we write f ( z ) = g ( τ ), then (3.12) implies(3.14) | g ( τ ) | ≤ C ( ǫ ) λ k for τ ∈ ∂A ′ Since f ( z ) dz k = h ( τ ) dτ k = g ( τ )( dzdτ ) k dτ k and dzdτ = zλiτ we see for λ small | h j ( τ ) | ≤ λ k (cid:12)(cid:12)(cid:12)(cid:12) dzdτ (cid:12)(cid:12)(cid:12)(cid:12) k = 1 λ k | z | k λ k | τ | k ≤ | τ | k where the last inequality follows from the fact 1 ≤ | z | ≤
2. Writing u j ( τ ) := h j ( τ ) τ k Thus we see | u j ( τ ) | ≤ τ ∈ ∂A ′ . The maximum principle now implies that | u j ( τ ) | ≤ τ ∈ A ′ . Since this is true for all X i , we see that any limit u ∞ must satisfy the sameinequality in the limit of the annuli, which is a punctured disk: | h ∞ ( τ ) | · | τ | k ≤ C . Thisshows h ∞ has at most a pole of order k .Moreover u (0) is the residue(3.15) u (0) = lim j →∞ π √− Z | τ | = r u j ( τ ) dττ Here 0 < r ≤ exp( −
1) is any fixed number (independent of j ).To summarize, we have now seen that a collar degenerates to a union of two punc-tured disks and so the limit of the η j is a pair of k forms, η ∞ = u ∞ ( τ ) (cid:0) dττ (cid:1) k and˜ η ∞ = ˜ u ∞ ( τ ′ ) (cid:0) dτ ′ τ ′ (cid:1) k where u and ˜ u are holomorphic in a neighborhood of the origin in C .There is one final condition that we need to check in order to verify that the limit is in H ( X ∞ , kK X ∞ ): Let R = exp( − r = exp( − π/ λ j ) and ǫ = exp( − π/λ j ) (so ǫ λ j /r = r ).We must show ˜ u (0) = ( − k u (0).(Here the inner white disk is | τ | ≤ ǫ/R ). To check this, let ˜ τ = ǫ j τ . Then Figure 1 remainsthe same, with τ replaced by ˜ τ and f ( z ) dz k = u j ( τ ) (cid:18) dττ (cid:19) k = u j ( ǫ j / ˜ τ )( − k (cid:18) d ˜ τ ˜ τ (cid:19) k := ˜ u j (˜ τ ) (cid:18) d ˜ τ ˜ τ (cid:19) k Now we see Z | τ | = r u j ( τ ) dττ = ( − Z | ˜ τ | = ǫ j /r u j ( ǫ j ˜ τ ) ( − d ˜ τ ˜ τ = ( − k Z | ˜ τ | = r ˜ u j (˜ τ ) d ˜ τ ˜ τ In the second integral, the factor of ( −
1) outside the integral is due to the fact that theorientation of the circle has been reversed and the ( −
1) inside the integral comes from thechange of variables. The second identity is a result of the fact that u (˜ τ ) is holomorphic on ǫ/R ≤ | τ | ≤ r | τ | = Rr ≤ | τ | ≤ R Figure 1. A ′ λ the annulus { ˜ τ ∈ C : ǫ j /r < ˜ τ < r } . Taking limits as j → ∞ we obtain ˜ u (0) = ( − k u (0).This establishes Theorem 3.1 when the X i are smooth.Now assume the X i are stable analytic curves, but not necessarily smooth. The Fenchel-Nielsen coordinates of X i determine a point [ X i ] ∈ ( R ≥ × R ) n . The simple observationwe need is that ( R > × R ) n ⊆ ( R ≥ × R ) n is dense so we may choose a smooth Riemannsurface ˜ X i such that [ X i ] ∈ ( R ≥ × R ) n is ǫ i close to [ X i ] where ǫ i → X i issmoothable). Now Corollary 2.1 implies that after passing to a subsequence, ˜ X i → X ∞ in the pointed Cheeger-Colding topology. We conclude that X i → X ∞ as well. Moreover,one easily sees that T ǫi and ˜ T ǫi have the same limit. This proves (3.6) and completes theproof of Theorem 3.1 (cid:3) Remark 3.1.
The proof of the log version Theorem 3.1 is almost the same. The onlyobservation we need is the following. If X is a compact Riemann surface and D = p + · · · + p n is a divisor supported on n points such that K X + D is ample, then X \ D has a unique metric ω such that Ric( ω ) = − ω and ω has cusp singularities at the points p j . Morover, just as in the case n = 0 , X has a pants decomposition. The only differenceis that we allow some of the length parameters to vanish, but this doesn’t affect the ar-guments. In particular, we can use the Fenchel-Nielson coordinates to find a limit of the ( X j , D j ) (after passing to a subsequence) and the T ǫj are defined exactly as before. Now suppose X i is a sequence of compact Riemann surfaces of genus g converginganalytically to a nodal curve X ∞ and let η i be a K¨ahler metric on X i is the same classas the K¨ahler-Einstein metric ω i . We have seen that ω i → ω ∞ , the K¨ahler-Einsteinmetric on X ∞ , in the pointed Cheeger-Colding sense. Let ˜ ω ∞ be a K¨ahler metric on X reg ∞ and assume ˜ ω i → ˜ ω ∞ in the pointed Cheeger-Colding sense. Let T i (˜ ω i ) : X i → P N bethe embedding defined by an orthonormal basis of H ( X i , K X i ) using the metric ˜ ω i onthe thick part of X i and define T ∞ (˜ ω ∞ ) : X ∞ → P N similarly. Thus the T i and T ∞ ofTheorem 2 can be written as T i ( ω i ) and T ∞ ( ω ∞ ) and in this notation, Theorem 2 says T i ( ω i ) → T ∞ ( ω ∞ ) Corollary 3.1.
After passing to a subsequence T i (˜ ω i ) → T ∞ (˜ ω ∞ ) Proof.
Since ˜ ω ∞ and ω ∞ are equivalent on the thick part of X ∞ , we see that T i (˜ ω i ) = γ i ◦ T i ( ω i )where γ i ∈ GL ( N + 1 , C ) has uniformly bounded entries as does γ − i . Thus after passingto a subsequence, γ i → γ ∞ ∈ GL ( N + 1 , C ) andlim i →∞ T i (˜ ω i ) = lim i →∞ γ i ◦ T i ( ω ∞ ) = γ ∞ ◦ T ∞ ( ω ∞ ) = T ∞ (˜ ω ∞ )Remark: The proof shows we only need to assume ˜ ω i → ˜ ω ∞ on the thick part of X ∞ .4. Existence of stable fill-in
Proof.
Let f : X → C = C \{ p , ..., p m } be a flat family of stable analytic curves ofgenus g ≥
2. We shall assume the fibers are smooth since except for some additionalnoation, the general case is proved in exactly the same way. We first observe that wecan find some completion (not necessarily nodal)
Y → C of the family X → C . To seethis let Ω X /C be the sheaf of relative differential forms (i.e. the relative canonical linebundle when X is smooth). Then the Hodge bundle f ∗ K X /C is a vector bundle over C of rank 3 g − f ∗ K ⊗ m X /C is a vector bundle E m of rank N m − m − g −
1) for m ≥
2. Choose E m → C an extension of the vector bundle E m → C to the curve C .For example, let U ⊆ C be any affine open subset over which E m is trivial and let s , ..., s N m be a fixed O ( U ) basis. Then if p j ∈ V ⊆ C is an affine open set such that V \{ p j } ⊆ U , then define E ( V ) to be the O ( V ) submodule of E ( V \{ p j } ) spanned by the s α .Once E is fixed, we choose m ≥ X ֒ → P ( E ) ⊆ P ( E ) be the canonicalimbedding. Then we define(4.1) Y ⊆ P ( E )to be the flat limit of X → C inside P ( E ) → C .Now we prove Theorem 1. To lighten the notation, we shall assume m = 1 and write C = C \{ } where 0 := p . Suppose t i ∈ C with t i → X t i → X ∞ where X ∞ is an stable analytic curve. We wish to show that thereexists a smooth curve ˜ C and a finite cover µ : ˜ C → C with the following property. If welet Σ = µ − (0) (a finite set) there exists a unique completion ˜ f : ˜ X → ˜ C of µ ∗ X → ˜ C \ Σwith X ∞ = p − (˜0) for all ˜0 ∈ Σ.Define Z = { ( t, z ) ∈ C × Hilb( P N m ) : z ∈ T t } where T t is the set of all Hilbert points [ T ( X t )]. Here T : X t → P N m ranges over the setof all Bergman imbeddings. In particular, T t ⊆ Hilb( P N m ) is a single G = SL ( N m + 1)orbit. We claim that Z ⊆ C × Hilb( P N m ) is a constructible subset. To see this, let U ⊆ C be an affine open subset and let σ , ..., σ N m be a fixed O ( U ) basis of E m ( U ). This basisdefines an imbedding(4.2) S : π − ( U ) → U × P N m given by x ( π ( x ) , σ ( x ) , ..., σ N m ( x )). Define H : U → Hilb( P N m ) by H ( t ) = Hilb( S ( X t ))and define the map f U : G × U → U × Hilb( P N m ) given by ( g, t ) ( t, g · H ( t ))Then f U is an algebraic map so its image is constructible. This shows Z | U is constructiblefor every affine subset U ⊆ C and hence Z is constructible.Now we fix 0 < ǫ < ǫ ( g ) and let W j = T j ( X t j ) where T j is the ǫ -Bergman imbedding.Then (3.6) implies T j ( X j ) = W j → T ∞ ( X ∞ ) = W ∞ , a stable algebraic curve in P N m . Let Z → C be the closure of Z in C × Hilb( P N m ) ⊆ C × P M . Here P M ⊃ Hilb( P N m ) is chosenso that there is a G action on P M which restricts to the G action on Hilb( P N m ). Then Z is a subvariety of C × Hilb( P N m ) whose dimension we denote by d . Let Z t the fiber of Z above t ∈ C . Then [ Y ∞ ] ∈ Z .To construct ˜ C we use the Luna Slice Theorem: There exists W ⊆ C M +1 a G [ Y ] invariant subspace such that [ Y ∞ ] ∈ P ( W ) ⊆ P M and such that the map P ( W ) × Lie( G ) → P M given by ( x, ξ ) exp( ξ ) x is a diffeomorphism of some small neighborhood U W × V ⊆ P ( W ) × Lie( G ) onto an openset Ω ⊆ P M , with U W ⊆ P ( W ) invariant under the finite group G [ Y ] . After shrinking U W if necessary, the intersection of a G orbit with U W \ [ Y ] is a finite set of order m | m where m = | G [ Y ] | . In other words, the quotient G [ Y ] \ U W parametrizes the G -orbits in P M thatintersect U W .Note that Ω contains ( t i , [ Y i ]) for infinitely many i so ( C × P ( W )) ∩ Z is a projectivevariety C of dimension at least one. Moreover, if we let C be the union of the componentsof C containing { } × [ Y ∞ ], then C contains infinitely many of ( t i , [ Y i ]) so the image of C → C contains infinitely many t i and thus C → C is surjective. On the other hand, C → C is finite of degree m (this follows from the construction of U ( W )).Let ˜ C ⊆ C be an irreducible component of C containing ( t i , [ Y i ]) for infinitely many i . Let H ⊆ G [ Y ∞ ] be the set of all σ ∈ G [ Y ∞ ] such that σ ( ˜ C ) = ˜ C . Then H has order d for some d | m and ˜ C → C is finite of degree d .Finally, we have ˜ C ⊆ Z ⊆ C × Hilb( P N m ). This gives us a map ˜ C → Hilb( P N m ). If wepull back the universal family we get a flat family ˜ X → ˜ C which extends X × ˜ C C . Thiscompletes the proof of Theorem 1.2. (cid:3) Uniqueness of the stable fill-in
Let π : X ∗ → ∆ ∗ ⊂ ∆ be an algebraic family of smooth curves genus g . We claim thatthere exists a unique stable analytic curve X such that X t → X in the Cheeger-Colding sense as t →
0. This will establish the uniqueness statement of Theorem 1.2, and sinceexistence was demonstrated in the previous section, it completes the proof.Let S : X ∗ → ∆ ∗ × P N m as in (4.2). For each t ∈ ∆ ∗ , the set σ t = ( σ ( t ) , ..., σ N m ( t ))is a basis of H ( X t , mK X t ). Let s t = ( s ( t ) , ..., s N m ( t )) be the orthonormal basis of H ( X t , mK X t ) obtained by applying the Gram-Schmidt process to the basis σ t and let T ǫt : X t → P N be the map T ǫt = T ǫs t . Here 0 < ǫ < ǫ ( g ) is fixed once and for all. Remark2.1 implies that t [ T ǫt ( X t )] defines a continuous function ∆ ∗ → Hilb. Let z : ∆ ∗ × SL ( N + 1 , C ) → ∆ ∗ × Hilband f : ∆ ∗ → ∆ ∗ × Hilbbe the maps z ( t, g ) = ( t, g · [ T t ( X t )]) and f ( t ) = z ( t, [ T t ( X t )]).Let F = Im( f ) ⊆ ∆ × Hilb and Z = Im z ⊆ ∆ × Hilb. Let π F : F → ∆ and π Z : Z → ∆be the projection maps and F = π − F (0), Z = π − Z (0). Observe that F ⊆ Hilb is closedand connected (this easily follows from the fact that ∆ ∗ is connected and Hilb is compactand connected). Moreover, Theorem 3.1 implies that every element of F is of the form T ǫs ( X ) for some stable analytic curve X and some basis s .Claim: F is contained in the U ( N + 1) orbit of [ X ].Assume the claim for the moment, and let’s show that it implies uniqueness. Supposethere exist subsequences t i , t ′ i ∈ ∆ ∗ such that X t i → X and X t ′ i → X ′ . We must showthat X ≈ X ′ , i.e. X and X ′ are isomorphic stable analytic curves. Theorem 3.1 impliesthere are bases s and s ′ such that [ T ǫs ( X )] , [ T ǫs ′ ( X ′ )] ∈ F so T ǫs ′ u ( X ′ ) ∈ U ( N +1) · T ǫs ( X ).Now Lemma 1.1 implies X ≈ X ′ . This gives uniqueness.The set U = SL ( N + 1 , C ) · [ T ǫs ( X )] ⊆ Z is open since dim Z = dim SL ( N + 1 , C ) andthe stabilizer of [ T ǫs ( X ] is finite. Lemma 1.1 implies(5.1) F ∩ U ⊆ U ( N + 1)[ T ǫs ( X )] ⊆ U Now U ( N + 1)[ T ǫs ( X )] is compact and F is connected, so F ∩ U = F . Thus the claimfollows from (5.1). (cid:3) References [BG] Berman, R. and Guenancia, H., “K¨ahler-Einstein metrics on stable varieties and log canon-ical pairs”, Geom. Funct. Anal. 24 (2014), no. 6, 1683–1730.[B] Bers, L., “Spaces of degenerating Riemann surfaces”, Discontinuous groups and Riemannsurfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), pp. 43–55. Ann. of Math.Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974.[B1] Bers, L., “On spaces of Riemann surfaces with nodes” Bull. Amer. Math. Soc. 80 (1974),1219–1222.[B2] Bers, L., “Deformations and moduli of Riemann surfaces with nodes and signatures”, Math.Scand. 36 (1975), 12–16 , 2943–2955[Ha] R. Hartshorne, “Algebraic Geometry”, Graduate Texts in Mathematics, 52. Springer-Verlag, New York, 1977[HK] Hubbard, J. H., and Koch, Sarah, “Analytic construction of the Deligne-Mumford com-pactification of the moduli space of curves”, J. Diff. Geom. 98 (2014), no. 2, 261–313[HM] J. Harris and I. Morrison, “Moduli of curves”, Graduate Texts in Mathematics, 187.Springer-Verlag, New York, 1998[LWX] C. Li, Wang, X. and Xu, C., “Degeneration of Fano K¨ahler-Einstein manifolds”,http://arxiv.org/abs/1411.0761[M] Mok, N., “Compactification of complete K¨ahler surfaces of finite volume satisfying certaincurvature conditions”, Ann. of Math. (2) 129 (1989), no. 2, 383–425.[MZ] Mok, N. and Zhong, J.Q. “Compactifying complete K¨ahler-Einstein manifolds of finitetopological type and bounded curvature”, Ann. of Math. (2) 129 (1989), no. 3, 427–470[O] Odaka, Y., “The GIT-stability of polarised varieties via discrepancy” Ann. Math., (2) 177645–661 (2013)[O2] Odaka, Y., “Tropical geometric compactification of moduli, I-Mg case.” Moduli of K-stablevarieties, 75–101, Springer INdAM Ser., 31, Springer, Cham, 2019.[P] H. Parlier, “A short note on short pants”, arxiv.org/pdf/1304.7515.pdf [PW] P. Petersen and G. Wei, “Analysis and geometry on manifolds with integral Ricci curva-ture bounds, II” Trans. of the AMS, Volume 353, Number , 457–478[V] Vakil, R., “Foundations of algebraic geometry”, book on author’s web page:http://math.stanford.edu/ ∼ vakil/216blog/FOAGjun1113public.pdf[DY] Yang, D., “Convergence of Riemannian manifolds with integral bounds on curvature I”,Ann. Scient. Ecole Norm. Sup. (4) 25 (1992) 77-105.[S] Song, J., “Degeneration of Kahler-Einstein manifolds of negative scalar curvature”arXiv:1706.01518[SSW] Song, J., Sturm, J. and Wang, X., “Riemannian geometry of Kahler-Einstein currents III:compactness of Kahler-Einstein manifolds of negative scalar curvature” arXiv:2003.04709[Y] Yau, S.T., “Survey on partial differential equations in differential geometry”, pp. 3–71, Ann.of Math. Stud., 102, Princeton Univ. Press, Princeton, N.J., 1982.[YSK] Yeung, Sai-Kee, “Compactification of K¨ahler manifolds with negative Ricci curvature”,Invent. Math. 106 (1991), no. 1, 13–25. ∗ Department of Mathematics, Rutgers University, Piscataway, NJ 08854 ∗∗ Department of Mathematics and Computer Science, Rutgers University, Newark,NJ 07102 ††