EEXACT LAGRANGIANS IN A n -SURFACE SINGULARITIES WEIWEI WU
Abstract.
In this paper we classify Lagrangian spheres in A n -surface singularities up toHamiltonian isotopy. Combining with a result of A. Ritter [27], this yields a completeclassification of exact Lagrangians in A n -surface singularities. Our main new tool is theapplication of the a technique which we call ball-swappings and its relative version. MSC classes:
Keywords: symplectic ball packing, Lagrangian isotopy, symplectomorphism groups1.
Introduction
One of the classical problems in sympletic topology is to understand the classification of exactLagrangians in a given exact symplectic manifold. As appealing as it is, the problem is ingeneral very difficult, even in its simplest form. In particular, Arnold’s nearby Lagrangianconjecture asserts that any exact Lagrangian in T ∗ M is Hamiltonian isotopic to the zerosection, which is still quite open. This line of questions have attracted many efforts involvinga long list of authors, among which we only mention a few very recent advances [1, 2, 33].Note that the proofs of above-mentioned works involved advanced techniques from Floertheory, and usually covers symplectic manifolds of arbitrary dimensions. While these deepmethods are amazingly powerful in determining the shape of exact Lagrangians couplingwith homotopy methods, a general construction of Hamiltonian isotopies is still missing indimension higher than 4. In contrast, one has more geometric tools available in dimension4, thus equivalence up to Hamiltonian classes is more approachable. See [9] for an exampleof a very nice application of foliation techniques to this problem, and [20, 10] for anotherapproach which is closer in idea to what we will use here.In the present paper, we investigate the classification of Lagrangian spheres in an A n -surfacesingularity. By definition, an A n -surface singularity is symplectically identified with thesubvariety { ( x, y, z ) : x + y + z n +1 = 1 } ⊂ ( C , ω std ) , endowed with the restricted K¨ahler form. Throughout the paper we will denote W as the A n − -surface singularity, which is the main object that we will investigate. It is by now well-known that W is identified symplectically with the plumbing of n − T ∗ S . We will Date a r X i v : . [ m a t h . S G ] M a y WU call the zero sections of these plumbed copies standard spheres . The following is our mainresult: Theorem 1.1.
Lagrangian spheres in W are unique up to Hamiltonian isotopy and La-grangian Dehn twists along the standard spheres. A fantastic result showed by A. Ritter in [27] says that, embedded exact Lagrangians in W are all Lagrangian spheres. We therefore obtain the following corollary, which completelyclassifies exact Lagrangians in A n surface singularities up to Hamiltonian isotopy: Corollary 1.2.
Exact Lagrangians in A n -surface singularities are isotopic to the zero sectionof a plumbed copy of T ∗ S , up to a composition of Lagrangian Dehn twists along the standardspheres. Such kind of classification seems desirable but rare in the literature, especially when thereexist smoothly isotopic but not Hamiltonian isotopic Lagrangians [30]. Notice that variousforms of partial results have been obtained previously. In particular, R. Hind in [9] provesTheorem 1.1 for the case of A and A . It was also known that the result is true up toequivalence of objects in the Fukaya category, using the deep computations in algebraicgeometry as well as the mirror symmetry of A n -surface singularities [11, 12]. This Floer-theoretic version already found interesting applications [18, 33]. In principle, Theorem 1.1along with computation of [13] on the symplectic side should recover corresponding resultsin [11, 12] on the mirror side. En route , we also prove the following result on the compactly supported symplectomorphismgroup of W : Theorem 1.3.
Any compactly supported symplectomorphism is Hamiltonian isotopic to acomposition of Dehn twists along the standard spheres. In particular, π ( Symp c ( W )) = Br n . This is a refinement of the results due to J. Evans [7, Theorem 4], which asserts that π ( Symp c ( W )) injects into Br n , and Khovanov-Seidel [13, Corollary 1.4], which proves inany dimension of A n -singularities, there is an injection from the other direction. Our resultshows that these two injections are in fact both isomorphisms in dimension 4.The paper is structured as follows. In Section 2 we set up the notation and describe twodifferent but closely related models for W and its compactification. We then reduce the maintheorems to problems in its compactification. Section 3 contains the proof of Proposition2.3, which implies Theorem 1.3, and section 4 contains the proof of Proposition 2.4, whichimplies Theorem 1.1. We conclude the article with some discussions on the ball-swappingsymplectomorphisms, which is the main technique involved in this article. Acknowledgements.
The author is deeply indebted to Richard Hind, who inspired the ideain this paper during his stay in Michigan State University. I warmly thank Dusa McDuff forexplaining many details of her recent preprint [22]. Many ideas involved in this work wereoriginally due to Jonny Evans in his excellent series of papers [6, 7], and the ball-swappingconstruction used here stems out from the beautiful ideas exhibited in Seidel’s lecture notes[31]. I would also like to thank the Geometry/Topology group of Michigan State Universityfor providing me a friendly and inspiring working environment, as well as supports for myvisitors.
XACT LAGRANGIANS IN A n -SURFACE SINGULARITIES 3 Two models of W In this section we recall two models of W and its compactifications due to J. Evans andI. Smith. Along the way we introduce various notations that we will use throughout thepaper.We first go through the compactification of the A n -singularities as a complex affine varietyfollowing [7].Let M be the blow-up of CP at { p i = [ ξ i , , } ni =1 , which is identified as { ([ x, y, z ] , [ a i , b i ]) : [ x, y, z ] ∈ CP , [ a i , b i ] ∈ CP , a k y = b k ( x − ξ k z ) , i = 1 , . . . , n } . Here ξ i denotes the i -th n -unit root. We therefore obtain a pencil structure of M , which isthe blow-up of the pencil of lines passing through [0 , ,
0] in CP . We denote: • the pencil as { P t } t =[ x,z ] ∈ CP , • C i is the exceptional curves of the blow-up at [ ξ i , , i = 1 , . . . , n , • C n +1 = { [ x, y, , [ x, y ] , . . . , [ x, y ] } , • C n +2 = { [ x, , z ] , [0 , , . . . , [0 , } .Here C n +1 plays the role of a generic fiber of the pencil which is a line, and C n +2 is the propertransform of the line passing through { p i } ni =1 , which is also a section of the pencil. Endow aK¨ahler form ω to M so that (cid:82) C n +1 ω = 1, (cid:82) C i ω = r := 1 /N for all 1 ≤ i ≤ n . Here we require N (cid:29) n to be a large integer. This is always possible by the construction of a symplecticblow-up [25].Let U = M \ ( C n +1 ∪ C n +2 ). Then U has a Lefschetz fibration structure induced from M .Lemma 7.1 of [7] showed that U is biholomorphic to the A n -singularity W . As a resultof [5, Lemma 2.1.6], U has a symplectic completion symplectomorphic to W . Therefore,one may obtain a symplectic embedding ι : U (cid:44) → W . Note that the Lefschetz fibration of U defined above coincides with that of W . In particular they have the same number ofLefschetz singularities and monodromies. Thus one may assume ι preserves the Lefschetzfibration structure. In particular, their Lefschetz thimbles, thus matching cycles coincide.If we identify C n +2 \ C n +1 with the base of the Lefschetz fibration of U , then the standardspheres are the matching cycles lying above the straight arcs connecting p i to p i +1 in thebase. We will use this interpretation of standard spheres throughout, but a more explicitsymplectic description is in order.The above compactification model is explicit in complex coordinates, but for our purpose ofconstructing symplectomorphisms we need to recall the following alternative model slightlygeneralized from [15, Example 4.25].For z = ( z , z ) ∈ C and real number R , let B ( z ; R ) = { ( z , z ) ∈ C : | z − z | + | z − z | < R } . Consider B = B (0 , √ π ) , B i = B ( i − n − n +1) √ π , √ Nπ ) for 1 ≤ i ≤ n . The circles {| z | = √ Nπ } overthe arcs γ i = { im ( z ) = 0 , Re ( z ) ∈ [ 2 i − n − n ) √ π , i − n + 1(1 + n ) √ π ] } ⊂ { z = 0 } , ≤ i ≤ n − , WU Figure 1.
Evans’s Construction
Figure 2.
Smith’s Constructionforms ( n −
1) Lagrangian tubes. Upon blowing up all B i , the boundary circles of the tubes arecollapsed to points on the exceptional spheres, and the Lagrangian tubes become matchingcycles between two singular fibers of the Lefschetz fibration described previously, thus givingthe standard spheres up to Hamiltonian isotopy.The relation between Evans’s and Smith’s construction is clear from the symplectic inter-pretation of blow-ups. In particular, since the complement of a line in CP is identifies as a4-ball, U is exactly the complement of the proper transform of { z = 0 } in the blow up of B (0 , √ π ) along B i . Notice the following facts: Lemma 2.1 ([7], Proposition 2.1) . Symp c ( U ) is weakly homotopic to Symp c ( W ) . Lemma 2.2.
Suppose Lagrangian spheres are unique up to compactly supported symplecto-morphisms in U , then the same holds in W . To see 2.2, notice that since U has the symplectization identified with W , any Lagrangian L ⊂ W is isotopic to one in U through the negative Liouville flow. The two lemmata reduceour main theorems to the compactified case. Concretely, they show that the following twopropositions imply the main theorems: XACT LAGRANGIANS IN A n -SURFACE SINGULARITIES 5 Proposition 2.3. π ( Symp c ( U )) is generated by the Dehn twists along L , . . . , L n , where L i are matching cycles of the Lefschetz fibration of U , for ≤ i ≤ n . Proposition 2.4.
Any pair of Lagrangian spheres
L, L (cid:48) ⊂ U are symplectomorphic, i.e.,there is a φ ∈ Symp c ( U ) such that φ ( L ) = L (cid:48) . The mapping class group of W In this section we introduce a technique of producing symplectomorphism alluded in [20, 3],which we call the ball-swapping , and try to address its relation with the Dehn twists.We start with a more general context. Suppose X is a symplectic manifold. Given twosymplectic ball embeddings: ι , : n (cid:97) i =1 B ( r i ) → X, where ι is isotopic to ι through a Hamiltonian path { ι t } . From the interpretation of blow-ups in the symplectic category [24], the blow-ups can be represented as X ι j = ( X \ ι j ( n (cid:97) i =1 B i )) / ∼ , for j = 0 , . Here the equivalence relation ∼ collapses the natural S -action on ∂B i = S . Now assumethat K = ι ( (cid:96) B i ) = ι ( (cid:96) B i ) as sets, then ι t defines a symplectic automorphism (cid:101) τ ι of X \ K ,which descends to an automorphism τ ι of X ι := X ι = X ι . We call τ ι a ball-swappingsymplectomorphism or ball-swapping on X ι . Notice it is not known to be true (nor false)that any two ball packings are Hamiltonian isotopic. D. McDuff gave a comfirmative answerto this question for the symplectic 4-manifolds with b + = 1 [21] which allows more freedom tocreate ball-swappings in the blow-ups of these manifolds, but the general case of the questionis still widely open. This construction is closely related to one in algebraic geometry, seediscussions in Section 5.As usual, we denote τ L the Lagrangian Dehn twist along L for a Lagrangian sphere L [31].To compare the ball-swapping, the Dehn twists in dimension 4 and the full mapping classgroup of W , we first need a local refinement of the following result due to Evans: Theorem 3.1 ([7], Theorem 1.4) . The compactly supported symplectomorphism group
Symp c ( W ) has weakly contractible connected components. Moreover, π ( Symp c ( W )) has an injective ho-momorphism into the pure braid group Br n with n -strands. We go over the main ingredients of Evans’ proof briefly, which indeed proves the conclusionfor
Symp c ( U ). Define a standard configuration { S i } ni =1 in M = CP n CP as:(i) each S i is disjoint from C n +1 ,(ii) [ S i ] = [ C i ],(iii) there exist J ∈ J ω , the set of almost complex structures compatible with ω , for whichall S i , C n +1 , C n +2 are J -holomorphic. WU (iv) There is a neighborhood ν of C n +2 such that S i ∩ ν = P t i ∩ ν , for t i = S i ∩ C n +2 .Proposition 7.4 of [7] showed that such configurations form a weakly contractible space C .The proof needs to ensure no bubble occurs from S i to apply Pinsonnault’s result [26, Lemma1.2]. This partly motivates our choice of areas of ω ( C i ).Let Conf ( n ) be the configuration space of n points on a disk. Proposition 7.2 of [7] showsone has the following homotopy fibration:Ψ : C → Conf ( n ) ,S = n (cid:91) i =1 S i (cid:55)→ { S ∩ C n +2 , . . . , S n ∩ C n +2 } . Here the disk is identified with C n +2 \ C n +1 . The fiber of this fibration is denoted as F .Explicitly, fix an unordered n -tuple of points [ x , . . . , x n ], x i ∈ ( D ), F is the space ofstandard configurations { S i } such that [ S i ∩ C n +2 ] ni =1 = [ x i ] ni =1 as unordered n -tuples. Theassociated long exact sequence thus gives the following isomorphism:(3.1) Br n = π ( Conf ( n )) ∼ −→ π ( F )Moreover, [7, Theorem 7.6] shows that:(3.2) π i ( F ) = 0 , for all i > , that is, the connected components of F are weakly contractible. The construction of theisomorphism (3.1) amounts to the following lemma: Lemma 3.2 ([7], Proposition 7.2) . Let α be a loop in Conf ( n ) . One may construct acompactly supported Hamiltonian on C n +2 \ C n +1 inducing α , then there is an extension of α to a Hamiltonian f α of M supported in a neighborhood of C n +2 , such that: • f α preserves C n +1 ∪ C n +2 and fixes C n +1 pointwisely, • f α preserves the set C . Take [ S , . . . , S n ] ∈ F and α a loop in Conf ( n ). Then (3.1) is given by [ f α ( S ) , . . . , f α ( S n )].Note that there is by no means a canonical group structure on π ( F ), so one should under-stand (3.1) as the Br n -action on π ( F ) is free and transitive. This explicit construction ofthe isomorphism (3.1) will be used later.We now look closer to the special case when n = 2 using Smith’s model. Let B = B (0; 1) , B + = B (( , ) , B − = B (( − , ), and (cid:101) C = { z = 0 } ∩ B . From the packing-blowup corre-spondence, B \ ( B + ∪ B − ∪ C ) is exactly U constructed previously for n = 2, N = . Wedenote this open symplectic manifold by U in case of confusions. Therefore, by Lemma 2.1and [29](3.3) Symp c ( U ) = Symp c ( T ∗ S ) = Z , and it is generated by the Lagrangian Dehn twist along the matching cycle described inSection 2. Now we consider a Hamiltonian isotopy ρ swapping B − and B + . Explicitly, XACT LAGRANGIANS IN A n -SURFACE SINGULARITIES 7 we choose a small number (cid:15) >
0, and a smooth function f : R → R such that f ( x ) = 0when x ≥ − (cid:15) and f ( x ) = 1 when x ≤ − (cid:15) . Then ρ is defined by the Hamiltonianfunction πf ( | z | + | z | ) · | z | . ρ thus defines a ball-swapping symplectomorphism τ ρ ∈ Symp c ( B CP ) with corresponding symplectic form. Denote C as the proper transformof (cid:101) C under this blow-up (which is C in the notation of general cases), and C + ( C − resp.)for the exceptional sphere by blowing up ρ ( B + ) ( ρ ( B − ) resp.). Then τ ρ preserves C andexchanges C + , C − as sets.Consider l := τ ρ | C : ( C, [ x + , x − ]) → ( C, [ x + , x − ]), where x ± = C ± ∩ C and [ x + , x − ] denotesthe unorder pair of points they form on (cid:101) C . l can be symplectically isotoped to id on C bya compactly supported Hamiltonian path l − t . We may then lift this isotopy to l − t × id ∈ Symp c ( D (1) × D ( (cid:15) )) for some small (cid:15) > l − . From our way of constructing the symplectomorphisms, it is clear that l − ◦ τ ρ = id ona neighborhood of C . Therefore, l − ◦ τ ρ ∈ Symp c ( U ). blow up blow upball-swap Untwist on by Figure 3.
Swapping two balls
Lemma 3.3. l − ◦ τ ρ is isotopic to a Lagrangian Dehn twist which generates π ( Symp c ( U )) . WU Proof.
Consider the action of l − ◦ τ ρ on ([ C + , C − ]) ∈ F . Since τ ρ exchanges C + and C − as sets, the action of τ ρ on the unordered pair [ C + , C − ] is actually trivial. Also, notice that l − ([ x + , x − ]) is the generator of π ( Conf (2)), from (3.1) we see that { ( l − ◦ τ ρ ) k ([ C + , C − ]) : k ∈ Z } contains exactly one point in each component of F . On the other hand, since τ L is the generator of π ( Symp c ( U )), l − ◦ τ ρ is compactly isotopic to τ mL for some integer m .Therefore, τ L also acts transitively on π ( F ). Now [7, Lemma 7.6] showed that the action Z = Symp c ( U ) is free on its orbit in π ( F ). Therefore, the action of τ L on π ( F ) is alsofree and transitive, so actions of the generators τ L and l − ◦ τ ρ have to match, that is,[ τ L ( C + ) , τ L ( C − )] = [ l − ◦ τ ρ ( C + ) , l − ◦ τ ρ ( C − )] ∈ π ( F ) up to a change of the orientation of L . This shows that τ L is Hamiltonian isotopic to l − ◦ τ ρ in Symp c ( U ), since the stabilizerof the action of Symp c ( U ) on F is weakly contractible by (3.2). (cid:3) Remark 3.4.
From the proof it is clear that the particular sizes of balls involved in Lemma3.3 is not relevant, as long as the K¨ahler packing is possible. By abuse of notation, we willdenote U by open symplectic manifolds obtained this way. Remark 3.5.
One may also construct a proof of Lemma 3.3 from a more classical algebro-geometric point of view. See discussions in Section 5.
We are now ready to return to the general case of U . From (3.1) we have a Br n -action on π ( F ), which is free and transitive. By comparing with the free action of π ( Symp c ( U )) onthe same space, Evans obtains the monomorphism in Theorem 3.1:(3.4) e : π ( Symp c ( U )) (cid:44) → Br n . This is precisely the identification taken in Lemma 3.3 in the case of U . Therefore, one mayinterpret Lemma 3.3 as showing that e is an isomorphism for n = 2. The following resultshows that e is an isomorphism for any n , and that π ( Symp c ( U )) is generated by Dehntwists along matching cycles, hence implying Proposition 2.3. Lemma 3.6.
Let T ⊂ π ( Symp c ( U )) be the subgroup generated by Lagrangian Dehn twistsof matching cycles. Then there exist an isomorphism κ : T → Br n , such that the followingdiagram commutes: (3.5) T ∼ κ (cid:41) (cid:41) (cid:31) (cid:127) c (cid:47) (cid:47) π ( Symp c ( U )) e (cid:15) (cid:15) Br n To free up the notations we define:
Definition 3.7.
Let f : B ( r ) ⊂ C → ( M, ω ) be a symplectic embedding, and Σ ⊂ M is asymplectic divisor. Then f and Σ is said to intersect normally if f − (Σ) = B ( r ) (cid:84) { z = 0 } . Proof of Lemma 3.6.
We adopt Smith’s model and notation from Section 2. Consider theblow-down of M \ C n +1 along C i , i ≤ n . The blow-down is identified with a symplectic ball B coming with the embedded balls B i resulted from C i for i ≤ n . Denote (cid:101) C n +2 ⊂ B as the XACT LAGRANGIANS IN A n -SURFACE SINGULARITIES 9 proper transformation of C n +2 under the blow-down. As long as r = ω ( C i ) is sufficientlysmall, it is clear that one has an embedded symplectic ball B i ( i +1) (cid:44) → B for 1 ≤ i ≤ n − • B i ( i +1) intersects (cid:101) C n +2 normally, • B i ∪ B i +1 ⊂ B i ( i +1) , • B i ( i +1) ∩ B k = ∅ , for any k (cid:54) = i, i + 1.For example, one may take B i ( i +1) = B ( i − n ( n +1) √ π , n ) √ π + √ Nπ ). From the local constructionLemma 3.3, one sees that the action of each generator of the braid group σ i on π ( F ) isexplicitly realized by a symplectomorphism coming from ball-swapping which is compactlysupported in B i ( i +1) . Moreover, these symplectomorphisms are isotopic to Lagrangian Dehntwists of matching cycles supported in B i ( i +1) , and such an identification is given preciselyby e ◦ c in (3.5) by the discussion preceding Lemma 3.6. Since both e and c are injective, allarrows in (3.5) are indeed isomorphisms. (cid:3) A symplectomorphism by ball-swappings
Constructing φ for Proposition 2.4. In this section, we will prove Proposition 2.4.The basic idea of constructing φ is again to use the ball-swapping in the compactifica-tion.Notice first that one may assume L and L (cid:48) are homologous in M = CP n CP . This followsfrom the classification of homology classes of Lagrangian spheres in rational manifolds [20,Theorem 1.4]. Since L, L (cid:48) are disjoint from C n +1 which is a line in M , their classes have tohave the form of [ C i ] − [ C j ] for i, j ≤ n . By certain Dehn twists along the standard Lagrangianspheres, one may assume [ L ] = [ L (cid:48) ] = [ C ] − [ C ] in M .Recall also the following result from [20]. Theorem 4.1 ([20], Theorem 1.1, 1.2) . Let b + ( M ) = 1 and GT ( A ) (cid:54) = 0 , L be a Lagrangiansphere. Then A has an embedded representative with minimal intersection with L . If A isrepresented by an embedded sphere, then any given representatives of A can be symplecticallyisotoped so that they achieve minimal intersections. Let
L, L (cid:48) ⊂ U = M \ ( C n +1 ∪ C n +2 ), Theorem 4.1 implies that one may isotope { C , . . . , C n } toanother standard configuration { S , . . . , S n } , which are disjoint from L . Extend this isotopyof spheres to a Hamiltonian isotopy Ψ t , then Ψ − t ( L ) isotopes L away from { C , . . . , C n } . Byperforming the same type of isotopy to L (cid:48) , we may assume { C , . . . , C n } are disjoint fromboth L and L (cid:48) .We now blow down along the set E = { C , . . . , C n } to obtain a set of balls B = { B , . . . , B n } ,and the resulting manifold is denoted as M = CP CP , where we have the proper trans-formations of L and L (cid:48) denoted as (cid:101) L and (cid:101) L (cid:48) . Note that by removing the proper transform of C n +1 and C n +2 we have M \ ( (cid:101) C n +1 ∪ (cid:101) C n +2 ) = U . One also have from Remark 3.4: Lemma 4.2 ([3], Lemma A.1 and the discussions following it) . Lagrangian spheres in U = M \ ( (cid:101) C n +2 ∪ (cid:101) C n +1 ) are unique up to Hamiltonian isotopy. Combining discussions above, there is a compactly supported Hamiltonian isotopy (cid:101) Φ t : U → U , such that (cid:101) Φ t ( (cid:101) L ) = (cid:101) L (cid:48) where (cid:101) Φ t = id near (cid:101) C n +1 ∪ (cid:101) C n +2 . Figure 4.
A pictorial proof of Proposition 2.4To obtain a compactly supported symplectomorphism on M \ ( C n +1 ∪ C n +2 ), it amounts toshowing the following lemma, whose proof will be given in the next section. Lemma 4.3.
There is (cid:101) φ ∈ Ham ( M ) , such that:(i) (cid:101) φ ( (cid:101) Φ ( B i )) = B i for i ≥ , (cid:101) φ ( (cid:101) C n +2 ) = (cid:101) C n +2 ,(ii) for some neighborhood N of (cid:101) C n +1 ∪ L (cid:48) , (cid:101) φ | N = id . Lemma 4.3 concludes the proof of Proposition 2.4 because (cid:101) φ ◦ (cid:101) Φ sends L to L (cid:48) and fixes B i for i ≥
3, and it clearly is compactly supported in M \ (cid:101) C n +1 . Therefore, it can be liftedto a compactly supported symplectomorphism φ of M \ C n +1 by blowing up the balls B i for i ≥
3, which preserves C n +2 . One then apply Lemma 3.2 again to obtain a symplectomor-phism φ which fixes C n +2 pointwisely. This is always possible because Ham c ( C n +2 \ C n +1 ) = Ham c ( D ) ∼ pt by the Smale’s theorem. Since the gauge group of the normal bundle of C n +2 \ ( C n +1 ∪ C n +2 ) is homotopic equivalent to M ap (( S, ∗ ) , SL ( R )) ∼ pt , by composinganother symplectomorphism fixing C n +2 pointwisely one may assume φ fixes also the normalbundle of C n +2 . All these adjustments can be made supported in a small neighborhood of C n +2 thus not affecting L (cid:48) . Therefore, φ indeed descends to a compactly supported symplec-tomorphism of U = M \ ( C n +1 ∪ C n +2 ), which concludes our proof. Remark 4.4.
For our purpose it suffices to prove (cid:101) φ ∈ Symp ( M ) in Lemma 4.3, which willbe slightly easier. From the proof it is clear that the lemma indeed holds for more generalcases of packing relative to a symplectic divisors, coupling with results in [22] and [3] , but werestrict ourselves for ease of expositions. Remark 4.5.
One recognizes that the braiding in the symplectomorphism group required forsending L to L (cid:48) comes exactly from the restriction of (cid:101) φ on C n +1 \ C n +2 , which swaps theshadows B i ∩ (cid:101) C n +2 by Dif f ( D ) . This nicely matches the pictures of Section 3. XACT LAGRANGIANS IN A n -SURFACE SINGULARITIES 11 Connectedness of ball packing relative to a divisor.
We prove Lemma 4.3 in thissection. The overall idea is not new. In the absence of (cid:101) C n +2 , this is just the ball-packingconnectedness problem in the complement of a Lagrangian spheres in rational manifolds.This was settled in [3] using McDuff’s non-generic inflation lemma [22, Lemma 4.3.3] and herideas in the original proof of connectedness of ball-packings [21, Section 3]. We will adaptthese ideas in our relative case and point out necessary modifications. We will continue touse notations defined in previous sections.Let ι : B i ( r ) (cid:44) → M be the inclusion and ι = (cid:101) Φ : B i (cid:44) → M . Fix some small δ > ι j : B i ( r ) → M to B i ( r + δ ) for j = 0 ,
1, so that theextensions still intersect (cid:101) C n +2 normally. Take two families of diffeomorphism φ sj , j = 0 ,
1, sothat the following holds:(1) φ j = id ,(2) φ sj | B ( sr i ) is a radial contraction from B ( sr i ) to B ( δ ), and identity near ∂B ( sr i + δ ),(3) φ sj ( (cid:101) C n +2 ) = (cid:101) C n +2 .This is not hard to achieve if we have B i intersecting (cid:101) C n +2 normally in the first place.The push-forward of ω by φ sj endows a family of symplectic forms { ω sj } ≤ s ≤ on M for j = 0 ,
1. Notice in our situation, (cid:101) Φ in Lemma 4.3 is compactly supported in U , whichimplies ι ( B i ( δ )) = ι ( B i ( δ )) for sufficiently small δ >
0. Therefore, performing a blow-upon ι j ( B i ( δ )) gives a family of symplectic forms { [ (cid:101) ω sj ] } on the same smooth manifold M without further identification. The resulting exceptional curves formed by blowing up B i as C i . Notice (cid:82) C i (cid:101) ω sj = π ( sr i ) . We claim: Lemma 4.6.
For any given s ∈ [0 , , (cid:101) ω s are isotopic to (cid:101) ω s via a family of diffeomorphism ψ s ( t ) : M → M such that ψ s ( t )( C i ) = C i and ψ s ( t )( L (cid:48) ) = L (cid:48) for any given s ∈ [0 , and ≤ i ≤ n + 2 .Proof. Notice that (cid:101) ω s is deformation equivalent to (cid:101) ω s by a family of symplectic forms { Ω t } t ∈ [0 , preserving the orthogonality of C i with C n +2 and constant near L (cid:48) . A by-now-standard ap-proach of correcting this deformation family into an isotopy of symplectic form, that is,a symplectic deformation by cohomologous symplectic forms, is done by inflation along J -holomorphic curves.To put L (cid:48) into this framework, we may perform a symplectic cut near it, which gives asymplectic ( −
2) sphere S (cf. [3, Remark 2.2]). The class of interests for inflation has theform [˜ ω s ] − (cid:15) (cid:80) ni =3 [ C i ] with suitably chosen (cid:15) . When raised to a sufficiently large multiple,this class has a nodal representative. So far the argument is standard. To inflate along thisnodal curve, one considers S as the union of all its underlying irreducible components, andinclude S and C i for 3 ≤ i ≤ n + 2 if necessary. Now we can follow the argument in [22,Proposition 1.2.9] and [23, Lemma 1.1] for this configuration S . In principle, we need afamily version of inflation here, see discussion of [23] for details. The outcome is an isotopyof symplectic forms from (cid:101) ω s to (cid:101) ω while keeping S and C i symplectic for 3 ≤ i ≤ n + 2.Lemma 4.6 is then an immediate consequence of the relative Poincare Lemma [4, Theorem2.3] and the relative Moser technique [24]. In particular, one first applies a family of dif-feomorphism g t supported in a neighborhood of ∪ n +2 i =3 C i preserving each of these divisors, so that g ∗ t ( (cid:101) Ω t ) | C i = (cid:101) Ω | C i . Now the relative Poincare Lemma implies that g ∗ t ( (cid:101) Ω t ) − (cid:101) Ω has aprimitive vanishing on ∪ n +2 i =3 C i and a neighborhood of L (cid:48) , where the Moser’s technique impliesthe desired property in Lemma 4.6. (cid:3) To finish the proof of Lemma 4.3, we already constructed a symplectomorphism ( φ s ) − ψ s φ s interpolating (cid:101) ω s and (cid:101) ω s for each s from Lemma 4.6. By blowing down C i , one obtains asymplectomorphism F s sending ι ( B i ( sr i )) to ι ( B i ( sr i )). From the smooth dependence ofthe family ψ s on s , one learns that ψ is an isotopy of M . By precomposing another isotopy h preserving C i for all 3 ≤ i ≤ n + 1 as in the proof of [21, Corollary 1.5], one may furtherassume ψ is identity near the C i for i ≤ n + 1. One thus obtains a family of ball-packingby blowing down along C , . . . , C n , which connects ι and ι through a normal intersectionfamily with (cid:101) C n +2 as desired.5. Concluding remarks on ball-swappings
The ball-swapping technique we used in this paper seems to have rich structures and couldbe of independent interests. This concluding remark summarizes several possibly interestingdirections of further study of this class of objects. • It was pointed out to the author that, it seems instructive to compare ball-swappingswith a closely related construction in algebraic geometry which is classical. Theauthor first learned about the following construction from Seidel’s excellent lecturenotes [31]. Consider C n and its 2-point configuration space Conf ( C n ). One mayassociate to this space a fibration E →
Conf ( C n ), where E b is the correspondingcomplex blow-up of at b ∈ Conf ( C n ). Seidel demonstrated in [31, Example 1.12]that, when n = 2, one may partially compactify this family to E by allowing the twopoints to collide, where the discriminant ∆ is a smooth divisor. A local normal disk D centered at p ∈ ∆ thus gives a sub-fibration over D , where the fiber over p is a surfacewith a single ordinary double point. Then [31, Lemma 1.11] shows the monodromyaround ∂D is precisely a Dehn twist when an appropriate K¨ahler form is endowed onthe fiber. But this monodromy is equally clearly a ball-swapping, which establishesthe relation between ball-swapping and the Dehn twists as monodromies in algebraicgeometry. This algebro-geometric point of view also provides another (more elegant)proof for Lemma 3.3, except now one needs to sort out details for the last step ofuntwisting on the removed divisor. In a Lefschetz fibration point of view, this untwistis equivalent to slowing down the Hamiltonian at infinity in the description of Seidel’sDehn twist. One may consult [28] for further details.However, the construction of ball-swapping is apriori richer than Dehn twists evenin dimension 4. Formally, given symplectic manifold M , consider the action ofHamiltonian group Ham ( M ) on the space of ball-embeddings Emb (cid:15) ( M ) = { φ : (cid:96) ni =1 B ( (cid:15) i ) → M } for (cid:15) = ( (cid:15) , . . . , (cid:15) n ). Take an orbit O of the action, the sta-blizer of the action descends to a subgroup of Symp ( M n CP ). Then isotopy classesof ball-swappings are the images of π ( Symp ( M n CP )) under the connecting mapfrom π ( Emb (cid:15) ( M )). From this formal point of view, the above algebro-geometricconstruction amounts to the ball-swappings when restricted to images of XACT LAGRANGIANS IN A n -SURFACE SINGULARITIES 13 π (Conf n ( M )) → π ( Emb (cid:15) ( M )) → π ( Symp ( M n CP )) , while the image of the second arrow forms the full ball-swapping subgroup. In general,the inclusion Conf n ( M ) → Emb (cid:15) ( M ) is not a homotopy equivalence even when n = 1and M is as simple as S × S with non-monotone forms [19] (but is still 1-connected!).In this particular example one already sees the sizes of packed balls come into play. Inhigher dimensions, the topology of space of ball-embedding in even C n is completelyopen. Therefore, it seems interesting to clarify the gap between the algebro-geometricconstruction and the full ball-swapping subgroup. • The ball-swapping symplectomorphisms seem particularly useful in problems involv-ing π ( Symp ( M )), when M is a rational or ruled manifold. From examples known todate and the algebro-geometric constructions above, it seems reasonable to speculatethat ball-swappings in dimension 4 is generally related to Lagrangian Dehn twists, atleast for those with b + = 1. A particular tempting question asks that, a blow-up at n balls on CP with generic sizes has a connected symplectomorphism group (becausethey do not admit any Lagrangian Dehn twists, see [20]). But to reduce the subgroupgenerated by ball-swapping into problems of braids, as in what appeared in this note(and also [31, 7]) seems to require independent efforts in finding an appropriate sym-plectic spheres, as well as a good control on the bubbles.Here we roughly sketch a viewpoint through ball-swapping to π ( Symp ( M )) when M = CP CP where all extra technicalities can be proved irrelavant. In this case,Evans [7] showed that π ( Symp ( M )) = Dif f ( S , ∗ ), where ∗ is a fixed set on S consisting of 5 points, which can be identified with a braid group on S . This coincideswith an observation of Seidel [31, Example 1.13]. Jonny Evans also explained to theauthor that these braids can be seen to be generated by Lagrangian Dehn twists,following from results of [7] .From the ball-swapping point of view, by blowing down the 5 exceptional spheresof classes E , . . . , E , we have 5 embedded balls B , . . . , B intersecting a 2 H -sphere C normally. Then the restriction of swapping these 5 balls on the 2 H -sphere givesprecisely a copy of the group Dif f ( S , ∗ ), where the 5 points now is actually the 5disks coming from intersections B i ∩ C . To get the actual statement of Evans, onestill needs to switch between blow-ups and downs and go through Evans’s proof toshow all contributions from other components (configuration space of curves involved,automorphisms of normal bundles, etc.) cancels.The ball-swapping also gives a way of seeing these braidings actually come fromLagrangian Dehn twists, but we will not give full details. A naive attempt is toinclude two embedded balls above to a larger ball normally intersecting C as inthe proof of Lemma 3.6. However, this cannot work for packing size restrictions.Instead, Evans shows that there is an embedded Lagrangian RP in the complementof a configuration of symplectic spheres consisting of classes { H, E , . . . , E } . Byremoving this RP , one can show that the remainder of M can be identified with asymplectic fibration over C , where a generic fiber is a disk, and there are 5 singularfibers consisting of the union of a E i -sphere and disk. One then choose a disk in C Private communications. separating the intersections of the E and E -spheres with C with other E i ’s, then thefibration restricted to this disk is a product of two D ’s blown-up 2 points, denoted as V . At this point one easily reproduces the proof of Lemma 3.6 with an identificationof the the symplectization of V \ C with T ∗ S . • In higher dimensions ball-swapping seems a new way of constructing monodromiesand quite symplectic in flavor. For example, when the ball-packing is sufficientlysmall, because of Darboux theorem one could essentially move the embedded balls asif they were just points. The question when such symplectomorphisms are actuallyHamiltonian seems intriguing.One particular situation in question is when there is only one embedded ball. Supposea small embedded ball moves along a loop which is nontrivial in π ( M ), is it true thatthe resulting ball-swapping always lies outside the Hamiltonian group? [15] explainedthe existence of a Lagrangian torus near a small blow-up. It is not difficult to finda symplectomorphism in the component of Symp ( M ) where the ball-swapping lies,so that this torus is invariant under it. Thus, in the sense of [32], one constructs afamily of objects in the Fukaya category of the mapping torus of M CP naturallyassociated to a ball-swapping. • In principle, one may extend the “swapping philosophy” to a more general backgroundof surgeries to obtain automorphisms, even in non-symplectic situations. As a simpleexample, consider a Hamiltonian loop of Lagrangian 2-torus in M , by which we meana path of Hamiltonians φ t ⊂ Ham ( M ) such that φ , ( L ) = L . One may perform aLuttinger surgery on L [14], which gives a new symplectic manifold M L . Then the“swapping” φ is lifted to an automorphism of M L .One may also choose any symplectic neighborhood of a symplectic submanifold, aslong as one has interesting loops of these objects. Naively thinking, when the sub-manifold is actually a divisor, the swapping symplectomorphism should always beHamiltonian, but the author has no proof for that. 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