aa r X i v : . [ m a t h . S G ] J un SYMPLECTIC BIRATIONAL GEOMETRY
TIAN-JUN LI & YONGBIN RUAN Dedicated to the occasion of Yasha Eliashberg’s 60-th birthday1.
Introduction
Birational geometry has always been a fundamental topic in algebraicgeometry. In the 80’s, an industry called Mori’s birational geometry pro-gram was created for the birational classification of algebraic manifolds ofdimension three. Roughly speaking, the idea of Mori’s program is to dividealgebraic varieties into two categories: uniruled versus non-uniruled. Theuniruled varieties are those containing a rational curve through every point.Even for this class of algebraic manifolds, the classification is usually noteasy. So one is content to carry out the classification of the more restrictiveFano manifolds and prove some structural theorems such as the Mori fiberspace structure of any uniruled variety with Fano fibers. For non-uniruledmanifolds, one wishes to construct a “minimal model” by a sequence of con-traction analogous to the blow-downs. One immediate problem is that acontraction of smooth variety often results a singular variety. This technicalproblem often makes the subject of birational geometry quite difficult. Onlyrecently, the minimal model program was carried out to a large extent inhigher dimensions in the remarkable papers [1] and [56].In the early 90’s, the second author observed that some aspects of this ex-tremely rich program of Mori can be extended to symplectic geometry via thenewly created Gromov-Witten theory [50]. Specifically, he extended Mori’snotion of extremal rays to the symplectic category and used it to studythe symplectomorphism group. A few years later, Koll´ar and the secondauthor showed that a smooth projective uniruled manifold carries a non-zero genus zero Gromov-Witten invariant with a point insertion. Shortlyafter, further deep relations between the Gromov-Witten theory and bira-tional geometry were discovered in [51], resulting in the speculation thatthere should be a symplectic birational geometry program. In the meantime the Gromov-Witten theory, together with the Seiberg-Witten theory,was applied with spectacular success to obtain basic structure theorems ofsymplectic 4-manifolds, especially the rational and ruled ones, cf. [43], [57],[36], [29].To be more specific, we define a genus 0 GW class as a nonzero degree 2homology class supporting nontrivial genus 0 GW invariants. Let us then , supported by NSF Grants. & YONGBIN RUAN ask the following question: what kind of structures of a symplectic manifoldare detected by its genus zero GW classes? What we would like to convey inthis article is that the answer is precisely the symplectic birational structure!
To start with consider the sweep out of all the pseudo-holomorphic rationalcurves in a GW class. The extreme case is that the sweep out is the wholespace. In this case the manifold is likely a uniruled manifold . In general,the sweep out is likely a possibly singular uniruled submanifold . It has beengradually realized that the symplectic birational geometry deals preciselywith these uniruled manifolds and uniruled submanifolds.In this article, we outline the main elements of this new program in sym-plectic geometry. Let us first mention that some technical difficulties inthe algebraic birational geometry are still present in our program, but wemight be able to treat them with more symplectic topological techniques.This is certainly the case for contractions. Recall that the goal of birationalgeometry is to classify algebraic varieties in the same birational class. Twoalgebraic varieties are birational to each other if and only if there is a bi-rational map between them. A birational map is an isomorphism betweenZariski open sets, but it is not necessarily defined everywhere. If a birationalmap is defined everywhere, we call it a contraction. A contraction changes alower dimensional uniruled subvariety only, hence we can view it as a surgery.Intuitively, a contraction simplifies a smooth variety, but as already men-tioned, it often produces a singular variety. Various other types of surgeriesare needed to deal with the resulting singularities. The famous ones are flipand flops which are much more subtle than contractions. We certainly cannot avoid some aspects of this issue in our program. A major problem inour program is then to see whether the flexibility in the symplectic categorycan produce many such kinds of surgery operations. In particular, we wouldlike to interpret and construct flips and flops symplectically.A new phenomenon in our program is that many obvious properties ofalgebraic birational geometry are no longer obvious in the symplectic cat-egory. Notably, the birational invariance of uniruledness in [13] is such anexample, where we have to draw newly developed powerful technology fromthe Gromov-Witten theory.However, this perspective makes the subject distinctively symplectic. Anddespite of these ‘old’ and ‘new’ obstacles, major progress has been maderecently in [13], [31], [32], [45].One eventual and remote goal of symplectic geometry is to classify sym-plectic manifolds. Symplectic birational geometry can be considered as thefirst step towards such a classification [51]. In addition, the author hopethat symplectic topological techniques and view points in this new programwill also bring some fresh insight to the birational classification of algebraicmanifolds.The article is organized as follows. We will set up symplectic birationalequivalence in section two. The transition as an extended symplectic bira-tional transformation will also be discussed. Section three is devoted to the
YMPLECTIC BIRATIONAL GEOMETRY 3 birational invariance of uniruled manifolds and its classifications. In sectionfour, we will discuss the dichotomy of uniruled submanifolds. In section five,we will briefly discuss speculations on minimal symplectic manifolds. Wedescribe various GW correspondences in section six. We finish the paperby several concluding remarks. We should mention that this article doesnot contain any proofs but provide appropriate references for the resultsdiscussed.Both author would like to thank Y. Eliashberg for the inspiration and en-couragement over so many years. It is our pleasure to dedicate this articleto the occasion of his 60-th birthday. Special thanks go to D. McDuff formany inspirational discussions and careful readings of manuscripts of ourpapers as well as many useful comments. Her influence on this subject iseverywhere. We appreciate the referee’s suggestions which made the articlemore readable. We are grateful to J. Hu for fruitful collaborations. Discus-sions with Y. P. Lee and D. Maulik on recent GW techniques in algebraicgeometry, with J. Dorfmeister on relative symplectic cones, and with WeiyiZhang on the geometry of uniruled manifolds are really helpful. We also ap-preciate the interest of V. Guillemin, H. Hofer, C. LeBrun, M. Liu, N. Mok,C. Voisin, C. Taubes, S. T. Yau, A. Zinger and many others. Finally wethank the algebraic geometry FRG group to organize a wonderful conferenceon this subject in Stony Brook.2.
Birational equivalence in symplectic geometry
For a long time, it was not really clear what is an appropriate notion ofbirational equivalence in symplectic geometry. Simple birational operationssuch as blow-up/blow-down were known in symplectic geometry for a longtime [12, 47]. But there is no straightforward generalization of the notion ofa general birational map in the flexible symplectic category. The situationchanged a great deal when the weak factorization theorem was establishedrecently (see the lecture notes [39] and the reference therein) that any bira-tional map between projective manifolds can be decomposed as a sequenceof blow-ups and blow-downs. This fundamental result resonates perfectlywith the picture of the wall crossing of symplectic reductions analyzed byGuillemin-Sternberg in the 80’s. Therefore, we propose to use their notionof cobordism in [12] as the symplectic analogue of the birational equivalence(see Definition 2.1). To avoid confusion with other notions of cobordism inthe symplectic category, we would call it symplectic birational cobordism .2.1.
Birational equivalence.
The basic reference for this section is [12].We start with the definition which is essentially contained in [12].
Definition 2.1.
Two symplectic manifolds ( X, ω ) and ( X ′ , ω ′ ) are birationalcobordant if there are a finite number of symplectic manifolds ( X i , ω i ) , ≤ i ≤ k , with ( X , ω ) = ( X, ω ) and ( X k , ω k ) = ( X ′ , ω ′ ) , and for each i , ( X i , ω i ) and ( X i +1 , ω i +1 ) are symplectic reductions of a semi-free Hamilton-ian S symplectic manifold W i (of 2 dimensions higher). TIAN-JUN LI & YONGBIN RUAN Here an S action is called semi-free if it is free away from the fixed pointset.There is a related notion in dimension 4 in [48]. However we remark thatthe cobordism relation studied in this paper is quite different from someother notions of symplectic cobordisms, see [4], [5], [8], [9].According to [12], we have the following basic factorization result. Theorem 2.2.
A birational cobordism can be decomposed as a sequence ofelementary ones, which are modeled on blow-up, blow-down and Z − lineardeformation of symplectic structure. A Z − linear deformation is a path of symplectic form ω + tκ , t ∈ I , where κ is a closed 2 − form representing an integral class and I is an interval. Itwas shown in [13] that Z − linear deformations are essentially the same asgeneral deformations.Observe that a polarization on a projective manifold, which is simply avery ample line bundle, gives rise to a symplectic form with integral class,well-defined up to isotopy. Together with the weak factorization theoremmentioned in the previous page, we then have Theorem 2.3.
Two birational projective manifolds with any polarizationsare birational cobordant as symplectic manifolds. Uniruled symplectic manifold
Basic definitions and properties.
Let us first recall the notion ofuniruledness in algebraic geometry.
Definition 3.1.
A projective manifold X (over C ) is called (projectively)uniruled if for every x ∈ X there is a morphism f : P → X satisfying x ∈ f ( P ) , i.e. X is covered by rational curves. A beautiful property of of a uniruled projective manifold ([18], [17]) isthat general rational curves are unobstructed, hence regular in the sense ofthe Gromov-Witten theory. It is this property which underlies the afore-mentioned result of Koll´ar and Ruan (stated here in a sharper form noticedby McDuff, cf. [31]).
Theorem 3.2.
A projective manifold is projectively uniruled if and only if h [ pt ] , [ ω ] p , [ ω ] q i XA > for a nonzero class A , a K¨ahler form ω and integers p, q . Here [ pt ] denotesthe fundamental cohomology class of X . In this article, for a closed symplectic manifold (
X, ω ), we denote itsgenus zero GW invariant in the curve class A ∈ H ( X ; Z ) with cohomologyconstraints α , · · · , α k ∈ H ∗ ( X ; R ) by(1) h α , · · · , α k i XA . YMPLECTIC BIRATIONAL GEOMETRY 5
To define it we need to first choose an ω − tamed almost complex structure J and consider the moduli space of J − holomorphic rational curves with k marked points in the class A . Via the evaluation maps at the markedpoints we pull back α i to cohomology classes over the moduli space, and theinvariant (1) is supposed to be the integral of the cup product of the pull-back classes over the moduli space. Due to compactness and transversalityissues the actual definition requires a great deal of work ([6], [38], [54], [53]).Intuitively the GW invariant (1) counts J − holomorphic rational curves inthe class A passing through cycles Poincar´e dual to α i . Definition 3.3.
Let A ∈ H ( X ; Z ) be a nonzero class. A is called a GWclass if there is a non-trivial genus zero GW invariant of ( X, ω ) with curveclass A . A is said to be a uniruled class if it is a GW class and moreover,there is a nonzero GW invariant of the form (2) h [ pt ] , α , · · · , α k i XA , where α i ∈ H ∗ ( X ; R ) . X is said to be (symplectically) uniruled if there is auniruled class. Remark 3.4.
It is easy to see that we could well use the GW invariantswith a disconnected domain to define this concept, subject to the requirementthat the curve component with the [ pt ] constraint represent nonzero classin H ( X ; Z ) . This flexibility is important for the proof of the birationalcobordism invariance. This notion has been studied in the symplectic context by G. Lu (see[34], [35]). Notice that, by [25], it is not meaningful to define this notionby requiring that there is a symplectic sphere in a fixed class through everypoint, otherwise every simply connected manifold would be uniruled.
Remark 3.5.
According to Theorem 3.2 a projectively uniruled manifold issymplectically uniruled, in fact strongly symplectically uniruled. Here X issaid to be strongly uniruled if there is a nonzero invariant of the form (2)with k = 3 . Obviously the only uniruled 2–manifold is S . In dimension 4 the converseis essentially true (see Theorem 4.3). While in higher dimensions it followsfrom [10] (see also [35]) that there are uniruled symplectic manifolds whichare not projective, and it follows from [52] that there could be infinitelymany distinct uniruled symplectic structures on a given smooth manifold.There are also descendant GW invariants which are variations of the GWinvariants with constraints of the form τ j i ( α i ). Here the class τ j i ( α i ) overthe moduli space is the cup product of the pull-back of the class α i and the j i − th power of a natural degree 2 class, which is the 1st Chern class of the(orbifold) line bundle over the moduli space whose fibers are the cotangentlines at the i − th marked point.It is very useful to characterize uniruledness using these more general GWinvariants ([13]). TIAN-JUN LI & YONGBIN RUAN Theorem 3.6.
A symplectic manifold X is uniruled if and only if there isa nonzero, possibly disconnected genus zero descendant GW invariant (3) h τ j ([ pt ]) , τ j ( α ) , · · · , τ j k ( α k ) i XA such that the curve component with the [ pt ] constraint has nonzero curveclass. In particular, Theorem 3.6 is used to establish the fundamental birationalinvariance property of uniruled manifolds in [13].
Theorem 3.7.
Symplectic uniruledness is invariant under symplectic blow-up and blow-down.
Constructions.
An important aspect of symplectic birational geome-try is the classification of uniruled manifolds. This remains to be a distantgoal. A more immediate problem is to construct more examples. Koll´ar-Ruan’s theorem shows that all the algebraic uniruled manifold is symplecticuniruled. Another class of example is from the following beautiful theoremof McDuff in [45].
Theorem 3.8.
Any Hamiltonian S -manifold is uniruled. Here a Hamil-tonian S -manifold is a symplectic manifold admitting a Hamiltonian S -action. A rich source of uniruled manifolds comes from almost complex uniruledfibrations. Suppose that π : X → B is a fibration (with possibly singularfibers) where X and B are symplectic manifolds. We call it an almost com-plex fibration if there are tamed J, J ′ for X, B such that π is almost complex.Symplectic fiber bundles over symplectic manifolds in the sense of Thurstonare almost complex fibrations. Lefschetz fibrations, or more generally, lo-cally holomorphic fibrations studied in [11] are also almost complex.Let ι : π − ( b ) → X be the embedding for a smooth fiber over b ∈ B . Wehave the following result in [31] by a direct geometric argument. Proposition 3.9.
Suppose that π : X → B is an almost complex fibra-tion between symplectic manifolds X, B . Then, for A ∈ H ( π − ( b ); Z ) and α , ..., α k ∈ H ∗ ( X ; R ) , (4) < [ pt ] , ι ∗ α , · · · , ι ∗ α k > π − ( b ) A = < [ pt ] , α , · · · , α k > Xι ∗ ( A ) . Corollary 3.10.
Suppose that π : X → B is an almost complex fibrationbetween symplectic manifolds X, B . If a smooth fiber is uniruled and homo-logically injective (over R ), then X is uniruled. The homologically injective assumption could be a strong one. Noticethat for a fiber bundle, the Leray-Hirsch theorem asserts that, under thehomologically injective assumption, the homology group of the total spaceis actually isomorphic to the product of the homology group of the fiberand the base. However, Corollary 3.10 can still be applied for all productbundles, and all projective space fibrations (more generally, if the rational
YMPLECTIC BIRATIONAL GEOMETRY 7 cohomology ring of a smooth uniruled fiber is generated by the restrictionof [ ω ]).Moreover, we were informed by McDuff that a Hamiltonian bundle ishomologically injective (or equivalently, cohomologically split) if (cf. [37])a) the base is S (Lalonde-McDuff-Polterovich), and more generally, acomplex blow up of a product of projective spaces,b) the fiber satisfies the hard Lefschetz condition (Blanchard), or its realcohomology is generated by H .Here is another variation. As in the case of a projective space, for auniruled manifold up to dimension 4, insertions of a uniruled class can allbe assumed to be of the form [ ω ] i , thus we also have Corollary 3.11.
If the general fibers of a possibly singular uniruled fibrationare -dimensional or − dimensional, then the total space is uniruled. This in particular applies to a 2 − dimensional symplectic conic bundle.A symplectic conic bundle is a conic hypersurface bundle in a smooth P k bundle. Holomorphic conic bundles are especially important in the theoryof 3 − folds. It is conjectured that a projective uniruled 3 − fold is eitherbirational to a trivial P − bundle or a conic bundle.Another important construction first analyzed by McDuff is the divisorto ambient space procedure. It is part of the dichotomy of uniruled divisorsand would be discussed in the next section (see Theorem 4.1).3.3. Geometry.
Recall that the symplectic canonical class K ω of ( X, ω )is defined to be − c ( T X, J ) for any ω − tamed almost complex structure J .Observe that for a uniruled manifold K ω is negative on any uniruled classby a simple dimension computation of the moduli space. In particular, K ω cannot be represented by an embedded symplectic submanifold. It leads tothe following intriguing question. Question 3.12.
Does a uniruled manifold of (real) dimension n have anegative K iω · [ ω ] n − i for some i ? On the other hand, we could ask if the canonical class K ω is negative inthe sense K ω = λ [ ω ] for λ >
0, is the symplectic manifold uniruled? Sucha manifold is known as a monotone manifold in the symplectic category,and it is the analogue of a Fano manifold. Fano manifolds are projectivelyuniruled by the famous bend-and-break argument of Mori. In fact, Fanomanifolds are even rationally connected.We also observe here that uniruled manifolds satisfy a simple ball packingconstraint. To state it let us introduce the notion of a minimal uniruled class,which is a uniruled class with minimal symplectic area among all uniruledclasses. This notion will also play a crucial role in subsection 4.1. Then itfollows from Gromov’s monotonicity argument that the size of any embeddedsymplectic ball is bounded by the area of a minimal uniruled class. The converse of this question should be compared with the Mumford conjecture: aprojective manifold is uniruled if and only if it has Kodaira dimension −∞ . TIAN-JUN LI & YONGBIN RUAN Rationally connected manifolds.
A projective manifold is ratio-nally connected if given any two points p and q there is a rational curveconnecting p and q . It is equivalent to “chain rational connected” wherethere is a chain of rational curves connecting p and q .The outstanding conjecture is: A projective manifold X is rationally con-nected if and only if there is a nonzero connected GW invariant of the form (5) < τ j ([ pt ]) , τ j ([ pt ]) , · · · , τ j k ( β k ) > XA = 0 for A = 0 . We define a symplectic manifold to be rationally connected if there is anonzero invariant of the form (5). Obviously, a rationally connected manifoldis uniruled.Whether symplectic rational connectedness is a birational property ap-pears to be a hard question (cf. [55]), though we believe it is possible toshow the invariance under certain types of blow-ups. Characterizing suchmanifolds is likewise more difficult. But at least we know that all such man-ifolds in dimension 4 are rational. Moreover, it is expected that symplecticmanifold containing a rationally connected symplectic divisor with certainstrong positivity is rationally connected (Initial progress has been made in[14]).Of course we can also similarly define N − rationally connectedness forany integer N ≥
2. Moreover, it is not hard to see that there is a parkingconstraint for N balls in terms of (the area of) a minimal N − rationallyconnected class. 4. Dichotomy of uniruled divisors
We have seen that up to birational cobordism symplectic manifolds arenaturally divided into uniruled ones and non-uniruled ones. In this sectionwe discuss uniruled submanifolds of codimension 2, which we simply callsymplectic divisors. One motivation comes from the basic fact in algebraicgeometry that various birational surgery operations such as contraction andflop have a common feature: the subset being operated on is necessarilyuniruled.Our key observation is that, as in the projective birational program, such auniruled symplectic divisor admits a dichotomy depending on the positivityof its normal bundle. If the normal bundle is non-negative in certain sense,it will force the ambient manifold to be uniruled. If the normal bundleis negative in certain sense, we can ‘contract’ it to simplify the ambientmanifold.We have a rather general result in the non-negative case. In the secondcase our progress is limited to simple uniruled divisors in 6–manifolds.4.1.
Dichotomy of uniruled divisor–non-negative half.
Suppose that ι : D → ( X, ω ) is a symplectic divisor (which we always assume to besmooth). Let N D be the normal bundle of D in X . Notice that N D is a YMPLECTIC BIRATIONAL GEOMETRY 9 − dimensional symplectic vector bundle and hence has a well defined firstChern class. We will often use N D to denote the first Chern class. Theorem 4.1.
Suppose D is uniruled and A is a minimal uniruled class of D such that (6) < ι ∗ α , · · · , ι ∗ α l , [ pt ] , β , · · · , β k > DA = 0 for α i ∈ H ∗ ( X ; R ) , β j ∈ H ∗ ( D ; R ) with β = [ pt ] , and k ≤ N D ( A ) + 1 . Then ( X, ω ) is uniruled. Here what matters in (6) is the number of insertions which do not comefrom X . There are situations where can simply take k = 1 hence only require N D ( A ) be non-negative. In particular, we have Corollary 4.2.
Suppose D is a homologically injective uniruled divisor of X and the normal bundle N D is non-negative on a minimal uniruled class.Then X is uniruled. We can ask whether the converse of Theorem 4.1 is also true. It is obviousin dimension 4. In higher dimension we could easily construct non-negativesingular uniruled divisors. The hard and important question is whether wecan they can be smoothed inside X .As mentioned in the previous section, Theorem 4.1 can also be viewedas a construction of uniruled manifolds, generalizing several early results ofMcDuff. We list some examples here, more examples can be found in [31]. : A deep result in dimension 4 is thatuniruled manifolds can be completely classified. Theorem 4.3. ( [42] , [29] , [30] , [36] , [57] ) A − manifold ( M, ω ) is uniruledif and only if it is rational or ruled. Moreover, the isotopy class of ω isdetermined by [ ω ] . Here symplectic 4 − manifold ( M, ω ) is called rational if its underlyingsmooth manifold M is S × S or P k P for some non-negative integer k .( M, ω ) is called ruled if its underlying smooth manifold M is the connectedsum of a number of (possibly zero) P with an S − bundle over a Riemannsurface.We need to analyze minimal uniruled classes and the corresponding in-sertions. Proposition 4.4. If A is a uniruled class of a − manifold, then A is repre-sented by an embedded symplectic surface, and A satisfies (i) K ω ( A ) ≤ − ,(ii) A ≥ , (iii) A · B ≥ for any class B with a non-trivial GW invariantof any genus. For P , let H be the generator of H with positive area. H is a uniruledclass and any uniruled class of the form aH with a >
0. Obviously, H isthe minimal uniruled class. The relevant insertion is ([ pt ] , [ pt ]). As [ pt ] is arestriction class, i.e. an α class, we can take k = 1. & YONGBIN RUAN Similarly, for the blow-up of an S − bundle over a surface of positivegenus, the fiber class is a uniruled class, and any uniruled class is a positivemultiple of the fiber class. The relevant insertion for the fiber class is [ pt ].Thus again we can take k = 1.It is easier to apply Theorem 4.1 in this case. Corollary 4.5.
Suppose ( X , ω ) contains a divisor D which is diffeomorphicto P or the blow-up of an S − bundle over a surface of positive genus. If thenormal bundle N D is non-negative on a uniruled class, then X is uniruled. For other M , the uniruled classes are not proportional to each other.Thus the minimality condition depends on the class of the symplectic formon M .We first analyze the easier case of an S − bundles over S . For S × S , byuniqueness of symplectic structures, any symplectic form is of product form.Let A and A be the classes of the factors with positive area. It is easy tosee that any uniruled class is of the form a A + b A with a ≥ , a ≥ A or A has the minimal area.For the nontrivial bundle S ˜ × S = P P , let F be the class of a fiberwith positive area and E be the unique − aF + bE is a uniruled class then b ≥ F · E = 1 , F · F = 0. And if b >
0, then a ≥ F is always the minimal uniruled class no matter what the symplecticstructure is.Since the relevant insertion for A , A and F is just [ pt ], we have Corollary 4.6.
Suppose D = S × S and the restriction of the normal bun-dle N D to the factor with the least area is non-negative, then X is uniruled.In the case of the non-trivial bundle, X is uniruled if the restriction of thenormal bundle N D to F is non-negative. The remaining M are connected sums of P with at least 2 P . It iscomplicated to analyze minimal uniruled classes in this case. In [31] it isshown that they are generated by the so called fiber classes. Higher dimensional case:
In higher dimension, it is still a remotegoal to classify all the uniruled symplectic manifolds. Instead of consideringan arbitrary uniruled symplectic divisor, we start with Fano hypersurfaces.When the divisor D ⊂ ( X, ω ) is symplectomorphic to a divisor of P n (for n ≥
4) of degree at most n , D is Fano and hence uniruled. Of course aparticular case is D = P n − . Since n ≥
4, by the Lefschetz hyperplanetheorem, b = 1 for D . According to Theorem 3.2, for a minimal uniruledclass A , we can take k to be equal to 1. Hence X is uniruled if N D = λ [ ω | D ]with λ ≥ §
2. Letus treat the case of a symplectic P k − bundle. Since the line class in the fiber YMPLECTIC BIRATIONAL GEOMETRY 11 is uniruled, and the relevant insertions can be taken to be ([ pt ] , [ ω | D ] k ), wehave Corollary 4.7.
Suppose D is a symplectic divisor of X . If D is a projectivespace bundle with the fiber class being the minimal uniruled class and normalbundle N D being non-negative along the fibers, then X is unruled. McDuff also considered the case of product P k − bundles in [44]. A naturalsource of such a D is from blowing up a ‘non-negative’ P k with a large trivialneighborhood. Suppose P k ⊂ X has trivial normal bundle. Then the blowup along P k has a divisor D = P k × P n − k − . The normal bundle of D alonga line in P k is trivial. Similar to the case of S × S , we can argue that thearea of this line is minimal among all uniruled class of D . In particular, asobserved by [44], a symplectic P with a sufficiently large product symplecticneighborhood can only exist in a uniruled manifold.In fact we can prove more. Corollary 4.8.
Suppose S is a uniruled symplectic submanifold whose min-imal uniruled class has area η and insertions all being restriction classes.If S has a trivial symplectic neighborhood of radius at least η . Then X isuniruled. Symplectic ‘blowing-down’ in dimension six.
Blowing up in di-mension 6 gives rise to a symplectic P with normal c = − S − bundle with normal c = − S − fibers. A natural questionis whether such a uniruled divisor always arises from a symplectic blow-up.In other words, we are interested in a criterion for blowing-down. A nicefeature is that by Theorem 4.3 the answer would also only depend on [ ω ].Moreover, as every symplectic structure on such a 4–manifold is K¨ahler, wecan apply algebro-geometric techniques to understand this problem.The case of P is simple: it can always be blown down just as in thecase of P with self-intersection − S − bundle over a Riemann surface Σ g of genus g , it is morecomplicated and perhaps more interesting. Topologically blowing down an S − bundle over Σ g is the same as topologically fiber summing with the pairof a P − bundle and an embedded P − bundle over Σ g with opposite normalbundle (see 6.1). To perform the fiber sum symplectically we also need tomatch the symplectic classes of the divisors. For this purpose we need tounderstand the relative symplectic cone of such a pair. We determine in [32]the relative K¨ahler cone for various complex structures coming from stableand unstable rank 3 holomorphic bundles over Σ g . Consequently we obtain Theorem 4.9.
Let ( X, ω ) be a symplectic manifold of dimension 6. Let D bea symplectic divisor which is an S − bundle over Σ g . Suppose N D ( f ) = − and N D ( s ) = d , where f is the fiber class of D , and s is a section classof D with square if D is a trivial bundle and square − if D is a non-trivial bundle. Further assume that [ ω | D ]( f ) = a and [ ω | D ]( s ) = b . Then a & YONGBIN RUAN symplectic fiber sum can be performed with a symplectic ( P , P ) / Σ g pair ifeither d ≥ , or(i) d < , g = 0 , b > [ − d +23 ] a ,(ii) d < , g ≥ , b > − d a . Here [ x ] denotes the largest integer bounded by x from above. This resultis optimal in the case of genus 0. For instance, we show that a symplectic S × S with normal c = − S can be blown downif the symplectic areas of the two factors are not the same. This pictureis consistent with the flop operation for projective 3–folds. On the otherhand, when g ≥
1, a symplectic S × Σ g with normal c = − S factor is atmost 3 times of that of the Σ factor. We would very much like to find outwhether the restriction on the areas is really necessary in the case of positivegenus. The picture would be rather nice if the restriction can eventually beremoved. On the other hand, it would be surprising, even intriguing, if itturns out there is an obstruction. We are also interested to see how muchof the 6–dimensional investigation can be carried out to higher dimensions.Another remaining issue is whether such a symplectic fiber sum is actuallya birational cobordism operation. This is because that a symplectic blow-ing down further requires the symplectic ( P , P ) / Σ g pair have an infinitysymplectic section, which is a symplectic surface of genus g . We are inves-tigating wether our more general fiber sums are equivalent to symplecticblowing down up to deformation. Either positive or negative answer wouldbe very interesting.5. Minimal symplectic manifolds
Minimality.
Motivated by the Mori program for algebraic 3–folds andunderstandings of symplectic 4–manifolds (and 2–manifolds), we discuss inthis section the notion of minimal manifolds in dimension 6.Let us first recall the notion of minimality by McDuff in dimension 4. Let E X be the set of homology classes which have square − X is smoothly minimal if E X is empty. Let E X,ω be the subset of E X which are represented by embedded ω − symplectic spheres. We say that ( X, ω ) is symplectically minimal if E X,ω is empty. When (
X, ω ) is non-minimal, one can blow down some of the sym-plectic − − manifold ( N, µ ), whichis called a (symplectic) minimal model of (
X, ω ) ([Mc]). We summarize thebasic facts about the minimal models in the following proposition.
Theorem 5.1 ([28], [30], [42], [57]) . Let X be a closed oriented smooth − manifold and ω a symplectic form on X compatible with the orientationof X .1. X is smoothly minimal if and only if ( X, ω ) is symplectically minimal.In particular the underlying smooth manifold of the (symplectic) minimalmodel of ( X, ω ) is smoothly minimal. YMPLECTIC BIRATIONAL GEOMETRY 13
2. If ( X, ω ) is not rational nor ruled, then it has a unique (symplectic) min-imal model. Furthermore, for any other symplectic form ω ′ on X compatiblewith the orientation of X , the (symplectic) minimal models of ( X, ω ) and ( X, ω ′ ) are diffeomorphic as oriented manifolds.3. If ( X, ω ) is rational or ruled, then its (symplectic) minimal models arediffeomorphic to CP or an S − bundle over a Riemann surface. Definition 5.2.
A symplectic 6–manifold is minimal if it does not containany rigid stable uniruled divisor.
Here, a uniruled divisor is stable if one of its uniruled classes A is a GWclass of the ambient manifold with K ω ( A ) ≤ −
1. And a uniruled divisor isrigid if none of its uniruled class is a uniruled class of the ambient manifold.We observe this definition also applies to manifolds of dimensions 2 and4. Every 2–manifold is obviously minimal as the only divisors are points.For a 4 manifold any uniruled divisor is S . Let A be the class of a stable S . Then K ω ( A ) ≤ −
1. On the other hand a rigid S must have K ω ( A ) < S with K ω ( A ) = − − K − negative) extremal rays for algebraic 3–folds give rise to non-trivial GW classes. One such an extremal ray arises from the line class ofa P divisor with normal c = −
2. To carry out the contraction one has toenlarge to the category of symplectic orbifolds.Now we single out an important class of minimal manifolds.
Definition 5.3.
We define a cohomology class α ∈ H ( X, Z ) to be nef if itis non-negative on all GW classes. Lemma 5.4.
Let ( X, ω ) be a manifold with nef K ω . Then ( M, ω ) is non-uniruled and minimal.Proof. The first statement is obvious as any uniruled class A of X satisfies K ω ( A ) ≤ − X as K ω ( A ) ≤ − A which is a uniruled class of a stable divisor as well as a GW class of X . (cid:3) A minimal model of a uniruled manifold is still uniruled and hence can notbe K ω − nef. The natural question is whether any minimal model of a non-uniruled manifold must have nef K ω . The first step towards this question & YONGBIN RUAN is to show that any GW class A with K ω ( A ) ≤ − K ω − nef manifoldsare related by an analogue of the K − equivalence. Two algebraic manifolds X, X ′ are K − equivalent if there is a common resolutions π : Z → X, π : Z → X ′ such that π ∗ K X = π ∗ K X ′ . K − equivalent manifolds have manybeautiful properties, in particular, they have the same betti numbers.5.2. Kodaira dimension.
The notion of Kodaira dimension has been de-fined for symplectic manifolds up to dimension 4 ([26], [47]). Whenever itis defined it is a finer invariant of birational cobordism then uniruledness.The Kodaira dimension of a 2–dimensional symplectic manifold (
F, ω ) isdefined as κ ( F, ω ) = −∞ if K ω · [ ω ] < K ω · [ ω ] = 0,1 if K ω · [ ω ] > κ ( F, ω ) = −∞ , , F is 0 , , ≥ κ ( F, ω ) = −∞ if and only if ( F, ω ) is uniruled.For a minimal symplectic 4 − manifold ( X, ω ) its Kodaira dimension isdefined in the following way ([21], [47], [26]): κ ( X, ω ) = −∞ if K ω · [ ω ] < K ω · K ω < , K ω · [ ω ] = 0 and K ω · K ω = 0 , K ω · [ ω ] > K ω · K ω = 0 , K ω · [ ω ] > K ω · K ω > . The Kodaira dimension of a non-minimal manifold is defined to be that ofany of its minimal models.Based on the Seiberg-Witten theory and properties of minimal models(cf. Theorem 5.1, [30], [24], [48]), [57]) it is shown in [26] that the Kodairadimension κ ( M, ω ) is well defined. In particular, we need to check that aminimal 4-manifold cannot have(7) K ω · [ ω ] = 0 , K ω · K ω > . We list some basic properties of κ ( X, ω ). It is also observed in [26] that,if ω is a K¨ahler form on a complex surface ( X, J ), then κ ( X, ω ) agrees withthe usual holomorphic Kodaira dimension of (
X, J ).(
X, ω ) has κ = −∞ if and only if it is uniruled.It is further shown in [26] that minimal symplectic 4 − manifolds with κ =0 are exactly those with torsion canonical class, thus they can be viewed as symplectic Calabi-Yau surfaces . Known examples of symplectic 4 − manifoldswith torsion canonical class are either K¨ahler surfaces with (holomorphic)Kodaira dimension zero or T − bundles over T . It is shown in [27] and [2]that a minimal symplectic 4 − manifold with κ = 0 has the rational homologyas that of K3 surface, Enriques surface or a T − bundle over T . YMPLECTIC BIRATIONAL GEOMETRY 15
Suppose (
X, ω ) is a minimal 6–dimensional manifold. we propose to defineits Kodaira dimension in the following way : κ ( X, ω ) = (cid:26) −∞ if one of K iω · [ ω ] − i is negative, k if K iω · [ ω ] − i = 0 for i ≥ k , and K iω · [ ω ] − i > i ≤ k .Notice that, as in dimension 4, there is the issue of well definedness of κ ( X, ω ). And this leads to some possible intriguing properties of of minimal6–manifolds, one of which is whether there is any minimal 6–manifold with(8) K ω · [ ω ] = 0 , K ω [ ω ] = 0 , K ω > . Correspondences in the Gromov-Witten theory
As the Gromov-Witten theory is built into the foundation of symplecticbirational geometry, it is natural that we use many techniques from theGromov-Witten theory such as localization and degenerations. It turns outthat we have to use very sophisticated Gromov-Witten machinery. Take thebirational invariance of the uniruledness as an example. The definition ofuniruledness requires only a single non-vanishing GW invariant. However,it is well-known that a single Gromov-Witten invariant tends to transformin a rather complicated fashion. On the other hand, it is often easier tocontrol the transformation of the Gromov-Witten theory as a whole. Oneoften phrases such a amazing phenomenon as a kind of correspondences.There are many examples such as the Donaldson-Thomas/Gromov-Wittencorrespondence [40], the crepant resolution conjecture [52] and so on.The correspondences appearing in our context are not as strong as theabove ones. Its first example is the “relative/absolute correspondence” con-structed by Maulik-Okounkov-Pandharipande ([41]). It is the generalizationto the situation of blow-up/down by the authors and Hu which underlies thebirational invariance of uniruledness. And Theorem 4.1 is proved by anothertechnical variation of the relative/absolute correspondences incorporatingdivisor invariants. Roughly speaking, a correspondence in this context is apackage to organize the degeneration formula in a very nice way.We restrict ourselves to genus GW invariants in this article.6.1.
Symplectic cut and the degeneration formula.
Symplectic cut along a submanifold.
Let (
X, ω ) be a closed symplec-tic manifold. Let S be a hypersurface having a neighborhood with a freeHamiltonian S − action. For instance, if there is a symplectic submanifoldin X , then the hypersurfaces corresponding to sphere bundles of the normalbundle have this property. Let Z be the symplectic reduction at the level S ,then Z is the S − quotient of S and is a symplectic manifold of 2 dimensionless. Compare with Question 3.12 & YONGBIN RUAN We can cut X along S to obtain two closed symplectic manifolds ( X + , ω + )and ( X − , ω − ) each containing a smooth copy of Z , and satisfying ω + | Z = ω − | Z ([20]).In particular, the pair ( ω + , ω − ) defines a cohomology class of X + ∪ Z X − ,denoted by [ ω + ∪ Z ω − ]. Let p be the continuous collapsing map p : X → X + ∪ Z X − . It is easy to observe that(9) p ∗ ([ ω + ∪ Z ω − ]) = [ ω ] . Let ι : D → X be a smooth connected symplectic divisor. Then we cancut along D , or precisely, cut along a small circle bundle S over D inside X .In this case, as a smooth manifold, X + = X , which we will denote by˜ X . Denote the symplectic reduction of S in ˜ X still by D . Notice however,the symplectic structure is different from the original divisor. And X − = P ( N D ⊕ C ), the projectivization of P ( N D ⊕ C ) . We will often denote itsimply by P D or P . Notice that P ( N D ⊕ C ) has two natural sections, D = P (0 ⊕ C ) , D ∞ = P ( N D ⊕ . The symplectic reduction of S in P D is the section D ∞ .In summary, in this case, X degenerates into ( ˜ X, D ) and ( P D , D ∞ ). Wealso denote ω − by ω P .More generally, we can cut along a symplectic submanifold Q of codimen-sion 2 k , or precisely, cut along a sphere bundle S over Q . Then X + is asymplectic blow up of X along Q and the symplectic reduction Z ⊂ X + isthe exceptional symplectic divisor. In this case X − = P ( N Q ⊕ C ), which isa P k bundle over Y .6.1.2. Degeneration formula.
Given a symplectic cut, there is a basic linkbetween absolute invariants of X and relative invariants of ( X ± , Z ) in [33](see also [15], and [23] in algebraic geometry). We now describe such aformula.Let B ∈ H ( X ; Z ) be in the kernel of p ∗ : H ( X ; Z ) −→ H ( X + ∪ Z X − ; Z ) . By (9) we have ω ( B ) = 0. Such a class is called a vanishing cycle. For A ∈ H ( X ; Z ) define [ A ] = A + Ker( p ∗ ) and(10) h τ d α , · · · , τ d k α k i X [ A ] = X B ∈ [ A ] h τ d α , · · · , τ d k α k i XB . At this stage we need to assume that each cohomology class α i is of theform(11) α i = p ∗ ( α + i ∪ Z α − i ) . Notice that our convention here is opposite to that in [13]
YMPLECTIC BIRATIONAL GEOMETRY 17
Here α ± i ∈ H ∗ ( X ± ; R ) are classes with α + i | Z = α − i | Z so that they give riseto a class α + i ∪ Z α − i ∈ H ∗ ( X + ∪ Z X − ; R ).The degeneration formula expresses h τ d α , · · · , τ d k α k i X [ A ] as a sum ofproducts of relative invariants of ( X + , Z ) and ( X − , Z ), possibly with dis-connected domains. In each product of relative invariants, what is relevantfor us are the following conditions: • the union of two domains along relative marked points is a stable genus0 curve with k marked points, • the total curve class is equal to p ∗ ( A ), • the relative insertions are dual to each other, • if α + i appears for i in a subset of { , · · · , k } , then α − j appears for j inthe complementary subset of { , · · · , k } .In the case of cutting along a symplectic submanifold it is easy to showthat all the invariants on the right hand side of (10) vanish except h τ d α , · · · , τ d k α k i XA . Thus the degeneration formula computes h τ d α , · · · , τ d k α k i XA in terms ofrelative invariants of ( X ± , Z ).6.2. Absolute/Relative, blow-up/down and divisor/ambient spacecorrespondences.
Giving a symplectic manifold (
X, ω ) we are interestedin determining its GW classes and uniruled classes. Suppose X has someexplicit symplectic submanifolds, then we could cut X and attempt to applythe degeneration formula to compute a given GW invariant. However, thisis often impractical, as we need to know all the relevant relative invariants of( X ± , Z ) and relative invariants are generally harder to compute themselves.Remarkably it is shown in [41] that, in case the submanifold D is a divisorand hence X + = X , the degeneration formula can be inverted to expressa (non-descendant) relative invariant of ( X, D ) in terms of invariants of X and relative invariants of ( P D , D ∞ ). We brief describe the strategy of proofin [41].The first idea is to associate a possibly descendant invariant of X toeach non-descendant relative invariant of ( X, D ) where absolute insertionsare kept intact and contact orders of relative insertions are replaced byappropriate descendant powers.Observe then relative GW invariants are linear on the insertions. So wecan choose a generating set I of non-descendant relative GW invariants bychoose bases of cohomology of X and D and require the absolute and relativeinsertions lie in the two bases.The next idea is to introduce a partial order on I with 2 properties.Firstly, given a relative invariant of ( X, D ), when applying the degenerationformula to the associated invariant of X , the given relative invariant is thelargest one among those relative invariants of ( X, D ) appearing in the for-mula and with nonzero coefficient. Recall that the right hand side of the & YONGBIN RUAN degeneration formula is a sum of products, for each product the relative in-variant of ( P D , D ∞ ) is considered to be the coefficient. Secondly, the partialorder is lower bounded in the sense there are only finitely many invariantsin I lower than any given relative invariant in I .Then inductively, any relative invariant in I can be expressed in terms ofinvariants of X and relative invariants of ( P D , D ∞ ).In [13] we slightly reformulate the absolute/relative correspondence as alower bounded and triangular (and hence invertible) transformation T in aninfinity dimension vector space, sending the relative vector v Irel determinedby all relative invariants in I to the v Iabs absolute vector determined by allthe associated invariants of X . Furthermore, if I pt is the subset of I suchthat one of the absolute insertions is a [ pt ] insertion, T still interchanges theabsolute and relative subvectors v ptabs and v ptrel determined by I pt .We further generalize the absolute/relative correspondence to the moregeneral cuts along submanifolds of arbitrary codimension to obtain the blow-up/down correspondence. Let ˜ X be a blow-up of X along a submanifold Q with exceptional divisor D . We can cut ˜ X along D as well as cut X along Q . It is important to observe that the + pairs of these 2 cuts areessentially the same as the pair ( ˜ X, D ), in particular, they have the samerelative invariants. Another important fact is that each invariant of ˜ X in v I pt abs ( ˜ X ) has a [ pt ] insertion, and the same is true for X . In fact, the converseis also true. Thus ˜ X is uniruled if and only if the absolute vector v I pt abs ( ˜ X ) isnonzero, and the same for X .We now explain why the birational invariance of uniruledness is an im-mediate consequence. Suppose the blow-up ˜ X is uniruled, then v ptabs ( ˜ X ) isnonzero. Hence the relative vector v ptrel ( ˜ X ) is nonzero by the absolute/relativecorrespondence. Apply now the blow-up/down correspondence to concludethat v ptabs ( X ) is nonzero. Therefore X is uniruled as well. Similarly we canobtain the reverse direction.In the case that the submanifold D is a divisor, another variation of theabsolute/relative correspondence, the so called sup-admissible correspon-dence is established in [31]. For this correspondence the relative vector isenlarged to include relative invariants of ( P D , D ∞ ) with curve classes in theimage of ι ∗ : H ( D ) → H ( M ).In the case the submanifold D is a uniruled divisor satisfying the con-dition of Theorem 4.1, it is further shown in [31] that the sup-admissiblecorrespondence can be restricted to the subvector with a relative [ pt ] inser-tion and with the curve class constrained to have symplectic area boundedabove by that of a minimal uniruled class of D . The proof is rather compli-cated. It involves the reduction scheme of relative invariants of P − bundleto invariants of the base in [41] as well as [3] to prove certain vanishingresults of relative invariants of ( P D , D ∞ ). To prove Theorem 4.1, we alsoneed a non-vanishing result of relative invariants of ( P D , D ∞ ) to get the divi-sor/ambient space correspondence. The outcome of this correspondence is a YMPLECTIC BIRATIONAL GEOMETRY 19 nonzero vector of invariants of X , each invariant containing a [ pt ] insertion.Hence X is uniruled. 7. Concluding remarks
Readers can clear sense that the subject of symplectic birational geometryis only at its beginning. Constructing uniruled manifolds by symplecticmethods remains to be a challenging problem. We have already mentionedthe issue of singularities: a divisoral contraction in dimension six alreadyintroduces orbifold singularities. We expect that our program can be carriedover to orbifolds.New areas of research include the dichotomy of higher codimension unir-uled submanifolds and transitions. We ponder whether it makes sense toview transitions as extended birational equivalences. Understanding thesequestions requires new ideas and technologies.7.1.
Dichotomy of uniruled submanifold of higher codimension.
Wehave a good knowledge of the dichotomy of uniruled symplectic divisors atleast in dimension six. It is natural that we want to expand our understand-ing to higher codimension uniruled submanifolds. There are many reasonsto believe that the higher codimension case is very different from the divisorcase. In algebraic geometry, this is where we encounter other more sub-tle surgeries such as flip and flop. In the symplectic category, this is whereGromov’s h-principle is very effective. Therefore, higher codimensional unir-uled submanifolds should provide a fertile ground for these two completelydifferent theories to interact.Corollary 4.8 strongly indicates that the size of a maximal neighborhoodshould plays an important role. Such a phenomenon was first observed inMcDuff [45]. One could also wonder whether convexity also plays a role(compare with [19]). It is also desirable to define stable and rigid uniruledsubmanifolds. Right now, this is largely an unknown and exciting territory.7.2.
Transition.
Recall that a transition in the holomorphic category inter-changes a resolution with a smoothing. Symplectically, a smoothing can bethought of as gluing with a neighborhood of a configuration of Lagrangianspheres (vanishing cycles). In particular, a simplest symplectic transition in-terchanges a symplectic submanifold with a Lagrangian sphere. However, atransition is in general not a birational operation. Thus an important ques-tion is to construct symplectic transitions which are birational cobordismoperations. Such transitions will enhance our ability to ‘contract’ stableuniruled divisors (submanifolds).For general transitions it was conjectured by the second author in [51] thatthe quantum cohomology behaves nicely. We could similarly ask whetheruniruledness is preserved under general transitions. Correspondences have & YONGBIN RUAN been very successful to keep track of the total transformation of Gromov-Witten theory under birational equivalences. A natural problem is to con-struct GW correspondence for transitions. This is a new territory for theGromov-Witten theory.The most famous example is the conifold transition. Geometrically, wereplace a holomorphic 2-sphere with a Lagrangian 3-sphere. The conifoldtransition plays an important role in the theory of Calabi-Yau 3-folds andstring theories. In this case one could partially verify the invariance ofuniruledness by [33].To build a full correspondence we may have to enlarge the usual Gromov-Witten theory to the so called open Gromov-Witten theory to allow holo-morphic curves with boundary in Lagrangian manifolds. This is the well-known open-closed duality in physics. It also raises an possibility to extendsymplectic birational geometry to the open birational geometry. For exam-ple, we can define a symplectic manifold to be open uniruled if it contains anonzero genus zero possibly open GW-invariant with a [ pt ] insertion. Sucha notion has already been studied in symplectic geometry (cf. [7]). Fur-ther investigation will greatly expand our horizon to understand symplecticbirational geometry.7.3. Final question.
We feel that symplectic birational geometry is aninteresting subject and have raised many questions. We finish this surveywith one more: what kind of structures of symplectic manifold are detectedby higher genus GW invariants ? References [1] C, Birkar, P. Cascini, C. Hacon, J. McKernan, Existence of minimal models for vari-eties of log general type, math. AG/0610203.[2] S. Bauer, Almost complex 4-manifolds with vanishing first Chern class, GT/0607714.[3] B. Chen, A. Li, Symplectic relative virtual localization, in preparation.[4] Y. Eliashberg, A. Givental, H. Hofer, Introduction to Symplectic Field Theory, Geom.Funct. Anal. 2000, Special Volume, Part II, 560-573.[5] J. Etnyre, K. Honda, On symplectic cobordisms, Math. Ann. 323(1)(2002), 31-39.[6] K. Fukaya, K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38(1999), 9333-1048.[7] K. Fukaya, Y-G. Oh, F. Ono, H. Otha, Lagrangian intersection Floer theory- Anomalyand Obstruction–, book to appear.[8] V. Ginzburg, Claculation of contact and symplectic cobordism groups, Topology31(1992), 767-773.[9] V. Guillemin, V. Ginzburg, Y. Karshon, Moment maps, cobordisms, and Hamiltoniangroup actions, American Mathematical Society, 2002.[10] R. Gompf, A new construction of symplectic manifolds, Ann. Math. 142(1995), 527-595.[11] R. Gompf, Locally holomorphic maps yield symplectic structures, Comm. Anal.Geom. 13 (2005), no. 3, 511–525.[12] V. Guillemin and S. Sternberg,
Birational equivalence in the symplectic category ,Invent. Math. 97(1989), 485-522.
YMPLECTIC BIRATIONAL GEOMETRY 21 [13] J. Hu, T. Li and Y. Ruan,
Birational cobordism invariance of symplectic uniruledmanifolds , arXiv:math/0611592, Invent. Math. 172(2008), 231-275.[14] J. Hu, Y. Ruan, Positive divisors in symplectic geometry, arXiv:0802.0590.[15] E. Ionel, T. Parker, Relative Gromov-Witten invariants, Ann. of Math.(2)157(2003),45-96.[16] J. Kollar, Low degree polynomial equations: arithmetic, geometry, topology, Euro-pean Congress of Mathematics, vol. I(Budapest, 1996),255-288. Progr. Math., 168,Birkhauser, Basel, 1998.[17] J. Kollar, Rational curves on algebraic varieties, Springer-Verlag, 1996.[18] Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model program,in Algebraic geometry(Sendai, 1985), Adv. Studies in Pure Math. 10, Kinokuniya,1987, 283-360.[19] H-H. Lai, Gromov-Witten invariants of blow-ups along submanifolds with convexnormal bundles, arXiv:0710.3968.[20] E. Lerman, Symplectic cuts, Math. Research Lett. 2(1995), 247-258.[21] C. LeBrun,
Four-manifolds without Einstein metrics , Math. Res. Lett. 3. (1996),133-147, dg-ga/9511015.[22] J. Li, Stable morphisms to singular schemes and relative stable morphisms, J. Diff.Geom. 57(2001),509-578.[23] J. Li, Relative Gromov-Witten invariants and a degeneration formula of Gromov-Witten invariants, J. Diff. Geom. 60(2002), 199-293[24] A. K. Liu, Some new applications of the general wall crossing formula, Math. Res.Letters 3 (1996), 569-585.[25] T. J. Li, Existence of embedded symplectic surfaces, Geometry and topology of man-ifolds, 203–217, Fields Inst. Commun., 47, Amer. Math. Soc., Providence, RI, 2005.[26] T. J. Li, Symplectic 4-manifolds with Kodaira dimension zero, J. Diff. Geom. 74(2006), 321-352[27] T. J. Li, Quaternionic vector bundles and Betti numbers of symplectic 4-manifoldswith Kodaira dimension zero, Internat. Math. Res. Notices, 2006 (2006), 1-28[28] T. J. Li, Smoothly embedded spheres in symplectic four-manifolds, Proc. Amer. Math.Soc. 127 (1999), 609-613[29] T. J. Li, A. Liu, Symplectic structures on ruled surfaces and a generalized adjunctioninequality, Math. Res. Letters 2 (1995), 453-471.[30] T. J. Li, A. Liu, Uniqueness of symplectic canonical class, surface cone and symplecticcone of 4 − manfolds with b + = 1, J. Differential Geom. Vol. 58 No. 2 (2001) 331-370.[31] T. J. Li, Y. Ruan, Uniruled symplectic divisors, math.SG/0711.4254.[32] T. J. Li, Y. Ruan, Symplectic divisor contractions in dimension 6, in preparation.[33] A. Li, Y. Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau3-folds, Invent. Math. 145(2001), 151-218.[34] G. Lu, Finiteness of the Hofer-Zehnder capacity of neighborhoods of symplectic sub-manifolds, IMRN vol. 2006, 1-33.[35] G. Lu, G. Lu, Symplectic capacities of toric manifolds and related results, NagoyaMath. J., 181(2006), 149-184.[36] F. LaLonde, D. McDuff, The classification of ruled symplectic 4-manifolds, Math.Res. Lett. 3 (1996), 769-778.[37] F. Lalonde, D. McDuff, Symplectic structures on fiber bundles, Topology 42 (2003),309-347.[38] J. Li, G. Tian, Virtual moduli cycles and Gromov-Witten invariants of general sym-plectic manifolds, Topics in symplectic 4-manifolds, 47-83, International Press, Cam-bridge, MA, 1998.[39] K. Matsuki, LECTURES ON FACTORIZATION OF BIRATIONAL MAPS,AG/0002084. & YONGBIN RUAN [40] D. Maulik, N. Nekrasov, A. Okounkov, R. Pandharipande, Gromov-Witten theoryand Donaldson-Thomas theory, I, math.AG/0312059.[41] D. Maulik, R. Pandharipande, A topological view of Gromov-Witten theory, Topology45(5)(2006), 887-918.[42] D. McDuff, The structure of rational and ruled symplectic 4 − manifold, J. Amer.Math. Soc., v.1. no.3. (1990), 679-710.[43] D. McDuff, Examples of simply-connected symplectic non-K¨ahler manifolds, J. Diff.Geom. 20(1984), 267-277.[44] D. McDuff, Symplectic manifolds with contact type boundaries, Invent. Math. 103(1991), 651-671.[45] D. McDuff, Hamitonian S manifolds are uniruled, arXiv:0706.0675.[46] D. McDuff, Singularities and positivity of intersections of J -holomorphic curves. Withan appendix by Gang Liu. Progr. Math., 117, Holomorphic curves in symplectic ge-ometry, 191–215, Birkhuser, Basel, 1994.[47] D. McDuff, D. Salamon, A survey of symplectic 4 − manifolds with b + = 1, TurkishJ. Math. 20 (1996), 47-60.[48] H. Ohta and K. Ono, Notes on symplectic 4-manifolds with b +2 = 1. II, Internat. J.Math. 7 (1996), no. 6, 755–770.[49] H. Ohta, K. Ono, Symplectic fillings of the link of simple elliptic singularities, J.Reine Angew. Math. 565 (2003), 183–205.[50] Y. Ruan, Symplectic topology on algebraic 3 − folds, J. Diff. Geom. 39 (1994), 215-227.[51] Y. Ruan, Surgery, quantum cohomology and birational geometry, Northern CaliforniaSymplectic Geometry Seminar, 183-198, Amer. Math. Soc. Transl. Ser.2, 196, Amer.Math. Soc., Providence, RI, 1999.[52] Y. Ruan, The cohomology ring of crepant resolution of orbifolds, Contemp Math 403,2006 (117-126)[53] Y. Ruan, Virtual neighborhoods and pseudoholomorphic curves, Turkish J. Math.,23(1999), 161-231.[54] N. Siebert, Gromov-Witten invariants for general symplectic manifolds, New trendsin algebraic geometry(Warwick, 1996), 375-424, Cambrdge Uni. Press, 1999.[55] C. Voisin, Rationally connected 3-folds and symplectic geometry, AG/0801.1396.[56] Y-T Siu, A general non-vanishing theorem and an analytic proof of the finite gener-ation of the canonical ring, math.AG/0610740.[57] C. Taubes, SW ⇒ Gr: From the Seiberg-Witten equations to pseudo-holomorphiccurves , J. Amer. Math. Soc. 9(1996), 845-918.
School of Mathematics, University of Minnesota, Minneapolis, MN 55455
E-mail address : [email protected] Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109
E-mail address ::