Circular spherical divisors and their contact topology
aa r X i v : . [ m a t h . S G ] J a n Circular spherical divisors and their contact topology
Tian-Jun Li, Cheuk Yu Mak, Jie MinJanuary 19, 2021
Abstract
This paper investigates the symplectic and contact topology associated to circularspherical divisors. We classify, up to toric equivalence, all concave circular sphericaldivisors D that can be embedded symplectically into a closed symplectic 4-manifoldand show they are all realized as symplectic log Calabi-Yau pairs if their complementsare minimal. We then determine the Stein fillability and rational homology type ofall minimal symplectic fillings for the boundary torus bundles of such D . When D isanticanonical and convex, we give explicit betti number bounds for Stein fillings of itsboundary contact torus bundle. Contents b + ( X ) = 1 . . . . . . . . . . . . . . . . . . . . 9 b + ( D ) ≥ D is rationally embeddable . . . . . . . . . . . . . 153.4 Classify rationally embeddable D . . . . . . . . . . . . . . . . . . . . . . . . . 163.5 Rationally embeddable D is anti-canonical . . . . . . . . . . . . . . . . . . . . 223.6 Anti-canonical D is rigid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 b + ( D ) = 0 Contact aspects 28 D . . . . . . . . . . . . . . . . . . . . . . . . . 325.4 Geography of Stein fillings for convex anti-canonical D . . . . . . . . . . . . . 36 Let X be a smooth rational surface and let D ⊂ X be an effective reduced anti-canonicaldivisor. Such pairs ( X, D ) are called anti-canonical pairs and has been extensively studiedsince Looijenga ([31]). Gross, Hacking and Keel studied the mirror symmetry aspects ofanti-canonical pairs in [16] and [17]. In particular, they proved Looijenga’s conjecture ondual cusp singularities in [16] and Torelli type results in [17] conjectured by Friedman. Thenotion of symplectic log Calabi-Yau pairs, as a symplectic analogue of anti-canonical pairs,was defined by the first and the second authors in [21], where they studied different notionsof deformation equivalence and enumerated minimal models.In this paper, we present a comprehensive study of circular spherical divisors, which aregeneralizations of symplectic log Calabi-Yau pairs. In particular, we are interested in thefollowing natural question on the embeddability and rigidity of circular spherical divisors.
Question 1.1.
Given a circular spherical divisor D , can we symplectically embed D into aclosed symplectic 4-manifold ( X, ω ) so that ( X, ω, D ) is a symplectic log Calabi-Yau pair? The Looijenga conjecture can be seen as an answer to this question for negative defi-nite divisor in terms of its dual (see Section 4.2). We give a complete answer (up to toricequivalence) for all other cases, i.e. when D is strictly negative semi-definite or b + ( D ) ≥ By a topological divisor, we mean a connected configuration of finitely many closed embeddedoriented smooth surfaces D = C ∪ · · · ∪ C r in a smooth oriented 4 dimensional manifold X (possibly with boundary or non-compact), satisfying the following conditions: (1) eachintersection between two components is positive and transversal, (2) no three componentsintersect at a common point, and (3) D does not intersect the boundary ∂X . Since we areinterested in the germ of a topological divisor, we usually omit X in the writing and justdenote the divisor by D . For each D denote by N D the neighborhood obtained by plumbingdisk bundles over the components C i and Y D = ∂N D the plumbed 3-manifold oriented as theboundary of N D . They are well-defined up to orientation-preserving diffeomorphisms ([38]).An intersection matrix is associated to each topological divisor. For a topological divisor D = ∪ ri =1 C i , we denote by [ C i ] the homology class of C i in H ( X ). Note that H ( N D ) isfreely generated by [ C i ]. The intersection matrix of D is the r by r square matrix Q D = ( s ij =2 C i ] · [ C j ]), where · is used for any of the pairings H ( X ) × H ( X ) , H ( X ) × H ( X ) , H ( X ) × H ( X, ∂X ). Via the Lefschetz duality for N D , the intersection matrix Q D can be identifiedwith the natural homomorphism Q D : H ( N D ) → H ( N D , Y D ). We use homology andcohomology with Z coefficient unless otherwise specified.In a symplectic 4-manifold ( X, ω ), a symplectic divisor is a topological divisor D embeddedin X , with each C i being symplectic and having the orientation positive with respect to ω .A symplectic log Calabi-Yau pair ( X, D, ω ) is a closed symplectic 4-manifold (
X, ω )together with a nonempty symplectic divisor D = ∪ C i representing the Poincare dual of c ( X, ω ). It’s an easy observation ([23]) that D is either a torus or a cycle of spheres. Inthe former case, ( X, D, ω ) is called an elliptic log Calabi-Yau pair . In the later case, it’scalled a symplectic Looijenga pair and it can only happen when (
X, ω ) is rational. Asa consequence, we have b + ( Q D ) = 0 or 1. We also remark that symplectic log Calabi-Yaupairs have vanishing relative symplectic Kodaira dimension (cf. [29],[26]).We call a topological divisor D consisting of a cycle of spheres a circular sphericaldivisor and a symplectic circular spherical divisor if such D is a symplectic divisor.For each circular spherical divisor D = ∪ ri =1 C i , we associate to it a self-intersection sequence( s i = [ C i ] ) ri =1 . Given any sequence ( s , . . . , s r ) with s i ∈ Z , it can always be realized asthe self-intersection sequence of a circular spherical divisor. Essentially there is no differencebetween a sequence and a circular spherical divisor (up to (anti-)cyclic permutation of thelabeling) and we denote by D = ( s , . . . , s r ) the circular spherical divisor with self-intersectionsequence ( s , . . . , s r ). We also denote by D = ( s ) the torus with self-intersection s . This doesnot cause any confusion as we always require that a circular spherical divisor has length atleast 2. We define the b + /b − /b of a divisor D to be the b + /b − /b of Q D , i.e. the number ofpositive/negative/zero eigenvalues.Circular spherical divisors generalize symplectic Looijenga pairs as they are not requiredto be embedded in a closed symplectic 4-manifold or to represent the Poincare dual of thefirst Chern class. They are the main objects of study in this paper and we are interested inthe following properties. A ciruclar spherical divisor D is called symplectically embed-dable if D admits a symplectic embedding into a closed symplectic 4-manifold ( X, ω ). Asymplectically embeddable D is called rationally embeddable if such ( X, ω ) can be chosento be a symplectic rational surface, i.e. X ∼ = CP l CP or S × S . A rationally embed-dable D is called anti-canonical if ( X, D, ω ) is a symplectic Looijenga pair for some (
X, ω ).An anti-canonical D is called rigid if for any symplectic embedding of D into a symplectic4-manifold ( X, ω ) with X − D minimal, ( X, D, ω ) is a symplectic Looijenga pair. Note thatthe complement of an anti-canonical D is by definition minimal.In terms of above terminalogy, Question 1.1 asks for a criterion to decide whether acircular spherical divisor is anti-canonical. The main theorem of our paper is the following. Theorem 1.2.
For a circular spherical divisor with b + ( D ) ≥ , being symplectically embed-dable, rationally embeddable, anti-canonical, and rigid are equivalent. In particualr when b + ( D ) ≥ D is not symplectically embeddable (Lemma 3.1). Com-bined with the classification of symplectically embeddable circular spherical divisors in Theo-rem 1.3, Theorem 1.2 gives a combinatorial characterization of anti-canonical circular spher-3cal divisor and answers Question 1.1 in the case b + ( D ) ≥
1. In this way, Theorem 1.2 canbe seen as a symplectic version of Looijenga conjecture for divisors with b + ( D ) ≥ b + ( D ) = 0) are allsymplectically fillable and not rigid. This contrasts Theorem 1.2, where divisors with b + ≥ D is strictly negative semi-definite (i.e. not negative defintie), we completely determinewhether D is anti-canonical or not, up to toric equivalence. These results give us examplesthat are symplecically embeddable but not anti-canonical, or anti-canonical but not rigid(Example 4.5 and 4.9). This shows that the assumption b + ≥ Y D of a plumbing of a cycle of spheres D is a topolog-ical torus bundle (cf. [38]). Also by − Y D we mean the negative of its induced boundaryorientation. Theorem 1.3.
A circular spherical divisor with b + ≥ is symplectically embeddable if andonly if it is toric equivalent to one of the following: (1) (1 , p ) or ( − , − p ) with p = 1 , , , in which case − Y D is elliptic, (2) (1 , , p ) with p ≤ , in which case − Y D is positive parabolic, (3) (0 , p ) with p ≤ , in which case − Y D is negative parabolic, (4) (1 , p ) with p ≤ − or (1 , − p , − p , . . . , − p l − , − p l ) blown-up (defined in Section3.4) with p i ≥ , l ≥ , in which case − Y D is negative hyperbolic.In particular, all − Y D above are distinct as oriented 3-manifolds.Remark . The reason why we look at the negative boundary − Y D is that induced con-tact structure from the divisor is a positive contact structure with respect to this negativeorientation. For details see Section 5.1. A topological divisor D is called concave (resp. convex ) if it admits a concave (resp. convex)plumbing (see Section 5.1). In Proposition 5.11, we show that a circular spherical divisor isconcave (resp. convex) if and only if b + ( D ) > Q D negative definite).If D has a concave (resp. convex) plumbing N D , there is a contact structure associated to D on the torus bundle as the negative (resp. positive) boundary of N D , which we will denoteby ( − Y D , ξ D ) (resp. ( Y D , ξ D )). This contact structure only depends on the topological divisor D and doesn’t vary with the symplectic structure ω on N D (Proposition 5.5). Furthermore,it is shown in [27] (see Proposition 5.7) that this contact structure is invariant under toricequivalence. So toric equivalence is a natural equivalence for circular spherical divisors fromthe perspective of contact topology.Symplectic fillability and Stein fillability of contact torus bundles have been extensivelystudied. Ding-Geiges showed in [4] every torus bundle admits infinitely many weakly but not4trongly symplectically fillable contact structures. Bhupal-Ozbagci constructed in [2] Steinfillings for all tight contact strutures on positive hyperbolic torus bundles. In [5] Ding-Listudied symplectic fillability and Stein fillability of some tight contact structures on negativeparabolic and negative hyperbolic torus bundles. For a large family concave circular sphericaldivisors (see Remark 1.7), Golla and Lisca investigated the topology of Stein and minimalsymplectic fillings of ( − Y D , ξ D ) in [13]. Note that ( Y D , ξ D ) is always symplectic fillable if D is convex, as N D provides a symplectic filling. In the case of elliptic log Calabi-Yau pairs,Ohta and Ono classified symplectic fillings of simple elliptic singularities up to symplecticdeformation in [39].Using Theorem 1.2 and 1.3, we determine the symplectic fillability of contact torus bundles( − Y D , ξ D ) for all concave circular spherical divisors D and study the topology of their fillings.In particular, the rational homology type of minimal symplectic fillings is unique, which iscompletely determined by D . Here by rational homology type , we mean the betti numbersand the intersection form on (co)homology over rational coefficients. To describe the bettinumbers, we recall that the charge q ( D ) = 12 − D − r ( D ) = 12 − r ( D ) − X s i introduced in [11], which is invariant under toric equivalence. Also note that the number b ( D ) of zero eigenvalues of Q D is also invariant under toric equivalence (Lemma 2.2). Theorem 1.5.
Let D be a concave circular spherical divisor. Then ( − Y D , ξ D ) is symplecticfillable if and only if D is toric equivalent to one in Theorem 1.3. Furthermore, we have that • all minimal symplectic fillings have a unique rational homology type with b = b ( Y D ) − , b = b + = 0 , b = 1 and b − = q ( D ) − b ( D ) ; • all minimal symplectic fillings have c = 0 ; • every minimal symplectic filling is symplectic deformation equivalent to a Stein filling; • there are at most finitely many (Stein) minimal symplectic fillings of ( − Y D , ξ D ) up tosymplectic deformation; Note that b ± , b completely determines the intersection form on (co)homology with ratio-nal coefficients. The following example of Golla-Lisca implies that integral intersection formsare not unique for fillings of every concave D , so our uniqueness of rational homology typeis sharp. Example 1.6 ([13]) . The divisor D = (1 , − , − , − , − , − , − admits two different symplectic embeddings in X = CP CP . Then we get two regularneighborhoods W , W of the two embedded divisors and two complements P , P of the neigh-borhoods. P and P are actually Stein fillings of ( − Y D , ξ D ) . Golla and Lisca showed theimage of H ( P i ; Z ) in H ( X ; Z ) are not isometric as integral lattices, which implies P and P have different intersection forms and thus not homotopic equivalent. emark . In [13], Golla and Lisca have investigated contact torus bundles ( − Y D , ξ D )arising from D in families (1), (3), (4) of Theorem 1.3 and proved that they are Stein fillable.They showed that all Stein fillings of ( − Y D , ξ D ) have c = 0, b = 0 and share the same bettinumbers. Moreover, up to diffeomorphism, for family (1) there is a unique Stein filling andfor family (3) and (4) there are finitely many Stein fillings. Many of their results also hold forminimal symplectic fillings. Our result deals with deformation types of minimal symplecticfillings and identifies Stein fillins with minimal symplectic fillings up to deformation. Weimprove their result by proving finiteness up to symplectic deformation, computing b ± , b andthus the rational intersection form for all minimal symplectic fillings and also generalizingthe results to the new family (2) of positive parabolic contact torus bundles. Remark . In light of the above remark, Theorem 1.5 is more interesting as a non-fillabilitycriterion and in particular gives many non-fillable contact torus bundles that are not previ-ously known. In literature, contact torus bundles are mostly presented as quotients of T × R or as Legendrian surgeries, which makes it difficult to compare these to contact torus bun-dles as boundaries of circular spherical divisors in our paper. We exhibit one example here.For a circular spherical divisor D = ( s , . . . , s r ) with s i ≥ − i , we can construct anopenbook decomposition for such contact torus bundles by [27], which corresponds to a worddecomposition of the monodromy by [43]. Then the circular spherical divisor( − , − , − , − , . . . , − | {z } n ) ∼ (0 , − n )has a word decomposition ( aba ) − a n . For n ≤ −
5, this contact structure is not Stein fillableby Theorem 1.5, which is previously unknown (see [43] Chapter 5, Example 3.(d)).When a symplectic circular spherical divisor D in ( X, ω ) is anticanonical and convex, itarises as a resolution of a cusp singularity and thus its boundary ( Y D , ξ D ) is Stein fillable.We study the geography of its Stein fillings and give restrictions on its betti numbers andEuler number in Proposition 5.21. Acknowledgments : This paper is inspired by the results and questions in Golla-Lisca[13]. All authors are supported by NSF grant 1611680.
In this section we discuss several aspects of circular spherical divisors. We first introducethe notion of toric equivalence for topological divisors in Section 2.1. All properties we areconcerned about in this paper will be invariant under toric equivalence, which makes it auseful reduction tool in the proofs. Then Section 2.2 reviews basic facts about torus bundles.Finally in Section 2.3 we give homological restrictions for a circular spherical divisor to beembedded in a closed manifold with b + = 1. In particular, Lemma 2.9 is essential to theproof of Theorem 1.3. Definition 2.1.
For a topological divisor D = ∪ C i , toric blow-up is the operation of addinga sphere component with self-intersection − between an adjacent pair of component C i and j and changing the self-intersection of C i and C j by − . Toric blow-down is the reverseoperation. D and D are toric equivalent if they are connected by toric blow-ups and toric blow-downs. D is said to be toric minimal if no component is an exceptional sphere (i.e. acomponent of self-intersection − ). Note that toric blow-ups and blow-downs can be performed in the symplectic category byadding an extra parameter of symplectic area and are thus operations on symplectic divisors.Also note that we could keep toric blowing down a circular spherical divisor until either itbecomes toric minimal or it has length 2. When a circular spherical divisor has length 2, wecannot further blow it down because it would result in a non-embedded sphere and thus nota topological divisor. So we exclude this case when we talk about toric blow-down.
Lemma 2.2.
The following are preserved under toric equivalence: (1) D being a circular spherical divisor, (2) b + ( Q D ) and b ( Q D ) (in particular the non-degeneracy of the intersection matrix Q D ), (3) the oriented diffeomorphism type of the plumbed 3-manifold Y D .Proof. (1) is obvious and (3) is part of Proposition 2.1 in [38].(2) follows from a direct computation. Let D = ( b , . . . , b r ) and D ′ = ( − , b − , b , . . . , b r −
1) be its toric blow-up. Then the intersection matrix Q D ′ is of the form − . . . b − . . .
00 1 b . . . ...... 11 0 . . . b r − . It’s easy to see that by a change of basis, Q D ′ is equivalent to ( − ⊕ Q D . So toric blow-uppreserves b + , b and increases b − by 1.Here is an example to illustrate how a self-intersection 0 component in a circular sphericaldivisor can be used to balance the self-intersection numbers of the two sides by performing atoric blow-up and a toric blow-down. Example 2.3.
The following three cycles of spheres are toric equivalent: −
20 2 − − − − . . . , k, , p, . . . ) to a toric equivalent divisor( . . . , k − n, , p + n, . . . ) a balancing move based at the 0-sphere.7 emma 2.4. Any circular spherical divisor is toric equivalent to a toric minimal one or ( − , p ) . If D = ( − , p ) , then Q D is degenerate only if p = − .Suppose D = ( s , . . . , s r ) is a toric minimal cycle of spheres. Then (1) b + ( Q D ) ≥ if and only if s i ≥ for some i . (2) Q D is negative definite if s i ≤ − for all i and less than − for some i . Q D is negativesemi-definite but not negative definite if s i = − for each i . (3) Q D is non-degenerate if either s ≥ and s i ≤ − for i ≥ , or s = s = 0 and s i ≤ − for i ≥ .Proof. By toric blow-down, any circular spherical divisor is toric equivalent to a toric minimalone or one of length 2. If D has length 2 and not toric minimal, then it is of the form ( − , p ).Then det( Q D ) = − p − p = − r = 2, Q D = (cid:18) s s (cid:19) , so Q D is non-degenerate if s ≥ s ≤ −
2. For the following, we assume r ≥ t i ≤ − i = 1 , . . . , r and p j , q j ∈ Z defined recursively by p − = 0 , p = 1 , p i +1 = − t i +1 p i − p i − ; q − = − , q = 0 , q i +1 = − t i +1 q i − q i − . Then A ( − t , . . . , − t j ) = (cid:18) p j q j − p j − − q j − (cid:19) ,with p j ≥ p j − + 1 ≥ q j ≥ q j − + 1 ≥ p j ≥ q j + 1 ≥ j . In [38], it was notclaimed that p j ≥ q j + 1 but it is a standard fact and can be verified by induction.We apply this observation to the chain of spheres with negative self-intersection. In thefirst case, the monodromy matrix is A ( − s , − t . . . , − t r − ) = A ( − t , · · · , − t r − ) (cid:18) − s − (cid:19) = (cid:18) − s p r − − q r − p r − s p r − + q r − − p r − (cid:19) , with s ≥ p r , q r as in the observation above. The trace is − s p r − − q r − − p r − ≤− q r − − p r − − ≤ − q r − + 1) ≤ −
2, so not equal to 2. In the second case, the monodromymatrix is A (0 , , − t , . . . , − t r − ) = (cid:18) − p r − − q r − p r − q r − (cid:19) . The trace is − p r − + q r − ≤ − p r − + q r − − ≤ −
2, so again cannot be 2.
Denote by T A an oriented torus bundle over S with monodromy A ∈ SL (2; Z ). A torusbundle T A is called elliptic if | tr A | <
2, parabolic if | tr A | = 2 and hyperbolic if | tr A | > T A is parabolic or hyperbolic, we call it positive (resp. negative) if tr A is positive (resp.negative). Also, − T A is orientation-preserving diffeomorphic to T A − .8o describe the plumbed 3-manifold Y D , we introduce the following matrix for a sequenceof integers ( t , · · · , t r ), A ( t , . . . , t r ) = (cid:18) t r − (cid:19) (cid:18) t r − − (cid:19) . . . (cid:18) t − (cid:19) ∈ SL ( Z ) . Lemma 2.5 (Theorem 6.1 in [38], Theorem 2.5 in [13]) . For a circular spherical divisor D = ( s , ..., s r ) , the plumbed 3-manifold Y D is the oriented torus bundle T A over S with mon-odromy A = A ( − s , . . . , − s r ) . Its homology is given by H ( Y D ; Z ) = Z ⊕ Coker ( A − I ) = Z ⊕ Coker ( Q D ) . The intersection matrix Q D is non-degenerate if the trace of A ( − s , . . . , − s r ) =2 . b + ( X ) = 1 In this subsection, we assume D = ∪ C i is a circular spherical divisor. Let r ( D ) denotethe number of components of D and r ≥ ( D ) the number of components with non-negativeself-intersection. Here are some restrictions on homologous components of D . Lemma 2.6.
For any D embedded in a smooth 4-manifold X , we have the following (1) At most three components are homologous in X . There are three homologous compo-nents only if r ( D ) = 3 . (2) There are a pair of homologous components only if r ( D ) ≤ . (3) If [ C i ] = [ C i +1 ] for some i then r ( D ) = 3 , s i = s i +1 = 1 , or r ( D ) = 2 , s i = s i +1 = 2 .Proof. Suppose there are m components homologous to a ∈ H ( X ) in D . Note that a ∈{ , , } because the divisor has only one cycle and the components are required to intersectpositively and transversally. If a = 1, then these components are all adjacent. In order toform exactly one cycle, m is at most 3. In particular, when m = 3, there cannot be othercomponents, i.e. r ( D ) = 3. If a = 0, there is another component C intersecting all thesecomponents. Again in order to form exactly one cycle, m is at most 2. Similarly if a = 2,they are adjacent and we must have m = r ( D ) = 2. This proves (1) and (3).Suppose C i , C j are a pair of homologous components in D . If they are adjacent, then[ C i ] · [ C j ] = 1 or 2 and r ( D ) ≤ C i must also intersect C j . There must be exactly two suchcomponents to form a cycle. So r ( D ) = 4 and this proves (2).Note that the above restrictions hold locally. When X is closed with b + ( X ) = 1, thereare various restrictions on components with non-negative self-intersection. Lemma 2.7.
Suppose D is embedded in a closed manifold X with b + ( X ) = 1 . (1) If C i and C j are not adjacent and s i ≥ , s j ≥ , then [ C i ] = ± [ C j ] and s i = s j = 0 . (2) r ≥ ( D ) ≤ . (3) r ≥ ( D ) = 4 only if r ( D ) = 4 , s i = 0 for each i and [ C ] = [ C ] , [ C ] = [ C ] . Suppose r ( D ) ≥ . If s i ≥ , s i +1 ≥ for some i , then [ C i ] = [ C i +1 ] and s i = s i +1 = 1 .This is only possible when r ( D ) = 3 .Proof. Since b + ( X ) = 1, by the light cone lemma (cf. [34]), any two disjoint componentswith non-negative self-intersection must be homologous up to sign and have self-intersection0. (2) and (3) follow from the (1).For (4), we can assume the two spheres are C and C . Since r ( D ) ≥
3, we have [ C ] · [ C ] =1. By toric blowing up the intersection point between C and C , we get two disjoint sphereswith classes [ C ′ ] = [ C ] − E and [ C ′ ] = [ C ] − E , where E is the exceptional class and[ C ′ ] = [ C ] − ≥
0, [ C ′ ] = [ C ] − ≥
0. Then by (1), we have [ C ′ ] = [ C ′ ] with [ C ′ ] = 0and thus [ C ] = [ C ] with [ C ] = 1. The fact that r ( D ) = 3 follows from (3) in Lemma 2.6. Remark . Note in (1) of Lemma 2.7, if C i and C j are symplectic spheres, we would have[ C i ] = [ C j ]. Lemma 2.9.
Suppose D has b + ( Q D ) = 1 and is embedded in a closed manifold X with b + ( X ) = 1 . Let k, p, p , p be integers such that k ≥ and p, p , p < . Up to cyclic andanti-cyclic permutations of D , we have the following. (1) If r ( D ) ≥ , then r ≥ ( D ) ≤ . When r ≥ ( D ) = 2 , s ≥ , s = 0 . (2) If r ( D ) = 4 and r ≥ ( D ) ≥ , then D = ( k, , p, , k + p ≤ and [ C ] = [ C ] . (3) If r ( D ) = 4 and r ≥ ( D ) = 2 , then the only possibilities of D are(i) (0 , p , , p ) , [ C ] = [ C ] ,(ii) ( k, , p , p ) , p + p + k ≤ . (4) If r ( D ) = 3 and r ≥ ( D ) = 3 , then the only possibilities of D are(i) (1 , , , [ C ] = [ C ] = [ C ] ,(ii) (1 , , , [ C ] = [ C ] ,(iii) ( k, , , k ≤ . (5) If r ( D ) = 3 and r ≥ ( D ) = 2 , then the only possibilities of D are(i) (1 , , p ) , [ C ] = [ C ] ,(ii) ( k, , p ) , p + k ≤ . (6) If r ( D ) = 2 and r ≥ ( D ) = 2 , then D is one in family F (2 ,
2) = { (4 , , (4 , , (3 , , (3 , , (2 , , (2 , , (2 , , (1 , , (1 , , (0 , } . (7) If r ( D ) = 2 and r ≥ ( D ) = 1 , then D = ( k, p ) . (8) If r ( D ) = 2 and r ≥ ( D ) = 0 , then D is one in family F (2 ,
0) = { ( − , − , ( − , − , ( − , − } . roof. Case (1):
Suppose r ( D ) ≥
5. If r ≥ ( D ) ≥ r ≥ ( D ) ≤ r ≥ ( D ) = 2, the two components must be adjacent bythe same reasoning. The claim that one of them has self-intersection 0 follows from (4) ofLemma 2.7 and the (3) of Lemma 2.6. Case (2):
The proof is similar when r ( D ) = 4 and r ≥ ( D ) ≥
3. In this case two suchcomponents are not adjacent, say C , C . By the (1) of Lemma 2.7, [ C ] = [ C ], s = 0 = s . Case (3):
Suppose r ( D ) = 4 and r ≥ ( D ) = 2. If two such components are not adjacent,we can assume them to be C , C , which satisfy [ C ] = [ C ] and s = s = 0 by the (1) ofLemma 2.7. If the two components are adjacent, we can assume them to be C , C . Noticethat [ C ] = [ C ] due to the (3) of Lemma 2.6. Now it follows from the 4th bullet of Lemma2.7 that either s = 0 or s = 0. Case (4):
Suppose r ( D ) = 3 = r ≥ ( D ). Since s i ≥ i , It is easy to see (i), (ii),(iii) give all the possibilities by (4) of Lemma 2.7. It’s easily checked by hand that ( k, , b + ≥ k ≥ Case (5): If r ≥ ( D ) = 2, apply (4) of Lemma 2.7 to the pair of components C i , C j with s i ≥ , s j ≥ Case (6)(7)(8):
Suppose r ( D ) = 2. Then we just check that the determinant of Q D = s s − ≤ b + ( D ) ≥ This section is devoted to the proof of Theorem 1.2 and 1.3. We start with the followingobservation on the embeddability of circular spherical divisors with b + ≥ Lemma 3.1.
A topological circular spherical divisor D cannot be symplectically embedded ina closed symplectic 4-manifold if b + ( D ) ≥ .Proof. Note that a divisor of form ( − , p ) always has b + ≤ D in closed symplectic 4-manifold ( X, ω ) is not toric minimal, then bytoric blow-downs, we can always get a toric minimal divisor D ′ embedded in ( X ′ , ω ′ ) with b + ( D ′ ) = b + ( D ) ≥ D is toric minimal. Then there is atleast one component C i in D with C i ≥ X, ω ) is rational orruled ([32]). This contradicts b + ( D ) ≥ b + ( X ) = 1 in this case.As a result, there is no symplectically embeddable circular spherical divisor with b + ≥ b + = 1. Next we show that the several propertiesof circular spherical divisors are preserved under toric equivalence, so it suffices to considerthe toric minimal divisors. Lemma 3.2.
A circular spherical divisor D being symplectically embeddable, rationally em-beddable, anti-canonical or rigid is preserved under toric equivalence.Proof. Since toric blow-ups and blow-downs can be realized by symplectic blow-ups and blow-downs when the divisor is symplectic and symplectic blow-up or blow-down of a symplectic11ational surface is still rational, it’s clear that being symplectically embeddable and rationallyembeddable is preserved.Let (
X, D, ω ) be a symplectic Looijenga pair. Blow up at a transverse intersection pointof D to get ( X ′ , ω ′ ) with the natural inclusion ι ∗ : H ( X ; Z ) → H ( X ′ ; Z ). Denote by D ′ the union of the proper transform of D and the exceptional curve E . Then D ′ is a toricblow-up of D with [ D ′ ] = [ D ] − [ E ]. So [ D ′ ] = ι ∗ [ D ] − [ E ] = ι ∗ ( − K X ) − [ E ] = − K X ′ and( X ′ , D ′ , ω ′ ) is also a symplectic Looijenga pair. The proof for toric blow-down is the samebut goes backwards.Let D be a circular spherical divisor and D ′ a toric blow-up of D with exceptional com-ponent E . Suppose D is rigid. For any symplectic embedding of D ′ into ( X ′ , ω ′ ), we canblow-down E , which is a symplectic exceptional sphere in ( X ′ , ω ′ ), to get a symplectic em-bedding of D into ( X, ω ). If X ′ − D ′ is minimal, then X − D is also minimal, since anyexceptional curve away from D would lift to an exceptional curve in X ′ − D ′ . Now that D is anti-canonical in ( X, ω ), we have D ′ is anti-canonical in ( X ′ , ω ′ ) by the previous para-graph. So D ′ is rigid. The same argument goes backwards and proves that D is rigid if D ′ isrigid.Notice that we have the following sequence of implications simply by their definitions.symplectically embeddable ⇐ rationally embeddable ⇐ anti-canonical ⇐ rigidTo prove Theorem 1.2, it suffices to prove the converse for every arrow above. We recallfacts about minimal models and deformation classes of symplectic log Calabi-Yau pairs inSection 3.1 and recollect some tools from pseudoholomorphic curves in Section 3.2. Thesewill be useful also in later sections. The proofs are distributed in Section 3.3 through 3.6, asindicated by the corresponding titles. The proof of Theorem 1.3 is contained in Section 3.4. Recall that a symplectic log Calabi-Yau pair (
X, D, ω ) is a closed symplectic 4-manifold(
X, ω ) together with a nonempty symplectic divisor D = ∪ C i representing the Poincare dualof c ( X, ω ). In this section, we review some facts about the minimal models and deformationclasses of symplectic log Calabi-Yau pairs studied in [21]. In particular, Theorem 3.6 gives alist of minimal models for anti-canonical circular spherical divisors and Corollary 3.9 servesas the source of finiteness for symplectic fillings in Theorem 1.5.We have introduced toric blow-ups and blow-downs in Definition 2.1. Here we introduceanother pair of operations on symplectic divisors used in the minimal reduction.
Definition 3.3. A non-toric blow-up of D is the proper transform of a symplectic blow-upcentered at a smooth point of D . A non-toric blow-down is the reverse operation whichsymplectically blows down an exceptional sphere not contained in D . Both toric and non-toric blow-ups/downs preserve the log Calabi-Yau condition and haveanalogues in the holomorphic category.
Definition 3.4.
A symplectic log Calabi-Yau pair ( X, D, ω ) is called minimal if ( X, ω ) isminimal, or ( X, D, ω ) is a symplectic Looijenga pair with X = CP CP . X, D, ω ). Such pair is actually a minimal symplectic log Calabi-Yau pair and is called aminimal model of (
X, D, ω ). Note that minimal models for a symplectic log Calabi-Yau pairare not unique.Various notions of equivalences have been introduced in the study of symplectic defor-mation classes of symplectic log Calabi-Yau pairs in [21], the following one is related tosymplectic deformation of fillings.
Definition 3.5.
Let ( X , D , ω ) and ( X , D , ω ) be pairs of 4-manifolds and symplecticdivisors in them. When X = X , they are said to be symplectic homotopic if ( D , ω ) and ( D , ω ) are connected by a family of symplectic divisors ( D t , ω t ) . ( X , D , ω ) and ( X , D , ω ) are said to be symplectic deformation equivalent if they are symplectichomotopic, up to an orientation preserving diffeomorphism. We recall here the deformation classes of minimal symplectic log Calabi-Yau pairs, all ofthem having length less than 5.
Theorem 3.6 ([21]) . Any minimal symplectic log Calabi-Yau pair ( X, D, ω ) is symplecticdeformation equivalent to one of the following. In particular, all circular spherical divisorslisted below are anti-canonical. • Case ( A ) : X is a symplectic ruled surface with base genus . D is a torus. • Case ( B ) : X = CP , c = 3 h . ( B D is a torus, ( B D consists of a h − sphere and a h − sphere, or ( B D consists of three h − spheres. The graphs in (B1), (B2), and (B3) are givenrespectively by • Case ( C ) : X = S × S , c = 2 f + 2 f , where f and f are the homology classes ofthe two factors. ( C D is a torus. ( C r ( D ) = 2 and [ C ] = bf + f , [ C ] = (2 − b ) f + f . ( C r ( D ) = 3 and [ C ] = bf + f , [ C ] = f , [ C ] = (1 − b ) f + f . ( C r ( D ) = 4 and [ C ] = bf + f , [ C ] = f , [ C ] = − bf + f , [ C ] = f .The graphs in (C1), (C2), (C3) and (C4) are given respectively by b − b b − b b − b Case ( D ) : X = C P C P , c = f + 2 s , where f and s are the fiber class and sectionclass with f · f = 0 , f · s = 1 and s · s = 1 . ( D D cannot be a torus because it would not be minimal. ( D r ( D ) = 2 , and either ([ C ] , [ C ]) = ( af + s, (1 − a ) f + s ) or ([ C ] , [ C ]) = (2 s, f ) . ( D r ( D ) = 3 and [ C ] = af + s, [ C ] = f, [ C ] = − af + s . ( D r ( D ) = 4 and [ C ] = af + s, [ C ] = f, [ C ] = − ( a + 1) f + s, [ C ] = f .The graphs in (D2), (D3) and (D4) are given respectively by a + 1 3 − a a + 1 01 − a a + 1 0 − a − Lemma 3.7 ([23]) . Each symplectic deformation class contains a K¨ahler pair.
In the holomorphic category, We have the following finiteness of deformation classes in[11].
Theorem 3.8 (Theorem 3.1 in [11]) . There are only finitely many deformation types ofanti-canonical pairs with the same self-intersection sequence.
Combining Lemma 3.7 and Theorem 3.8, we obtain the following finiteness of symplecticdeformation classes.
Corollary 3.9 ([23]) . There are only finitely many symplectic deformation types of symplecticlog Calabi-Yau pairs with the same self-intersection sequence.
In this subsection, we recall some useful notions in the theory of maximal surfaces ([29]) andpseudo-holomorphic curves in dimension 4 ([33]). These will be used frequently in the restof the paper to study the minimality of divisor complements.
Definition 3.10.
Suppose F ⊂ ( X, ω ) is a symplectically embedded surface without spherecomponents. F is called maximal if [ F ] · E = 0 for any exceptional class E . The notion of maximal surfaces can be thought of as a relative version of minimality asfollows. By Proposition 4.1 of [40], for any exceptional class E , there exists an almost complexstructure J such that both F and an embedded representative S of E are J -holomorphic. Bypositivity of intersection, we have [ F ] · E ≥
0. In particular, [ F ] · E = 0 if and only if F and S are disjoint. As a consequence, we get the following lemma.14 emma 3.11. Suppose F ⊂ ( X, ω ) is an embedded symplectic surface without sphere com-ponents. Then F is maximal if and only if X − F is minimal. Similar to Proposition 4.1 of [40], McDuff and Opshtein gave a criterion on the existenceof embedded pseudo-holomorphic curves relative to a pseudo-holomorphic normal crossingdivisor.
Definition 3.12.
Let D = ∪ C i be an ω − orthogonal symplectic divisor in ( X, ω ) . An excep-tional class e ∈ H ( X ; Z ) is called D − good if e · [ C i ] ≥ for all i . Lemma 3.13. (Theorem 1.2.7 of [33]) Let D be an ω -orthogonal symplectic divisor. There isa non-empty space J ( D ) of ω -tamed almost complex structures making D pseudo-holomorphicsuch that for any D − good exceptional class e , there is a residual subset J ( D, e ) ⊂ J ( D ) sothat e has an embedded J -holomorphic representative for all J ∈ J ( D, e ) . So for a D -good exceptional class e , we have e · [ D ] ≥
0. In particular, e is D − good if e · [ C i ] = 0 for every component C i of D . So E is disjoint from D if and only if e · [ C i ] = 0for all i .We recall the following definition from [18]. Definition 3.14.
A homology class b ∈ H ( X ; Z ) is said to be stable if b for any ω -tamealmost complex structure J , it can be represented by a J -holomorphic curve. In particular, by [32] we see that any exceptional class e is stable. The following lemmasays any symplectic surface class of non-negative self-intersection pairs non-negatively withany exceptional class. Lemma 3.15 (Lemma 3.9 of [18]) . Suppose a ∈ H ( X ; Z ) with a ≥ is realized by aconnected embedded symplectic surface, then a · b ≥ for any stable class b . D is rationally embeddable Actually we only need to consider circular spherical divisors with b + = 1 in a symplecticrational surface by the following lemma. Lemma 3.16.
Let D be a symplectic circular spherical divisor in ( X, ω ) with b + ( Q D ) = 1 ,then ( X, ω ) is rational.Proof. Since being a symplectic rational surface is preserved under blow-up and blow-down,we could assume that D is toric minimal or of the form ( − , p ) , p > −
4. If s i ≥ i ,then ( X, ω ) is rational as it contains a positive symplectic sphere ([32]).Now we assume D is toric minimal and s i ≤ i . By Lemma 2.4, we must have s i = 0 for some i in order for b + ( Q D ) ≥ X, ω ) must rational or ruled. Withoutloss of generality, we assume s = 0. If X is irrational ruled with π : X → B , then [ C ]must be the fiber class. So [ C ] must contain a positive multiple of the section class as[ C ] · [ C ] ≥
1. Then π | C : C → B has positive degree so that g ( C ) ≥ g ( B ) ≥
1, which iscontradiction. 15uppose D is of the form ( − , p ) , p > −
4. When p ≥
0, it follows from the same argumentas above. The case ( − , ǫ − ǫ = − , ,
1, needs a different argument. Let D = C ∪ C with [ C ] = −
1, [ C ] = ǫ − C ] · [ C ] = 2. Blow down C to get X ′ such that X = X ′ CP and C becomes an immersed nodal symplectic sphere C ′ with self-intersection2 + ǫ . Smoothing the singularity of C ′ we obtain a smoothly embedded symplectic torus T with self-intersection 2+ ǫ ≥
1. By Proposition 4.3 of [25], (
X, ω ) is rational or ruled. Suppose X is irrational ruled, then H ( X ; Z ) is generated by { f, s, e , . . . , e k } where f is the class ofa fiber, s is the class of a section and e i ’s are exceptional classes. Note that all exceptionalclasses in X are of the form e i or f − e i . By Lemma 6.1 of [6], we have [ C ] = bf + P ± e i and thus [ C ] · [ C ] = ±
1, which is a contradiction. D In this section, we derive some restrictions on a symplectic circular spherical divisor embeddedin a symplectic rational surface. In particular, we give a complete list of rationally embeddablecircular spherical divisors up to toric equivalence in Proposition 3.21.
Lemma 3.17.
For a symplectic circular spherical divisor D in a symplectic rational surface ( X, ω ) we have that [ D ] ≤ .Proof. We can symplectically smooth out D to obtain a symplectic torus T with [ T ] = [ D ].By Proposition 3.14 in [29], [ T ] ≤ [ K ω ] if T is maximal. If, in addition, [ T ] = 9, T mustsit in X = CP by Theorem 6.10 in [39]. If the torus T is not maximal, we can performblow-downs away from T to get a maximal T ′ ⊂ ( X ′ , ω ′ ). Notice that [ T ′ ] ≤ [ K ω ′ ] ≤ T ] = [ T ′ ] ≤
9, and [ T ] = 9 only if X ′ = CP . Therefore we have the conclusion[ D ] = [ T ] ≤ D ] becomes an upper bound on r ( D ) when s i ≥ − i .Combined with the homological classification in Lemma 2.9, we can determine exactly whensuch D is rationally embeddable. Lemma 3.18.
Let D be a circular spherical divisor with s i ≥ − for all i . Then it isrationally embeddable if and only if it is toric equivalent to (1 , , or ( − , k ) , − ≤ k ≤ or (1 , k ) , ≤ k ≤ or one in the family F (2 , of Lemma 2.9 (6).Proof. Let D = ( s , . . . , s r ) be a circular spherical divisor. In the case r ≥
3, we mightassume D is toric minimal and s i ≥
0. Suppose D is embedded in symplectic rational surface( X, ω ), then 9 ≥ [ D ] = P ri =1 ( s i + 2) ≥ r , thus r ≤
4. By Lemma 2.9, we have that D canbe one of the following: (0 , , , , (1 , , , (1 , , , ( k, , , ≤ k ≤
2. Note that (0 , , , , , − ,
0) using the balancing move in Example 2.3 and thus toricequivalent to (1 , , , ,
0) is toric equivalent to (4 ,
1) and ( k, ,
0) to ( k + 2 , r = 2, we cannot assume D to be toric minimal. Again by Lemma 2.9, wehave that D is ( k, − , − ≤ k or one of (4 , , (4 , , (3 , , (3 , , (2 , , (2 , , (2 , , (1 , , (1 , , (0 , ≥ D = k + 3 because X contains at least oneexceptional class. So k ≤
5. They are all rationally embeddable as blow-ups of the minimalmodels.Given two circular spherical divisors
D, D ′ of length l , we say D is blown-up if D canbe obtained from non-toric blowing up D , for some toric blow-up D of (1 , , D = (1 , − p , − p , . . . , − p l − , − p l ) being blown-up in our definition is equivalent to thedual cycle of ( − p , . . . , − p l ) being embeddable in the sense of [13]. In Theorem 3.1 (iii) of[13], it is proved that minimal symplectic fillings of the boundary contact torus bundles ofsuch divisors always have vanishing first Chern class. Their proof translates to the followinglemma in our setting. Lemma 3.19 ([13]) . If D = (1 , − p + 1 , − p , ...., − p l − , − p l + 1) with p i ≥ and l ≥ , thenit is rationally embeddable, anti-canonical and rigid if and only if it is blown-up.Proof. If D is of such form and is blown-up, then D is realized by blowing up three lines ingeneral position inside CP and is rationally embeddable (Lemma 2.4 of [13]). The conversewas actually contained in the proof of Theorem 3.1 (iii) of [13] but was not explicitly written intheir theorem. We recall their arguments here for readers’ convenience. Let D be embedded ina symplectic rational surface ( X, ω ) with D = (1 , − p , . . . , − p l − , − p l ), we blow up D to D ′ in X ′ = CP M CP with M ≥ D ′ = (1 , − p , − p , . . . , − p l − − , − , − p l ). We choosean ω -tame almost complex structure J on X that makes all the symplectic spheres in D ′ J -holomorphic. Let S ′ be the irreducible component in D ′ with [ S ′ ] = − p l and let ˜ D ′ = D ′ − S ′ be the symplectic string with intersection sequence (1 , − p , − p , . . . , − p l − − , − X ′ to CP such that ˜ D ′ blows down to the union of two lines L ∪ L ′ ⊂ CP . At each blow-down, the almost complexstructure J descends. Since the complement of D ′ is minimal, the exceptional divisors weblow down either intersect D ′ once or contained in D ′ . During this process, S ′ blows downto a smoothly embedded symplectic sphere intersecting both L and L ′ exactly once, hence S ′ blows down to a line. So D is a blow-up of (1 , ,
1) corresponding to three lines in CP ,which exactly means D is blown-up. Lemma 3.20. (5 + p, − p ) with p ≥ − is not rationally embeddable if p = − .Proof. Suppose D = (5 + p, − p ) is symplectically embedded in a symplectic rational surface( X, ω ). Observe that if p ≥ X cannot be CP because there is no second homologyclass with negative self-intersection in CP . Also, (3 ,
2) cannot be embedded into CP eitherbecause there is no second homology class with self-intersection 2 or 3. So we could assume X = CP l CP for some l ≥ f C is an odd sphere with [ C ] = 2 x + 1 ≥
3, we use the CP l CP model with l ≥
1, where H ( X ; Z ) = Z { h, e , . . . , e l } . Then we have[ C ] = ( x + 1) h − xe , [ C ] = ah − be − l X i =2 b i e i , [ C ] = a − b − l X i =2 b i = 5 − (2 x + 1) = − x + 4 , [ C ] · [ C ] = ( x + 1) a − xb = 2 . The equation with [ C ] implies that a − b ≥ − x + 4. Any solution to ( x + 1) a − xb = 0is of the form ( a, b ) = ( ux + 2 , u ( x + 1) + 2) for an integer u . We have that a − b = ( ux + 2) − ( u ( x + 1) + 2) = − u (2 ux + u + 4) = ( − x − u + 42 x + 1 u ) =: F ( u ) . First we check that u = 0. If u = 0, then ( a, b ) = (2 , g ( C ) = 0is ( a − a − − b ( b − − P b i ( b i −
1) = 0. Since y ( y − ≥ y , we have( a − a − ≥ b ( b − . Clearly ( a, b ) = (2 ,
2) violates this inequality.Since 0 < x +1 < u + x +1 u ) for an integer u are smallest when u = 0 or −
1. We have F ( u ) ≤ F ( −
1) = − x + 3 < − x + 4 for u ≤ − F ( u ) ≤ F (1) = − x − < − x + 4 for u ≥
1. But this contradicts a − b ≥ − x + 4. If C is an even sphere with [ C ] = 2 x ≥
6, we can use the ( S × S ) l CP modelwith l ≥
0, where H ( X ; Z ) = Z { s, f, e , . . . , e l } . Then we have[ C ] = s + xf, [ C ] = as + bf − l X j =1 b j E j , [ C ] = 2 ab − l X j =1 b j = 5 − x, [ C ] · [ C ] = ax + b = 2 . The equation with [ C ] implies that 2 ab ≥ − x + 5. Any solution to ax + b = 2 is of theform ( u, − ux + 2) for an integer u . Similarly, we have that2 ab = 2 u ( − ux + 2)= − xu + 4 u = − x ( u + 42 x u ) =: G ( u ) . u = 0. If u = 0, then ( a, b ) = (0 , g ( C ) = 0 is 2( a − b − − P b i ( b i −
1) = 0 Since y ( y − ≥ y , we have( a − b − ≥ . Clearly ( a, b ) = (0 ,
2) violates this inequality.Since 0 < x <
2, we have G ( u ) ≤ G ( −
1) = − x + 4 < − x + 5 for u ≤ − G ( u ) ≤ G (1) = − x − < − x + 5 for u ≥
1. But this contradicts 2 ab ≥ − x + 5. Proposition 3.21.
Any rationally embeddable circular spherical divisor with b + ≥ is toricequivalent to one in the list: (1) (1 , , p ) with p ≤ . (2) (1 , p ) with p ≤ . (3) (0 , p ) with p ≤ . (4) (1 , − p , − p , . . . , − p l − , − p l ) with p i ≥ , l ≥ . (5) s i ≥ − for all i . (6) ( − , − , ( − , − Proof.
The proof is a case-by-case analysis based on the length of the divisor. Since arationally embeddable circular spherical divisor must have b + ≤
1, Lemma 2.9 applies here.
Case 1 : r ( D ) = 2. • When r ≥ ( D ) = 2, the divisors are listed in (6) of Lemma 2.9. They all belong to (5)of the list. • When r ≥ ( D ) = 1, they are of the forms ( k, p ) with k ≥ p < k + p ≤ k + p = 5, then by Lemma 3.20 the only rationally embeddable one is(1 ,
4) belonging to (2). Now suppose k + p ≤
4. If k ≤
1, the divisor ( k, p ) belongs to (2)or (3) of the list. If k ≥
2, we can toric blow up the pair ( C , C ), and if necessary, applysuccessive toric blow-ups to the pairs of the proper transform of C and the exceptionalspheres to get ¯ D with ¯ s = 1 , ¯ s = − , ¯ s i ≤ − i ≥
3. Then ¯ D belongs to (4) of thelist. • When r ≥ ( D ) = 0, by (8) of Lemma 2.9, there are only 3 circular spherical divi-sors with b + ( Q D ) = 1: ( − , − , ( − , −
2) and ( − , − − , −
1) belongs to (5) and( − , − , ( − , −
3) belong to (6) of the list.
Case 2 : r ( D ) = 3 and D is toric minimal (if not, reduce to the r = 2 case). • When r ≥ ( D ) = 3, it belongs to (5) of the list.19 When r ≥ ( D ) = 2, by (5) of Lemma 2.9, we can assume that D = (1 , , p ≤ D = ( s ≥ , , s ) with s + s ≤
2. The former case is already in the list. In the lattercase, we can apply the balancing move as in Example 2.3 based at C to decrease s to¯ s = 0 and increase s to ¯ s = s + s . Denote the new divisor by ¯ D . By toric blowingup the pair ( ¯ C , ¯ C ) and contracting the proper transform of ¯ C , ¯ C , we can reduce thelength of the divisor to 2. So D is toric equivalent to one in the list. • When r ≥ ( D ) = 1, if s = 1, then it belongs to (4) of the list. If s ≥
2, toric blow up thepair C , C , and if necessary, apply successive toric blow-ups to the pairs of the propertransforms of C and the exceptional spheres to get ¯ D with ¯ s = 1 , ¯ s = − , ¯ s i ≤ − i ≥ D is not toric minimal), so ¯ D belongs to (4) of the list.If s = 0, apply the balancing move based at C to increase s to ¯ s = 0 (whiledecreasing s r to ¯ s r − s ). Notice that r ≥ ( ¯ D ) = 2 , ¯ s = ¯ s = 0, and ¯ D toric minimal.We treated this case above. Case 3 : r ( D ) = 4 and D is toric minimal. • When r ≥ ( D ) = 4, it is (0 , , ,
0) by (3) of Lemma 2.7. Note that any divisor of form(0 , , , p ) is toric equivalent to the divisor (1 , , p + 1) by toric blowing up ( C , C ) andcontracting the proper transforms of C , C . • When r ≥ ( D ) = 3, by (2) of Lemma 2.9, ( s ) = ( k, , p,
0) with k ≥ , p < − k . Usingthe balancing move based at C , ( k, , p,
0) is toric equivalent to (0 , , k + p,
0) and thustoric equivalent to (1 , , k + p + 1). • When r ≥ ( D ) = 2, by (3) of Lemma 2.9 we have (0 , p , , p ) , p i < k ≥ , , p , p ) , p i < , k + p + p ≤ k, , p , p ) we apply the balancing move based at C to transform D to¯ D with ¯ s = ¯ s = 0 , ¯ s = s + s = k + p ≤ , ¯ s = s = p ≤ −
2. By blowing up thepair ( ¯ C , ¯ C ) and contracting the proper transforms of ¯ C , ¯ C , we get a divisor ¯ D ′ oflength 3.For the case (0 , p , , p ) , p i <
0, using the balancing move based at C , it is toricequivalent to (0 , , , p + p ) and thus toric equivalent to (1 , , p + p + 1). • When r ≥ ( D ) = 1, if s = 1 and s i ≤ − i ≥
2, then it belongs to (4) of the list.If s ≥
2, toric blow up the pair C , C , and if necessary, apply successive toric blow-upsto the pairs of the proper transforms of C and the exceptional spheres to get ¯ D with¯ s = 1 , ¯ s = − , ¯ s i ≤ − i ≥
3, so ¯ D belongs to (4) of the list.If s = 0, apply the balancing move based at C to increase s to ¯ s = 0 (whiledecreasing s r to ¯ s r − s ). Notice that r ≥ ( ¯ D ) = 2 , ¯ s = ¯ s = 0, and ¯ D toric minimal.We have treated this case above. Case 4 : r ( D ) ≥ D is toric minimal.This is proved by induction. Suppose we have proved the case r ( D ) ≤ n with some n ≥ D is not assumed to be toric minimal. The case r ( D ) = n + 1 and D not toric minimal20ollows directly from induction hypothesis by toric blow-down. We will verify the case where r ( D ) = n + 1 and D toric minimal.We have r ≥ ( D ) ≥ s ≥
0. By (1) ofLemma 2.9, r ≥ ( D ) ≤ • When r ≥ ( D ) = 2, by (1) of Lemma 2.9, we can assume that s ≥ s = 0. Apply thebalancing move based at C to transform to ¯ D with ¯ s = ¯ s = 0 , ¯ s = s + s , ¯ s i = s i ≤ − ≤ i ≤ n + 1. Toric blow up the pair ( ¯ C , ¯ C ) and then contract theproper transforms of ¯ C and ¯ C to get ¯ D ′ with ¯ s ′ = 1 , ¯ s ′ = ¯ s + 1 , ¯ s ′ n = ¯ s n +1 + 1. Since r ( ¯ D ′ ) = r ( D ) − n , the induction hypothesis applies. • Suppose r ≥ ( D ) = 1 and we may assume s ≥
0. If s = 1, then it belongs to (4) ofthe list. If s ≥
2, toric blow up the pair C , C , and if necessary, apply successive toricblow-ups to the pairs of the proper transforms of C and the exceptional spheres to get¯ D with ¯ s = 1 , ¯ s = − , ¯ s i ≤ − i ≥
3, which belongs to (4) of the list.If s = 0, apply the balancing move based at C to increase s to ¯ s = 0 (whiledecreasing s n +1 to ¯ s n +1 = s n +1 + s < − s , s n +1 < − r ≥ ( ¯ D ) =2 , ¯ s = ¯ s = 0, and ¯ D is toric minimal. We have treated this case above. Proof of Theorem 1.3.
The theorem is basically Proposition 3.21 with the exception of a fewcases. We still need to show that the list in Proposition 3.21 is toric equivalent to the list inthe theorem and every circular spherical divisor in the theorem can indeed be symplecticallyembedded in a rational manifold. This is done by Lemma 3.18, Lemma 3.19 and simplyobserving the following toric equivalences (denoted by ∼ ):( − , k ) ∼ (1 , − , − , . . . , −
2) with k − − , k ≥ , ∼ (3 , − , ∼ (1 , , , ∼ (1 , , − D = ( s , . . . , s r ), its negative boundary − Y D is a torus bundlewith monodromy A ( − s , . . . , − s r ) − . So their monodromy can be easily calculated. Forfamily (4) where D = (1 , p ) or (1 , − p , − p , . . . , − p l − , − p l ), by smoothly blowing downthe (+1) component, it has the same boundary Y D as ( − p , . . . , − p l ) with l ≥ p i ≥ p j ≥
3, which is negative hyperbolic by Theorem 6.1 of [38]. Note that iftr( A ) < −
2, then tr( A − ) < −
2. So − Y D is also negative hyperbolic.Finally we show all 3-manifolds in the list are distinct. For families (1)(2)(3), the mon-odromies are all distinct. For family (4), as we mentioned above, the family of torus bundles Y D are the same as the family of torus bundles obtained as boundaries of ( − p , . . . , − p l ) with l ≥
1. Note that by Proposition 6.3 of [38], the divisors ( p , . . . , p l ) are in bijection with theset of conjugacy classes A ( p , . . . , p l ) with trace ≥
3. Since the classification of conjugacyclasses of A ∈ SL (2 , Z ) with trace ≤ − ≥ −
1, we conclude that the divisors ( − p , . . . , − p l ) are in bijection with set ofconjugacy classes A ( − p , . . . , − p l ) with trace ≤ −
3. Hence all torus bundles in family (4) aredistinct. 21 .5 Rationally embeddable D is anti-canonical Lemma 3.22.
A rationally embeddable D = ( s , . . . , s r ) with s i ≥ − for all i is anti-canonical and rigid.Proof. Let D be symplectically embedded in a symplectic rational surface ( X, ω ) such thatits complement is minimal. We could symplectically smooth D to a symplectic torus T with[ T ] = [ D ] = P ( s i + 2) >
0. It suffices to prove T is maximal. Then by [39], T actuallyrepresents c ( X, ω ), i.e.
P D ([ D ]) = P D ([ T ]) = c ( X, ω ).Suppose T is not maximal, then there exists exceptional class E such that [ T ] · E = 0.Note that if s i ≥
0, we have [ C i ] · E ≥
0. If s i = −
1, then [ C i ] is an exceptional class. Notethat [ C i ] = E because [ C i ] · [ D ] = 1. Then we have [ C i ] · E ≥ E is D − good and there is an almost complex structure J such that D is J − holomorphic and E has an embedded J − holomorphic representative S . Since [ D ] · E = [ T ] · E = 0, D and S are disjoint, contradicting the minimality of thecomplement of D . Proposition 3.23.
Any rationally embeddable circular spherical divisor with b + = 1 is anti-canonical.Proof. It suffices to show that any rationally embeddable circular spherical divisor listed inProposition 3.21 is anti-canonical.(4) is anti-canonical by Lemma 3.19 and (5) is anti-canonical by Lemma 3.22.(1),(2) and (3) are realized respectively as non-toric blow-ups of minimal models (B3),(B2)and (D2) in Theorem 3.6.(6) is also realized as non-toric blow-ups of minimal models (B2) or (C2) or (D2). D is rigid Lemma 3.24. ( − , − and ( − , − are rigid.Proof. Suppose D = ( − , −
3) is a symplectic circular spherical divisor in a symplectic rationalsurface X . Denote the components of D by A, B , where A = − , B = −
3. Smooth D toget a symplectic torus T with T = 0. Note that we must have K X ≤
0. Otherwise thesubspace in H ( X ; Z ) spanned by 3[ A ] + 2[ B ] and − K X has intersection form (cid:18) − K X ) (cid:19) and is positive definite, contradicting b + ( X ) = 1. So we have that X must be CP l CP with l ≥ l = 9, we have that T = K X = 0 and K X · T = 0. By light cone lemma, we have T is proportional to − K X . Since T · A = ( A + B ) · A = 1 = − K X · A , we actually have[ D ] = [ T ] = − K X , i.e. D is anti-canonical.When l ≥
10, suppose T is maximal. By Proposition 3.14 of [29], we must have ( K X + T ) ≥
0. However, ( K X + T ) = K X + 2 K X · T + T = K X = 9 − l <
0. So T cannotbe maximal. By Theorem 3.21 of [29], exceptional classes orthogonal to [ T ] are pairwiseorthogonal and above discussion implies that there are at least l − Q , . . . , Q l − . 22ow consider the blowup class ˜ K = K X − Q − Q − · · · − Q l − . Then we have T = 0 , ( − ˜ K ) = 0 , and T · ( − ˜ K ) = 0 . By the light cone lemma, we have T and − ˜ K are proportional. Pairing both with A , we have T · A = 1 = ( − ˜ K ) · A . Therefore we conclude that T = − ˜ K = − K X l + Q + · · · + Q l − .Now we have1 = ( A + B ) · A = T · A = ( − K X l + Q + · · · + Q l − ) · A = 1 + X Q i · A. Since both Q i , A are stable classes, we must have Q i · A ≥ Q i · A = 0 for all i . Thisimplies Q i · B = Q i · ( T − A ) = 0. So each Q i is D − good and has an embedded symplecticrepresentative in the complement of D .So D has minimal complement only if l = 9, where D is anti-canonical.The proof for ( − , −
2) follows the exact same line as the proof for ( − , − D = ( − , −
2) is a symplectic circular spherical divisor embedded in a symplectic rationalsurface (
X, ω ) and T is the symplectic torus we get from smoothing D . Note that we musthave K X ≤
1. Otherwise the subspace spanned by [ T ] and − K X has intersection form (cid:18) − K X ) (cid:19) and is positive definite, contradicting b + ( X l ) = 1. So X must be CP l CP with l ≥
8. We could show that D has minimal complement only if l = 8. When D hasminimal complement, the symplectic torus T we get from smoothing D is maximal and T = 1 >
0. By [39], T represents c ( X, ω ), i.e.
P D ([ D ]) = c ( X, ω ). Proposition 3.25.
Anti-canonical circular spherical divisors with b + = 1 are rigid.Proof. It suffices to show anti-canonical circular spherical divisors of forms listed in Proposi-tion 3.21 are rigid.(6) is rigid by Lemma 3.24.(5) is rigid by Lemma 3.22.(4) is rigid by Lemma 3.19.(3) follows from Theorem 3.5 in [13].The case of p ≤ − s i ≥ − , ∀ i and follows from (5).Now we want to show the divisor (1 , , p ) is rigid. By McDuff, ( X, ω ) is rational and itmust be CP l CP . By (4) and (5) of Lemma 2.9, we may assume that [ C ] = [ C ] = h and[ C ] = h − P a i e i , where { h, e , . . . , e l } is a basis of H ( X ; Z ). Adjunction formula for C says P ( a i − a i ) = 0, which implies each a i is 0 or 1. If a k = 0 for some k , then [ C j ] · e k = 0for j = 1 , ,
3. Then e k is D − good and there is a symplectic sphere representing e k in thecomplement of D , contradicting the minimality. b + ( D ) = 0 This section is devoted to the study of embeddability and rigidity of negative semi-definitecircular spherical divisors. The main result of this section is that all such divisors are sym-23lectically embeddable and not rigid. We also classify anti-canonical strictly negative semi-definite divisors up to toric equivalence and show that the condition b + ≥ In this subsection, we study strictly negative semi-definite circular spherical divisors, whichmeans they are negative semi-definite but not negative definite. Recall that by Lemma3.2, being symplectically embeddable, rationally embeddable, anti-canonical and rigid are allpreserved by toric equivalence, so it suffices to consider toric minimal divisors. By Lemma2.4 any strictly negative semi-definite divisor is toric equivalent to D = ( − , −
4) or D n = ( − , . . . , − | {z } n ) , n ≥ . We first introduce another convenient operation on circular spherical divisors.
Definition 4.1.
Let D = ( s , . . . , s r ) be a circular spherical divisor. Then D ′ = ( s ′ , . . . , s ′ r − ) is a smoothing of D at the intersection ( s i , s i +1 ) if s ′ j = s j for j < i , s ′ j = s j +1 for j > i , and s ′ i = s i + s i +1 + 2 . For a symplectic circular spherical divisor D = ∪ C i ⊂ ( X, ω ), we can symplecticallysmooth the intersection point of C i and C i +1 to get a symplectic embedding of D ′ intothe same symplectic 4-manifold ( X, ω ) ([28]). Clearly smoothing of a symplectically em-beddable/rationally embeddable divisor is still symplectically embeddable/rationally embed-dable. Also because the symplectic smoothing does not change the total homology class, i.e.[ D ] = [ D ′ ] ∈ H ( X ; Z ), being anti-canonical is also preserved. Note that any smoothing of D n +1 is D n for n ≥
2, which makes smoothing a convenient reduction tool in the negativesemi-definite case.To find symplectic embeddings of D n , we also need to recall some basic facts of ellipticsurfaces ([1]). An elliptic fibration of a complex surface X is a proper holomorphic map f : X → S to a smooth curve S , such that the general fiber is a non-singular elliptic curve.An elliptic surface is a surface admitting an elliptic fibration. Following the classification ofsingular fibers by Kodaira, a singular fiber of I n type is a cycle of n self-intersection ( − n ≥
2, and a rational nodal curve with self-intersection 0 for n = 1. Werestrict ourselves to elliptic fibrations that are relatively minimal, have at least one singularfiber and have a global section. Such an elliptic fibration f : X → S has Kodaira dimensionkod( X ) = −∞ if and only if χ ( X ) + 2 g ( S ) − <
0. The elliptic surface X has even first Bettinumber b ( X ) = 2 g ( S ) ([41]), and in particular X is Kahler. As a Kahler surface, it is notsymplectically rational or ruled if and only if it has non-negative Kodaira dimension ([20]). Lemma 4.2.
Let D be a strictly negative semi-definite circular spherical divisor. Then D is symplectically embeddable. In particular, D symplectically embeds into a symplectic 4-manifold which is not symplectically rational.Proof. It suffices to consider D n with n ≥
1. The circular spherical divisor D can be obtainedby blowing up from the minimal model (1 ,
4) and thus is symplectically embeddable.24or each n ≥
2, we can find m ≥ m − ≥ n . By Theorem 2.3 of [42], for each m ≥
1, there exists a minimal elliptic surface X over P , with a maximal singular fiber of type I m − , m + 1 singular fibers of type I . This X is an elliptic modular surface by Theorem2.4 of [42] and has a global section by Definition 4.1 of [41]. It has χ ( X ) = [ 2 m + 12 ] ≥ X with a Kahler form ω . Then ( X, ω )is not symplectically rational or ruled and is symplectically minimal. The I m − singularfiber gives a symplectic embedding of D m − . Symplectically smoothing D m − gives asymplectically embedding of D n into ( X, ω ).Note that a singular fiber of type I in X is an immersed symplectic sphere with one nodalpoint and self-intersection 0. Blowing up the nodal point gives a symplectic embedding of D = ( − , −
4) into X CP with some blow-up symplectic form. Lemma 4.3.
If a circular spherical divisor D admits a symplectic embedding into ( X, ω ) which is not symplectically rational surface, then D is not rigid.Proof. By blowing down exceptional spheres in the complement of D , we get a symplecticembedding of D into ( X ′ , ω ′ ) with X ′ − D minimal. But ( X ′ , ω ′ ) is not rational because( X, ω ) is not, so D cannot be anti-canonical in ( X ′ , ω ′ ). Proposition 4.4.
The circular spherical divisor D n is anti-canonical if n ≤ and is notanti-canonical if n ≥ . D n is not rigid for all n ≥ .Proof. The circular spherical divisor D , D can be obtained by blowing up from the minimalmodel (1 , D from (1 , ,
1) and D n from (0 , , ,
0) for 4 ≤ n ≤
9. So for 1 ≤ n ≤
9, each D n is anti-canonical.Recall the charge q ( D ) = 12 − r ( D ) − P s i . A Toric blow-up preserves the charge anda non-toric blow-up increases the charge by 1. Since every minimal model has q ( D ) ≥ D ′ is obtained from a minimal model throughtoric and non-toric blow-ups, we must have q ( D ′ ) ≥
0. Since there is no ( − D n , there must be at least one non-toric blow-up. For a divisor of the form( − , . . . , − | {z } n − , − , there is only one way to toric blow down up to (anti-)cyclic permutations, which brings itto ( − , n − n = 10 ,
11, ( − , n −
4) is not one of the minimal models. We concludethat there must be at least two non-toric blow-ups and thus q ( D n ) ≥
2. and thus q ( D n ) ≥ q ( D n ) = 12 − n . So we conclude that D n is not anti-canonical for all n ≥ n = 10. For D n to be anti-canonical, it must be obtainedfrom one of the minimal models via toric and non-toric blow-ups. So the minimal modelscannot have any component less than −
2. Also note that (1 , ,
1) must blow up to (1 , , − , ,
1) must blow up to (3 , − , p, − p ) , − ≤ p ≤ p, , − p ) , − ≤ p ≤
1; 253) ( p, , − p, , ≤ p ≤ n = 10, suppose D is obtained from a minimal model in case i above for i = 1 , , − i toric blow-ups to reach the correct length and then i − D .It is easy to see that D n is not rigid by Lemma 4.2 and Lemma 4.3.Combining Lemma 4.2 and Proposition 4.4, we now get non-examples to Theorem 1.2 inthe strictly negative semi-definite case. Example 4.5.
The circular spherical divisor D n is • anti-canonical but not rigid, for n ≤ , • symplectically embeddable but not anti-canonical, for n ≥ . It seems generally difficult to obstruct or construct rational embeddings of D n . The onlyobstruction we know is that D n cannot be embedded in CP k CP for k <
9. This is becauseif D n is embedded in X = CP k CP , we can smooth it to a symplectic torus T with T = 0,which implies c ( X ) · T = 0. Then by light cone lemma, we must have c ( X ) ≤
0, implyingthat k ≥
9. Note that all anti-canonical D n must embed in CP CP . This tempts us tomake the following conjecture. Conjecture 4.6. D n is rationally embeddable if and only if D n is anti-canonical. A cusp singularity is the germ of an isolated, normal surface singularity such that the ex-ceptional divisor of the minimal resolution is a cycle of smooth rational curves D meetingtransversely ([7]). Note that D is a negative definite circular spherical divisor. Converselywhen D is negative definite, it arises as the resolution of a cusp singularity by Mumford-Grauert criterion ([15],[37]).A cusp singularity is called rational if its minimal resolution D is realized as the anti-canonical divisor of a rational surface, i.e. D is anti-canonical. Every cusp singularity has adual cusp singularity with its minimal resolution ˇ D to be the dual cycle of D . For a negativedefinite toric minimal divisor D = ( − a , − , . . . , − | {z } b , . . . , − a k , − , . . . , − | {z } b k ) , with a i ≥ b i ≥
0, its dual cycle is explicitly given byˇ D = ( − b − , − , . . . , − | {z } a − , . . . , − b k − , − , . . . , − | {z } a k − ) . Looijenga proved in [31] that if the cusp singularity with exceptional divisor D is smoothable,then there exists anti-canonical pair ( Y, ˇ D ). He conjectured that the converse is also true,26hich was proved in [16] and [9]. So the Looijenga conjecture gives an answer to Question1.1 for the negative definite circular spherical divisors in terms of its dual cycle. Theorem 4.7 (Looijenga conjecture [31],[16],[9]) . A toric minimal negative definite circularspherical divisor D is anti-canonical if and only if the cusp singularity corresponding to itsdual cycle ˇ D is smoothable. It is a hard (if not harder) problem to determine whether a cusp singularity is smoothable.So it is still interesting to understand when negative definite circular spherical divisors aresymplectically embeddable, rationally embeddable, anti-canonical or rigid.Proposition 4.4 implies that any toric minimal negative definite D with length ≤ D n with n ≤
9. Another useful obstructionto negative definite anti-canonical divisors is that they must have q ( D ) ≥ b + ≥ D is negative definite, the plumbing N D can be equipped with a convex symplecticstructure such that N D is a symplectic filling. Then D is symplectically embeddable bycapping off the filling with a symplectic cap ([10], [8]).More concretely, as in Lemma 2.4, any toric minimal negative definite circular sphericaldivisor D can be obtained from some D n through non-toric blow-ups. Since by Lemma 4.2 D n admits a symplectic embedding into ( X, ω ) which is not symplectic rational, then byblowing up D admits a symplectic embedding into some ( X ′ , ω ′ ) which is also not symplecticrational. By Lemma 4.3, we get the following result. Proposition 4.8.
Every negative definite circular spherical divisor is symplectically embed-dable and is not rigid.
Similar to Example 4.5, we can find negative definite spherical divisors that violatesTheorem 1.2. These examples show that the condition b + ≥ Example 4.9.
The negative definite circular spherical divisor D = ( − , . . . , − | {z } n − , − , for n ≥ , is not anti-canonical since q ( D ) < . But it is symplectically embeddable byProposition 4.8. Also divisor D = ( − , − n ) for n ≥ is negative definite and anti-canonical,but not rigid. Similar to the strictly negative semi-definite case, rational embeddability of negative def-inite divisors seems difficult to study. The only obstruction we know is that they do notembed in CP k CP with k <
10. If D embedds in X = CP k CP , we get a symplectictorus T in X with T < D . The torus can be made to be J -holomorphicby choosing a suitable tame almost complex structure J . But such torus doesn’t exist in CP k CP for k ≤ D must also embed in CP k CP for k ≥
10. Then we can ask if Conjecture 4.6 holds fornegative definite divisors.
Question 4.10.
Let D be a negative definte circular spherical divisor. If D is rationallyembeddable, is it anti-canonical? Contact aspects
This section is devoted to the study of contact topology related to circular spherical divisors.Section 5.1 reviews the relation between topological divisors and contact structures, andshows that the induced contact structure only depends on the toric equivalence class of thedivisor. Then 5.2 explains that the trichotomy of circular spherical divisors into b + > D is negative definiteand raise a conjecture related to Looijenga conjecture. In this section, we review some results about the convexity of divisor neighborhoods and theinduced contact structure on the boundary.Let D be a symplectic divisor in symplectic 4-manifold ( W, ω ) (not necessarily closed). Aclosed regular neighborhood of D is called a plumbing of D . A plumbing N D of D is calleda concave/convex plumbing if it is a strong symplectic cap/filling of its boundary andsuch D is called a concave/convex symplectic divisor. A concave plumbing is also called a divisor cap of its boundary. Let Q D be the intersection matrix of D and a = ([ C i ] · [ ω ]) ∈ ( R + ) r be the area vector of D . A symplectic divisor D is said to satisfy the positive (resp.negative) GS criterion if there exists z ∈ ( R + ) r (resp. ( R ≤ ) r ) such that Q D z = a .The GS criterion provides a way to tell when the divisor neighborhood is convex orconcave. Theorem 5.1 ([24]) . Let D ⊂ ( W, ω ) be an ω -orthogonal symplectic divisor. Then D has aconcave (resp. convex) plumbing if ( D, ω ) satisfies the positive (resp. negative) GS criterion. Note that a symplectic divisor can always be made ω -orthogonal by a local perturbation([14]). A necessary condition for D to have concave or convex plumbing is ω being exact onthe boundary Y D . To determine the exactness of ω | Y D , it suffices to check the following localcriterion. Lemma 5.2 ([24]) . ω | Y D is exact if and only if there is a solution for z to the equation Q D z = a , where a = ([ ω ] · [ C ] , . . . , [ ω ] · [ C r ]) is the area vector. In particular, this holds if Q D is non-degenerate. One can also check by simple linear algebra that the above condition is preserved undertoric equivalence. If ω | Y D is exact, then there is the following dichotomy depending on whether D is negative definite. Theorem 5.3 ([24]) . Let D ⊂ ( W, ω ) be a symplectic divisor. (1) If Q D is negative definite, then D has a convex plumbing. (2) If Q D is not negative definite and ω restricted to the boundary of D is exact, then ω can be locally deformed through a family of symplectic forms ω t on W keeping D symplectic and such that ( D, ω ) is a concave divisor. W, ω ),it actually does not rely on it. Suppose D is only a topological divisor with intersection matrix Q D such that there exists z, a satisfying the positive (resp. negative) GS criterion Q D z = a .Then Theorem 5.1 actually constructs a compact concave (resp. convex) symplectic manifold( N D , ω ( z )) such that D is ω ( z ) − orthogonal symplectic divisor in N D and a is the ω ( z ) − areavector of D .Now that ( N D , ω ( z )) is a convex or concave neighborhood of D , the Liouville vector fieldinduces a contact structure on the boundary which we call ξ D . Remark . In this paper we always require a contact structure to be positive, i.e. α ∧ dα > α . Note that Y D is oriented as the boundary of N D . In the case of convexneighborhood, ( Y D , ξ D ) is a positive contact manifold. But if N D is a concave neighborhood,we have α ∧ dα < − Y D , ξ D )is the correct positive contact manifold when N D is concave.The following uniqueness result implies that the symplectic structure ω ( z ) may vary with z but the induced contact structure on the boundary only depends on D . Proposition 5.5 ([24], cf. [35]) . Suppose D is an ω − orthogonal symplectic divisor which sat-isfies the positive/negative GS criterion. Then the contact structures induced on the boundaryare contactomorphic, independent of choices made in the construction and independent of thesymplectic structure ω , as long as ( D, ω ) satisfies positive/negative GS criterion.Moreover, if D arises from resolving an isolated normal surface singularity, then thecontact structure induced by the negative GS criterion is contactomorphic to the contactstructure induced by the complex structure. This motivates us to consider the notion of convexity for topological divisors. A topolog-ical divisor D is called concave (resp. convex ) if there exists z ∈ ( R + ) r (resp. z ∈ ( R ≤ ) r )such that a = Q D z ∈ ( R + ) r . Then there is a contact manifold ( − Y D , ξ D ) (resp. ( Y D , ξ D ))and its symplectic cap (resp. filling) ( N D , ω ( z )) containing D as a symplectic divisor. Onecan check by simple linear algebra that being concave (resp. convex) is preserved by toricblow-up (see for example Lemma 3.8 of [24]).When D is negative definite, ( Y D , ξ D ) is contactomorphic to the contact boundary ofsome isolated surface singularity ([15]) and is called a Milnor fillable contact structure. Aclosed 3-manifold Y is called Milnor fillable if it carries a Milnor fillable contact structure.For every Milnor fillable Y , there is a unique Milnor fillable contact structure ([3]), i.e. thecontact structure ξ D only depends on the oriented homeomorphism type of Y D instead of D when D is negative definite.In light of this uniqueness result, it is natural to ask if similar results hold when D isconcave. The first and the third authors formulated the following question in [27]. Question 5.6 ([27]) . Suppose D and D are concave divisors with − Y D ∼ = − Y D . Sup-pose either b + ( Q D ) = b + ( Q D ) or ξ D , ξ D both symplectically fillable, then is ( − Y D , ξ D ) contactomorphic to ( − Y D , ξ D ) ? Since the torus bundles in Theorem 1.3 are all distinct, we get a positive answer to thequestion if we restrict ourselves to fillable concave circular spherical divisors.As a first step towards this quesition, we have the following invariance result.29 roposition 5.7 ([27]) . The contact structure induced by GS construction is invariant undertoric equivalence.
In light of Proposition 5.7, we see that toric equivalence is a natural equivalence ondivisors. For the study of contact structures and symplectic fillings, it suffices to considertoric minimal divisors. In particular it is used in the proof of Theorem 1.5 (see Theorem5.18).
In this subsection, we characterize the convexity of circular spherical divisors. We startwith the following lemma about more general topological divisors. By a subdivisor D ′ of atopological divisor D , we mean the union of a subset of the components so that D ′ is stillconnected. Lemma 5.8.
A topological divisor D is concave if it contains a concave subdivisor D ′ .Proof. From D ′ we can always get back to D by successively adding components to D ′ , sowithout loss of generality we could assume D = ∪ ri =1 C i can be obtained from D ′ = ∪ r − i =1 C i by adding exactly one component C r .Note that the intersection matrix Q D ′ = ( q i,j ) of D ′ is a ( r − × ( r −
1) symmetric matrixwith nonnegative off-diagonal entries. Then we could write Q D = q ,r Q D ′ ... q r − ,r q r, . . . q r,r − q r,r , where q i,r = q r,i ≥ i = 1 , . . . , r − z ′ ∈ ( R + ) r − such that a ′ = Q D ′ z ′ ∈ ( R + ) r − and let z be a vector in ( R + ) r suchthat z i = z ′ i for i = 1 , . . . , r −
1. Let a = Qz . Then we have a i = a ′ i + q i,r z r ≥ a ′ i > i ≤ r − a r = P i ≤ r − q r,i z ′ i + q r,r z r . Since there is at least one strictly positive q r,i , wehave P i ≤ r − q r,i z ′ i >
0. If q r,r ≥
0, then for any z r > a r >
0. If q r,r <
0, then wecan choose z r ∈ (0 , − P i ≤ r − q r,i z ′ i q r,r ) and get a r > Corollary 5.9.
A topological divisor with at least two components and at least one componentwith nonnegative self-intersection must be concave.Proof.
Such a divisor would have a subdivisor with intersection matrix looking like Q = (cid:18) a nn b (cid:19) , where a ≥ n ≥
1. It suffices to find z ∈ ( R + ) such that Qz ∈ ( R + ) . If b ≥
0, we cantake z = (1 , T . If b <
0, we can take z = ([ − bn + 1] , T . So this subdivisor is concave andthus the original divisor is concave by Lemma 5.8.30eneralizing the symplectic Kodaira dimension of a symplectic 4-manifold, the followingcontact Kodaira dimension for contact 3-manifolds was proposed in [22], based on the typeof symplectic cap it admits. Definition 5.10 ([22], [25]) . Let ( P, ω ) be a concave symplectic 4-manifold with contactboundary ( Y, ξ ) . ( P, ω ) is called a Calabi-Yau cap of ( Y, ξ ) if c ( P, ω ) is a torsion class, andit is called a uniruled cap of ( Y, ξ ) if there is a contact primitive β on the boundary such that c ( P, ω ) · [( ω, β )] > .The contact Kodaira dimension of a contact 3-manifold ( Y, ξ ) is defined in terms of unir-uled caps and Calabi-Yau caps. Precisely, Kod ( Y, ξ ) = −∞ if ( Y, ξ ) has a uniruled cap, Kod ( Y, ξ ) = 0 if it has a Calabi-Yau cap but no uniruled caps,
Kod ( Y, ξ ) = 1 if it has noCalabi-Yau caps or uniruled caps.
We have the following characterization of convexity for circular spherical divisors andcontact Kodaira dimension of their boundaries.
Proposition 5.11.
Let D be a circular spherical divisor and Q D its intersection matrix. (1) If Q D is negative definite, then D is convex. In addition, if D is anti-canonical in ( X, ω ) , then ( Y D , ξ D ) has Kod ( Y D , ξ D ) ≤ . (2) If b + ( Q D ) > , then D is concave. In addition, if D is symplectically embeddable,then ( − Y D , ξ D ) has Kod ( − Y D , ξ D ) = −∞ . (3) If b + ( Q D ) = 0 and Q D is not negative definite, then D is neither concave or convex.Proof. Since being concave or convex is preserved under toric blow-up, we could assume D iseither toric minimal or of the form ( − , p ) and make use of the classification in Lemma 2.4. Case (1) : Suppose Q D is negative definite. Then for any a ∈ ( R + ) r , there is a unique so-lution z ∈ ( R − ) r such that Q D z = a . So D is convex. Since D is anti-canonical, let ( X, ω, D )be a symplectic Looijenga pair. Then by Theorem 5.3 there is a convex neighborhood N D of D in ( X, ω ) and P = X − N D is a symplectic cap of Y D with vanishing c , i.e. a Calabi-Yaucap. It follows that Kod ( Y D , ξ D ) ≤ Case (2) : Suppose b + ( Q D ) >
0. If D is toric minimal, there exists a component C j withpositive self-intersection. By Corollary 5.9, D must be concave. If D is of the form ( − , p ),we must have p ≥ −
3. Then for z = (7 , T , we have Q D z = (1 ,
14 + 4 p ) T ∈ ( R + ) , i.e. D is concave. If D is symplecticaly embeddable, then D is anti-canonical by Theorem 1.2.Let ( X, ω, D ) be a symplectic Looijenga pair and assume ω | Y D is exact, then by Theorem 5.3there is a concave neighborhood N D of D in ( X, ω ) with boundary ( − Y D , ξ D ), up to a localsymplectic deformation. For any contact primitive α of ω | Y D , we have c ( N D ) · [( ω, α )] = c ( X ) | N D · [( ω, α )] = D · [( ω, α )] = D · [ ω ] >
0, i.e. N D is a uniruled cap for ( − Y D , ξ D ).Now we show that ω | Y D is exact. By Lemma 5.2, it suffices to show that for any a ∈ ( R + ) r ,there is a solution z ∈ R r to the equation Qz = a . And it suffices to show it for all D listedin Theorem 1.3. By Lemma 2.5, we only need to look at the positive parabolic case, because Q D is nondegenerate in all other cases. Let a = ( a , a , a ) = ([ ω ] · [ C ] , [ ω ] · [ C ] , [ ω ] · [ C ]) forthe divisor (1 , , p ). First if p = 1, then by (4) of Lemma 2.9, we have [ C ] = [ C ] = [ C ] and31hus a = a = a . Then z = z = z = a Q D z = a . If p = 1, then againby (4) and (5) of Lemma 2.9, we have [ C ] = [ C ] and thus a = a . Solving the equation Q D z = a is thus equivalent to solving (cid:18) p (cid:19) (cid:18) z z (cid:19) = (cid:18) a a (cid:19) , which is solvable as p = 1. Case (3) : Suppose b + ( Q D ) = 0 and Q D is not negative definite. If D is toric minimal, it isa cycle of self-intersection − i th-entry of Q D z is ( z i +1 − z i ) − ( z i − z i − )for all i , where the index is taken to be modulo r . So it is easy to see that for any z ∈ R r , Q D z cannot have all entries being positive. If D is of the form ( − , p ), we must have p = − Q D z = ( − ( z − z ) , z − z )) T can not be in ( R + ) for any z . So D cannot beconvex or concave. D In this section we prove Theorem 1.5, which is splitted into Theorem 5.18 and 5.19. We startby recalling the notions of maximal divisors and Donaldson divisors. They will be used totransit between minimal filling and minimal complement of divisors.
Definition 5.12.
For a symplectic divisor D = ∪ C i , we extend the notion and also call theformal sum ˜ D = P z i C i , z i ∈ Z a symplectic divisor. Then a symplectic divisor ˜ D = P z i C i in a closed symplectic 4-manifold ( X, ω ) is called a maximal divisor if [ ˜ D ] · e > for anyexceptional class e . Maximal divisors are natural extensions of maximal surfaces introduced in Section 3.2.They are intimately related to divisor caps through the notion of Donaldson divisors.
Definition 5.13.
Let ˜ D = P z i C i be a symplectic divisor in a symplectic cap ( P, ω ) of ( Y, ξ ) .Then ˜ D is called a Donaldson divisor if [ ˜ D ] ∈ H ( P ; Z ) is the Lefschetz dual of c [( ω, α )] for some c > , where α is a contact form for ξ such that ([ ω, α ]) is a rational class. Lemma 5.14 ([24]) . Let ( D, ω ) be a symplectic divisor satisfying positive GS criterion Qz = a and N D a concave plumbing of D with contact form α on the boundary. Then P D ([( ω, α )]) = P z i [ C i ] ∈ H ( N D ; R ) . So for any divisor cap ( N D , ω ( z )), there is a Donaldson divisor ˜ D = P z i C i . The impor-tance of a Donaldson divisor lies in the fact that it is maximal when we close up the cap witha minimal symplectic filling. Lemma 5.15 ([25]) . Let ( P, ω P ) be a symplectic cap of ( Y, ξ ) and let α P be a contact 1-form for ξ . Assume that [( ω P , ω P )] is a rational class and let [ ˜ D ] be the Lefschetz dual of c [( ω P , α P )] for some c > . Then for any minimal symplectic filling ( W, ω W ) of ( Y, ξ ) andany symplectic exceptional class e in ( X, ω ) = (
P, ω P ) ∪ ( W, ω W ) , we have [ ˜ D ] · e > . Next we prove the fillability result in Theorem 1.5 and study the topology of the minimalfillings. We start by collecting some homological information related to a cycle of spheres ina closed 4-manifold X with b ( X ) = 1, which will also be useful in Section 5.4.32 emma 5.16. Let X = U ∪ Y V be a closed 4-manifold obtained by gluing U and V alongtheir common boundary Y . Suppose b ( U ) = 0 and b ( X ) = 0 . Then we have b ( V ) = 0 and b ( V ) + b ( V ) = b ( U ) + b ( U ) = b ( Y ) .Proof. Since H ( X ) → H ( Y ) is an isomorphism, we have 0 → H ( V ; Q ) ⊕ H ( U ; Q ) → H ( X ; Q ) = 0 and thus b ( V ) = 0. Note that b ( V ) is the kernel dimension of the map H ( V ; Q ) → H ( V, Y ; Q ). Then we have the long exact sequence0 → Q b ( V ) → H ( V ; Q ) → H ( V, Y ; Q ) → H ( Y ; Q ) → H ( V ; Q ) → , which gives b ( V ) + b ( V ) = b ( Y ) by Lefschetz duality.Recall that the charge q ( D ) = 12 − D − r ( D ) is invariant under toric equivalence. Lemma 5.17 (cf. Theorem 2.5 and Theorem 3.1 in [13]) . Let D be a cycle of spheres in aclosed 4-manifold X with b ( X ) = 1 and W = X − N D , where N D is a plumbing of D . Thenwe have H ( N D ) = Z r ( D ) = H ( N D ) , H ( N D ) = H ( N D ) = Z , H ( N D ) = H ( N D ) = 0 . If Q D is nonsingular, then we have b ( Y D ) = 1 , b ( V ) = 0 , b ( V ) = 1 , b ( V ) = b ( X ) + 1 − r ( D ) . If ( X, D, ω ) is a symplectic Looijenga pair, then e ( V ) = q ( D ) and σ ( V ) = 4 − q ( D ) − b + ( D ) − b ( D ) .Proof. The homology and cohomology of N D are straightforward to compute since N D de-formation retracts to D .As in Theorem 2.5 of [13], we have H ( Y D ) = Z ⊕ Coker ( Q D ). Now suppose Q D isnonsingular. Then b ( Y ) = 1. By Mayer-Vietoris sequence H ( Y ; Q ) → H ( N D ; Q ) ⊕ H ( V ; Q ) → , we get that b ( Y ) ≥ b ( V ). So b ( V ) = 0 and then b ( V ) = 1. Since e ( V ) = e ( X ) − e ( N D ) = 2 + b ( X ) − r ( D ) and by definition e ( V ) = 1 + b ( V ), we have b ( V ) = b ( X ) + 1 − r ( D ).If ( X, D, ω ) is a symplectic Looijenga pair, then e ( X ) = 12 − D and σ ( X ) = D −
8. Note e ( N D ) = r ( D ) and σ ( N D ) = b + ( D ) − b − ( D ). By additivity, we get e ( V ) = 12 − D − r ( D ) = q ( D ) and σ ( V ) = D − − σ ( N D ) = D −
12 + r ( D ) − b + ( D ) − b ( D ) + 4 = 4 − q ( D ) − b + ( D ) − b ( D ). Theorem 5.18.
Let D be a concave circular spherical divisor. Then ( − Y D , ξ D ) is symplecticfillable if and only if D is toric equivalent to one of Theorem 1.3. Moreover, there are finitelymany minimal symplectic fillings. All minimal symplectic filling ( W, ω W ) of such ( − Y D , ξ D ) have c ( W, ω W ) = 0 . They all have b + ( W ) = b ( W ) = 0 , b ( W ) = 1 , b ( W ) = b ( Y D ) − , b − ( W ) = q ( D ) − b ( D ) and hence a unique rational homology type. In particular, for D in case (1)(3)(4), we have b − ( W ) = q ( D ) − , b ( W ) = q ( D ) − and b ( W ) = 1 ; • for D in case (2) with p = 1 , we have b − ( W ) = 0 , b ( W ) = 1 and b ( W ) = 2 ; • for D in case (2) with p < , we have b − ( W ) = q ( D ) − , b ( W ) = q ( D ) and b ( W ) = 1 .Proof. If D is toric equivalent to one in Theorem 1.3, then it admits an anticanonical sym-plectic embedding into a symplectic rational surface ( X, ω ). By Proposition 5.11, up to localsymplectic deformation, there is a concave symplectic neighborhood N D of D with contactboundary ( − Y D , ξ D ), then X − Int ( N D ) is a symplectic filling of ( − Y D , ξ D ).Now suppose ( − Y D , ξ D ) is symplectic fillable and let ( W, ω W ) be any minimal symplecticfilling. Let z, a be a pair of vectors satisfy the positive GS criterion Q D z = a . Then we have adivisor cap ( N D , ω ( z )) of ( − Y D , ξ D ). Glue ( W, ω W ) with ( N D , ω ( z )) to get a closed manifold( X, ω ). By Lemma 3.1 and 3.16, we have that b + ( D ) = 1 and X is a rational surface and D is toric equivalent to one listed in Theorem 1.3.Let ˜ D = P z i C i be the Donaldson divisor in N D , then ˜ D is a maximal divisor in X . Forany exceptional class e in X , we have ˜ D · e >
0. If X − D is not minimal, then there is anexceptional curve E ⊂ X − D . Then [ E ] · [ C i ] = 0 for all C i in D . So [ ˜ D ] · [ E ] = P z i [ C i ] · [ E ] =0, which contradicts the maximality of ˜ D . So we conclude that X − D is minimal. Because D is a rationally embeddable circular divisor with b + ( D ) = 1, D is rigid and thus ( X, D, ω ) isa symplectic Looijenga pair. So we must have c = b + = b = 0 for ( W, ω W ). The finitenessof symplectic deformation types follows from Corollary 3.9.Now we compute other betti numbers of W . By Lemma 5.17, we have e ( W ) = q ( D )and σ ( W ) = 2 − q ( D ) − b ( D ). Since b + ( W ) = 0, we have that b − ( W ) = − σ ( W ) = q ( D )+ b ( D ) −
2. By Lemma 2.2, b ( D ) is invariant under toric equivalence. Then Proposition5.5 implies that both q ( D ) and b ( D ) depend only on ( − Y D , ξ D ). So all minimal symplecticfillings have the same b ± ( W ) , b ( W ) and a unique rational homology type.When Q D is nonsingular, we have by Lemma 5.17 that b ( W ) = 0, b ( W ) = b ( X ) + 1 − r ( D ) = 10 − D − r ( D ) + 1 = q ( D ) − b − ( W ) = q ( D ) − Q D is singular, we have D is toric equivalent to (1 , , p ) with p ≤
1. By Lemma5.16, we still have1 − b ( W ) + b ( W ) + b − ( W ) = e ( W ) = q ( D ) and b ( W ) − b ( Y ) + b ( W ) = 0 . These two equations imply that b ( W ) = 12 ( b ( Y ) + q ( D ) − b − ( W ) −
1) and b ( W ) = 12 ( b ( Y ) − q ( D ) + b − ( W ) + 1) . Note again that toric equivalence preserves b + ( D ) , b ( D ) by Lemma 2.2. • When p = 1 we have b ( Y D ) = 3, b ( D ) = 2, and thus b − ( W ) = 0. Then b ( W ) = 1and b ( W ) = 2. • When p < b ( Y D ) = 2, b ( D ) = 1, and thus b − ( W ) = q ( D ) −
1. Then b ( W ) = b ( W ) = 1. 34ext we identify minimal symplectic fillings with Stein fillings up to symplectic deforma-tion equivalence. This finishes the proof of Theorem 1.5. Theorem 5.19.
For a symplectic Looijenga pair ( X, D, ω ) with b + ( Q D ) = 1 , there exists aK¨ahler Looijenga pair ( X, D, ω ) in its symplectic deformation class such that D is the supportof an ample line bundle. Then every minimal symplectic fillings of ( − Y D , ξ D ) is symplecticdeformation equivalent to a Stein filling.Proof. Let (
X, D, ω ) be a symplectic Looijenga pair. By Lemma 3.7 there is an holomorphicLooijenga pair ( X ′ , D ′ , ω ′ ) symplectic deformation equivalent to ( X, D, ω ). By Proposition5.11, ω ′ is exact on ∂P ( D ) and there is another symplectic form ω deformation equivalent to ω ′ such that Q D z = a , where z = ( z , . . . , z r ) ∈ ( R + ) r and a = ([ ω ] · [ C ] , . . . , [ ω ] · [ C r ]). Itmeans that ˜ D = r P i =1 z i [ C i ] pairs positively with all [ C i ] and in particular ˜ D > X, D ) isa generic pair, which means there is no smooth rational curve of self-intersection − D (Definition 1.4 of [17]). Any algebraic curve C disjoint from D is in particularorthogonal to ˜ D . By Hodge index theorem, we must have [ C ] ≤
0. Then C is a self-intersection − C cannot be a self-intersection − X, D ). We claim that there is no elliptic curve in the complement of D of self-intersection0. Suppose there exists such elliptic curve T . Since b + ( Q D ) = 1, we can assume there is aself-intersection 0 component C of D (possibly after toric blow-down and non-toric blow-up).Using light cone lemma, we get that [ T ] = λ [ C ] for some λ >
0. However, C intersects othercomponents of D non-trivially, and so would T , which is contradiction. Any algebraic curvethat intersects D but not contained in D has positive pairing with P ri =1 z i [ C i ]. Also, bythe choice of P ri =1 z i [ C i ], it pairs positively with any irreducible curve in D . Therefore, byNakai-Moishezon criterion, P ri =1 z i [ C i ] is an ample divisor and the support is D . So X − D is an affine surface and provides a Stein filling of ( − Y D , ξ D ).As shown in the proof of Theorem 5.18, every minimal symplectic filling gives rise to asymplectic Looijenga pair ( X, D, ω ), which is symplectic deformation equivalent to a Kahlerpair (
X, D, ω ) as above. So the minimal symplectic filling is symplectic deformation equiva-lent to a Stein filling.
Remark . A similar result for elliptic log Calabi-Yau pairs was obtained by Ohta andOno in [39]. Their results were stated for links of simple elliptic singularities, but actuallyconcerns symplectic torus of positive self-intersection. We summarize their results as follows.Let D be a torus with 0 < [ D ] ≤ D ] = 8, then the minimal symplectic filling of( − Y D , ξ D ) is unique up to diffeomorphism. In the case [ D ] = 8, there are two diffeomorphismtypes of minimal symplectic fillings. All these minimal fillings have c = 0 and b + = 0. When[ D ] ≥
10, there is a unique minimal symplectic filling up to diffeomorphism, but we don’thave c = 0 in this case. 35 .4 Geography of Stein fillings for convex anti-canonical D When Q D is negative definite, the circular spherical divisor D corresponds to resolutionsof cusp singularities. Then ( Y D , ξ D ) is a Milnor fillable contact structure and is thus Steinfillable ([15], see also [3]).The elliptic log Calabi-Yau pairs arise from resolutions of simple elliptic singularities. ByTheorem 2 in [39], any simple elliptic singularity has either one or two minimal symplecticfillings up to diffeomorphism, arising either from a smoothing or the minimal resolution.It is then natural to ask what kind of finiteness holds for symplectic fillings of ( Y D , ξ D )when D is convex anti-canonical circular spherical divisor. Since ( Y D , ξ D ) has non-positivecontact Kodaira dimension (Proposition 5.11), the Betti numbers of exact fillings of ( Y D , ξ D )are bounded by Theorems 1.3 and 1.8 in [25]. Here we provide explicit Betti number boundsfor their Stein fillings. Proposition 5.21.
Let D be a convex anti-canonical circular spherical divisor, i.e. Q D negative definite. Let ( U, ω U ) be a Stein filling of ( Y D , ξ D ) , then (1) U is negative definite with b ( U ) = 1 , (2) or ( b +2 ( U ) , b ( U ) , b ( U )) = (1 , , , c ( U ) = 0 , b − ( U ) = 21 − q ( D ) ≤ , and ≤ q ( D ) ≤ , (3) or ( b +2 ( U ) , b ( U ) , b ( U )) = (2 , , , c ( U ) = 0 , b − ( U ) = 22 − q ( D ) ≤ , and ≤ q ( D ) ≤ .In particular, when q ( D ) ≥ , U must be negative definite.Proof. Since D is anti-canonical, there exists ( X, ω ) such that (
X, D, ω ) is a symplecticLooijenga pair. Let V = X − N D be the symplectic cap obtained as the complement of N D and X U = U ∪ V the closed symplectic manifold obtained by capping off the filling U with V . Since U is Stein and in particular a 2-handlebody, we have 1 = b ( Y D ) ≥ b ( U ). TheMayer-Vietoris sequence H ( U ) ⊕ H ( V ) → H ( X U ) → b ( X U ) ≤ b ( U ) + b ( V ) ≤ b ( V ). By Lemma 5.17, b ( V ) = 0, so b ( X U ) ≤
1. Since V is a Calabi-Yau capas c ( V ) = 0, X U must be a symplectic non-minimal rational or minimal Calabi-Yau surfaceby Lemma 2.8 of [25]. It then follows from b ( X U ) ≤ b + ( X U ) = 1 , b ( X U ) = 0 and X U is non-minimal rational or a minimal integralhomology Enrique surface,(2) or b + ( X U ) = 3 , b ( X U ) = 0 and X U is a minimal integral homology K b ( X U ) = 0 and by Lemma 5.16, b ( U ) + b ( U ) = b ( Y D ) = 1. ByLemma 5.17, we also have e ( V ) = q ( D ) and σ ( V ) = σ ( V ) = 4 − q ( D ) − b + ( D ) − b ( D ) = − q ( D ) + 4 since D negative definite. By additivity, we get b + ( U ) − b − ( U ) = σ ( U ) = σ ( X U ) + q ( D ) − ,b + ( U ) + b − ( U ) = e ( U ) − b ( U ) = b ( X U ) + 2 − q ( D ) − b ( U ) . b + ( U ) = b + ( X U ) − b ( U ) − ,b − ( U ) =3 − q ( D ) − b ( U ) + b − ( X U ) . Note that if b + ( X U ) = 3, then c ( X U ) = 0 and thus c ( U ) = 0. By Lemma 4.3 of [36], wehave that q ( D ) ≥ • If b ( U ) = 1, then b + ( X U ) ≥
2, which implies that X U must be an integer homology K3with b + ( X U ) = 3 and b − ( X U ) = 19. Then we have c ( U ) = 0, b ( U ) = 0, b + ( U ) = 1, b − ( U ) = 21 − q ( D ) ≤
18 and 3 ≤ q ( D ) ≤ • If b ( U ) = 0 and b + ( X U ) = 3, we have c ( U ) = 0, b ( U ) = 1, b + ( U ) = 2, b − ( U ) =22 − q ( D ) ≤
19 and 3 ≤ q ( D ) ≤ • If b ( U ) = 0 and b + ( X U ) = 1, we have U is negative definite, b ( U ) = 1, and e ( U ) = b − ( U ) = 3 + b − ( X U ) − q ( D ).Finally, we discuss the potential implication of Proposition 5.21 for Stein fillings of cuspsingularities. By Looijenga conjecture, a cusp singularity is smoothable if and only if it hasa rational dual. Note that a smoothing of a cusp singularity gives a Stein filling of ( Y D , ξ D )with b + = 1. In light of this, we make the following symplectic/contact analogue of theLooijenga conjecture. Conjecture 5.22.
If a cusp singularity does not have a rational dual, then it admits onlynegative definite Stein fillings.
By Theorem 4.5 of [36], if q ( D ) ≥
22 then D doesn’t have a rational dual. Then Proposi-tion 5.21 proves the conjecture for rational cups singularities with q ( D ) ≥
23. In particular,given any negative-define anti-canonical divisor, if we perform 20 non-toric blow-ups (to in-crease the charge while remaining negative-definite and anti-canonical), the resulting contacttorus bundle has only negative definite Stein fillings.
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School of Mathematics, University of Minnesota, Minneapolis, MN, US
E-mail address : [email protected] School of Mathematics, University of Edinburgh, Edinburgh, UK
E-mail address : [email protected] School of Mathematics, University of Minnesota, Minneapolis, MN, US
E-mail address : [email protected]@umn.edu