Elliptic diffeomorphisms of symplectic 4-manifolds
aa r X i v : . [ m a t h . S G ] J u l ELLIPTIC DIFFEOMORPHISMSOF SYMPLECTIC -MANIFOLDS VSEVOLOD SHEVCHISHIN AND GLEB SMIRNOV
Abstract.
We show that symplectically embedded ( − -tori give rise to certain ele-ments in the symplectic mapping class group of -manifolds. An example is given wheresuch elements are proved to be of infinite order. Contents
0. Introduction 11. Construction of the elliptic twist. 31.1. Elliptic twist 31.2. The Abreu-McDuff framework. 51.3. Symplectic economics. 62. Elliptic geometrically ruled surfaces 62.1. General remarks. 62.2. Classification of complex surfaces ruled over elliptic curves. 82.3. One family of ruled surfaces over elliptic base. 112.4. Embedded curves and almost-complex structures. 112.5. Rulings and almost-complex structures. 122.6. Diffeomorphisms. 142.7. Vanishing of elliptic twists. 163. Blow up once 183.1. Rational ( − -curves. 183.2. Straight structures. 203.3. Refined Gromov invariants. 223.4. Loops M B . 233.5. Loops in J st . 243.6. Let’s twist again. 25References 250. Introduction
Let ( X, ω ) be a closed symplectic 4-manifold. Denote by π S ymp ( X, ω ) the group ofsymplectic diffeomorphisms of X modulo symplectic isotopy. Let us consider the forgetfulhomomorphism π S ymp ( X, ω ) → π D iff ( X ) . Mathematics Subject Classification.
Primary:
Here π D iff ( X ) denotes the smooth mapping class group for X . It is known this homo-morphism is not necessary injective. If Σ is a smooth Lagrangian sphere in X , then thereexists a symplectomorphism T Σ : X → X , called symplectic Dehn twist along Σ , such that T is smoothly isotopic to the identity. In his thesis [SeiTh], Seidel proved that in manycases T is not symplectically isotopic to the identity. He than proved that for certain K surfaces containing two Lagrangian spheres Σ and Σ , the element T has infiniteorder, and hence the forgetful homomorphism has infinite kernel. The reader is invitedto look at [Sei2] for a detailed description of symplectic Dehn twists.Somewhat later, Biran and Giroux introduced different symplectomorphisms, namelythe fibered Dehn twists, among which one can find smoothly yet not symplectically trivialmaps. In fact, Seidel’s Dehn twist is a particular case of a fibered Dehn twist. Supposethat that X admits a separating contact type hypersurface P carrying a free S -actionin P × [0 , that preserves the contact form on P . Then one can define the fibered Dehntwist as T P : P × [0 , → P × [0 , , ( x, t ) → ( x · [ f ( t ) mod π ] , t ) , where a function f : [0 , → R equals π near t = 0 and near t = 1 . As T P is asymplectomorphism of P × [0 , that is the identity near the boundary of P × [0 , , itcan be extended to be a symplectomorphism of the whole X . We refer the reader to[R-D-O, U] for an extensive study of fibered Dehn twists.Given that it is easy to find a separating contact hypersurface, fibered Dehn twistsmake an effective tool to construct symplectomorphisms of a given -manifold (and of ahigher-dimensional manifold, for that matter.) But even though a plethora of results hasbeen obtained in symplectic mapping class groups (see e.g. [Ab-McD, Bu, Anj, Anj-Gr,Anj-Lec, Ev, La-Pin, LiJ-LiT-Wu, Ton, Sei1, Sei3, Wen]), it remains hard to detect non-triviality of symplectomorphisms.In this paper we introduce and study a new type of symplectomorphisms for 4-manifolds.In short, our construction is as follows. Let ( X, ω ) be a symplectic -manifold whichcontains a symplectically embedded torus C ⊂ X of self-intersection ( − . In particular, µ := R C ω > . We construct a family of symplectic forms ω t on X in the cohomologyclass [ ω t ] such that R C ω t = µ − t . We show that such a family exists for t large enough for C to have a negative symplectic area.For each t we construct an ω t -symplectomorphism T C : X → X , called the elliptic twistalong C . As smooth maps, those symplectomorphisms T C for different t are isotopic, sowe can think of T C as a single diffeomorphism defined up to isotopy.We then study whether or not these elliptic twist are symplectically isotopic to theidentity. It appears that it is so in the case when R C ω t > . In particular, T C is alwayssmoothly isotopic to the identity. As we shall see below, it is not so in the case R C ω t ,and T C could be non-trivial.Let Y be a tubular neighbourhood of C in X . Then ∂Y is a separating contact hyper-surface in X , which carries a free S -action. One can pick a symplectic form e ω on X such that ω | X − Y = e ω | X − Y and R C e ω . We conjecture that, for ( X, e ω ) , the fiberedDehn twist associated to ∂Y is symplectically isotopic to T C .Our first result is an example of a -manifold X and a ( − -torus C in it, where theelliptic twist T C turns out to be always symplectically trivial. LLIPTIC DIFFEOMORPHISMS OF SYMPLECTIC -MANIFOLDS 3 Theorem 0.1.
Let ( X, ω ) be a symplectic ruled 4-manifold diffeomorphic to the totalspace of the non-trivial S -bundle over T ; we denote it by S ˜ × T for short. Then i) there is a symplectic form ω on X which admits an ω -symplectic ( − -torus C ⊂ X ,and the elliptic twist T C is well-defined. ii) the forgetful homomorphism π S ymp ( X, ω ) → π D iff ( X ) is injective for any sym-plectic form ω . In particular, the elliptic twist T C is always symplectically isotopic to theidentity. The injectivity property claimed in part ii) was proved previously by McDuff for S × T ,see [McD-B]. We thus cover the remaining non-spin case and, therefore, prove the so-calledsymplectic isotopy conjecture for elliptic ruled surfaces, see Problem 14 in [McD-Sa-1].The main result of this note shows that it is possible for an elliptic twist to contributenontrivially to a symplectic mapping class group.
Theorem 0.2.
Let Z be S ˜ × T CP . There exist a symplectic form ω on Z and three ω -symplectic ( − -tori C , C , and C in Z such that the elliptic twists T C i are well-definedand none of them is symplectically isotopic to the identity; each T C i has infinite order inthe symplectic mapping class group. Our proof follows closely to the ideas introduced by Abreu-McDuff in [Ab-McD] andMcDuff in [McD-B].The main technique we use in the proof is Gromov’s theory of pseudoholomorphiccurves. This theory involves various Banach manifolds and constructions with them.Dealing with them we often pretend to be in the finite-dimensional case. We refer thereader to the book [Iv-Sh-1] and articles [Iv-Sh-2, Iv-Sh-3] for a comprehensive analyticsetup to Gromov’s theory of pseudoholomorphic curves. Of course, reader is free to addressto any of numerous alternative sources and expositions of the theory such as [McD-Sa-3]or the seminal paper [Gro].
Acknowledgements.
We are deeply indebted to Boris Dubrovin, Yakov Eliashberg,and Viatcheslav Kharlamov for a number of useful suggestions which were crucial for thepresent exposition of this paper. Part of this note was significantly improved during ourstay at the University of Pisa and the Humboldt University of Berlin, and we are verygrateful to Paolo Lisca and Klaus Mohnke for numerous discussions and for the wonderfulresearch environment they provided. We also would like to thank Rafael Torres for readingthe manuscript and pointing out certain inconsistencies. Special thanks to Dasha Alexeevafor sending us a preprint of her thesis. Finally, we are grateful to the referee for apositive comment on our paper, for her/his constructive and thorough criticism, and forthe tremendous amount of work he/she did reviewing the manuscript. The second authorwas supported by an ETH Fellowship.1.
Construction of the elliptic twist.
Elliptic twist.
Let ( X, ω ) be a symplectic -manifold, and let C be an embeddedsymplectic ( − -torus in X . We let Ω( X, ω ) to denote the space of symplectic forms on X that are isotopic to ω , and let J ( X, Ω) to denote the space of almost-complex structuresfor which there exists a taming form in Ω( X, ω ) . VSEVOLOD SHEVCHISHIN AND GLEB SMIRNOV
Pick an almost-complex structure J ∈ J ( X, Ω) for which C is pseudoholomorphic. Onethinks of J as a point of the subspace D [ C ] ⊂ J ( X, Ω) of those almost-complex structureswhich admit a smooth pseudoholomorphic curve in the class [ C ] . In what follows, we referto D [ C ] as the elliptic divisorial locus for the class [ C ] . The term divisorial locus is takenfrom the fact that in some neighbourhood of J the subspace D [ C ] locally behaves as asubmanifold of J ( X, Ω) of real codimension , see e.g. [Iv-Sh-1].Let ∆ ⊂ J ( X, Ω) be a small disc transverally intersecting D [ C ] precisely at J , and let J : [0 , → ∆ be the boundary of ∆ . We will make the following assumption: ( A ) There exists a class ξ ∈ H ( X ; R ) , ξ · [ C ] such that every J ( t ) ∈ ∂ ∆ is tamed bysome symplectic form θ t , [ θ t ] = ξ .One can arrange θ t so that they depend smoothly on t . Moser isotopy then gives usa path of diffeomorphisms f t : X → X , f ∗ t θ t = θ . Now f is a symplectomorphism of ( X, θ ) . We call f the elliptic twist along C and use the notation T C for it.As we will explain below (see §1.2 ), for T C to give a non-trivial element in π S ymp ( X, θ ) ,it is necessary that [ J ( t )] ∈ π ( J ( X, Θ)) is non-trivial; here Θ stands for the space ofsymplectic forms on X that are isotopic to θ . We emphasize that J ( t ) is contractible in J ( X, Ω) ; thus, T C is trivial for ( X, ω ) .Assumption ( A ) always holds, though we do not prove it in the full generality. But weshall consider a series of -manifolds for which the assumption is easy to verify. Let ( X, ω ) be a symplectic -manifold, and let C be a symplectic torus of self-intersection number . Take an ω -tamed almost-complex structure on X for which C becomes pseudoholo-morphic, and then perturb this structure slightly to make it integrable in some tubularneighbourhood of C . More precisely, we want a sufficiently small neighbourhood of C toadmit an elliptic fibration with C being a multiple fiber of multiplity m > .Let T be an elliptic curve C / Z τ ⊕ Z τ , where ( τ , τ ) form a basis for C ( u ) as a realvector space, and let ∆ be a complex disc with a local parameter z . The neighbourhoodof C in X is biholomorphic to the quotient ∆ × T / ∼ , where ( z, u ) ∼ ( ze πi/m , u + τ /m ) .Here C is given by the equation { z = 0 } .Blowing-up X at a point (0 , u ) , we get a manifold Z which contains a smooth ellipticcurve in the class [ C ] − E (the strict transform of C .) Here E stands for the homologyclass of the exceptional line. Unless z = 0 , the blow-up of X at ( z , u ) does not containsuch a curve, since it contains one in the class m [ C ] − E (the strict transform of { z = z } ,which we denote by C m .)Pick a taming symplectic form ω on Z . Clearly, the form satisfies Z [ C ] − E ω > . Let Z ( t ) be the blow-up of X at ( R e it , u ) . Observe that the complex structures on Z ( t ) are ω -tamed for R sufficiently small. Using the deflation (see §1.3 ) along C m , wedeform ω on Z ( t ) into a symplectic form θ t for which Z [ C m ] θ t = ε LLIPTIC DIFFEOMORPHISMS OF SYMPLECTIC -MANIFOLDS 5 for ε positive arbitrary small. Implementing deflation does not violate the taming con-dition for Z t . Being performed in a small neighbourhood of C m , the deflation does notaffect the symplectic area of C , see [Bu]. Since Z [ C ] − E θ t = ε − ( m − Z [ C ] ω, one may take m sufficiently large to make the area of [ C ] − E as negative as desired. Wehave now verified ( A ) for the family Z ( t ) .1.2. The Abreu-McDuff framework.
Let D iff ( X ) be the identity component of thediffeomorphism group of X and Ω( X, ω ) be the space of all symplectic forms on X thatare isotopic to ω . We have a natural transitive action of D iff ( X ) on Ω( X, ω ) . So we geta principle fiber bundle S ymp ( X, ω ) ∩ D iff ( X ) → D iff ( X ) → Ω( X, ω ) , (1.1)where the last arrow stands for the map ϕ : D iff ( X ) → Ω( X, ω ) with ϕ : f f ∗ ω. To shorten notation, we put: S ymp ∗ ( X, ω ) := S ymp ( X, ω ) ∩ D iff ( X ) Following [Kh], we consider an exact sequence of homotopy groups . . . → π ( D iff ( X )) ϕ ∗ −−→ π (Ω( X, ω )) ∂ −→ π ( S ymp ∗ ( X, ω )) −→ π ( D iff ( X )) . Let J ( X, Ω) be the space of those almost-complex structures J on X for which thereexists a taming symplectic form ω J ∈ Ω( X, ω ) . It is easy to see that J ( X, Ω) is connected.Let J be some ω -tamed almost-complex structure. It was shown by McDuff, see Lemma2.1 in [McD-B], that there exists a homotopy equivalence ψ : Ω( X, ω ) → J ( X, Ω) forwhich the diagram D iff ( X ) Ω( X, ω ) J ( X, Ω) ϕν ψ (1.2)commutes. Here ν : D iff ( X ) → J ( X, Ω) is given by ν : f f ∗ J .Following the fundamental idea of Gromov’s theory [Gro] we study the space J ( X, Ω) rather than Ω( X, ω ) . We see from the following diagram . . . −−−→ π ( D iff ( X )) ϕ ∗ −−−→ π (Ω( X, ω )) ∂ −−−→ π ( S ymp ∗ ( X, ω )) −−−→ id y ψ ∗ y . . . −−−→ π ( D iff ( X )) ν ∗ −−−→ π ( J ( X, Ω)) , (1.3)that each loop in J ( X, Ω) contributes non-trivially to the symplectic mapping class groupof X , provided this loop does not come from D iff ( X ) . We will use diagram (1.3) to proveboth Theorem 0.1 and
Theorem 0.2 . The reader is referred to [McD-B] for more extensivediscussion of the topic.In what follows we work with a slightly bigger space J k ( X, ω ) of C k -smooth almost-complex structures. Here and below “ C k -smoothness” means some C k,α -smoothness with VSEVOLOD SHEVCHISHIN AND GLEB SMIRNOV < α < and k natural sufficiently large. The reason to do this is that the space J k ( X, ω ) is a Banach manifold, while the space of C ∞ -smooth structures J ( X, Ω) is merely Fr´echet.What we prove for π i ( J k ( X, ω )) works perfectly for π i ( J ( X, Ω)) because the inclusion J ( X, Ω) ֒ → J k ( X, ω ) induces the weak homotopy equivalence π i ( J ( X, Ω)) → π i ( J k ( X, ω )) .1.3. Symplectic economics.
Here we give a brief description of the inflation techniquedeveloped by Lalonde-McDuff [La-McD, McD-B], and a generalization of this proceduregiven by Bu¸se, see [Bu].
Theorem 1.1 (Inflation) . Let J be an ω -tamed almost complex structure on a symplectic -manifold ( X, ω ) that admits an embedded J -holomorphic curve C with [ C ] · [ C ] > .Then there is a family ω s , s > , of symplectic forms that all tame J and have cohomologyclass [ ω s ] = [ ω ] + s PD ([ C ]) , where PD ([ C ]) is Poincar´e dual to [ C ] . For negative curves a somewhat reverse procedure exists, called negative inflation ordeflation.
Theorem 1.2 (Deflation) . Let J be an ω -tamed almost complex structure on a symplectic -manifold ( X, ω ) that admits an embedded J -holomorphic curve C with [ C ] · [ C ] = − m .Then there is a family ω s of symplectic forms that all tame J and have cohomology class [ ω s ] = [ ω ] + s PD ([ C ]) for all s < ω ([ C ]) m . Elliptic geometrically ruled surfaces
General remarks.
A complex surface X is called ruled if there exists a holomorphicmap π : X → Y to a Riemann surface Y such that each fiber π − ( y ) is a rational curve; if,in addition, each fiber is irreducible, then X is called geometrically ruled. A ruled surfaceis obtained by blowing up a geometrically ruled surface. Note however that a geometricallyruled surface need not be minimal (the blow up of CP , denoted by CP CP , is theunique example of a geometrically ruled surface that is not a minimal one). Unlessotherwise noted, all ruled surfaces are assumed to be geometrically ruled. One can speakof the genus of the ruled surface X , meaning thereby the genus of Y . We thus haverational ruled surfaces, elliptic ruled surfaces and so on.Up to diffeomorphism, there are two total spaces of orientable S -bundles over a Rie-mann surface: the product S × Y and the non-trivial bundle S ˜ × Y . The product bundleadmits sections Y k of even self-intersection number [ Y k ] = 2 k , and the non-trivial bun-dle admits sections Y k +1 of odd self-intersection number [ Y k +1 ] = 2 k +1 . We will choosethe basis Y = [ Y ] , S = [ pt × S ] for H ( S × Y ; Z ) , and use the basis Y − = [ Y − ] , Y + = [ Y ] for H ( S ˜ × Y ; Z ) . To simplify notations, we denote both the classes S and Y + − Y − , whichare the fiber classes of the ruling, by F . Further, the class Y + + Y − , which is a class for abisection of X , will be of particular interest for us, and will be widely used in forthcomingcomputations; we denote this class by B . Throughout this paper we will freely identifyhomology and cohomology by Poincar´e duality. LLIPTIC DIFFEOMORPHISMS OF SYMPLECTIC -MANIFOLDS 7 Clearly, we have [ Y k ] = Y + k F and [ Y k +1 ] = Y + + k ( Y + + Y − ) . This can be seenby evaluating the intersection forms for these 4-manifolds on the given basis: Q S × Y = (cid:18) (cid:19) , Q S ˜ × Y = (cid:18) − (cid:19) . Observe that these forms are non-isomorphic. That is why the manifolds S × Y and S ˜ × Y are non-diffeomorphic. One more way to express the difference between them is tonote that the product S × Y is a spin 4-manifold, but S ˜ × Y is not spin. Note that afterblowing up one point, they become diffeomorphic: S × Y CP ≃ S ˜ × Y CP .This section is mainly about the non-spin elliptic ruled surface S ˜ × T . When studyingthis manifold we sometimes use the notations T + and T − instead of Y + and Y − for thestandard homology basis in H ( S ˜ × T ; Z ) .From the viewpoint of complex geometry every such X is a holomorphic CP -bundleover a Riemann surface Y whose structure group is P Gl (2 , C ) . Biholomorphic classifi-cation of ruled surfaces is well understood, at least for low values of the genus. Belowwe recall a part of the classification of elliptic ruled surfaces given by Atiyah in [At-2];this being the first step towards understanding the almost-complex geometry of thesesurfaces. We also provide a short summary of Suwa’s results: i) an explicit constructionof a complex analytic family of ruled surfaces, where one can see the jump phenomenonof complex structures, see §2.3 , ii) an examination of those complex surfaces which areboth ruled and admit an elliptic pencil, see Theorem 2.3 .In what follows we will use a formula for the first Chern class of a geometrically ruledsurface. In terms of Y , S , Y ± , it becomes c ( S × Y ) = 2 Y + χ ( Y ) S , c ( S ˜ × Y ) = (1 + χ ( Y )) Y + + (1 − χ ( Y )) Y − . (2.1)The symplectic geometry of ruled surfaces has been extensively studied by many authors[Li-Li, Li-Liu-1, Li-Liu-2, AGK, Sh-4, H-Iv]. Ruled surfaces are of great interest fromthe symplectic point of view mainly because of the following significant result due toLalonde-McDuff, see [La-McD, McD-6]. Theorem 2.1 (The classification of ruled 4-manifolds) . Let X be oriented diffeomorphicto a minimal rational or ruled surface, and let ξ ∈ H ( X ) . Then there is a symplecticform (even a K¨ahler one) on X in the class ξ iff ξ > . Moreover, any two symplecticforms in the class ξ are diffeomorphic. Thus all symplectic properties of ruled surfaces depend only on the cohomology classof a symplectic form.Our main interest is to study symplectic ( − -tori in X and the corresponding elliptictwists. It is easy to prove that, except for S ˜ × T , there are no symplectic ( − -tori inruled surfaces. For a suitable symplectic form the homology class T − ∈ H ( S ˜ × T ; Z ) canbe represented by a symplectic ( − -torus, but none of the other classes of H ( S ˜ × T ; Z ) can.Let ( X, ω µ ) be a symplectic ruled 4-manifold ( S ˜ × T , ω µ ) , where ω µ is a symplecticstructure of the cohomology class [ ω µ ] = T + − µ T − , µ ∈ ( − , . By Theorem 2.1 ( X, ω µ ) is well-defined up to symplectomorphism. As promised in the introduction, we will provethat π S ymp ∗ ( X, ω µ ) is trivial. Here and in §2.7 we abbreviate Ω( X, ω µ ) to Ω µ . VSEVOLOD SHEVCHISHIN AND GLEB SMIRNOV
Given µ > , the elliptic divisorial locus is contained in J ( X, Ω µ ) . Thus, each looplinked to the locus is contractible in J ( X, Ω µ ) . As such, we do not expect any non-trivial elliptic twists in this case. Following McDuff [McD-B], we will show that thegroup S ymp ∗ ( X, ω µ ) coincides with a group of certain diffeomorphisms, see Lemma 2.15 ;the latter group can be proved to be connected by standard topological techniques, see
Proposition 2.11 .When µ , the elliptic divisorial locus D T − is no longer included in J ( X, Ω µ ) . Thegeometry of this divisorial locus is studied below in §2.7 , and particularly it is proved that: i) Assumption ( A ) is satisfied for each loop linked to D T − ; hence, ( X, ω µ ) admits certainelliptic twists, see Lemma 2.13 . ii) The symplectic mapping class group S ymp ∗ ( X, ω µ ) isgenerated by elliptic twists coming from D T − . iii) Each of them is symplectically isotopicto the identity, see
Lemma 2.18 .2.2.
Classification of complex surfaces ruled over elliptic curves.
Here we verybriefly describe possible complex structures on elliptic ruled surfaces and study some oftheir properties.Let X be diffeomorphic to either S × Y or S ˜ × Y . The Enriques-Kodaira classificationof complex surfaces (see e.g.[BHPV]) ensures the following:(1) Every complex surface X of this diffeomorphism type is algebraic and hence K¨ahler.(2) Every such complex surface X is ruled, i.e. there exists a holomorphic map π : X → Y such that Y is a complex curve, and each fiber π − ( y ) is an irreducible rational curve.Note that, with the single exception of CP × CP , a ruled surface admits at mostone ruling.It was shown by Atiyah [At-2] that every holomorphic CP -bundle over a curve Y with structure group the projective group P Gl (2 , C ) admits a holomorphic section, andhence the structure group of such bundle can be reduced to the affine group Aff (1 , C ) ⊂ P Gl (2 , C ) .All of what was said so far applied for any ruled surface, irrespective of genus. Keep inmind, however, that everything below is for genus one surfaces. It was Atiyah who gave aclassification of ruled surfaces with base an elliptic curve. The description presented hereis taken from [Sw]. Theorem 2.2 (Atiyah) . Every holomorphic CP -bundle with structure group P Gl (2 , C ) over an elliptic curve is isomorphic to preciesly one of the following: i) a bundle associated to a principal C ∗ -bundle of nonpositive degree, ii) a bundle A , defined below, having structure group Aff (1 , C ) , and iii) a bundle A Spin , having structure group
Aff (1 , C ) . We shall proceed with a little discussion of these bundles: i) We first describe those
P Gl (2 , C ) -bundles whose structure group reduces to C ∗ . Let y ∈ Y be a point on the curve Y , and let { V , V } be an open cover of Y such that V = Y \ { y } and V is a small neighbourhood of y , so the domain V ∩ V =: ˆ V is apunctured disc. We choose a multivalued coordinate u on Y centered at y . LLIPTIC DIFFEOMORPHISMS OF SYMPLECTIC -MANIFOLDS 9 A surface X k associated to the line bundle O ( k y ) (or if desired, a C ∗ -bundle) can bedescribed as follows: X k := (cid:0) V × CP (cid:1) ∪ (cid:0) V × CP (cid:1) / ∼ , where ( u, z ) ∈ V × CP and ( u, z ) ∈ V × CP are identified iff u ∈ ˆ V , z = z u k .Here z , z are inhomogeneous coordinates on the two copies of CP . Clearly, the biholo-morphism ( u, z ) → ( u, z − ) , ( u, z ) → ( u, z − ) maps X k to X − k . Thus it is sufficient toconsider only values of k that are nonpositive.There is a natural C ∗ -action on X k via g · ( z , u ) := ( gz , u ) , g · ( z , u ) := ( gz , u ) for each g ∈ C ∗ . The fixed point set of this action consists of two mutually disjoint sections Y k and Y − k defined respectively by z = z = 0 and z = z = ∞ . We have [ Y k ] = k and [ Y − k ] = − k .It is very well known that any line bundle L of degree deg ( L ) = k = 0 is isomorphicto O ( k y ) for some y ∈ Y . Thus all the ruled surfaces associated with line bundles ofnon-zero degree k are biholomorphic to one and the same surface X k . On the other hand,the parity of the degree of the underlying line bundle is a topological invariant of a ruledsurface. More precisely, a ruled surface X associated with a line bundle L is diffeomorphicto Y × S for deg ( L ) even, and to Y ˜ × S for deg ( L ) odd. ii) Again, we start with an explicit description of the ruled surface X A associated withthe affine bundle A . Let { V , V , ˆ V } be the open cover of Y as before, u be a coordinateon Y centered at y , and z , z be fiber coordinates. Define X A := (cid:0) V × CP (cid:1) ∪ (cid:0) V × CP (cid:1) / ∼ , where ( z , u ) ∼ ( z , u ) for u ∈ ˆ V and z = z u + u − .There is an obvious section Y defined by the equation z = z = ∞ , but in contrastto C ∗ -bundles, the surface X A contains no section disjoint from that one. This can beshown by means of direct computation in local coordinates, but one easily deduce thisfrom Theorem X A , whoseproof is given in [Sw], see Theorem 5 . Theorem 2.3.
The surface X A associated with the affine bundle A admits an ellipticfibration over CP ; the general fiber is a smooth elliptic curve in the class T + + 2 T − and there are three multiple fibers each having the property that the underlying reducedsubvariety is a smooth elliptic curve in the class T + + T − . There are no other singularfibers. The following corollary will be used later. The reader is invited to look at [McD-D] forthe definition of the Gromov invariants and some examples of their computation.
Corollary 2.4. Gr ( Y + + Y − ) = 3 . Proof.
There are no smooth curves in X A , other than the multiple fibers, that are inthe class T + + T − . Each multiple fiber contributes ± to Gr ( T + + T − ) , for their normalbundles are holomorphically non-trivial, see § 1.7 in [McD-D]. If the complex structureis integrable, then each non-multiple-covered torus should appear with sign (+1) , see[Tb]. (cid:3) Based on this theorem, Suwa then gives another construction of X A . We mention thisconstruction here because it appears to have interest for the sequel.Let Y ∼ = C / Z τ ⊕ Z τ be an elliptic curve, and u be a multivalued coordinate on Y .Define X A to be a quotient space of CP × Y / G , where G ∼ = Z ⊕ Z is generated by thefollowing involutions ( z, u ) → (cid:16) − z, u + τ (cid:17) , ( z, u ) → (cid:18) z , u + τ (cid:19) . The surface obtained is elliptic ruled and is non-spin; see [Sw], where the latter is provedby constructing a section for X A of odd self-intersection number, see also Exercises 6.13and 6.14 in [McD-Sa-1].The elliptic fibration of X A mentioned in Theorem 2.3 comes from the G -invariantfunction f ( z, u ) = 12 (cid:18) z + 1 z (cid:19) , whose values are regular for all but three points of CP . For a regular point, when z = {− , , ∞} , the fiber f − ( z ) is an elliptic curve in the class T + + T − ) , whereas eachof the three singular fibers is a curve in the class T + + T − .There is an obvious action of the complex torus T ∼ = Y on CP × Y by translations.This action commutes with that of G . Hence, T acts also on X A . As the function f is T -invariant, so are the fibers f − ( z ) , z ∈ CP of our elliptic fibration; they are, in fact,simply the orbits of the action. Although T acts effectively on X A , it does not act freely;the isotropy groups of the singular fibers correspond to the three pairwise different ordertwo subgroups of T . For instance, for ( z, u ) ∈ f − ( ∞ ) , the stabilizer is z → − z .As each fiber f − ( z ) is the torus, it gives a homomorphism H ( f − ( z ); Z ) → H ( X A ; Z ) between the two copies of Z . To see what this homomorphism is for the multiplefibers, we regard X A as a ruled surface over Y ′ ∼ = C / Z ( τ / ⊕ Z ( τ / . Then themultiple fibers appear as bisections, double covering of Y ′ . Note that there is a one-to-one correspondence between the double covering of Y ′ and the index subgroups of H ( Y ′ ; Z ) . This implies that, for the singular fibers f − ( z ) , z = {− , , ∞} , the images of H ( f − ( z ); Z ) → H ( X A ; Z ) correspond to three pairwise different index subgroups of H ( X A ; Z ) ∼ = H ( Y ′ ; Z ) . iii) The ruled surface associated to A Spin is diffeomorphic to S × T , thus it will notbe discussed in this note, but see [Sw].Summarizing our above observations, we see that X ∼ = S ˜ × T admits countably manydiffeomorphism classes of complex structures. These structures are as follows: • The structures J ∈ J − k , k > , such that the ruled surface ( X, J ) contains asection of self-intersection number − k ; these are all biholomorphic to X − k . • The type A structures J ∈ J A such that the ruled surface ( X, J ) contains no sectionsof negative self-intersection number but does contain a triple of smooth bisections;these are all biholomorphic to X A . LLIPTIC DIFFEOMORPHISMS OF SYMPLECTIC -MANIFOLDS 11 One family of ruled surfaces over elliptic base.
Here is a construction of aone-parametric complex-analytic family p : X → C of non-spin elliptic ruled surfaces,such that the surfaces p − ( t ) , t = 0 , are biholomorphic to X A and p − (0) ∼ = X − .As before, we take a point y on Y , let u be a coordinate of the center y , and put { V , V , ˆ V } to be an open cover for Y such that V := Y \ { y } , V is a small neighbourhoodof y , and ˆ V := V ∩ V . Further, let ∆ be a complex plane, and let t be a coordinate on it.We construct the complex 3-manifold X by patching ∆ × V × CP and ∆ × V × CP in such a way that ( t, z , u ) ∼ ( t, z , u ) for u ∈ ˆ V and z = z u + tu − .The preimage of and under the natural projection p : X → ∆ are biholomorphicrespectively to X − and X A . In fact, it is not hard to see that for each t = 0 , the surface p − ( t ) is biholomorphic to X A as well. One way to prove this is to use the C ∗ -action on X g · ( t, z , u ) := ( tg, gz , u ) , g · ( t, uz , u ) := ( tg, gz , u ) for each g ∈ C . This proves even more than we desired, namely, that there exists a C ∗ -action on X suchthat for each g ∈ C ∗ we get a commutative diagram X · g −−−→ X p y y p C −−−→ · g C , where X · g −→ X denotes the biholomorphism induced by g ∈ C ∗ .The construction of the complex-analytic family X is due to Suwa, see [Sw], thoughthe existence of the C ∗ -action was not mentioned in Suwa’s paper. Let us summarize hisresult in a theorem.2.4. Embedded curves and almost-complex structures. In Section 2.2 the classifi-cation for non-spin elliptic ruled surfaces was given. It turns out that this classificationcan be extended to the almost-complex geometry of S ˜ × T .Let X be diffeomorphic to S ˜ × T , and let J ( X ) be the space of almost-complex struc-tures on X that are tamed by some symplectic form; the symplectic forms need not bethe same. Here we use the short notation J for J ( X ) .Given k > , let J − k ( X ) (we will abbreviate it to J − k ) be the subset of J ∈ J consisting of elements that admit a smooth irreducible J -holomorphic elliptic curve in theclass T + − k F . It is well known that J − k forms a subvariety of J of real codimension · (2 k − , see e.g. Corollary 8.2.3 in [Iv-Sh-1].Further, define J A ( X ) (or J A , for short) be the subset J ∈ J of those element forwhich there exists a smooth irreducible J -holomorphic elliptic curve in the class B .By straightforward computations one can show that the sets J − k are mutually disjoint,and each J − k is disjoint from J A . Further, it is not hard to see that J − ⊂ J A and J − k +1) ⊂ J − k , where J − k is for the closure of J − k . A less trivial fact is that J = J A G J − G J − G J − . . . , (2.2)which can be also stated as follows. Proposition 2.5 (cf.
Lemma 4.2 in [McD-B]) . Let ( X, ω ) be a symplectic ruled 4-manifolddiffeomorphic to S ˜ × T . Then every ω -tamed almost-complex structure J admits a smoothirreducible J -holomorphic representative in either B or T + − k F for some k > . Proof.
The proof is analogous to that of
Lemma 4.2 in [McD-B]. Observe that theexpected codimension for the class B is zero. By Lemma 2.4 we have Gr ( T + + T − ) > .Hence, J A is an open dense subset of J , and, thanks to the Gromov compactness theorem,for each J ∈ J the class B has at least one J -holomorphic representative, possiblysingular, reducible or having multiple components.By virtue of Theorem 2.7 , no matter what J was chosen, our manifold X admits thesmooth J -holomorphic ruling π by rational curves in the class F .Since B · F > , it follows from positivity of intersections that any J -holomorphicrepresentative B of the class B must either intersect a J -holomorphic fiber of π or mustcontain this fiber completely. a) First assume that B is irreducible. Then it is of genus not greater than because ofthe adjunction formula. This curve is of genus because every spherical homology classof X is proportional to F . We now can apply the adjunction formula one more time toconclude that B is smooth, i.e. J ∈ J A . b) The curve B is reducible but contains no irreducible components which are the fibersof π . Then it contains precisely two components B and B , since B · F = 2 . Both thecurves B and B are smooth sections of π , and hence [ B i ] = T + + k i F , i = 1 , . Since [ B ]+[ B ] = B , it follows that k + k = − , and hence either k or k is negative. Thus wehave that either B or B is a smooth J -holomorphic section of negative self-intersectionindex. c) If some of the irreducible components of B are in the fibers class F , then onecan apply arguments similar to that used in a) and b) to prove that the part B ′ of B which contains no fiber components has a section of negative self-intersection index as acomponent. (cid:3) Rulings and almost-complex structures.
Let X be a ruled surface equippedwith a ruling π : X → Y , and let J be an almost-complex structure on X . We shall saythat J is compatible with the ruling π : X → Y if each fiber π − ( y ) is J -holomorphic.We wish to express our thanks to D. Alexeeva [Al] for sharing her proof of the followingstatement. Proposition 2.6.
Let J ( X, π ) be the space of almost-complex structures on X compatiblewith π . i) J ( X, π ) is contractible. ii) Any structure J ∈ J ( X, π ) , as well as any compact family J t ∈ J ( X, π ) , is tamedby some symplectic form. Proof. i) Let be J ( R , R ) be the space of linear maps J : R → R such that J = − id and J ( R ) = R , i.e. it is the space of linear complex structures preserving R . Inaddition, we assume R and R are both oriented and each J ∈ J ( R , R ) induces thegiven orientations for both R and R . We now prove the space J ( R , R ) is contractible. LLIPTIC DIFFEOMORPHISMS OF SYMPLECTIC -MANIFOLDS 13 Indeed, let us take J ∈ J ( R , R ) . Fix two vectors e ∈ R and e ∈ R \ R . Thevectors e and J e form a positively oriented basis for R . Therefore J e is in the upperhalf-plane for e . Further, the vectors e , J e , e , J e form a positively oriented basis for R . Therefore J e is in the upper half-space for the hyperplane spanned on e , J e , e .We see that the space J ( R , R ) is homeomorphic to the direct product of two half-spaces, and hence it is for sure contractible.To finish the proof of i) we consider the subbundle V x := Ker d π ( x ) ⊂ T x X, x ∈ X , ofthe tangent bundle T X of X . Every J ∈ J ( X, π ) is a section of the bundle J ( T X, V ) → X whose fiber over x ∈ X is the space J ( T x X, V x ) . Since the fibers of J ( T X, V ) arecontractible; it follows that the space of section for J ( T X, V ) is contractible as well. ii) Again, we start with some linear algebra. Let V be a 2-subspace of W ∼ = R , andlet J ∈ J ( W, V ) . Choose a 2-form τ ∈ Λ ( W ) such that the restriction τ | V ∈ Λ ( V ) of τ to V is positive with respect to the J -orientation of V , i.e. τ ( ξ, J ξ ) > . Clearly, thesubspace H := Ker τ ⊂ W is a complement to V . Further, let σ ∈ Λ ( V ) be any 2-formsuch that σ | V vanishes, but σ | H does not. If H is given the orientation induced by σ ,then the J -orientation of W agrees with that defined by the direct sum decomposition W ∼ = V ⊕ H . We now prove that J is tamed by τ + K σ for
K > sufficiently large.It is easy to show that there exists a basis e , e ∈ V, e , e ∈ H for W such that J takesthe form J = − −
10 0 0 −
10 0 1 0 . The matrix Ω of τ + K σ with respect to this basis is block-diagonal, say
Ω = − K σ + . . . − K σ + . . . for σ > .It remains to check that the matrix Ω J is positive definite, i.e. ( ξ, Ω J ξ ) > . A matrix ispositive definite iff its symmetrization is positive definite. It is straightforward to checkthat Ω J + (Ω J ) t is of that kind for K large enough.Let us go back to the ruled surface X . The theorem of Thurston [Th] (see also Theorem6.3 in [McD-Sa-1]) ensures the existence of a closed 2-form τ on X such that the restric-tions of τ to each fiber π − ( y ) is non-degenerate. Choose an area form σ on Y . By thesame reasoning as before, any J ∈ J ( X, π ) is tamed by τ + K π ∗ σ for K large enough. (cid:3) The following theorem by McDuff motivates the study of compatible almost-complexstructures, see
Lemma 4.1 in [McD-B].
Theorem 2.7.
Let X be an irrational ruled surface, and let J ∈ J ( X ) . Then there existsa unique ruling π : X → Y such that J ∈ J ( X, π ) . Diffeomorphisms.
Let X be diffeomorphic to either T × S or T ˜ × S , and let π : X → Y be a smooth ruling. Further, let Fol ( X ) be the space of all smooth foliationsof X by spheres in the fiber class F (the class F generates π ( X ) and, therefore, it is theonly class that can be the fiber class of an S -fibration.)The group D iff ( X ) acts transitively on Fol ( X ) as well as the group D iff ( X ) acts transi-tively on a connected component Fol ( X ) of Fol ( X ) . This gives rise to a fibration sequence D ∩ D iff ( X ) → D iff ( X ) → Fol ( X ) , where D is the group of fiberwise diffeomorphisms of X . By the definition of D thereexists a projection homomorphism τ : D → D iff ( T ) such that for every F ∈ D we have acommutative diagram X F −−−→ X π y y π T −−−→ τ ( F ) T , which induces the corresponding commutative diagram for homology H ( X ; Z ) F ∗ −−−→ H ( X ; Z ) π ∗ y y π ∗ H ( T ; Z ) −−−→ τ ( F ) ∗ H ( T ; Z ) . Notice that τ ( F ) is isotopic to the identity if only if τ ( F ) ∗ = id . Since π ∗ is an isomor-phism, it follows that the subgroup D ∩ D iff ( X ) of D is mapped by τ to D iff ( T ) , so weend up with the restricted projection homomorphism τ : D ∩ D iff ( X ) → D iff ( T ) . Since we shall exclusively be considering this restricted homomorphism, we use the samenotation τ for this.Given an isotopy f t ∈ D iff ( T ) , f = id , one can lift it to an isotopy F t ∈ D ∩ D iff ( X ) , F = id such that τ ( F t ) = f t . This immediately implies that the inclusion Ker τ ⊂ D ∩ D iff ( X ) induces an epimorphism π ( Ker τ ) → π ( D ∩ D iff ( X )) . (2.3)Because of this property we would like to look at the group Ker τ in more detail, but firstintroduce some useful notion.Let X be a smooth manifold, and let f be a self-diffeomorphism X . Define the mappingtorus T ( X, f ) as the quotient of X × [0 , by the identification ( x, ∼ ( f ( x ) , . Forthe diffeomorphism f to be isotopic to identity it is necessary to have the mapping torusdiffeomorphic to T ( X, id ) ∼ = X × S .Let us go back to the group Ker τ that consists of bundle automorphisms of π : X → T .Let F ∈ Ker τ be a bundle automorphism of π , and let γ be a simple closed curveon T . By F γ denote the restriction of F to π − ( γ ) ∼ = S × S . The mapping torus T ( π − ( γ ) , F γ ) is either diffeomorphic to S × T or S ˜ × T . In the later case we shall saythat the automorphism F is twisted along γ . LLIPTIC DIFFEOMORPHISMS OF SYMPLECTIC -MANIFOLDS 15 Lemma 2.8.
Let X be diffeomorphic to either T × S or T ˜ × S , and let F ∈ Ker τ . Then F is isotopic to the identity through Ker τ iff T contains no curve for F to be twistedalong. Proof.
The -torus T has a cell structure with one cell, F can be isotopically deformed to id over the 0-skeleton of T . The obstructionfor extending this isotopy to the 1-skeleton of T is a well-defined cohomology class c ( F ) ∈ H ( T ; Z ) ; the obstruction cochain c ( F ) is the cochain whose value on a 1-cell e equals 1if F is twisted along e and 0 otherwise. It is evident that c ( F ) is a cocycle.By assumption c ( F ) = 0 . Consequently there is an extension of our isotopy to anisotopy over a neighbourhood of the 1-skeleton of T , but such an isotopy always can beextended to the rest of T . (cid:3) A short way of represent the issue algebraically is by means of the obstruction homo-morphism c : Ker τ → H ( X ; Z ) defined in the lemma; any two elements F, G ∈ Ker τ are isotopic to each other through Ker τ iff c ( F ) = c ( G ) . Lemma 2.9.
Let X be diffeomorphic to S × T , and let F ∈ Ker τ , then T contains nocurve for F to be twisted along. This means that the obstruction homomorphism vanishes. Proof.
The converse would imply that the mapping torus T ( X, F ) is not spin, but T ( X, id ) ∼ = S × T × S is a spin 5-manifold. (cid:3) The following result is due to McDuff [McD-B], but the proof follows by combining
Lemma 2.9 with
Lemma 2.8 . Proposition 2.10.
Let X be diffeomorphic to S × T , then the group D ∩ D iff ( X ) isconnected. In what follows we need a non-spin analogue of this Proposition for the case of ellipticruled surfaces.
Proposition 2.11.
Let X be diffeomorphic to S ˜ × T , then the group D ∩ D iff ( X ) isconnected. Proof.
Fix any cocycle c ∈ H ( X ; Z ) ∼ = Z ⊕ Z , then we claim there exists F ∈ Ker τ such that c ( F ) = c and, moreover, F is isotopic to id through diffeomorphisms D∩ D iff ( X ) .It follows from Suwa’s model, see §2.2 , that the automorphism group for the complex ruledsurface X A contains the complex torus T as a subgroup. By construction, it is clear that T is a subgroup of D ∩ D iff ( X ) . Besides that, the 2-torsion subgroup T ∼ = Z ⊕ Z of T is a subgroup of Ker τ . We trust the reader to check T is mapped isomorphically by theobstruction homomorphism to H ( T ; Z ) .The algebra behind this argument is expressed by a commutative diagram T i −−−→ Ker τ j −−−→ D ∩ D iff ( X ) y y y π ( T ) i ∗ −−−→ π ( Ker τ ) j ∗ −−−→ π ( D ∩ D iff ( X )) , where i ∗ is an isomorphism, j ∗ ◦ i ∗ is zero, and therefore j ∗ is zero as well. But we alreadyknow that j ∗ is an isomorphism, and hence π ( D ∩ D iff ( X )) is trivial. (cid:3) Vanishing of elliptic twists.
Here is the part where a proof of
Theorem 0.1 comes.We split it into a few pieces. Let X be the symplectic ruled 4-manifold ( S ˜ × T , ω µ ) , [ ω µ ] = T + − µ T − , and let Ω µ be the space of symplectic forms on X that are isotopic to ω µ . Here we work with the connected component of J ( X ) that contains J ( X, Ω µ ) ; thesame applies to J A and J − k . Lemma 2.12. J A ⊂ J ( X, Ω µ ) for every µ ∈ ( − , . Proof.
For every J ∈ J A we take any symplectic form ω such that J is ω -tamed. Theninflate ω along the classes Y + − Y − and Y + + Y − , and then rescale it. (cid:3) Lemma 2.13. J A = J ( X, Ω µ ) for every µ ∈ ( − , . Proof.
It is clear that J ( X, Ω µ ) does not contain the structures J − k for µ ∈ ( − , ,and hence by (2.2) and Lemma 2.12 the proof follows. (cid:3)
This means that there is no topology change for the space J ( X, Ω µ ) when µ is beingvaried in ( − , . In particular, π ( J ( X, Ω µ )) = π ( J A ( X )) for µ ∈ ( − , . Lemma 2.14. J − k ⊂ J ( X, Ω µ ) iff µ ∈ (cid:18) − k , (cid:19) . Proof.
The “only if” part is obvious, while the “if” can be proved by deflating along Y + − k ( Y + − Y − ) and inflating along Y + − Y − . (cid:3) Combining
Lemma 2.12 with
Lemma 2.14 , as well as the fact that the higher codimensionsubmanifolds J − k , k > do not affect the fundamental group of J ( X, Ω µ ) , we see thatthere is no topology change in π ( J ( X, Ω µ )) as µ increased within (0 , , i.e. we have π ( J ( X, Ω µ )) = π ( J ( X )) for µ ∈ (0 , . (2.4)Diagram (1.3) implies that the symplectic mapping class group is the cokernel of ν ∗ : π ( D iff ( X )) → π ( J ( X )) which we now show is trivial. Lemma 2.15. ν ∗ is an epimorphism. Proof.
Though the map ν : D iff ( X ) → J ( X ) is not a fibration, it can be extended toone; namely, to D iff ( X ) → J ( X ) → Fol ( X ) , where the last arrow is a homotopy equivalence, see Theorem 2.7 and
Proposition 2.6 .Thus, we end up with the homotopy exact sequence . . . → π ( D iff ( X )) → π ( J ( X )) → π ( D ∩ D iff ( X )) . If X is of genus 1, the group π ( D ∩ D iff ( X )) is trivial by Propositions (cid:3)
The following corollary will not be used in the remainder of the paper, but it is a verynatural application of
Lemma 2.15 . LLIPTIC DIFFEOMORPHISMS OF SYMPLECTIC -MANIFOLDS 17 Corollary 2.16.
The space J ( X ) is homotopy simple. In other words, π ( J ( X )) isabelian and acts trivially on π n ( J ( X )) . By virtue of (1.3) and (2.4),
Lemma 2.15 immediately implies
Proposition 2.17. π ( S ymp ∗ ( X, ω µ )) = 0 for every µ ∈ (0 , . In order to compute the group π ( S ymp ∗ ( X, ω µ )) for µ ∈ ( − , it is necessary to knowbetter the fundamental group of J A . The space J A is the complement to (the closure of)the elliptic divisorial locus D T − in the ambient space J ( X ) . We denote by i the inclusion i : J A ( X ) → J ( X ) . By Lemma 2.15 every loop J ( t ) ∈ π ( J A ) can be decomposed into a product J ( t ) = J ( t ) · J ( t ) , where J ( t ) ∈ Im ν ∗ , and J ( t ) ∈ Ker i ∗ .By Lemma 2.13 , for µ ∈ ( − , , J A = J ( X, Ω µ ) . In particular, assumption ( A ) issatisfied for each loop in J A . Thus, each loop in J A , that lies in Ker i ∗ , could contributedrastically to π ( S ymp ∗ ( X, ω µ )) via the corresponding elliptic twists. But this will nothappen, because the following holds. Lemma 2.18.
Ker i ∗ ⊂ Im ν ∗ . Proof.
Choose some J ∗ ∈ D T − , and let ∆ be a -disc which intersects D T − transversallyat the single point J ∗ . Denote by J ( t ) the boundary of ∆ . By Lemma 2.19 one simplyneeds to show that the homotopy class of J ( t ) comes from the natural action of D iff ( X ) on J A , and the lemma will follow.If J ∗ is integrable, then one can choose ∆ such that J ( t ) is indeed an orbit of the actionof a certain loop in D iff ( X ) , see the description of the complex-analytic family constructedin §2.3 . Thus it remains to check that every structure J ∗ ∈ D T − can be deformed to beintegrable through structures on D T − . This will be proved by Lemma 2.20 below. (cid:3)
Lemma 2.19.
Let x, y ∈ J A , and let H ( t ) ∈ J A , t ∈ [0 , be a path joining themsuch that H (0) = x, H (1) = y . If a loop J ( t ) ∈ π ( J A , y ) , t ∈ [0 , lies in the im-age of π ( D iff ( X ) , id ) → π ( J A , y ) , then H − · J · H ∈ π ( J A , x ) lies in the image of π ( D iff ( X ) , id ) → π ( J A , x ) . Proof.
Without loss of generality we assume that there exists a loop f ( t ) ∈ π ( D iff , id ) such that J ( t ) = f ∗ ( t ) J (0) . Let H s be the piece of the path H that joins the points H (0) = x and H ( s ) . To prove the lemma it remains to consider the homotopy J ( s, t ) := H − s · f ∗ ( t ) H ( s ) · H s , where J (1 , t ) = H − · J · H and J (0 , t ) = f ∗ ( t ) H (0) . (cid:3) Lemma 2.20.
Every connected component of J − contains at least one integrable struc-ture. Proof.
Take a structure J ∈ J − , and denote by C the corresponding smooth ellipticcurve in the class [ C ] = T − . Let π : X → C be the ruling such that J ∈ J ( X, π ) , see Theorem 2.7 . Apart from the section given by C , we now choose one more smooth section C of π such that C is disjoint from C ; the section C need not be holomorphic, but besmooth. We claim that there exists a unique C ∗ -action on X such that (a) it is fiberwise, i.e. this diagram X · g −−−→ X p y y p C −−−→ · g C , (2.5)commute for each g ∈ C ∗ ,(b) it acts on the fibers of π by means of biholomorphisms, and(c) it fixes both C and C .The complement X − C is a C -bundle with C being the zero-section; we keep thenotation π for the projection X − C → C . This bundle inherits the C ∗ -action describedabove. Let U (1) be the unitary subgroup of C ∗ . The (0 , -part of a U (1) -invariantconnection on the C -bundle π : X − C → C defines a ¯ ∂ -operator which associated tosome holomorphic structure J on X − C , see Chapter 0, § 5 in [Gr-Ha]. As a complexstructure, J agrees with J on the fibers of π .To every holomorphic C ∗ -bundle one canonically associates a CP -bundle. Hence, thereis a unique extension J to a complex structure J ∈ J ( X, π ) such that C becomesholomorphic.When restricted to the bundle T X | C , J coincides with J . By Proposition 2.6 there is asymplectic form ω taming both structures J and J . Given a symplectic curve, say C , in X , and an almost-complex structure, say J , defined along C (i.e. on T X | C ) and tamedby ω . There exists an ω -tamed almost-complex structure on X which extends the givenone. Moreover, such an extension is homotopically unique. In particular, one can alwaysconstruct a family J t joining J and J such that C stays J t -holomorphic, and the lemmais proved. (cid:3) Summarizing the results of
Lemma 2.15 and
Lemma 2.18 we obtain
Lemma 2.21. π ( D iff ( X )) → π ( J A ( X )) is epimorphic. Again, it is implied by diagram 1.3 that the following holds.
Proposition 2.22. π ( S ymp ∗ ( X, ω µ )) = 0 for every µ ∈ ( − , . Together with
Proposition 2.17 , this statement covers what is claimed in
Theorem 0.1 .3.
Blow up once
Rational ( − -curves. Let ( Z, ω ) be a symplectic ruled 4-manifold diffeomorphicto S ˜ × Y CP . Here we study homology classes in H ( Z ; Z ) that can be represented bya symplectically embedded ( − -sphere. Given a symplectically embedded ( − -sphere A , it satisfies [ A ] = − , K ∗ ([ A ]) = 1 . (3.1)A simple computation shows that there are only two homology classes satisfying (3.1),namely, [ A ] = E and [ A ] = F − E .The following lemma will be used in the sequel, often without any specific reference. LLIPTIC DIFFEOMORPHISMS OF SYMPLECTIC -MANIFOLDS 19 Lemma 3.1.
Let ( Z, ω ) be a symplectic ruled 4-manifold diffeomorphic to S ˜ × Y CP .Then for every choice of ω -tamed almost-complex structure J , both the classes E and F − E are represented by smooth rational J -holomorphic curves. Proof.
Given an arbitrary ω -tamed almost-complex structure J , the exceptional class F − E is represented by either a smooth J -holomorphic curve or by a J -holomorphiccusp-curve A of the form A = P m i A i where each A i stands for a rational curve occuringwith the multiplicity m i > . Clearly, we have < Z A i ω < Z A ω. (3.2)Because K ∗ ( F − E ) = 1 , there exists at least one irreducible component of the curve A ,say A , with K ∗ ([ A ]) > .Note that spherical homology classes in H ( Z ; Z ) are generated by F and E . Hence,we have [ A ] = p F − q E , which implies in particular that [ A ] = − q , with equalityiff [ A ] = p F . But the latter is prohibited by (3.2) because Z E ω > . Therefore, we have [ A ] − . Further, one may use the adjunction formula to obtainthat A is a smooth rational curve with [ A ] = − and K ∗ ( A ) = 1 . Note that it is notpossible for A to be in the class F − E because of (3.2). Hence, we have [ A ] = E .Take another irreducible component, say A . If A does not intersect A , then [ A ] = p F , which contradicts (3.2). Thus A intersects A , positively. Hence, [ A ] = p F − q E for q positive. The same argument works for the other irreducible components A , A , . . . of the curve A . But note that [ A ] · [ A ] < , and hence there are no other components of A , except A and A . We thus have m [ A ] = F − ( m + 1) E for m , m > . The class F − ( m + 1) E is primitive, and hence m = 1 . Further, this class cannot be representedby a rational curve, which can be easily checked using the adjunction formula. We thusproved the lemma for the class F − E ; the case of E is analogous. (cid:3) This lemma leads to the following generalization of
Theorem 2.7 for ruled but notgeometrically ruled symplectic 4-manifolds.
Lemma 3.2.
Let ( Z, ω ) be a symplectic ruled 4-manifold diffeomorphic to S ˜ × Y CP ,and let J be an ω -tamed almost-complex structure. Then Z admits a singular ruling givenby a proper projection π : Z → Y onto Y such that i) there is a singular value y ∗ ∈ Y such that π is a spherical fiber bundle over Y − y ∗ ,and each fiber π − ( y ) , y ∈ Y − y ∗ , is a J -holomorphic smooth rational curve in the class F ; ii) the fiber π − ( y ∗ ) consists of the two exceptional J -holomorphic smooth rational curvesin the classes F − E and E . Proof.
It follows from
Lemma 3.1 that Z admits J -holomorphic ( − -curves E and E ′ in the classes E and F − E , respectively. We have to show that for each point p ∈ Z ,except for those on E and E ′ , there exists a smooth J -holomorphic sphere in the class F that passes through p . Such a sphere would necessarily be unique due to positivity ofintersections. To get such a curve for a generic (by Gromov compactness, for every) point p ∈ Z it suffices to show Gr ( F ) = 0 . But this follows from the Seiberg-Witten theory, see[McD-Sa-2].Having a J -holomorphic curve passing through p ∈ Z , we have to prove it is smooth.Along the same lines as Lemma 3.1 , one shows that the only non-smooth J -holomorphiccurve in the class F is E ∪ E ′ . (cid:3) Straight structures.
Let Z ∼ = S ˜ × T CP be a complex ruled surface, and let E be a smooth rational ( − -curve in E ∈ H ( Z ; Z ) . The blow-down of E from Z , which isa non-spin geometrically ruled genus one surface, will be denoted by X . The surface Z issaid to be a type A surface if X is biholomorphic to the surface X A , see §2.2 .Let p ∈ X be the image of E under the contraction map. Recall that X A containsthe triple of bisections, which are smooth elliptic curves in the class B ∈ H ( X ; Z ) . Thesurface Z is called straight type A surface if there is no bisection passing through p in X . Inother words, a straight type A surface contains a triple of smooth curves in the homologyclass B , while a non-straight type A surface contains a smooth elliptic ( − -curve in theclass B − E ∈ H ( Z ; Z ) . We remark that it follows from Theorem 2.3 that straight type A surfaces can be characterized as those for which there exists a smooth elliptic ( − -curvein the homology class B − E ∈ H ( Z ; Z ) .Let π be the ruling of X , and let S be the fiber of π that passes through p . When Z is type A , there are three bisections B i ⊂ X , each of which intersects S at preciselytwo distinct points. The following result was established in §2.2 , see the construction ofSuwa’s model. Lemma 3.3.
There exists a complex coordinate s on S such that the intersection points B i ∩ S are as follows: B ∩ S = { , ∞} , B ∩ S = {− , } , B ∩ S = {− i, i } . (3.3)We then claim Lemma 3.4.
There exists a complex-analytic family Z → CP of type A surfaces Z s parametrized by s ∈ CP . When s equals one of the exceptional values { , ∞} , {− , } , {− i, i } , the surface Z s is not a straight type A surface, while for other parameter values, Z s isstraight type A . Proof.
Pick a fiber F of the ruling of X ∼ = X A . Consider the complex submanifold F × CP ⊂ X × CP , and denote by S the diagonal in F × CP . We construct Z as theblow-up of X × CP along S . The -fold Z forms the complex-analytic family Z → S that was claimed to exist in the lemma. (cid:3) The notion of the straight type A complex structure can be generalized to almost-complex geometry as follows. Choose a tamed almost-complex structure J ∈ J ( Z ) . Wewill call J straight type A , or simply straight , if each J -holomorphic representative in theclass B ∈ H ( Z ; Z ) is smooth. Clearly, the space of straight structures J st ( Z ) is an opendense submanifold in J ( Z ) . Instead of J ( Z ) or J st ( Z ) we write J and J st for short.This definition of straightness is motivated by the following lemma the proof of which is LLIPTIC DIFFEOMORPHISMS OF SYMPLECTIC -MANIFOLDS 21 left to the reader because it is similar to the proof of Proposition 2.5 , (but the modifiedversion of
Theorem 2.7 given by
Lemma 3.2 should be used).Let s : [0 , → S be a loop in S , and let Z ( t ) → s ( t ) be the restriction of Z → S to s ( t ) . Because s ( t ) is contractible inside the sphere S , we can think of Z ( t ) as a familyof type A complex structures on Z . Each structure J ( t ) is straight iff s ( t ) does not passthrough any of points (3.3). The following choice of s ( t ) will be used in §3.4 s ( t ) = εe πit . (3.4) Lemma 3.5.
Let ( Z, ω ) be a symplectic ruled 4-manifold diffeomorphic to S ˜ × T CP ,and let J be an ω -tamed almost-complex structure. Then every J -holomorphic represen-tative in the class B is either irreducible smooth or contains a smooth component in oneof the classes B − E , T + − k F , k > . Similarly to
Proposition 2.5 this lemma leads to a natural stratification of the space J of tamed almost-complex structures. Namely, this space can be presented as the disjointunion J = J st G D T − G D B − E + . . . , where D T − and D B − E , which are submanifolds of real codimension 2 in J , are the ellipticdivisorial locus for respectively the classes T − and B − E . Here we omitted the terms ofreal codimension greater than 2, because they do not affect the fundamental group of J .Coming to the symplectic side of straightness, we claim that if a symplectic form ω on Z satisfies the period conditions Z T − ω < , Z B − E ω < , then J ( Z, ω ) ⊂ J st . Moreover, a somewhat inverse statement holds, at least for integrablestructures. Lemma 3.6 (cf. §1.1 ) . Every complex straight type A structure is tamed by a symplecticform satisfying the period conditions. Moreover, every compact family of straight type A structures is tamed by a family of cohomologous forms satisfying the period conditions. Proof.
We first check that a complex straight type A surface Z has a taming symplecticform θ such that θ satisfies the period conditions.If Z is type A then it is the surface X A ∼ = S ˜ × T blown-up once. Since X A admits asymplectic structure which satisfies the first period condition, then so does Z . Further,the second period condition can be achieved by means of deflation along a smooth ellipticcurve in the class B − E ; such a curve indeed exists thanks to the straightness of Z .We let K to denote the parameter space for our family Z t , and let θ t , t ∈ K be ataming symplectic form on Z t that satisfies the period condition. For every point t ′ ∈ K ,let U t ′ ∈ K be a sufficiently small neighbourhood of t ′ ∈ K such that for each t ∈ U t ′ θ t ′ tames the complex structure in Z t .As K is compact, one may take a finite subcover U t i , t i ∈ I of K . The forms θ I are notnecessarily cohomologous because they may have different integrals on the homology class E . Set ε t i := R E θ t i , t i ∈ I , and ε := min ε t i . We now deflate ( Z t i , θ t i ) along the homology class E to get R E θ t i = ε . Thanks to this deflation the forms θ I become cohomologous andstill do satisfy the period conditions.Finally, set ˆ θ ( t ) := P I ′ ρ t i ( t ) θ t i , where the functions ρ t i = ρ t i ( t ) is a partition of unityfor the finite open cover U t i , t i ∈ I of K . What remains is to verify that Z t is tamed by ˆ θ ( t ) for every t ∈ K . Pick some t ∗ ∈ K , then there are but finitely many charts U t , . . . , U t p that contains the point t ∗ ∈ K . Then ˆ θ ( t ∗ ) = ρ t ( t ∗ ) θ t + . . . + ρ t p ( t ∗ ) θ t p . Since each of θ t , . . . , θ t p tames Z t ∗ , then so does ˆ θ ( t ∗ ) . (cid:3) From this we have verified assumption ( A ) for the family of straight structures givenby (3.4).3.3. Refined Gromov invariants.
In this subsection, we work with an almost-complexmanifold ( Z, J ) equipped with a straight structure J ∈ J st , i.e. every J -holomorphiccurve of class B ∈ H ( Z ; Z ) in Z is smooth. We also note that such a curve is notmultiply-covered, because the homology class B is primitive. The universal modulispace M ( B ; J st ) of embedded non-parametrized pseudoholomorphic curves of class B isa smooth manifold, and the natural projection pr : M ( B ; J st ) → J st is a Fredholm map,see [Iv-Sh-1, McD-Sa-3]. Given a generic J ∈ J st , the preimage pr − ( J ) is canonicallyoriented zero-dimensional manifold, see [Tb] where it is explained how this orientation ischosen. The cobordism class of pr − ( J ) is independent of a generic J , thus giving us awell-defined element of Ω SO = Z ; the number is equal to Gr ( B ) . Corollary 2.4 states that Gr ( B ) = 3 , and hence Z contains not one but several curves inthe class B . Once we restricted almost-complex structures to those with the straightnessproperty, the following modification of Gromov invariants can be proposed: given the im-age G of a certain homomorphism Z → H ( Z ; Z ) , instead of counting pseudoholomorphiccurves C such that [ C ] = B , we will count curves C such that [ C ] = B and the embedding i : C ֒ → Z satisfies Im i = G . The definitions of Gromov invariants Gr ( B , G ) , moduli space M ( B , G ; J st ) , and so forth are completely analogous to those in “usual” theory of Gromovinvariants.Suppose J is an integrable straight type A structure, then the complex surface ( Z, J ) contains precisely 3 smooth elliptic curves C , C , and C in the homology class B . Wedenote by G k the subgroup of H ( Z ; Z ) generated by cycles on C k ; these subgroups G k are pairwise distinct, see §2.2 .It is clear now that the space M ( B ; J st ) is disconnected and can be presented as theunion M ( B ; J st ) = G k =1 M ( B , G k ; J st ) . We define the moduli space of bisections to be the fiber product M B = { ( x , x , x ) | x k ∈ M ( B , G k ; J st ) , pr ( x ) = pr ( x ) = pr ( x )) } . Similarly to M ( B ; J st ) , the moduli space M B is a smooth manifold and the projection pr : M B → J st is a smooth map. We close this section by stating an obvious property ofthe projection map that we shall use in the sequel. Lemma 3.7.
The projection map pr : M B → J st is a diffeomorphism, when is restrictedto the subset of integrable straight type A complex structures. LLIPTIC DIFFEOMORPHISMS OF SYMPLECTIC -MANIFOLDS 23 Loops M B . The map ν : D iff ( Z ) → J st defined by D iff ( Z ) ν −→ J ( Z, ω ) : f → f ∗ J, can be naturally lifted to a map D iff ( Z ) → M B . Indeed, take a point s ∈ M B , whichis a quadruple [ J, B , B , B ]( s ) consisting of an almost-complex structure J ( s ) ∈ J st on Z and a triple of smooth J ( s ) -holomorphic elliptic curves B ( s ) , B ( s ) , and B ( s ) in Z .Then one can define D iff ( Z ) ν −→ M B : f → [ f ∗ J, f ( B ) , f ( B ) , f ( B )] . Here we construct an element of π ( M B ) that does not lie in the image of ν ∗ : π ( D iff ( Z )) → π ( M B ) .To start, we consider the tautological bundle Z ∼ = M B × Z over M B whose fiber overa point x ∈ M B is the almost-complex manifold ( Z, J ( s )) . By Lemma 3.1 every almost-complex manifold ( Z, J ( s )) contains a unique smooth rational ( − -curve S ( s ) in the class F − E . Thus, one associates to Z an auxiliary bundle S whose fiber over x ∈ M B is therational curve S ( s ) . Note that each B i ( s ) intersects S ( s ) at precisely 2 distinct pointsdenoted by P i, and P i, . Hence we can mark out 3 distinct pairs of points ( P i, , P i, ) , i = 1 , , on each fiber S ( s ) of S . Besides that, every ( Z, J ( s )) contains a unique smoothrational curve E ( s ) in the class E . The curve E ( s ) intersects S ( s ) at precisely one point,say Q ( s ) . This point Q does not coincide with any of the point P i, , P i, , because J ( x ) isassumed to be a straight one. Therefore S can be considered as a fiber bundle over M B whose fiber is the rational curve S ( s ) with 7 distinct marked points, partially ordered as ( { P , , P , } , { P , , P , } , { P , , P , } , Q ) (3.5)As such, there is an obvious map λ : M B → M sending S ( s ) to the corresponding point in the moduli space of points in CP , partiallyordered as (3.5). Notice that the space M is also the moduli space of points on C ,partially ordered as ( { P , , P , } , { P , , P , } , { P , , P , } ) . One considers M as a quotient Conf ( C ) / Aff (1 , C ) , where Conf ( C ) is the configuration space of sextuples ( z , . . . , z ) ∈ C , z i = z j with the identifications ( z , z , . . . ) ∼ ( z , z , . . . ) , ( . . . , z , z , . . . ) ∼ ( . . . , z , z , . . . ) , ( . . . , z , z ) ∼ ( . . . , z , z ) . The homotopy exact sequence for
Conf ( C ) → M reads Z ∼ = π ( Aff (1 , C )) −→ π ( Conf ( C )) −→ π ( M ) −→ π ( Aff (1 , C )) . Let δ be the element of π ( Conf ( C )) coming from π ( Aff (1 , C )) . It is known that δ generates the center of π ( Conf ( C )) (and even the center of a larger group, the braidgroup on strands.)Let γ : [0 , → M be the loop given by ( P , ( t ) , P , ( t ) , P , ( t ) , P , ( t ) , P , ( t ) , P , ( t ) , Q ( t )) = (0 , ∞ , , − , i, − i, ε e πit ) with respect to some inhomogeneous coordinate on CP . Introducing the transformation z → εe πit z − εe πit − z , in which Q ( t ) = ∞ , one lifts γ to Conf ( C ) as z ( t ) = − εe πit , z ( t ) = − εe πit , z ( t ) = 1 , z ( t ) = − ,z ( t ) = εe πit i − εe πit − i , z ( t ) = − εe πit i + 1 εe πit + i . It is not hard to show that the homology class of this loop is non-zero in H ( Conf ( C )); R ) and not a multiple of δ . Following [Ar], one can prove this by integrating the differentialform α := 12 πi d z − d z z − z − πi d z − d z z − z , for which R δ α = 0 yet R γ α = − . As such, one obtains: [ γ ] = 0 in H ( M ; R ) .Using the family Z → CP from Lemma 3.4 , we get a loop s : [0 , → M B with λ ( s ( t )) = γ ( t ) . The class [ s ] ∈ π ( M B ) does not lie in Im ν ∗ , as for it were, that wouldimply that [ s ] ∈ Ker λ ∗ . (Here we used the inclusion Im ν ∗ ⊂ Ker λ ∗ following from the factthat λ is D iff ( Z ) -invariant.)If γ was given either by ( P , ( t ) , P , ( t ) , P , ( t ) , P , ( t ) , P , ( t ) , P , ( t ) , Q ( t )) = (0 , ∞ , εe πit , − , i, − i, , or ( P , ( t ) , P , ( t ) , P , ( t ) , P , ( t ) , P , ( t ) , P , ( t ) , Q ( t )) = (0 , ∞ , , − , i + εe πit , − i, , then a similar argument would work to get another non-contractible loop in M B .3.5. Loops in J st . Here we construct an element of π ( J st ) that does not lie in the imageof ν ∗ : π ( D iff ( Z )) → π ( J st ) .Let J ( t ) be the loop of integrable structures with pr ( s ( t )) = J ( t ) for the loop s ( t ) constructed in §3.4 . Lemma 3.8.
The class [ J ] ∈ π ( J st ) does not lie in Im ν ∗ . Proof.
Assume the contrary, i.e. that there exists a family f : [0 , → D iff ( Z ) , f (0) = f (1) = id such that J ( t ) is homotopic to e J ( t ) := f ( t ) ∗ J (0) . We join J ( t ) and e J ( t ) with a tube T ⊂ J st . By Sard-Smale theorem we can arrange that T is transverseto pr . Thus, the preimage pr − ( T ) is a smooth orientable surface that bounds s ( t ) ∪ e s ( t ) .Note that e s ( t ) := pr − ( e J ( t )) is connected thanks to Lemma 3.7 . It follows that [ e s ] = [ s ] in H ( M B ; Z ) . This is a contradiction, as [ s ] does not lie in Ker λ ∗ , whereas λ itself isconstant on e s ( t ) . (cid:3) LLIPTIC DIFFEOMORPHISMS OF SYMPLECTIC -MANIFOLDS 25 Let’s twist again.
Here we outline the proof of
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Faculty of Mathematics and Computer Science, University of Warmia and Mazury,ul. S l oneczna 54, 10-710 Olsztyn, Poland E-mail address : [email protected] ETH Z ¨urich, R¨amistrasse 101, 8092 Z¨urich, Switzerland
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