Pair-of-pants decompositions of 4-manifolds diffeomorphic to general type hypersurfaces
aa r X i v : . [ m a t h . DG ] F e b PAIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDSDIFFEOMORPHIC TO GENERAL TYPEHYPERSURFACES
YUGUANG ZHANG
Abstract.
In this paper, we show that a smooth 4-manifold diffeomor-phic to a complex hypersurface in CP of degree d > d ( d − copies of 4-dimensional pair-of-pants andcertain subsets of K3 surfaces. Introduction P of real dimension two, or complex dimension one, isdefined as the complement set of three generic points in CP , i.e. P = CP \{ , , ∞} . There is a unique complete Riemannian metric g on P with Gaussian curvature − g ( P ) = 2 π . There is afibration structure P → Y from P to a graph of Y-shape with genericfibres S , and one singular fibre of shape ⊖ over the vertex.If Σ is a compact Riemann surface of genus g >
2, then there is an opensubset Σ o ⊂ Σ consisting exactly of 2 g − \ Σ o is the disjoint union of 3 g − χ (Σ) = − π g − X Vol g ( P ) = 1 − gπ Vol g ( P ) . Figure 1.1.
Fibration P → Y, and two graphs associ-ated with pair-of-pants decompositions of Riemann surfacesof genus 2. (cid:0)(cid:0)(cid:0) ✈ Y P ⊖ = ✚✙✛✘✚✙✛✘✚✙✛✘✈ ✈ ✈ ✈ The combinatoric structure of the decomposition Σ = Σ o ∪ (Σ \ Σ o ) is rep-resented by a cubic graph, i.e. any vertex is of shape Y, where each pair-of-pants corresponds to one vertex, and any edge associates with a circle inΣ \ Σ o .In [40], Mikhalkin has generalised the notion of pair-of-pants to the caseof any even dimension, and proved that a smooth complex hypersurface in CP n +1 , and more general toric manifolds, admits pair-of-pants decomposi-tions. [24] studied pair-of-pants decompositions for real 4-dimensional man-ifolds from the topology perspective. It is shown in [24] that every finitelypresented group is the fundamental group of a 4-manifold admitting pair-of-pants decompositions. In this paper, we study some differential/algebraicgeometry aspects of pair-of-pants decompositions for general type hypersur-faces in CP .A pair-of-pants P of real dimension 4, equivalently complex dimension2, is defined as the complement of 4 general positioned lines D in CP , i.e. P = CP \ D (cf. [40]), where D can be chosen as D = { [ z , z , z ] ∈ CP | ( z + z + z ) z z z = 0 } . Equivalently, P = { ( w , w ) ∈ ( C ∗ ) | w + w = 0 } . If the compact pair-of-pants is defined as ¯ P = CP \ ˜ D , where ˜ D denotesthe union of small tubular open neighbourhoods of the 4 generic lines, then¯ P ⊂ P , and the interior int( ¯ P ) is diffeomorphic to P . The boundary ∂ ¯ P consists of 6 copies of 2-torus T and 4 copies of the total space F ofthe trivial S -bundle over P . By composing with the fibration P → Y, F → Y is a fibration with generic fibres T , and one singular fibre of shape ⊖ × (cid:13) over the vertex of Y.As in the case of Riemann surfaces, P carries a natural complete Einsteinmetric of finite volume. Many works were devoted to generalise Yau andAubin’s theorem ([69, 70, 8]) on the Calabi conjecture of projective manifoldswith negative first Chern class to the quasi-projective case under variousconditions (cf. [16, 32, 33, 62] etc.). In the current case, P is a quasi-projective manifold, and the log-canonical divisor K CP + D = H is ample,where H denotes the hyperplane class. Since D has only simple normal AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 3
Figure 1.2. P = CP \ D . ❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) z = 0 z = 0 z = 0 z + z + z = 0crossing singularities, Theorem 1 in [32] shows that there is a completeK¨ahler-Einstein metric ω on P with Ricci curvature − π ( K CP + D ) = 2 π .If X is a smooth complex hypersurface in CP of degree d >
5, then thecanonical divisor K X of X is very ample, and more precisely the canonicalbundle O X ( K X ) ∼ = O CP ( d − | X by the adjunction formula. X is a minimalalgebraic surface of general type. The Yau-Aubin-Calabi theorem assertsthat there is a unique K¨ahler-Einstein metric with Ricci curvature − π K X t = 2 π d ( d − . When we deform the complex structure of X , the K¨ahler-Einstein metric varies along the deformation. Therefore, if M is a smooth 4-manifold diffeomorphic to X , K¨ahler-Einstein metrics on M are not unique, and form a moduli space in a certain sense, which is asituation analogous to the case of Riemann surfaces of positive genus.The Mikhalkin’s theorem in [40] asserts that X admits a pair-of-pantsdecomposition consisting of d copies of pair-of-pants, i.e. there is an opendense subset X o ⊂ X diffeomorphic to the disjoint union of d copies ofpair-of-pants P . The number of pair-of-pants is bigger than that we expectas shown in the following example.We consider a family of hypersurfaces X t in CP of degree 5 defined by X i =0 z i + t ( z + z + z + z ) Y j =0 z j = 0 , [ z , z , z , z ] ∈ CP , where t ∈ [0 , ∞ ). When t → ∞ , X t tends to the singular variety X givenby ( z + z + z + z ) z z z z = 0 , which is the union of 5 generic hyperplanes in CP , and the regular locus X o consists of 5 copies of pair-of-pants P . We can diffeomorpically embed X o ֒ → X t for t ≫
1, and obtain a decomposition of X t with 5 copies of pair-of-pants. And 5 < = 125. Furthermore, the number 5 is the intersectionnumber K X t .The following theorem shows that the number of pair-of-pants can bereduced to the expected one, if we allow components of other types appearingin the decomposition. YUGUANG ZHANG
Theorem 1.1.
Let M be a 4-manifold diffeomorphic to a hypersurface X in CP of degree d > , and p g be the geometric genus of X . i) There is an open subset M o ⊂ M such that M o is diffeomorphic tothe disjoint union of d ( d − copies of pair-of-pants P , i.e. M o = d ( d − a P . ii) There is a fibration λ : M o \ M o → B onto a graph B with genericfibres 2-torus T and finite singular fibres of shape ⊖ × (cid:13) , where M o denotes the closure of M o in M . iii) M \ M o is the union of subsets of the K3 surface Y including ( C ∗ ) and C × C ∗ . Furthermore, there is an open subset M ′ ⊂ M \ M o admitting a smooth embedding ι : M ′ ֒ → Y such that the closure ofthe image is the K3 surface, i.e. ι ( M ′ ) = Y . iv) Any generic fibre T of λ represents a non-trivial homological classof M , i.e. = [ T ] ∈ H ( M, R ) . Moreover, if ̟ is a symplectic form representing c ( O CP (1)) | X , then Z T ̟ = 0 . v) If χ ( M ) is the Euler number of M , and τ ( M ) denotes the signatureof M , then χ ( M ) + 3 τ ( M ) = d ( d − π Vol ω ( P ) , where ω is the complete K¨ahler-Einstein metric with Ricci curvature − on P . vi) There is a family X t , t ∈ [0 , ∞ ) , of degree d hypersurfaces in CP ,and there are embeddings Ψ t : X t → CP p g − with O X t ( K X t ) =Ψ ∗ t O CP pg − (1) , such that Ψ t ( X t ) converges to a singular variety X in CP p g − as analytic subsets, when t → ∞ , and the regular locus of X is diffeomorphic to M o . In this theorem, we also regard certain subsets of the complex torus ( C ∗ ) , C × C ∗ , and the K3 surface as basic building blocks besides the pair-of-pants. It has been explored to decompose general type hypersurfaces intoCalabi-Yau components in [39]. The pair-of-pants part M o is an analogue ofhyperbolic pieces, and the complement M \ M o plays a similar role as graphmanifolds in the decompositions of 3-manifolds.The conclusions ii) and iv) resemble the facts that hyperbolic pieces areglued with other parts along incompressible tori in the 3-dimensional case,and the circles in the pair-of-pants decompositions of Riemann surfaces rep-resent non-trivial classes in the fundamental group. For example, CP ad-mits a decomposition CP = ¯ P ∪ ˜ D , and the generic fibre T in ∂ ¯ P repre-sents the zero class in H ( CP , R ). At least in this case, iv) plays a similar AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 5 role as those incompressible tori in the 3-dimensional decomposition. Un-fortunately, iv) is much weaker than either cases as the homological groupis used.The assertion v) expresses the Hitchin-Thorpe inequality via the volumeof the K¨ahler-Einstein metric on the pair-of-pants, and generalises (1.1). vi)shows that the expected pair-of-pants appear in a classical algebro-geometricway. What happens is that the d ( d − copies of pair-of-pants obtainedin i) converge nicely to the regular locus of X , while those K3 componentsare crushed into the singularities of X .We are motivated by a paper [14] of Cheeger and Tian about the collaps-ing of Einstein 4-manifolds, and we recall the relevant results in [14]. Let g i be a sequence of Einstein metrics on a 4-manifold M with Ricci curvatureRic(g i ) = − g i and volume Vol g i ( M ) ≡ const., which could be obtained byapplying Yau and Aubin’s theorem on the Calabi conjecture to hypersurfaces X of degree d >
5. Theorem 10.5 of [14] shows that by passing to a subse-quence and choosing certain base points, ( M, g i ) converges to K ` ν =1 ( M ν , g ∞ ),when i → ∞ , in the pointed Gromov-Hausdorff sense, andVol g i ( M ) = K X ν =1 Vol g ∞ ( M ν ) , where each M ν admits at most finite isolated orbifold points as singularities,and g ∞ are complete negative Einstein metrics in the orbifold sense. Conse-quently, for i ≫
1, g i induces a thick-thin decomposition M = M ∪ ( M \ M ),where the thick part ( M , g i ) is close to the pointed Gromov-Hausdorff limit K ` ν =1 ( M ν , g ∞ ), and the thin part M \ M supports metrics with bounded Riccicurvature and volume collapsing. Moreover, g i has bounded sectional cur-vature on a subset M ′ ⊂ M \ M by Theorem 0.8 of [14], and therefore M ′ admits an F -structure of positive rank in the Cheeger-Gromov sense (cf.[12, 13]).In Theorem 1.1, M o is expected to be the thick part, the complement M \ M o should be the thin part, and the fibration M o \ M o → B is an exampleof F -structure of positive rank. The limiting behaviour of negative K¨ahler-Einstein metrics along degenerations of algebraic manifolds with negativefirst Chern class has been studied under various hypotheses (cf. [58, 48, 49,50, 71, 53] and the references in them), which can certainly be applied tothe current situation. Especially, [50] has explored the connection betweenpair-of-pants degenerations and the convergence of K¨ahler-Einstein metrics.We outline the paper briefly. Section 2 presents a variant of Theorem1.1, which shows a correlation between the appearance of pair-of-pants andthe negativity of scalar curvature of metrics allowed on 4-manifolds in thecurrent case. In Section 3, we give some further remarks about the geometryof pair-of-pants. We prove the assertions i), ii), iii), and v) of Theorem 1.1 YUGUANG ZHANG in Section 4, which depends on Mikhalkin’s work [40] and tropical geometry.Finally, Section 5 proves the conclusions iv) and vi) of Theorem 1.1.In this paper, we use the Riemannian geometric convention of scalar cur-vature instead of the K¨ahler geometry one, i.e. the scalar curvature of ametric on a Riemann surface equals to the twice of the Gaussian curvature.2.
Yamabe invariant
In the case of Riemann surfaces, (1.1) shows a strong correlation betweenthe appearance of pair-of-pants and the negativity of scalar curvature ofmetrics allowed on the Riemann surfaces. However this information is lostin v) of Theorem 1.1. The goal of this section is to present a variant of v)in Theorem 1.1, which restores this correlation.First, we recall the definition of the Yamabe invariant. Let M be a smoothcompact real manifold of dimension n . For any conformal class c on M , theYamabe constant of c is defined as σ ( M, c) = inf g ∈ c (cid:0) Vol − nn g ( M ) Z M R g dv g (cid:1) , where R g denotes the scalar curvature of g. The supremum σ ( M ) = sup c ∈ C σ ( M, c)is a diffeomorphism invariant, and is called the Yamabe invariant of M (cf.[51, 31]), where C denotes the set of conformal classes on M . If n = 2, theGauss-Bonnet formula implies that the Yamabe invariant equals to 2 πχ ( M ).When n = 3, a Perelman’s theorem (Section 8.2 in [46], and see alsoProposition 93.9 in [30]) answers a conjecture of Anderson (cf. [4]), andsays that if σ ( M ) <
0, then there is an open subset M ⊂ M admitting acomplete hyperbolic metric g with sectional curvature −
1, and the Yamabeinvariant is realised by the volume of g, i.e. σ ( M ) = − g ( M ) . Actually, Perelman used the ¯ λ -invariant introduced in [45], which was provedto equal to the Yamabe invariant when σ ( M ) Theorem 2.1 (Theorem 7 in [36]) . Let M be a smooth 4-manifold diffeo-morphic to a minimal projective surface X of general type, i.e. the canonicaldivisor K X of X is nef and big K X > . Then the Yamabe invariant σ ( M ) = − q π K X = − p Vol ω ( X ) , where ω is a K¨ahler-Einstein metric with Ricci curvature − on an opensubset X ⊂ X . AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 7
An alternative proof.
We estimate the upper bound and the lower bound ofthe Yamabe invariant σ ( M ), and show that they match. The upper boundfollows LeBrun’s argument via the Seiberg-Witten equation with a minortwist, and we present it here for the completeness. The difference is toprove the lower bound where we use the K¨ahler-Ricci flow instead of theoriginal approach in [36].First, we recall the reinterpretation of the Yamabe invariant due to Lottand Kleiner [30]. For any 4-manifold M , (93.6) in [30] asserts that if σ ( M )
0, then(2.1) σ ( M ) = sup g ∈M ˇ R (g) q Vol g ( M ) , where ˇ R (g) denotes the minimal of the scalar curvature of g, i.e.ˇ R (g) = inf x ∈ M R (g)( x ) . If c is a Spin c structure on M , and S ± c denotes the Spin c bundle, then theSeiberg-Witten equation is introduced in [67], and reads D A φ = 0 , F + A = q ( φ ) , for an unknown positive spinor φ and an unknown U (1)-connection A on thedeterminant bundle L of c , where D A denotes the Dirac operator, F A is thecurvature of A , and q ( φ ) is a quadratic form of φ satisfying | q ( φ ) | = | φ | .The Seiberg-Witten invariant SW M ( c ) is a diffeomorphism invariant definedvia the moduli space of the solutions ( φ, A ) module U (1)-gauge changes (cf.[67, 42]). For example, if M is diffeomorphic to a minimal K¨ahler surface X of general type, and L is the anti-canonical bundle L = O X ( K − X ), then SW M ( c ) = 0 by [35], which implies that for any Riemannian metric g, theSeiberg-Witten equation has a solution ( φ, A ). Since c ( L ) = K X > F + A and φ are not identical to zero.The Weitzenb¨ock formula says4 ∇ ∗ A ∇ A φ + R (g) φ + | φ | φ = 0 . By producing with φ and integration, we haveˇ R (g) Z M | φ | dv g Z M (4 |∇ A φ | + R (g) | φ | ) dv g = − Z M | φ | dv g , and by the Schwarz inequality,ˇ R (g) q Vol g ( M ) − sZ M | φ | dv g = − s Z M | F + A | dv g − q π c ( L ) . Therefore, M does not admit any Riemannian metric of positive scalar cur-vature, and the Yamabe invariant σ ( M )
0. We obtain the upper bound σ ( M ) − q π K X , YUGUANG ZHANG by taking the supremum and (2.1) (See [19] for another proof via the Perel-man’s ¯ λ -functional).Now we prove the lower bound by considering the K¨ahler-Ricci flow(2.2) ∂ω t ∂t = − Ric( ω t ) − ω t , t ∈ [0 , T ) , on X with an initial K¨ahler metric ω . This version of K¨ahler-Ricci flow wasintensely studied in recent years for the differential geometric understandingof the minimal model programme (cf. [65, 63, 54, 64] and the references inthem). See also [20, 21] for some interactions between the real Ricci flowand the Seiberg-Witten equation.Let X can be the canonical model of X , and π : X → X can be the con-traction map. Note that X can is a 2-dimensional normal variety with onlyfinite A-D-E singularities, and − c ( X ) = c ( O X ( K X )) = π ∗ α for an ampleclass α on X can . In [65, 63], it is proved that the solution ω t exists for a longtime t ∈ [0 , ∞ ), i.e. T = ∞ , and ω t converges to a semi-positive current ω presenting − πc ( X ) with bounded local potential functions when t → ∞ .Furthermore, ω is a smooth K¨ahler-Einstein metric with Ricci curvature − π − ( X ocan ), where X ocan is the regular locus of X can , and2 π K X = 2 π c ( X ) = Vol ω ( X ocan ) . The evolution equation of scalar curvature is ∂R t ∂t = ∆ t R t + 2 | Ric t | + R t = ∆ t R t + 2 | Ric ot | − ( R t + 4)(cf. Lemma 2.38 in [15]), where R t = R ( ω t ), Ric t = Ric( ω t ) and Ric ot =Ric t + ω t , which satisfies | Ric ot | = | Ric t | + R t + 2 by R t = 2tr ω t Ric t . Bythe maximal principle, the minimal ˇ R t of R t satisfies ∂ ˇ R t ∂t > − ( ˇ R t + 4) , and therefore we obtain ˇ R t > − − Ce − t → − , for a constant C > t .Since the Chern-Weil theory shows that the cohomological classes evolveas ∂ [ ω t ] ∂t = − πc ( X ) − [ ω t ] , we solve the ordinary differential equation and obtain[ ω t ] = − πc ( X ) + e − t (2 πc ( X ) + [ ω ]) . Thus the volumesVol ω t ( X ) = 12 Z X ω t → π c ( X ) = 2 π K X , AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 9 when t → ∞ . We obtain the lower bound of the Yamabe invariant σ ( M ) > lim t →∞ ˇ R t p Vol ω t ( X ) = − q π K X , which is the same as the upper bound. We obtain the conclusion by letting X = π − ( X ocan ). (cid:3) The Riemann-Roch theorem asserts K X = 2 χ ( X ) + 3 τ ( X )for any compact complex surface X . The canonical divisor K X of a hy-persurface X in CP of degree d > X is of minimalgeneral type. We reach a variant of Theorem 1.1 by using Theorem 2.1,which shows a correlation between the negativity of the Yamabe invariantand the appearance of pair-of-pants. Theorem 2.2.
Under the setup of Theorem 1.1, the Yamabe invariant of M is σ ( M ) = − p d ( d − Vol ω ( P ) , where ω is the K¨ahler-Einstein metric with Ricci curvature − on the pair-of-pants P . Equivalently, the number of pair-of-pants in M o equals to π (max { , − σ ( M ) } ) . Finally, we remark that the assertions ii) and iv) in Theorem 1.1 providea certain constraint on the Yamabe invariant. The proof can be appliedto more broader scenarios, and therefore we present a general result, whichmight have some independent interests.
Theorem 2.3.
Let M be a compact 4-dimensional symplectic manifold, and ̟ be a symplectic form on M . If there is a Lagrangian submanifold Σ in M such that Σ is a Riemann surface of genus g > , and represents anon-trivial homological class in M , i.e. = [Σ] ∈ H ( M, R ) , then M does not admit any Riemannian metric of positive scalar curvature.Furthermore, if M is minimal, then χ ( M ) + 3 τ ( M ) > , and the Yamabeinvariant (2.3) σ ( M ) − p π (2 χ ( M ) + 3 τ ( M )) . If M is a finite covering of an oriented manifold ¯ M , then (2.3) also holdsfor ¯ M .Proof. The Weinstein neighbourhood theorem (cf. [68]) says that there is atubular neighbourhood of Σ in M diffeomorphic to a neighbourhood of thezero section in the cotangent bundle T ∗ Σ. Since the first Chern number of T ∗ Σ is the minus of the Euler number of Σ, i.e. R Σ c ( T ∗ Σ) = − χ (Σ), theself-intersection number Σ · Σ = 2 g − > Let g be a Riemannian metric compatible with ̟ , i.e. g( · , · ) = ̟ ( · , J · ) foran almost complex structure J compatible with ̟ . Then ̟ is a self-dual2-form with respect to g, i.e. ∗ ̟ = ̟ , where ∗ denotes the Hodge staroperator of g. We identify H ( M, R ) with the sum of spaces of self-dualand anti-self-dual harmonic 2-forms via the Hodge theory, i.e. H ( M, R ) = H + ( M ) ⊕ H − ( M ). If A ∈ H ( M, R ) is the Poincar´e dual of [Σ], then A = α + β such that dα = dβ = 0, ∗ α = α , ∗ β = − β , and0 Σ · Σ = Z M ( α + β ) = Z M α + Z M β . Thus Z M α > − Z M β = Z M β ∧ ∗ β > . Note that if α = 0, then β = 0 and A = 0, which is a contradiction. Thus α = 0, and Z Σ α = Z M α ∧ ( α + β ) = Z M α = 0 . Since Z Σ ̟ = 0 ,̟ and α are linearly independent in the space of self-dual harmonic 2-forms H + ( M ). Therefore, b +2 > SW M ( c ) = 0, for a certain Spin c structure c . Thesame argument as in the proof of Theorem 2.1 proves that there does notexist any Riemannian metric of positive curvature on M , and σ ( M ) M is minimal, then σ ( M ) − p π c ( L ), since c ( L ) =2 χ ( M ) + 3 τ ( M ) > L is the determinant bundle of the Spin c structure c .Finally, if M → ¯ M is a finite ν -sheets covering, then √ νσ ( ¯ M ) σ ( M ), νχ ( ¯ M ) = χ ( M ), and ντ ( ¯ M ) = τ ( M ). We obtain the conclusion. (cid:3) Note that if we replace the Lagrangian condition in this theorem by theexistence of a local T -fibration satisfying iv) in Theorem 1.1, then the self-intersection number T · T = 0, and the same argument proves the sameconclusion. Certain submanifolds provide obstructions for the existence ofRiemannian metrics of positive scalar curvature by Schoen-Yau [52] andGromov-Lawson [25]. Unlike these earlier pioneer works, the obstruction inTheorem 2.3 is the constraint provided by the Seiberg-Witten theory. Ofcourse, we should ideally use the techniques as in [52, 25] to prove thatthe existence of pair-of-pants under some topological hypotheses implies thenegativity of the Yamabe invariant. AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 11 Remarks on pair-of-pants
In this section, we give some remarks on pair-of-pants, which are notdirectly used in the proof of the main theorem, but provide a better under-standing of the geometry of pair-of-pants.First, we recall the existence of complete K¨ahler-Einstein metrics withnegative Ricci curvature on quasi-projective manifolds, which was studiedby many authors under various hypotheses (cf. [16, 32, 33, 62] etc.). Let(
X, D ) be a pair of log K¨ahler surface such that the divisor D has onlysimple normal crossing singularities and the log canonical divisor K X + D is ample. Then there is a complete K¨ahler-Einstein metric ω on X \ D withfinite volumeVol ω ( X \ D ) = 2 π ( K X + D ) , and Ric( ω ) = − ω, by Theorem 1 in [32]. Since ( K X + D ) ∈ Z , the smallest possible value ofthe volumes of complete K¨ahler-Einstein metrics constructed by this methodis 2 π , i.e. ( K X + D ) = 1.We apply this result to ( CP , D ) with D being the sum of 4 generalpositioned lines, i.e. CP \ D = P , and obtain a complete K¨ahler-Einsteinmetric ω on the 4-dimensional pair-of-pants P with(3.1) Ric( ω ) = − ω, and Vol ω ( P ) = 2 π , since K CP = − H and D = 4 H , where H denotes the hyperplane class.Note that complete K¨ahler-Einstein metrics with the numerical data (3.1)are certainly not unique. For example, if D ′ is a smooth curve of degree 4,then there is a complete K¨ahler-Einstein metric on CP \ D ′ satisfying (3.1),and CP \ D ′ is not diffeomorphic to P . However, the pair-of-pants is stillspecial as the sum of 4 generic lines is the most degenerated reduced curveof degree 4.We provide a criterion for 4-dimensional pair-of-pants via the numericaldata (3.1) in the following proposition. Proposition 3.1.
Let X be a smooth minimal projective surface with van-ishing first Betti number, i.e. b ( X ) = 0 , and D be a reduced effectivedivisor on X with only simple normal crossing singularities. Assume thatlog canonical divisor K X + D is ample, and D is a most degenerated divi-sor in the sense that the number of irreducible components of D is biggeror equal to the number of irreducible components of any reduced divisor D ′ linearly equivalent to D . Then ( K X + D ) = 1 , if and only if X = CP and X \ D is the pair-of-pants, i.e. X \ D = P .Proof. We only need to prove that ( K X + D ) = 1 implies X = CP and X \ D = P .Note that ( K X + D ) · D ∈ Z , ( K X + D ) · K X ∈ Z and( K X + D ) = ( K X + D ) · D + ( K X + D ) · K X = 1 . By the adjunction formula for singular curves in surfaces, ( K X + D ) · D =2( g ( D ) − , where g ( D ) is the virtual genus of D . Since K X + D is ample,( K X + D ) · D >
0, which implies ( K X + D ) · D >
2, and( K X + D ) · K X < . Hence K X is not nef, and X is either CP or a minimal ruled surface overa curve by the classification of minimal surfaces (cf. Theorem 4 in Chapter10 of [22]).Assume that X is a ruled surface over a curve C . Since the first Bettinumber of X is zero, C = CP (cf. Lemma 13 in Chapter 5 of [22]), and X is a Hirzebruch surface, i.e. X = P ( O CP ⊕ O CP ( − µ )). Let s be the0-section with s = − µ , µ >
0, and l be the fibre class which satisfies l = 0and s · l = 1. If K X + D = as + bl for a ∈ Z and b ∈ Z , then( K X + D ) · l = a > , ( K X + D ) · s = − µa + b > , by the ampleness. Thus1 = ( K X + D ) = − µa + 2 ab > µa > , and µ = 0. We obtain a contradiction 1 = 2 ab > X = CP , which implies K X = − H and D = 4 H , where H is the hyperplane class. We obtain the conclusion since the sum of 4 genericlines is the most degenerated curve of degree 4. (cid:3) Orbifold singularities, especially the A-D-E singularities, appear naturallyin the study of minimal models for algebraic surfaces, and limit spaces of Ein-stein 4-manifolds in the Gromov-Hausdorff convergence (cf. [3, 43]). If sin-gularities present, we certainly have examples of complete K¨ahler-Einsteinmetrics in the orbifold sense with volume less than 2 π . Example 3.2.
We consider the Z -action on CP by permutating the homo-geneous coordinates, i.e. γ · [ z , z , z ] = [ z , z , z ] for the generator γ of Z .There are three fixed points [1 , , , [1 , ε , ε ] , [1 , ε, ε ] where ε = exp π √− ,and the volume form dw ∧ dw w w is preserved where w = z z − and w = z z − .Note that D = { [ z , z , z ] ∈ CP | z z z ( z + z + z ) = 0 } is Z -invariant,and [1 , ε , ε ] , [1 , ε, ε ] ∈ D by 1+ ε + ε = 0. Thus Z acts on the pair-of-pants P with only one fixed point [1 , , P / Z is an orbifoldwith one isolated singularity. Note that the class c ( O CP ( K CP + D )) is alsoinvariant, and the complete K¨ahler-Einstein metric ω on P descends to acomplete orbifold metric on P / Z , denoted still as ω , which satisfiesRic( ω ) = − ω, and Vol ω ( P / Z ) = 2 π . We remark that the K¨ahler-Einstein metric ω on P is not complex hy-perbolic as follows. Note that the Euler number of P is one (cf. Proposition AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 13 χ ( P ) = 18 π Z P ( R
24 + | W + | + | W − | ) dv ω , where W ± denotes the self-dual/anti-self-dual Weyl curvature of ω (cf. [9]).Since ω is K¨ahler, 24 | W + | = R , and we obtain(3.2) Z P | W − | dv ω = 163 π . Hence W − is not identical to zero, and ω is not a complex hyperbolic metric.Furthermore, (3.2) indicates that we might not expect that complex hyper-bolic manifolds admit sensible pair-of-pants decompositions, but we wouldrather view them as another basic negative pieces because of the Mostowrigidity.Many works have been done towards certain decompositions of smooth4-manifolds and canonical metrics from various perspectives, for example,metric geometry, topology, algebraic geometry and symplectic geometry etc.See [7, 5, 6, 59, 60, 61, 17, 18, 11, 38] and the references in them, andespecially [6, 60, 17, 18] for some expository discussions. In this paper, weexplore decompositions for 4-manifolds of a very specific type, i.e. projectivehypersurfaces of general type, while keeping in mind the general questionswhether there is a geometrisation theory for 4-manifolds, and if it exists atall, what the zoo of fundamental 4-manifolds might consists of.Next, we remark that the product of two 2-dimensional pair-of-pants, P × P , deforms to a singular variety containing two copies of pair-of-pants P . Therefore, we may not regard P × P as a fundamental building block. Example 3.3.
There is a degeneration f : X → C satisfying that X issmooth, ( f − (0)) o = P a P , and f − (1) = P × P , where ( f − (0)) o denotes the regular locus of f − (0). Proof.
Let ¯ X = { ( w , w , w , w ) ∈ ( C ) × C ∗ | w w = w w } , and f : ¯ X → C be given by ( w , w , w , w ) w . Note that ¯ X is smoothsince the defining equation defines one ordinary double point that is notin ( C ) × C ∗ . If X w = f − ( w ), then X is a singular variety given by w w = 0, i.e. X = { ( w , w ) ∈ C × C ∗ } ∪ { ( w , w ) ∈ C × C ∗ } ⊂ C × C ∗ , and X = { ( w , w , w ) ∈ C × C ∗ | w = w w = 0 } = { ( w , w ) ∈ ( C ∗ ) } . If we let D = { ( w , w , w , w ) ∈ ¯ X | w + w + w = 0 } , then the regular locus of X \D is the Zariski open subset { ( w , w ) ∈ ( C ∗ ) | w + w = 0 } ∪ { ( w , w ) ∈ ( C ∗ ) | w + w = 0 } , i.e. the regular locus of X \D consists of two copies of pair-of-pants P . By1 + w + w + w w = (1 + w )(1 + w ), X \D = { ( w , w ) ∈ ( C ∗ ) | w = − , w = − } = P × P . We obtain the conclusion by setting X = ¯ X \D . (cid:3) If D is a divisor in CP consisting of d > K CP + D ) = ( d − , and we expect that CP \ D deforms to ( d − -copiesof pair-of-pants P . Thus we might not obtain more sensible building blocksby removing lines from the pair-of-pants. D is an example of arrangements oflines, and one technique to construct general type surfaces is to use branchedcoverings of CP along arrangements of lines (cf. [29]). We work out theconcrete case of d = 5, and the proof would become more transparent byusing the techniques from toric geometry. Example 3.4.
We assume that D = { [ z , z , z ] ∈ CP | z z z ( z + z + z )( a z + a z + a z ) = 0 } for certain generic chosen a , a , a ∈ C . Then there is a degeneration f : X → C such that( f − (0)) o = a P , and f − (1) = CP \ D, where ( f − (0)) o denotes the regular locus of f − (0). Proof.
We consider a family of Veronese embeddings Ψ w : CP ֒ → CP givenby Z = wz , Z = z z , Z = z z , Z = z z , Z = wz , Z = wz , where w ∈ C ∗ , and [ Z , Z , Z , Z , Z , Z ] are homogeneous coordinates of CP . The image is given by the equations Z Z = w Z , Z Z = w Z , Z Z = w Z ,wZ Z = Z Z , wZ Z = Z Z , wZ Z = Z Z . When w →
0, Ψ w ( CP ) converges to a singular variety X with 4 irreduciblecomponents, i.e. X = X ∪ X ∪ X ∪ X , where X = { [ Z , Z , Z , , , ∈ CP } , X = { [0 , Z , Z , Z , , ∈ CP } ,X = { [0 , Z , , Z , Z , ∈ CP } , X = { [0 , , Z , Z , , Z ] ∈ CP } . We let D = { [0 , , , Z , Z , Z ] ∈ CP } , D = { [ Z , , Z , , , Z ] ∈ CP } , D = { [ Z , Z , , , Z , ∈ CP } , and define D ′ ⊂ CP by a Z + ( a + a ) Z + ( a + a ) Z + ( a + a ) Z + a Z + a Z = 0 . AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 15
Figure 3.1.
Intersection complex of X in Example 3.4. Z Z Z Z Z Z ❅❅❅❅❅ ❅❅❅ Note that D ′ ∩ X i , i = 1 , , ,
4, is a generic line in CP . If we regard CP asa toric manifold and ( C ∗ ) ⊂ CP , then ( C ∗ ) \ ( D ′ ∩ X i ) is a pair-of-pants P , and does not intersect with any D j , j = 0 , ,
2. Furthermore,Ψ ( D ) = Ψ ( CP ) ∩ ( D ′ ∪ D ∪ D ∪ D ) . We obtain the conclusion by letting X = ( X [ w ∈ C ∗ Ψ w ( CP )) \ ( D ′ ∪ D ∪ D ∪ D ) ⊂ CP × C , and f be the projection to w . (cid:3) We finish this section by showing that there is a pair-of-pants in theoriginal Godeaux surface.
Example 3.5.
We consider the Fermat quintic X = { [ z , z , z , z ] ∈ CP | z + z + z + z = 0 } , which is a smooth minimal surface of general type with K X = 5. If Z actson CP by ε · [ z , z , z , z ] = [ z , εz , ε z , ε z ] where ε = exp π √− , then Z acts on X freely. The quotient ˜ X = X/ Z is a smooth surface of generaltype with K X = 1, called a Godeaux surface (cf. [47]). The geometric genus p g ( ˜ X ) = 0 and the fundamental group π ( ˜ X ) = Z . Theorem 3.6. If M is diffeomorphic to the Godeaux surface ˜ X , then thereis an open subset M o ⊂ M diffeomorphic to the pair-of-pants P , and theYamabe invariant σ ( M ) = − p Vol ω ( P ) = −√ π . Furthermore, if υ : M ′ → M is the universal covering of M , then there is a2-torus T in M ′ such that i) υ ( T ) ⊂ M o , ii) T represents a non-trivial homological class in M ′ , i.e. = [ T ] ∈ H ( M ′ , R ) , iii) the self-intersection number T · T = 0 , and Z T ̟ = 0 , for a symplectic form ̟ on M ′ .Proof. If H is the hyperplane given by h = z + z + z + z = 0, then ε i H is defined by h i = z + ε i z + ε i z + ε i z = 0 for i = 0 , · · · ,
4. Let X = H ∪ εH ∪ ε H ∪ ε H ∪ ε H , which is defined by h h h h h = 0.We claim that the regular locus of X consists of 5 copies of pair-of-pants, i.e. for any i , ε i H \ S j = i ε j H is a pair-of-pants P . Since Z acts on X and switches the irreducible components, we only need to prove that H \ ( εH ∪ ε H ∪ ε H ∪ ε H ) is a pair-of-pants. Note that ε ( H ∩ εH ∩ ε H ∩ ε H ) = H ∩ ε H ∩ ε H ∩ ε H , ε ( H ∩ εH ∩ ε H ∩ ε H ) = H ∩ εH ∩ ε H ∩ ε H ,and ε ( H ∩ εH ∩ ε H ∩ ε H ) = H ∩ εH ∩ ε H ∩ ε H . Thus we only needto prove that H ∩ εH ∩ ε H ∩ ε H is empty. The coefficient matrix of theequations h = h = h = h = 0 is the Vandermonde matrix, and thedeterminant is equal to ε ( ε − ( ε + 1) ( ε + ε + 1) = 0. We obtain theclaim.If we let X t = { [ z , z , z , z ] ∈ CP | z + z + z + z + th h h h h = 0 } , t ∈ (1 , ∞ ) , then X t is invariant under the Z -action, and X t converges to X in CP asvarieties when t → ∞ . Note that [1 , , , , [0 , , , , [0 , , , , [0 , , ,
1] arethe fixed points when Z acts on CP , and 1 + tε j = 0. Thus the Z -actionon X t is free. For t ≫
1, there is an open subset X ot ⊂ X t diffeomorphicto the regular locus of X , which consists of 5 copies of pair-of-pants, and X ot is invariant under the Z -action. Therefore the quotient X ot / Z is apair-of-pants in X t / Z .The rest of the assertion seems a corollary of Theorem 1.1. However wegive a direct calculation proof since the degeneration used here is differentfrom the family of varieties in the proof of Theorem 1.1.We choose new homogeneous coordinates Z , Z , Z , Z on CP such that ε i H is defined by Z i = 0, i = 0 , , ,
3. Equivalently, Z i = z + ε i z + ε i z + ε i z . Then ε H is given by a Z + a Z + a Z + a Z = 0, and X t isdefined by¯ f t = Z Z Z Z ( a Z + a Z + a Z + a Z ) + t − P = 0 , where a Z + a Z + a Z + a Z = z + ε z + ε z + ε z , and P isa homogeneous polynomial of degree 5. If w j = Z j /Z , j = 1 , ,
3, then ε j H ∩ { Z = 0 } is given by w j = 0, j = 1 , ,
3, and ε H ∩ { Z = 0 } is givenby a + a w + a w + a w = 0. We let f t = ¯ f t /Z = w w w ( a + a w + a w + a w ) + t − P ′ , where P ′ = P /Z . Note that a = 0 since, otherwise, (0 , ,
0) solves w i = 0, i = 1 , ,
3, and a + a w + a w + a w = 0, which is a contradiction.We consider the hyperplane ε H ∩ { Z = 0 } ⊂ C , and ε H ∩ { Z =0 }\ ( ε H ∪ ε H ∪ ε H ) is the pair-of-pants P belonging to the irreducible AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 17 component ε H . We choose a 2-torus T = { ( w , w ) ∈ P || w | = r , | w | = r } ⊂ ε H, for certain r , r ∈ R \{ } . If θ i , i = 1 ,
2, is the angle of w i , i.e. w i = r i exp √− θ i , then θ and θ are angular coordinates on T . If V is a smallneighbourhood of T in CP , then X t ∩ V is given by f t = 0 and ε H ∩ V is given by w = 0. We choose V such that V does not intersect with εH , ε H , and ε H .Since the meromorphic formΩ = Z Z Z Z dw ∧ dw ∧ dw ¯ f t w w w has only a simple pole along X t , the Poincar´e residue formula gives a holo-morphic 2-form Ω t = res X t (Ω) on X t , which defines a non-trivial cohomo-logical class, i.e. 0 = [Ω t ] ∈ H ( X t , C ) . Note that Ω t = dw ∧ dw ∂f t ∂w = w w d log( w ) ∧ d log( w ) ∂f t ∂w , on X t ∩ V . We calculate ∂f t ∂w = w w ( a + a w + a w ) + 2 a w w w + t − ∂P ′ ∂w , and thus Ω t = d log( w ) ∧ d log( w ) a + a w + a w + 2 a w + t − w − w − ∂P ′ ∂w . We choose r and r such that | a | r + | a | r ≪ | a | since a = 0, and r r > ǫ > ǫ >
0. Since | a + a w + a w + a w | >ǫ ′ > w ∈ V and a constant ǫ ′ >
0, the equation f t = 0 shows that ǫǫ ′ | w | X t ∩ V | < | P ′ | t − → , when t →
0. For any isotopic embedding φ t : P ֒ → X t with φ ∞ = id, Z φ t ( T ) Ω t → Z T dθ ∧ dθ a + a w + a w = Z T ( a + a r e −√− θ + a r e −√− θ ) dθ ∧ dθ | a + a r e √− θ + a r e √− θ | = 0 , when t → ∞ . Thus for t ≫ φ t ( T ) ⊂ φ t ( P ) ⊂ X ot , and0 = [ φ t ( T )] ∈ H ( X t , R ) . By varying r and r , we obtain that the self-intersection number φ t ( T ) · φ t ( T ) = 0. If ̟ is a toric symplectic form on CP , then ̟ is preserved by the Z -action, and ̟ | T ≡
0. Therefore Z φ t ( T ) ̟ = Z T ̟ = 0 . Note that if υ : X t → X t / Z denotes the quotient map, then υ : φ t ( P ) → M o is a diffeomorphism. We obtain the conclusion since M is diffeomorphicto X t / Z , and M ′ = X t . (cid:3) The difference between the assertion ii) in this theorem and iv) in Theo-rem 1.1 is that the torus υ ( T ) is homological zero since Ω t is not preservedby the Z -action, and the geometric genus p g ( ˜ X ) = 0. However υ − ( υ ( T ))consists of 5 copies of T in the universal covering, and each connected com-ponent represents a non-trivial class in M ′ . The assertion ii) is also differentfrom the case of P ⊂ CP , where CP is simply connected, and numericalLagrangian 2-tori in P are homological zero. Moreover, Theorem 2.3 canbe applied to the current case, and shows that the assertions ii) and iii) ofTheorem 3.6 imply directly the Yamabe invariant σ ( M )
0. Therefore wewould like to think that Theorem 3.6 provides a sensible decomposition ofthe Godeaux surface.4.
Pair-of-pants vs K3 surfaces
The goal of this section is to prove the assertions i), ii), iii), and v) ofTheorem 1.1, which depends heavily on Mikhalkin’s paper [40] and tropicalgeometry (cf. Chapter 1 of [26] and [41]). We review the relevant facts first.4.1.
Amoebas. ( C ∗ ) denotes the complex torus of dimension 3, and forany m = ( m , m , m ) ∈ Z , w m = w m w m w m where w , w , w are coor-dinates on ( C ∗ ) . Let X o be the hypersurface in ( C ∗ ) defined by a Laurentpolynomial(4.1) f ( z ) = X m ∈ S a m w m = 0 , where a m ∈ C , and S ⊂ Z is finite. The Newton polytope ∆ ⊂ R of X o is defined as the convex hull of m ∈ S such that a m = 0. We consider thelog-map Log : ( C ∗ ) → R , Log( w ) = (log | w | , log | w | , log | w | ) . The amoeba of X o is defined in [23] (see also [41, 44] for analytic treatments)as the image A = Log( X o ) ⊂ R . Let v : ∆( Z ) → R be a function where ∆( Z ) = ∆ ∩ Z is the set of latticepoints of ∆. v induces a rational polyhedral subdivision T v of ∆ as follows.If ˜∆ is the upper convex hull of { ( m, v ( m )) | m ∈ ∆( Z ) } in ∆ × R , i.e.˜∆ = { ( a, b ) ∈ ∆ × R |∃ b ′ ∈ Cov { ( m, v ( m )) | m ∈ ∆( Z ) } , b > b ′ } , AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 19 then T v is the set of the images of proper faces under the projection ˜∆ → ∆(cf. Chapter 1 in [26]). The discrete Legendre transform L v : R → R of v is defined as(4.2) L v ( x ) = max m ∈ ∆( Z ) { l m ( x ) } , l m ( x ) = h m, x i − v ( m ) , where h· , ·i is the Euclidean metric. Then L v is a convex piecewise linearfunction.The non-smooth locus Π v of L v is called the tropical hypersurface definedby v , or the non-archimedean amoeba, which is a balanced polyhedral com-plex dual to the subdivision T v of ∆ (cf. Proposition 2.1 in [40]). Π v is astrong deformational retract of the amoeba A (cf. [41, 44]). Furthermore,there is a one-to-one corresponding between the set of k -dimensional cellsˇ ρ ⊂ Π v , k = 0 , ,
2, and the set of (3 − k )-dimensional cell ρ in T v , i.e.(4.3) { k − cells ˇ ρ ⊂ Π v } = { (3 − k ) − cells ρ ∈ T v } , via ˇ ρ ρ, and the cell ˇ ρ is unbounded if and only if ρ ⊂ ∂ ∆. The vertices of T v areone-to-one corresponding to the connected components of the complementof Π v . Two cells ρ ⊂ ρ ∈ T v if and only if ˇ ρ ⊃ ˇ ρ in Π v .We deform X o via the Viro’s patchworking polynomials (cf. [66]). Thepatchworking family X ot , t ∈ (1 , ∞ ), is defined by f t ( w ) = X m ∈ S a m t − v ( m ) w m = 0 . If the deformed log map Log t : ( C ∗ ) → R is defined by(4.4) Log t ( w ) = (cid:16) log | w | log t , log | w | log t , log | w | log t (cid:17) , then we have a family of the amoebas A t = Log t ( X ot ) ⊂ R . Corollary 6.4 in [40] asserts that A t → Π v , when t → ∞ , in the Hausdorff sense. Therefore, Π v is a good approximation of A t when t ≫ H o = { ( w , w , w ) ∈ ( C ∗ ) | w + w + w = 0 } = { ( w , w ) ∈ ( C ∗ ) | w + w = 0 } , then H o is the 4-dimensional pair-of-pants, i.e. H o = P . The Newtonpolytope ∆ is the standard simplex in R , i.e.∆ = { ( x , x , x ) ∈ R | x + x + x , x i > , i = 1 , , } . If v ≡
0, then the corresponding tropical hypersurface Π v is the non-linearlocus of L v ( x ) = max { , x , x , x } . Figure 4.1.
Tropical 2-dimensional pair-of-pants P . P (cid:0)(cid:0)(cid:0) x x Π v =0 ❅❅❅ (0 ,
0) (1 , ,
1) ∆
Figure 4.2. Π v vs T v .Π v (cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) ✈✈ ✈ ✈ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) ❅❅❅❅❅❅❅❅❅❅❅❅ ❅❅❅❅❅❅❅❅❅ ❅❅❅❅❅❅ ❅❅❅ ∆ T v ✈ ✈✈ ✈ Decompositions.
Let M be a smooth manifold diffeomorphic to ahypersurface X in CP of degree d >
5. We regard CP as a toric manifoldby ( C ∗ ) ⊂ CP . For any positive integer d , the corresponding polytope ∆ d of ( CP , O CP ( d )) is the standard simplex in R , i.e.(4.5) ∆ d = { ( x , x , x ) ∈ R | x + x + x d, x i > , i = 1 , , } . If ∆ d ( Z ) = ∆ d ∩ Z , then for any m ∈ ∆ d ( Z ), w m = w m w m w m is viewedas a section of O CP ( d ).Let ∆ od be the convex hull of int(∆ d ) ∩ Z , i.e. ∆ od = Cov(int∆ d ∩ Z ),and ∆ od ( Z ) = ∆ od ∩ Z . Note that ∆ od is a translation of the simplex ∆ d − ,i.e.(4.6) ∆ od = (1 , ,
1) + ∆ d − . If p g ( d ) denotes the geometric genus of X in CP , then p g ( d ) = ( d − d − d − ♯ ∆ od ( Z ) , by the Noether’s formula χ ( X ) = 12(1 − p g ( d )) − K X , since X is simplyconnected, K X = d ( d − and the Euler number χ ( X ) = d − d + 6 d .Let v : ∆ d ( Z ) → R be a function such that the induced subdivision T v isa uni-modular lattice triangulation, i.e. any 3-cell in T v is a simplex withEuclidean volume . Since the Euclidean volume of ∆ d is d , T v has d v has d vertices. v defines a patchworking family AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 21
Figure 4.3.
A decomposition of projective curve of degree4 in CP , which consists of 4 copies of pair-of-pants P , and3 copies of (0 , × S ⊂ C ∗ . (cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) ✈✈ ✈ ✈ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) ✈✈ ✈ ✈ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) X t , t ∈ (1 , ∞ ), given by(4.7) f t ( w ) = X m ∈ ∆ d ( Z ) t − v ( m ) w m = 0 . Since any smooth hypersurface of degree d is diffeomorphic to the samedifferential manifold, a smooth X t is diffeomorphic to M . We let X ot = X t ∩ ( C ∗ ) .In [40], Mikhalkin proved that M admits a pair-of-pants decomposition. Theorem 4.1 ([40]) . There is a fibration λ : X ot → Π v satisfying: i) For any vertex ˇ q ∈ Π v , let U ˇ q ⊂ Π v be the interior of the star ofthe barycentre triangulation of Π v , called a primitive piece associatedwith ˇ q . Then U ˇ q ∩ U ˇ q ′ is empty if ˇ q = ˇ q ′ , and λ − ( U ˇ q ) is diffeomorphicto the pair-of-pants P . ii) a ˇ q ∈{ vertices in Π v } λ − ( U ˇ q ) ⊂ X ot is open dense. Therefore X t admits a pair-of-pants decompositionconsisting of d copies of P . iii) If a point x belongs to the interior of a 2-cell in Π v , then the fibre f − ( x ) is a torus T , and if x is in the interior of a 1-cell of Π v ,then f − ( x ) has the shape of ⊖ × (cid:13) . The goal of Theorem 1.1 is to reduce the number of pair-of-pants from d to d ( d − by showing that those extra pair-of-pants form certain subsetsof K3 surfaces. To decompose X t into a union of subsets of Calabi-Yaumanifolds was previously studied by Leung and Wan in [39].We remark that there is an alternative way to view the pair-of-pantsdecomposition in the current case. Assume that any vertex ˇ q ∈ Π v is anintegral vector, i.e. ˇ q ∈ Z , and for any m ∈ ∆ d ( Z ), v ( m ) ∈ Z . The pair (∆ d , v ) induces a toric degeneration X → C with a relative ample line bundle O X / C ( d ) (cf. Chapter 1 of [26] and [27]) satisfying:i) Any generic fibre X w , w = 0, is CP , and the restriction of O X / C ( d )on X w is O CP ( d ).ii) The central fibre X is a singular variety consisting of d copies of CP as irreducible components. The singular set of X belongs tothe union of toric boundary divisors of CP .iii) There is a one-to-one corresponding between 3-cells ρ of T v and irre-ducible components X ,ρ of X . Furthermore, the pair (∆ d , T v ) canbe regarded as the intersection complex of X .iv) For any m ∈ ∆ d ( Z ), Z m = w v ( m )0 w m defines a section of O X / C ( d )such that the restriction of Z m on the irreducible component X ,ρ is identical to zero if m does not belong to ρ . Furthermore, if { m , m , m , m } = ρ ∩ ∆ d ( Z ), then Z m i , i = 0 , , ,
3, are homoge-neous coordinates on CP . Z m are generalised theta functions in theGross-Siebert sense (cf. [28]).v) There is a divisor D ⊂ X such that the intersection of D with anygeneric fibre X w is the toric boundary divisor of X w . Moreover, if X o denotes the regular locus of X , then X o \D is the disjoint unionof d copies of ( C ∗ ) .The patchworking polynomial (4.7) reads X m ∈ ∆ d ( Z ) Z m = 0 , which defines a subvariety of X over C , denoted as ˜ X → C . When w = t − ,the fibre ˜ X w = X t , and therefore ˜ X can be regarded as an extension of thepatchworking family. For any irreducible component X ,ρ in the central fibre,˜ X ∩ X ,ρ is given by Z m + Z m + Z m + Z m = 0 , where { m , m , m , m } = ρ ∩ ∆ d ( Z ), and ˜ X ∩ X ,ρ ∩ ( X o \D ) is the pair-of-pants P . Thus ˜ X ∩ ( X o \D ) consists of d copies of pair-of-pants P , andcan be diffeomorphically embedded into X t for t ≫ Proofs of i), ii), iii), and v) in Theorem 1.1.
We fix a function v : Z → R by(4.8) v ( m ) = 4 X i =1 m i + (2 m + 2 m + 3 m ) , for any m = ( m , m , m ) ∈ Z , which is the restriction of a positive de-fined quadratic form ˜ v on R = Z ⊗ Z R . v induces a Delaunay polyhedraldecomposition Del v of R (cf. [2]). Since v has a Z -symmetry by switching x and x , Del v is invariant under the Z -action. By Lemma 1.8 of [2], Del v is obtained by projecting the facets of the convex hull of countable points in AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 23 the paraboloid { ( m, v ( m )) ∈ Z × R | m ∈ Z } ⊂ { ( x, ˜ v ( x )) ∈ R × R | x ∈ R } .If ρ ∈ Del v is a 3-cell, then there is a linear affine function ℓ ρ : R → R ,called the supporting function of ρ , such that ℓ ρ ( m ) = v ( m ) for m ∈ ρ ∩ Z ,and ℓ ρ ( m ) < v ( m ) for m ∈ ( R \ ρ ) ∩ Z .It is well-known that Del v is invariant under the translation by m ∈ Z as follows. Since, for any m = ( m , m , m ) ∈ Z , v ( m ′ + m ) = v ( m ′ ) + X i =1 (8 m i + 2(2 m + 2 m + 3 m )) m ′ i + v ( m ) ,v ( m ′ + m ) − v ( m ′ ) is a linear affine function of m ′ , and v ( m ′ + m ) inducesthe same decomposition Del v , i.e. { m + ρ | cells ρ ∈ Del v } = Del v . Lemma 4.2.
For any integer ˜ d > , the restriction of Del v on ∆ ˜ d is aunimodular subdivision T v of ∆ ˜ d , which has ˜ d simplices of dimension 3.Proof. Table 1 shows the restriction of the decomposition Del v on the unitcube [0 , , which consists of 6 simplices of dimension 3. We let p =(0 , , p = (1 , , p = (0 , , p = (0 , , p = (1 , , p =(1 , , p = (0 , ,
1) and p = (1 , , p i p j p k p l be the convex hullof points p i , p j , p k , p l , i.e. p i p j p k p l = Cov( p i , p j , p k , p l ). Table 1.
Decomposition Del v on [0 , . p p p p p p p p v p p p p x + 8 x + 13 x p p p p x + 16 x + 21 x − p p p p x + 20 x + 25 x −
12 –12 4 8 13 24 29 33 49 p p p p x + 16 x + 25 x −
12 –12 8 4 13 24 33 29 49 p p p p x + 20 x + 29 x −
16 –16 4 4 13 24 33 33 53 p p p p x + 28 x + 37 x −
32 –32 –4 –4 5 24 33 33 61
We need to verify that the supporting functions do not support any latticepoints outside the cube. Because of the Z -translation symmetry and Z -reflection symmetry of Del v , we check the nearby points p ′ = (0 , − , p ′ =(1 , − , p ′ = (1 , − , p ′ = (0 , − , p ′ = (0 , , − p ′ = (1 , , − p ′ = (1 , , −
1) and p ′ = (0 , , − Z -translation invariance of Del v implies that all 3-cells of Del v aresimplices with Euclidean volume . The restriction of Del v on the hyper-planes given by x + x + x = ˜ d, x i = 0 , i = 1 , , v on ∆ ˜ d ,for any ˜ d >
1, is a subdivision on ∆ ˜ d . Since the Euclidean volume of ∆ ˜ d is ˜ d , ∆ ˜ d contains ˜ d simplices. (cid:3) By using this lemma, we obtain the triangulation T v on ∆ d , and therestriction of T v on ∆ is the restriction of Del v on ∆ . The Z -translation Table 2.
Values of the functions on nearby lattice points. p ′ p ′ p ′ p ′ p ′ p ′ p ′ p ′ v p p p p x + 8 x + 13 x –8 0 13 5 –13 –5 3 –5 p p p p x + 16 x + 21 x − p p p p x + 20 x + 25 x −
12 –32 –16 9 –7 –37 –21 –1 –17 p p p p x + 16 x + 25 x −
12 –28 –8 17 –5 –37 –17 –1 –21 p p p p x + 20 x + 29 x −
16 –36 –16 13 –7 –45 –25 –5 –25 p p p p x + 28 x + 37 x −
32 –60 –32 5 –23 –69 –41 –13 –41
Figure 4.4.
Decomposition Del v on [0 , . ❍❍❍❍❍❍(cid:0)(cid:0)(cid:0)✟✟✟✟✟✟✟✟✟❍❍❍❅❅❅ ❍❍❍(cid:0)(cid:0)(cid:0)✟✟✟✟✟✟❍❍❍❅❅❅ ❍❍❍(cid:0)(cid:0)(cid:0)✟✟✟✟✟✟❍❍❍❅❅❅✟✟✟ ❍❍❍❍❍❍(cid:0)(cid:0)(cid:0)✟✟✟✟✟✟❍❍❍✟✟✟ (cid:0)(cid:0)(cid:0) ❍❍❍✟✟✟✟✟✟❍❍❍❅❅❅ (cid:0)(cid:0)(cid:0) ✟✟✟❍❍❍❅❅❅ (cid:0)(cid:0)(cid:0) invariance of Del v implies that for any m ∈ ∆ od ( Z ), the restriction of T v on m − (1 , ,
1) + ∆ is the translation of the restriction of T v on ∆ .Again Π v denotes the tropical variety of the restriction of v on ∆ d , i.e.the non-linear locus of the discrete Legendre transform (4.2) of v | ∆ d , whichis dual to T v . The restriction of v on any m − (1 , ,
1) + ∆ induces the sametropical variety Π K dual to the restriction of T v on ∆ by the Z -translation. Lemma 4.3.
Let T o be the set of 3-cells ρ ∈ T v such that either ρ belongs to ∆ od , i.e. ρ ⊂ ∆ od , or ρ shares a 2-face ρ ′ with the boundary ∂ ∆ od , i.e. ρ ′ ∈ T v , ρ ′ = ρ ∩ ∂ ∆ od , and dim ρ ′ = 2 . Then T o consists of d ( d − simplices, i.e. ♯ T o = d ( d − . Proof.
Note that ∆ od = (1 , ,
1) + ∆ d − , and the boundary ∂ ∆ od = (1 , ,
1) + ∂ ∆ d − is the union of some 2-cells in T v . The restriction of T v on (1 , ,
1) +∆ d − consists of ( d − simplices of dimension 3, and ∂ ∆ d − has 4( d − T v . For a 2-cell ρ ′ ⊂ ∂ ∆ od , there is a unique 3-cell ρ in T v such that ρ ⊃ ρ ′ and ρ does not belong to ∆ od . Therefore ♯ T o = ( d − + 4( d − = d ( d − . AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 25
Figure 4.5.
Illustration of Π v vs Π K via cases of curves,where Π is dual to a triangulation T of ∆ ⊂ R , and Π isdual to the restriction of T on ∆ . (cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) ⇐ =Π (cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Π (cid:3) Lemma 4.4.
For any m ∈ ∆ od ( Z ) , let T m be the set of 3-cells ρ ∈ T v suchthat ρ ⊂ m − (1 , ,
1) + ∆ . Then [ m ∈ ∆ od ( Z ) { (1 , , − m + ρ | ρ ∈ T m \T o } = { − cells ρ ∈ T v | ρ ⊂ ∆ } . Proof.
Note that∆ = ∆ ∪ ((1 , ,
0) + ∆ ) ∪ ((0 , ,
0) + ∆ ) ∪ ((0 , ,
1) + ∆ ) , and if a 3-cell ρ ∈ T v belongs to∆ ∪ (( d − , ,
0) + ∆ ) ∪ ((0 , d − ,
0) + ∆ ) ∪ ((0 , , d −
3) + ∆ ) , then ρ shares at most one point with ∆ od ( Z ). For example, if ρ ⊂ ∆ , thenfor any x ∈ ρ , x + x + x
3, while for an x ∈ ∆ od , x + x + x > x = (1 , , ρ does not belong to T o .It is clear that (1 , , − m + ρ ⊂ ∆ for any ρ ∈ T m . If ρ is a 3-cell of T v and ρ ⊂ ∆ , then ρ belongs to one of ∆ , (1 , ,
0) + ∆ , (0 , ,
0) + ∆ or(0 , ,
1) + ∆ . For example, if ρ ⊂ (1 , ,
0) + ∆ without loss of generality,then ( d − , ,
0) + ρ ⊂ ( d − , ,
0) + ∆ . We obtain the conclusion by thefact that ∆ ⊂ ∆ , ( d − , ,
0) + ∆ ⊂ ( d − , ,
0) + ∆ , (0 , d − ,
0) + ∆ ⊂ (0 , d − ,
0) + ∆ and (0 , , d −
3) + ∆ ⊂ (0 , , d −
4) + ∆ . (cid:3) Proofs of i), ii), iii), and v) in Theorem 1.1.
First, we recall the construc-tion in Section 4 of [40]. Let P = CP \ ( ˜ H ∪ ˜ H ∪ ˜ H ∪ ˜ H ), where˜ H i denotes a tubular neighbourhood of the hyperplane H i . The interiorint( P ) is diffeomorphic to P . The boundary ∂ P admits a stratification ∂ P = ∂ P ∪ ∂ P by Proposition 2.24 of [40], where ∂ P is the disjointunion of 6 copies of T , and ∂ P consists of 4 connected components whereeach component is diffeomorphic to the total space of a trivial S -bundle on Figure 4.6.
Illustrations of Lemma 4.3 and Lemma 4.4 via2-dimensional cases. ❅❅❅❅❅❅❅❅❅❅ ❅❅❅❅❅❅❅❅❅ ❅❅❅❅❅❅❅❅ ❅❅❅❅❅❅ ❅❅❅❅❅ ❅❅❅❅❅ ∆ d r r r rrrr rr rrr r r rrr r rr rr rr r rr r ❅❅❅❅❅❅❅❅❅❅ ❅❅❅❅❅❅ ❅❅❅❅❅ ❅❅❅ ∆ od the 2-dimensional pair-of-pants P , i.e. S × P under a certain trivialisa-tion. Moreover ∂ P ⊂ ∂ P . The boundary ∂ P is obtained by gluing theclosures of components of ∂ P along T components of ∂ P .Proposition 4.6 in [40] shows that there exists a proper submanifold Q n ⊂ ( C ∗ ) n +1 , n = 0 , ,
2, such thati) Q is a point.ii) Q n is isotopic to the hyperplane H o = { ( w , · · · , w n +1 ) ∈ ( C ∗ ) n +1 | w + · · · + w n +1 = 0 } ⊂ ( C ∗ ) n +1 . Note that the pair-of-pants P n is bi-holomorphic to H o .iii) Q n is invariant under the symmetric group S n +2 -action on ( C ∗ ) n +1 ,which interchanges the homogeneous coordinates on CP n +1 ⊃ ( C ∗ ) n +1 .iv) For a ̺ ≪ − Q n ∩ ( C ∗ ) n +1 ̺ = Q n − × C ∗ ̺ , where ( C ∗ ) n +1 ̺ = { ( w , · · · , w n +1 ) ∈ ( C ∗ ) n +1 | log | w n +1 | < ̺ } and C ∗ ̺ = { w ∈ C ∗ | log | w | < ̺ } . Furthermore, Q n ∩ ( C ∗ ) n +1 ̺ is invariantunder the translation w n +1 cw n +1 for 0 < c < Q n = Q n ∩ Log − ( R ∆) where R ∆ = { ( x , · · · , x n +1 ) ∈ R n +1 | x i > − R, i = 1 , · · · , n + 1 , n +1 X i =1 x i R } , for R ≫
1, then ¯ Q n is diffeomorphic to P n as manifolds with corners.For each 3-cell q ∈ T v , we take a copy ¯ Q q of P , which is identifiedwith ¯ Q ⊂ ( C ∗ ) . Q q denotes the interior of ¯ Q q , and ˇ q is the vertex of Π v corresponding to q . Each connected component of ∂ ¯ Q q = ∂ P correspondsto a 2-cell of T v , which is a face of q . If ρ is a 2-cell of T v such that ρ = q ∩ q ′ ,we consider the corresponding components F q and F q ′ of ∂ ¯ Q q and ∂ ¯ Q q ′ respectively. We glue ¯ Q q and ¯ Q q ′ along the closures of F q and F q ′ . Moreprecisely, if we assume that both F q and F q ′ are given by log | w | = − R via certain transforms of the symmetry group S , then we attach F q and AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 27
Figure 4.7. ¯ Q , and P = Q → Y. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ¯ Q ✚✙✛✘ = F q ′ by the map ( w , w , w ) ( w , w , ¯ w ), i.e. gluing the 2-dimensionalpair-of-pants canonically, and reversing the orientation of the S -fibre. Insuch way, we obtain a manifold Q with boundary, i.e. Q = [ all 3 − cells q ∈T v ¯ Q q . The boundary ∂ Q is obtained by gluing the closures of connected compo-nents F of ∂ ¯ Q q corresponding to 2-cells of T v in ∂ ∆ d . Note that any two F and F ′ are glued along a certain T component. Thus we have strati-fied structure ∂ Q = ∂ Q ∪ ∂ Q where ∂ Q consists of finite T , and eachconnected component of ∂ Q is a S -bundle.Let W o = Q\ ∂ Q , and let W be the space obtained by collapsing the S fibres of ∂ Q (cf. Section 4 in [40]). Note that W is a differential manifoldsince the collapsing locally coincides with collapsing the boundary of ¯ P in CP . Theorem 4 of [40] proves that W o is diffeomorphic to X ot ⊂ X t ,and W is diffeomorphic to X t . Thus X t is decomposed into d copies ofpair-of-pants.Let M o = a q ∈T o Q q , which is the disjoint union of d ( d − copies of pair-of-pants P by Lemma4.3. Note that q intersects with the boundary ∂ ∆ d at most one point, i.e. avertex of q , which corresponds to a connected component of the complementof Π v in R . Therefore any non-zero dimensional face ρ ⊂ q does not belongto the boundary ∂ ∆ d , and the closure M o ⊂ W o . Furthermore, M o \ M o = [ q ∈T o ∂ ¯ Q q . Note that there is a fibration P → Y from P to a graph of Y-shapewith generic fibres S and one singular fibre of shape ⊖ . For each ¯ Q q ,any component F of ∂ ¯ Q q admits a fibration F → Y to a Y -shape graphY with generic fibres T and one singular fibre of shape ⊖ × (cid:13) . Since ∂ ¯ Q q consists of 6 copies of T , we have a fibration ∂ ¯ Q q → B q where B q is a graph consisting of 4 vertices and 6 edges with generic fibres T and 6singular fibres ⊖ × (cid:13) . If we have glued ¯ Q q and ¯ Q q ′ along the components F q and F q ′ of ∂ ¯ Q q and ∂ ¯ Q q ′ respectively, then we glue the closures of thecorresponding graphs Y q and Y q ′ , and obtain a graph B . Moreover, thereis a fibration M o \ M o → B with generic fibres T and finite singular fibresof shape ⊖ × (cid:13) .We have proved i) and ii) of Theorem 1.1, and now we prove the statementiii).We apply the above construction to the restriction of T v on ∆ and Π K ,and obtain a manifold Y diffeomorphic to the K3 surface. More precisely,for any m ∈ ∆ od ( Z ), we consider m − (1 , ,
1) + ∆ . If we let Q m = [ q ∈T m ¯ Q q ⊂ Q , where T m is defined in Lemma 4.4, then as above, Q m is a manifold withboundary ∂ Q m , which is obtained by gluing the closures of connected com-ponents F of ∂ ¯ Q q corresponding to 2-cells in m − (1 , ,
1) + ∂ ∆ . Since therestriction of T v on m − (1 , ,
1) + ∆ is the translated to the restriction of T v on ∆ , Q m is independent of the choice of m , and we obtain Y by collapsingthe boundary ∂ Q m as the above construction. And Y is a K3 surface.If Q ′ m = [ q ∈T m \T o ¯ Q q ⊂ Q m , then we collapse the closures of the components of ∂ ¯ Q q corresponding to2-cells in ∂ ∆ d ∩ ∂ ( m − (1 , ,
1) + ∆ ). Note that if q shares only one 2-cell(or two 2-cells) with ∂ ∆ d , then the interior of the collapsed ¯ Q q is ( C ∗ ) ( C × C ∗ respectively). We obtain a manifold Y m with boundary, and denotethe interior as Y om = Y m \ ∂Y m , which satisfies Y om ⊃ a q ∈T m \T o Q q . Then Y om can be regarded as a subset both in Y and W , i.e. Y om ⊂ Y and Y om ⊂ W , and can have many connected components including ( C ∗ ) and C × C ∗ .Since S m ∈ ∆ od ( Z ) ( T m \T o ) S T o consists of all 3-cells of T v , W = M o [ [ m ∈ ∆ od ( Z ) Y om , and Y = [ m ∈ ∆ od ( Z ) Y om , by regarding Y om ⊂ Y and Lemma 4.4. Note that Y om does not intersect with Q q , for any m ∈ ∆ od ( Z ) and q ∈ T o , when they are regarded as subsets in W ,and thus Y om ∩ M o is empty. We obtain iii) by choosing an open dense subset M ′ ⊂ Y such that any connected component of M ′ belongs to a certain Y om . AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 29
Finally the Riemann-Roch formula shows that2 χ ( X ) + 3 τ ( X ) = K X = d ( d − , and we obtain v) by (3.1). (cid:3) We remark that many arguments in this section may be carried out inthe more general frame work, the Gross-Siebert programme (cf. [27, 28]).We expect to generalise Theorem 1.1 to more general cases, and leave it tothe future research. 5.
Degenerations
The final section finishes the proof of Theorem 1.1. We are continuing touse the notations and the conventions in Section 4.Recall that X t is the patchworking family given by the polynomial(5.1) f t ( w ) = X m ∈ ∆ d ( Z ) t − v ( m ) w m , t ∈ (1 , ∞ ) , where v is defined by (4.8), which is a family of degree d hypersurfaces in CP . If [ z , z , z , z ] are the homogeneous coordinates on CP , then w i = z i /z , i = 1 , ,
3. Note that the canonical bundle O X t ( K X t ) = O CP ( d − | X t by the adjunction formula, and is very ample.For any m = ( m , m , m ) ∈ ∆ o ( Z ), since m + m + m d −
1, and m i > i = 1 , , t − v ( m ) w m dw ∧ dw ∧ dw w (1 , , f t = t − v ( m ) w m − w m − w m − dw ∧ dw ∧ dw f t is a meromorphic 3-form on CP , and has a simple pole along X t . ThePoincar´e residue formula gives a holomorphic 2-formΩ m = t − v ( m ) Res X t w m dw ∧ dw ∧ dw w (1 , , f t on X t , i.e.Ω m = t − v ( m ) w m dw ∧ dw w (1 , , ∂f t ∂w = − t − v ( m ) w m dw ∧ dw w (1 , , ∂f t ∂w = t − v ( m ) w m dw ∧ dw w (1 , , ∂f t ∂w on X t . Note that Ω m represents a non-trivial cohomological class, i.e.0 = [Ω m ] ∈ H ( X t , C ) . Proof of iv) in Theorem 1.1.
Let ρ be a 3-cell in T v such that ρ ∩ ∆ od ( Z ) is not empty, and let m ∈ ρ ∩ ∆ od ( Z ). If m ′ ∈ ρ ∩ ∆ d ( Z ) is adifferent vertex of ρ , i.e. m = m ′ , then the 1-cell m, m ′ ⊂ ρ connecting m and m ′ corresponds a 2-cell Π m,m ′ in Π v . For any x ∈ int(Π m,m ′ ), l m ( x ) = l m ′ ( x ) > l m ′′ ( x ) , for any m ′′ ∈ ∆ d ( Z ) \ m, m ′ , where l m ( x ) = h m, x i − v ( m ). We assume that m − m ′ = 0 without loss of generality. By Corollary 6.4 in [40], the amoebas A t converges to Π v when t → ∞ ,and furthermore, Theorem 5 of [40] asserts that a certain normalisation of X ot converges to a limit in ( C ∗ ) . We take a close look at the limit in thecurrent case.If V is a small open neighborhood of a point in int(Π m,m ′ ) such that V ∩ Π v ⊂ int(Π m,m ′ ), then V ∩ Π v belongs to the hyperplane given by theequation l m ( x ) − l m ′ ( x ) = h m − m ′ , x i − v ( m ) + v ( m ′ ) = 0 . Let θ i be the angle of w i , i.e. w i = exp( x i log( t ) + √− θ i ) , i = 1 , , . We regard ( θ , θ , θ ) as angular coordinates on T , and identify ( C ∗ ) = R × T by w ( x, θ ), where x = ( x , x , x ) and θ = ( θ , θ , θ ). Equiva-lently, we regards ( C ∗ ) as the quotient of C by √− Z . Lemma 5.1. If H t : ( C ∗ ) → R × T is defined by w ( x, θ ) , then H t ( X ot ) ∩ ( V × T ) converges to W V in the Hausdorff sense, when t
7→ ∞ ,where W V = { ( x, θ ) ∈ V × T | l m ( x ) = l m ′ ( x ) , h m − m ′ , θ i = π + 2 π Z } , and h m, θ i = m θ + m θ + m θ .Proof. Note that the log map Log t is the composition of H t with the pro-jection R × T → R . For any x ∈ V ∩ A t and w ∈ Log − t ( x ) ∩ X ot , t − v ( m ) | w m | = t l m ( x ) , and the patchworking polynomial (5.1) says that0 = e √− h m,θ i + t l m ′ ( x ) − l m ( x ) e √− h m ′ ,θ i + X m ′′ ∈ ∆ d ( Z ) \ m,m ′ t l m ′′ ( x ) − l m ( x ) e √− h m ′′ ,θ i , which is the defining equation of X ot ∩ ( V × T ). When t → ∞ , since t l m ′′ ( x ) − l m ( x ) →
0, we obtain e √− h m,θ i + t l m ′ ( x ) − l m ( x ) e √− h m ′ ,θ i → , and t l m ′ ( x ) − l m ( x ) → . The limiting equations are h m − m ′ , θ i = π + 2 π Z , and l m = l m ′ , which define W V . If we define a neighbourhood Ξ t of W V in V × T by theinequality (cid:12)(cid:12)(cid:12) e √− h m,θ i + t l m ′ ( x ) − l m ( x ) e √− h m ′ ,θ i (cid:12)(cid:12)(cid:12) < x ′ ∈ V (cid:16) X m ′′ ∈ ∆ d ( Z ) \ m,m ′ t l m ′′ ( x ′ ) − l m ( x ′ ) (cid:17) , then X ot ∩ ( V × T ) ⊂ Ξ t , and converges to W V in the Hausdorff sense. (cid:3) We have an isotopic embedding φ t : W V → X ot with φ ∞ = id, which iscertainly not unique. One way to obtain φ t is to integrate a vector field ( u, ϑ )satisfying ∂ ˜ f t ∂t + h ∂ ˜ f t ∂x , u i + h ∂ ˜ f t ∂θ , ϑ i = 0 , where ˜ f t = t − l m ( x ) f t , since a direct AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 31 calculation shows that | ∂ ˜ f t ∂x | > | m ′ − m | log( t ), and | ∂ ˜ f t ∂θ | > | m ′ − m | ,for t ≫ T m,m ′ = { θ ∈ T |h m − m ′ , θ i = π + 2 π Z } , then W V = ( V ∩ Π v ) × T m,m ′ . Lemma 5.2.
For any isotopic embedding φ t : W V → X ot with φ ∞ = id and x ∈ V ∩ Π m,m ′ , Z φ t ( { x }× T m,m ′ ) Ω m = 0 , and consequently, φ t ( { x } × T m,m ′ ) represents a non-zero class in H ( X t , R ) for t ≫ , i.e. = [ φ t ( { x } × T m,m ′ )] ∈ H ( X t , R ) . Furthermore, if ̟ is a toric symplectic form on CP , then Z φ t ( { x }× T m,m ′ ) ̟ = 0 . Proof.
A direct calculation shows that w ∂f t ∂w = m t − v ( m ) w m + m ′ t − v ( m ′ ) w m ′ + X m ′′ ∈ ∆ d ( Z ) \ m,m ′ m ′′ t − v ( m ′′ ) w m ′′ = ( m − m ′ ) t − v ( m ) w m + m ′ f t + X m ′′ ∈ ∆ d ( Z ) \ m,m ′ ( m ′′ − m ′ ) t − v ( m ′′ ) w m ′′ = ( m − m ′ ) t − v ( m ) w m + m ′ f t + o ( t ) . Note that on V , t − v ( m ′′ ) | w m ′′ | t − v ( m ) | w m | = t ( l m ′′ − l m )(Log t ( w )) → , and thus | o ( t ) | t − v ( m ) | w m | → , when t → ∞ . Therefore Ω m | Log − t ( V ) ∩ X t = d log( w ) ∧ d log( w ) m − m ′ + t v ( m ) w − m o ( t ) , and Z φ t ( { x }× T m,m ′ ) Ω m → Z T m,m ′ dθ ∧ dθ m ′ − m = 0 , when t → ∞ . We obtain the first conclusion.Since ( C ∗ ) = R × T → R is a Lagrangian fibration with respect to ̟ ,i.e. ̟ | { x }× T ≡
0, we have Z φ t ( { x }× T m,m ′ ) ̟ = Z { x }× T m,m ′ ̟ = 0 . Figure 5.1.
Illustrations of R ∆, ˜ Q , and Q . (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Q ˜ Q Log= ⇒ R ∆ ❅❅❅❅❅❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0) (cid:3) We continue to prove Theorem 1.1.
Proof of iv) in Theorem 1.1.
Let Q ⊂ ( C ∗ ) be the submanifold constructedin Proposition 4.6 of [40]. Recall that the last assertion of Proposition 4.6of [40] says that for a certain ̺ ≪ − Q n ∩ ( C ∗ ) n +1 ̺ = Q n − × C ∗ ̺ , where n = 1 ,
2, and ( C ∗ ) n +1 ̺ = { ( w , · · · , w n +1 ) ∈ ( C ∗ ) n +1 | log | w n +1 | < ̺ } , and Q n ∩ ( C ∗ ) n +1 ̺ is invariant under the translation w n +1 cw n +1 for 0 < c < Q = Q ∩ { ( w , w , w ) ∈ ( C ∗ ) | log | w | < ̺, log | w | < ̺ } = { ( w , w , w ) ∈ ( C ∗ ) | log | w | < ̺, log | w | < ̺, w ≡ − } , where we choose w ≡ − Q ⊂ C ∗ . If we consider the identifi-cation ( C ∗ ) = R × T , via x = ( x , x , x ) ∈ R , θ = ( θ , θ , θ ) ∈ T , and w i = exp( x i + √− θ i ), i = 1 , ,
3, then˜ Q = { x ∈ R | x = 0 , x < ̺, x < ̺ } × { θ ∈ T | θ = π } . If Π denotes the tropical hyperplane of the pair-of-pants H o defined by1 + w + w + w = 0, i.e. the non-smooth locus of max { , x , x , x } ,then Π = { ( x , x , x ) ∈ R | x = 0 , x , x } is the 2-cell in Πcorresponding to the 1-cell (0 , , , (1 , ,
0) in the standard simplex ∆. Sincethe Log map is the projection of R × T → R , Log( ˜ Q ) ⊂ Π ,Log t ( ˜ Q ) = { x ∈ R | x = 0 , x < ̺ log( t ) , x < ̺ log( t ) } ⊂ Π , and S t> Log t ( ˜ Q ) = int(Π ). Furthermore, if V is an open subset such that V ∩ Π ⊂ int(Π ), then the same argument as in the proof of Lemma 5.1shows that H t ( H o ∩ ( V × T )) converges to (Π ∩ V ) × { θ ∈ T | θ = π } ⊂ ˜ Q in the Hausdorff sense.By the S -symmetries of Π and Q , there is an open subset ˜ Q ⊂ Q consisting of 6 connected components, and each component is a copy of ˜ Q by passing to the S -action on ( C ∗ ) . Moreover, S t> Log t ( ˜ Q ) is the unionof the interior of 2-cells of Π.Now, we recall the proof of Theorem 4 in [40], which is in Subsection 6.6of [40]. Note that ASL (3 , Z ) acts on Z by A ( m ) j = P a ji m i + b j for any A ∈ ASL (3 , Z ), and acts on ( C ∗ ) by w m w A ( m ) . For any 3-cell q in T v , there is an open neighbourhood V q ⊂ R of the vertex o ∈ Π, and an A q ∈ ASL (3 , Z ) such that ˇ q = A q ( o ) where ˇ q ∈ Π v is corresponding vertexof q , and Π v = [ all 3 − cells q ∈T v A q ( V q ∩ Π) . Then Q ot = [ all 3 − cells q ∈T v A q (( V q × T ) ∩ H t ( Q ))is diffeomorphic to W o , and is isotopic to X ot for t ≫ W o = Q ot , we have M o ⊂ Q ot , and Q q ⊂ A q (( V q × T ) ∩ H t ( Q )) for any q ∈ T o . If T is a generic fibre of λ : M o \ M o → B ,then T is a generic fibre of λ q : ∂ ¯ Q q = ¯ Q q \ Q q → B q for a certain q ∈ T o .Recall that ¯ Q q = A q (( R ∆ × T ) ∩ H t ( Q )) where R ∆ = { x ∈ R | x i > − R, i = 1 , , , x + x + x R } for a certain R >
0, and R ∆ ⊂ V q .The boundary ∂ ¯ Q q = A q (( ∂ ( R ∆) × T ) ∩ H t ( Q )) consists of 4 copies of S -bundles over ¯ P gluing along 6 copies of T . Note that for any 2-face ρ ⊂ ∂ ( R ∆), for example given by x = − R , the intersection( ρ × T ) ∩ H t ( Q ) ⊂ Q × { w ∈ C ∗ | log | w | = − R } = Q × S , and is diffeomorpic to ¯ P × S . The fibration λ q is obtained by gluing( ρ × T ) ∩ H t ( Q ) → ρ ∩ Π along boundaries, and ρ ∩ Π is the Y-shapegraph.Let T be a generic fibre of λ : M o \ M o → B . By letting t ≫
1, we cantake A − q ( T ) as a fibre of ( ρ × T ) ∩ H t ( ˜ Q ) → ρ ∩ Log t ( ˜ Q ), which is therestriction of the T -fibration ˜ Q → Log t ( ˜ Q ). By passing to the S -action,we can assume that A − q ( T ) = { ( x, θ ) ∈ R × T | θ = π, x = − R, x = − R , x = 0 } . We apply Lemma 5.2 to T ⊂ A q (( V q × T ) ∩ H t ( ˜ Q )), and obtain theconclusion iv). (cid:3) Proof of vi) in Theorem 1.1.
Note that { Ω m | m ∈ ∆ o ( Z ) } is a basisof H ( X t , K X t ), and defines an embedding(5.2) Ψ t : X t → CP N , N = ♯ ∆ o ( Z ) − p g − , such that O X t ( K X t ) = Ψ ∗ t O CP N (1). If(5.3) Z m = t − v ( m ) w m , m ∈ ∆ o ( Z ) , then we embed X ot in ( C ∗ ) N ⊂ CP N via Z m /Z (1 , , , m ∈ ∆ o ( Z ), and theclosure of X ot in CP N is the image Ψ t ( X t ). We regard Z m , m ∈ ∆ o ( Z ),as the homogeneous coordinates of CP N . Furthermore, Z m are generalisedtheta functions in the sense of [28].For any 2-cell ρ in ∂ ∆ od of the subdivision T v , let X ρ = { Z m = 0 , m ∈ ∆ od ( Z ) \ ρ } ⊂ CP N , and for any 3-cell ρ ⊂ ∆ od , let X ρ = { Z m + Z m + Z m + Z m = 0 , { m , m , m , m } = ∆ od ( Z ) ∩ ρ , and Z m = 0 , m ∈ ∆ od ( Z ) \ ρ } ⊂ CP N . We define a singular variety(5.4) X = (cid:16) [ all 3 − cells ρ ⊂ ∆ od X ρ (cid:17) [ (cid:16) [ all 2 − cells ρ ⊂ ∂ ∆ od X ρ (cid:17) , which has d ( d − irreducible components by Lemma 4.3, and each com-ponent is the complex projective plane, i.e. X ρ = CP . Lemma 5.3.
The regular locus X o of X consists of d ( d − copies ofpair-of-pants, i.e. X o = d ( d − ` P .Proof. If X ρ is the corresponding component of a 3-cell ρ ⊂ ∆ od , then theintersection with another component X ρ ′ is a hyperplane { Z m = 0 } ∩ X ρ without loss of generality, where { m , m , m , m } = ∆ od ( Z ) ∩ ρ and { m , m , m } ⊂ ρ ∩ ρ ′ . Thus the regular locus X oρ = { [ Z m , Z m , Z m , Z m ] ∈ X ρ | Z m Z m Z m Z m = 0 } is a pair-of-pants, i.e. X oρ = P . More precisely, if we regards X ρ as theprojective plane { [ Z m , Z m , Z m ] ∈ CP } , then Z m i = 0, i = 1 , ,
3, givetwo coordinates axis, and the infinite line, and − Z m = Z m + Z m + Z m = 0defines the fourth line.If X ρ corresponds to a 2-cell ρ ∈ ∂ ∆ od , then the intersection with otherthree components X ρ ′ corresponding three different 2-cells ρ ′ ∈ ∂ ∆ od is givenby Z m Z m Z m = 0 where { m , m , m } = ∆ od ( Z ) ∩ ρ , which is the unionof coordinates axis and the infinite line. Since there is a 3-cell ˜ ρ ⊂ ∆ od suchthat ρ ⊂ ∂ ˜ ρ , X ρ ∩ X ˜ ρ is given by − Z m = Z m + Z m + Z m = 0 where m is the vertex of (˜ ρ \ ρ ) ∩ ∆ od ( Z ). Hence the regular locus X oρ of X ρ is apair-of-pants. (cid:3) Lemma 5.4.
Let ρ be a 3-cell of T v such that ρ ⊂ ∆ od , { m , m , m , m } = ρ ∩ ∆ od ( Z ) , and ℓ ρ be the supporting function of ρ . i) For any m ∈ ∆ d ( Z ) , w m = t ℓ ρ ( m ) Z a m Z a m Z a m Z a m , where m = a m + a m + a m + a m , and a + a + a + a = 1 . AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 35 ii) If m ∈ ∆ od ( Z ) , then Z b m Z b m Z b m Z b m Z m = t ℓ ρ ( m ) − v ( m ) Z a + b m Z a + b m Z a + b m Z a + b m , where b i = max { , − a i } , i = 0 , , , . iii) If ρ ′ is a 3-cell sharing a 2-face with ρ , i.e. ρ ∩ ρ ′ is a 2-cell in T v ,and { m , m , m , m } = ρ ′ ∩ ∆ d ( Z ) , then w m + m = w ǫ m + ǫ m + ǫ m , and w m Z m = t ℓ ρ ( m ) Z ǫ m Z ǫ m Z ǫ m , where two elements of { ǫ , ǫ , ǫ } equal to , and one element is ,e.g. ǫ = ǫ = 1 and ǫ = 0 .Proof. Note that ρ is a standard simplex generated by a Z -basis e , e , e ∈ Z , i.e. ρ = { x e + x e + x e ∈ R | x i > , i = 1 , , , x + x + x } and Z = Z · e + Z · e + Z · e . For example, e = (1 , − , e = (1 , , e = (0 , − ,
1) generate a simplex in T v .If we regard m as the origin, then e i = m i − m , i = 1 , ,
3, is the Z -basis.Therefore, for any m ∈ ∆ d ( Z ), m = m + a e + a e + a e where a i ∈ Z .Since w m j = w m w e j , j = 0 , , ,
3, we have w m = w m Y i =1 ( w e i ) a i = t v ( m ) Z m Y i =1 ( t v ( m i ) − v ( m ) Z m i /Z m ) a i . Let ℓ ρ ( x ) = h n ρ , x i + b ρ be the supporting function of the cell ρ , i.e. ℓ ρ ( m j ) = v ( m j ), j = 0 , , ,
3, and v ( m ) > ℓ ρ ( m ) for any m ∈ ∆ d ( Z ) \ ρ . Then v ( m i ) − v ( m ) = h n ρ , e i i , and ℓ ρ ( m ) = v ( m ) + h n ρ , X i =1 a i e i i . Thus w m = t ℓ ρ ( m ) Z a m Z a m Z a m Z a m , where a = 1 − P i =1 a i .If we write m = ǫ m + ǫ m + ǫ m + ǫ m , then m − m = P i =1 ǫ i ( m i − m ). Note that ρ ′ belongs to the cube generated by m i − m , i.e. 0 ǫ i i = 1 , ,
3, and ρ ∩ ρ ′ belongs to the hyperplane { P i =1 x i ( m i − m ) | x + x + x = 1 } . Thus ǫ = 1 − ǫ − ǫ − ǫ <
0, and P i =1 ( m i − m ) belongs toa 3-cell sharing no faces with ρ . We assume that ǫ = − ǫ = ǫ = 1, and ǫ = 0 without loss of generality. Therefore w m + m = w ǫ m + ǫ m + ǫ m , Figure 5.2. m ∈ ρ vs m ∈ ρ ′ , and w m Z m = t ℓ ρ ( m ) Z m Z m . ❍❍❍(cid:0)(cid:0)(cid:0)✟✟✟✟✟✟❍❍❍❅❅❅ ✈ m m (cid:0)(cid:0)(cid:0)✟✟✟ m m ❅❅❅ ρρ ′ m m ✟✟✟❅❅❅ ρm m ❍❍❍ and we obtain the conclusion. (cid:3) Note that if d = 5, then the canonical bundle O X t ( K X t ) = O CP (1) | X t ,and X t ⊂ CP is the canonical embedding. Thus vi) in Theorem 1.1 holdsas shown in the introduction. Now we assume d >
5. Then ∆ od contains( d − T v . Lemma 5.5.
Let ρ ⊂ ∆ od be a 3-cell in T v , and ρ ′ be another 3-cell sharinga 2-face with ρ , i.e. ρ ∩ ρ ′ is a 2-cell in T v . Denote { m , m , m , m } = ρ ∩ ∆ d ( Z ) and { m , m , m , m } = ρ ′ ∩ ∆ d ( Z ) . i) If ρ ′ ⊂ ∆ od , then there are integers c > , c > , c > such that Z c m Z c m Z c m (cid:16) X m ∈ ∆ od ( Z ) Z m (cid:17) + o ( t ) = 0 , where o ( t ) is a polynomial. ii) Assume that m ∈ ∂ ∆ d . If ρ ′′ ⊂ ∆ od is another 3-cell such that m isnot a vertex of ρ ′′ , i.e. ρ ′′ ∩ ∆ d ( Z ) = { m , m , m , m } and m = m j , j = 5 , , , , then there are integers c > , c > , c > , c > such that Z c m Z c m Z c m Z c m Z m (cid:16) X m ∈ ∆ od ( Z ) Z m (cid:17) + o ( t ) = 0 , for a polynomial o ( t ) . iii) Both o ( t ) and o ( t ) are polynomials of Z m with the coefficients tend-ing to zero when t → ∞ , i.e. o ( t ) → , and o ( t ) → . Proof.
The patchworking polynomial reads f t = X m ∈ ∆ d ( Z ) t − v ( m ) w m = X m ∈ ∆ od ( Z ) Z m + X m ′ ∈ ∂ ∆ d ∩ Z t − v ( m ′ ) w m ′ = 0 , by ∆ od ( Z ) = int(∆ d ) ∩ Z .Assume that there is another 3-cell ρ ′ ⊂ ∆ od such that ρ ∩ ρ ′ is the 2-cell with vertices m , m , m . If m denotes the other vertex of ρ ′ , then by AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 37
Lemma 5.4, w m + m = w ǫ m + ǫ m + ǫ m , or equivalently, Z m Z m = t ℓ ρ ( m ) − v ( m ) Z ǫ m Z ǫ m Z ǫ m , for some 0 ǫ i i = 1 , ,
3, and t − v ( m ′ ) w m ′ = t ℓ ρ ( m ′ ) − v ( m ′ ) Z a m Z a m Z a m Z a m = t ℓ ρ ′ ( m ′ ) − v ( m ′ ) Z b m Z b m Z b m Z b m , for certain a i and b i , i = 0 , , ,
3, where m ′ = a m + P i =1 a i m i = b m + P i =1 b i m i , and a + b = 0. We choose the monomial with the power of Z m or Z m being non-negative in the above expression, i.e. if a > t − v ( m ′ ) w m ′ = t ℓ ρ ( m ′ ) − v ( m ′ ) Z a m Z a m Z a m Z a m . Equivalently, we choose either the cell ρ or ρ ′ such that the only possiblepoles of w m ′ are along { Z m i = 0 } , i = 1 , ,
3. Thus there are c > , c > , c > Z c m Z c m Z c m (cid:16) X m ∈ ∆ od ( Z ) Z m (cid:17) + X m ′ ∈ ∂ ∆ d ∩ Z t ℓ ( m ′ ) − v ( m ′ ) M m ′ = 0 , where M m ′ is a monomial, and either ℓ = ℓ ρ or ℓ = ℓ ρ ′ . Let o ( t ) = X m ′ ∈ ∂ ∆ d ∩ Z t ℓ ( m ′ ) − v ( m ′ ) M m ′ , which satisfies o ( t ) → t → ∞ , since t ℓ ρ ( m ′ ) − v ( m ′ ) and t ℓ ρ ′ ( m ′ ) − v ( m ′ ) have negative powers.Now we assume that m ∈ ∂ ∆ d . By Lemma 5.4, Z m (cid:16) X m ∈ ∆ od ( Z ) Z m + X m = m ′ ∈ ∂ ∆ d ∩ Z t − v ( m ′ ) w m ′ (cid:17) + t ℓ ρ ( m ) − v ( m ) Z ǫ m Z ǫ m Z ǫ m = 0 . If ρ ′′ ⊂ ∆ od is another 3-cell such that m is not a vertex of ρ ′′ , i.e. ρ ′′ ∩ ∆ d ( Z ) = { m , m , m , m } and m = m j , j = 5 , , ,
8, then by Lemma 5.4, t − v ( m ′ ) w m ′ = t ℓ ρ ′′ ( m ′ ) − v ( m ′ ) Z a ′ m Z a ′ m Z a ′ m Z a ′ m . The same argument as above shows that there are integers c > c > c > c > Z c m Z c m Z c m Z c m Z m (cid:16) X m ∈ ∆ od ( Z ) Z m (cid:17) + o ( t ) = 0 , where o ( t ) = X m ′ ∈ ∂ ∆ d ∩ Z t ℓ ′′ ( m ′ ) − v ( m ′ ) M ′ m ′ → , when t → ∞ , M ′ m ′ are monomials of Z m , Z m , Z m , Z m , Z m , Z m , Z m , Z m ,and either ℓ ′′ = ℓ ρ or ℓ ′′ = ℓ ρ ′′ . (cid:3) Proof of vi) in Theorem 1.1.
When t → ∞ , Ψ t ( X t ) converges to a analyticsubset X ∞ of dimension 2 in CP N by passing to a subsequence if necessary(cf. [10]). The convergence is in the sense of analytic spaces. Recall thatan analytic subset Y is locally defined by finite holomorphic functions u =0 , · · · , u k = 0, and the function sheaf O Y on Y is locally given by thequotient O CP N / ( u , · · · , u k ). Note that nilpotent functions are allowed. Ananalytic subset Y ′ ⊂ Y is a subset of Y with surjections O Y → O Y ′ . A familyof subsets Y t converges to Y , if Y t is locally given by u t = 0 , · · · , u tk = 0, and u tj → u j , j = 1 , · · · , k , when t → ∞ . For example, if Y is defined by x = 0in C , and Y ′ is given by x = 0, then Y ′ ⊂ Y via C [ x, y ] / ( x ) → C [ x, y ] / ( x ).We could view Y as two copies of Y ′ stacking together, i.e. Y = 2 Y ′ . Afamily Y t , defined by x ( x + t − y ) = 0, converges to Y .For any m ∈ ∆ od ( Z ), let H m be the hyperplane in CP N defined by Z m = 0,and for any 3-cell ρ ⊂ ∆ od of T v , let ˜ X ρ be the subset given by Z m ′ = 0 forall m ′ ∈ ∆ od ( Z ) \ ρ , which is the projective space of dimension 3, i.e.˜ X ρ = { [ Z m , Z m , Z m , Z m ] ∈ CP } = \ m ′ ∈ ∆ od ( Z ) \ ρ H m ′ , where { m , m , m , m } = ρ ∩ ∆ od ( Z ). Furthermore, if ρ ′ ⊂ ∆ od is an another3-cell, then ˜ X ρ ∩ ˜ X ρ ′ = ˜ X ρ ∩ H m ′′ , where m ′′ ∈ ( ρ ′ \ ρ ) ∩ ∆ od ( Z ).The image Ψ t ( X t ) satisfies the equations in Lemma 5.4 and Lemma 5.5.If ρ ⊂ ∆ od with vertices { m , m , m , m } , then ii) of Lemma 5.4 assertsthat X ∞ satisfies Z b m Z b m Z b m Z b m Z m = 0 for all m ∈ ∆ od ( Z ) \ ρ , and thus,as analytic subset, X ∞ ⊂ ˜ X ρ ∪ b ′ H m ∪ b ′ H m ∪ b ′ H m ∪ b ′ H m , for certain b ′ i > i = 0 , , ,
3. Since ˜ X ρ ⊂ H m for all m ∈ ∆ od ( Z ) \ ρ , weobtain X ∞ ⊂ [ all 3 − cells ρ ⊂ ∆ od ˜ X ρ = ˜ X, as analytic subsets. Note that ˜ X is a subvariety, and consists of ( d − irreducible components. Each irreducible component of ˜ X is a copy of CP .We consider two 3-cells ρ and ρ ′ in ∆ od , which share a 2-face, i.e. ρ ∩ ρ ′ isa 2-cell in T v . Let { m , m , m , m } = ρ ∩ ∆ d ( Z ) and { m , m , m , m } = ρ ′ ∩ ∆ d ( Z ). By Lemma 5.5, X ∞ satisfies Z c m Z c m Z c m (cid:16) X m ∈ ∆ od ( Z ) Z m (cid:17) = 0 , for certain c i > i = 1 , ,
3, and thus, X ∞ ∩ ˜ X ρ is given by Z c m Z c m Z c m (cid:16) Z m + Z m + Z m + Z m (cid:17) = 0 . AIR-OF-PANTS DECOMPOSITIONS OF 4-MANIFOLDS 39
Consequently, X ∞ ∩ ˜ X ρ ⊂ X ρ ∪ c ( H m ∩ ˜ X ρ ) ∪ c ( H m ∩ ˜ X ρ ) ∪ c ( H m ∩ ˜ X ρ ) , and X ∞ ∩ ˜ X ρ ∩ H m ⊂ X ρ ∩ H m , i.e. a line in the projective plane ˜ X ρ ∩ H m .Therefore, the plane ˜ X ρ ∩ H m = ˜ X ρ ′ ∩ H m corresponding to the 2-cell ρ ∩ ρ ′ is not a irreducible component of X ∞ , and only intersects with X ∞ at mosta line.Assume that ρ = ρ ∩ ∂ ∆ od is a 2-cell in T v . If the vertex m ⊂ ρ doesnot belong to ρ , then X ρ is given by Z m = 0, i.e. X ρ = ˜ X ρ ∩ H m . If ρ ′′ ⊂ ∆ od is another 3-cell such that m is not a vertex of ρ ′′ with ρ ′′ ∩ ∆ d ( Z ) = { m , m , m , m } , then X ∞ ∩ ˜ X ρ satisfies that Z c m Z c m Z c m Z c m Z m (cid:16) Z m + Z m + Z m + Z m (cid:17) = 0 , for certain c j > j = 5 , , , X ∞ ∩ ˜ X ρ ⊂ X ρ ∪ X ρ ∪ c ′ ( H m ∩ ˜ X ρ ) ∪ c ′ ( H m ∩ ˜ X ρ ) ∪ c ′ ( H m ∩ ˜ X ρ ) , for certain c ′ j >
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