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«Published in the proceedings of the 1994 Symplectic Topology program at the Newton Institute, ed S. Donaldson and C. Thomas, Cambridge University ...»

-- [ Page 1 ] --

J-curves and the classification of rational and

ruled symplectic 4-manifolds

Fran¸ois Lalonde∗

c

Universit´ du Qu´bec ` Montr´al

e e a e

(flalonde@math.uqam.ca)

Dusa McDuff†

State University of New York at Stony Brook

(dusa@math.sunysb.edu)

November 1995

Published in the proceedings of the 1994 Symplectic Topology program

at the Newton Institute, ed S. Donaldson and C. Thomas,

Cambridge University Press (1996), 3–42.

In this paper, we present a complete classification of rational and ruled symplectic 4-manifolds up to symplectomorphism as well as describing ways of recognizing these manifolds. The proofs are essentially self-contained.

1 Introduction The classification of symplectic structures on CP2 has followed a quite simple story: Gromov showed in [2] that the existence of an embedded sympectic 2sphere in the homology class [CP1 ] of a line implies that the symplectic structure is diffeomorphic to the standard K¨hler structure on CP2, and Taubes showed a in [37] that such a sphere always exists if the space is diffeomorphic to CP2.

Both proofs are due to spectacular advances in the application of elliptic PDEs methods to symplectic 4-manifolds.

The classification of rational ruled symplectic 4-manifolds, that is the classication of symplectic S 2 -fibrations over a 2-sphere, was established by Gromov [2] in a rather special case, and was extended to all cases by McDuff [14]. Her ∗ Partially supported by NSERC grants OGP 0092913 and CPG 0163730, and FCAR grant ER-1199.

† Partially supported by NSF grant DMS 9401443.

proof was based on two new ideas: the realization that in 4-dimensions cuspcurves do not affect the behaviour of the evaluation map for generic J, and the construction of symplectic sections of a rational ruled 4-manifold with a method suggested by Eliashberg.

However, the classification of symplectic structures on spaces diffeomorphic to S 2 -bundles over Riemann surfaces of genus 0 is more complicated. There have been several attempts to solve this problem: partial results (including a complete solution of the problem for ruled surfaces over the torus) were obtained by McDuff in [14, 21, 23] and by Lalonde in [5]. Actually the main difficulty, as we will see below, is not to derive the existence of an embedded symplectic 2-sphere. It is to classify the symplectic S 2 -fibrations on a given S 2 -bundle over a Riemann surface of genus 0. This required more work, both on the symplectic and elliptic aspects of the problem. The complete classification finally appeared in our recent note [7]. It is based on a simpler and more general geometric argument which reduces the classification of the irrational ruled manifolds to the rational ones, up to the computation of some Gromov invariant on the given irrational ruled manifold. The first step of this reduction – the “cutting and pasting” argument – appeared in McDuff [14], and the second step of the reduction – the transformation of a symplectic deformation into a genuine isotopy via a one-parameter family of symplectic submanifolds – was worked out by Lalonde in [5]. The computation of the relevant Gromov invariant is based on the equivalence of the Gromov and Seiberg-Witten invariants, and is easily reduced to the calculation of a wall-crossing number at the reducible solutions of the Seiberg-Witten equations.

The second theme of this article is the characterization of rational and ruled symplectic manifolds. While developing the theory of J-holomorphic spheres, McDuff discovered that any symplectic 4-manifold which contains a symplectically embedded 2-sphere C of nonnegative self-intersection is the blow-up of a rational or ruled manifold. Taubes’s work now applies to show that this remains true if we just assume that C is smoothly embedded. Another beautiful new result was recently proved by Liu [10]. He showed that Gompf’s conjecture holds: namely, any minimal symplectic 4-manifold whose canonical class K is such that K 2 0 is irrational ruled. As a corollary, it follows that any minimal symplectic 4-manifold which admits a metric of positive scalar curvature is rational or ruled: see Liu [10] and Ohta–Ono [31].

2 Statement of the classification theorems We assume that the reader is familiar with blowing-up and blowing-down in the symplectic category: see for instance [27] or [5], §II, for a convenient summary.

Recall that there are only two S 2 -bundles over any given Riemann surface B, the trivial bundle π : B × S 2 → B and the nontrivial bundle π : MB → B. By analogy with the complex case, such a bundle is said to be a ruled symplectic manifold if the total space is equipped with a symplectic form ω which is nondegenerate on each fiber. In this case, we also say that ω is compatible with the given ruling π. Similarly, we say that a symplectic 4-manifold (M, ω) is rational if it can be obtained from the standard CP2 by a sequence of blowups and blow-downs. (This is slightly unfortunate terminology, since in other contexts the word rational is used to denote a rationality condition on some cohomology class, for instance on the class [ω] or on some Maslov class.) An exceptional sphere in a symplectic 4-manifold is a symplectically embedded 2-sphere of self-intersection −1.1 It was shown in [17, 14] that these spheres behave very much like exceptional curves in complex surfaces. In particular, they can be blown down and replaced by a ball of appropriate size.





(In fact, if the weight ω(E) of the exceptional curve E is πr2 the ball should have radius r.) If we define a minimal symplectic 4-manifold to be one which contains no exceptional spheres, one can reduce every symplectic 4-manifold to a minimal one by blowing down a maximal collection of disjoint exceptional spheres. Moreover, just as in the complex case, this minimal reduction is unique unless it it is rational or ruled: see [18].

It is also convenient to consider manifolds M which are such that the complement M −C of a symplectic 2-submanifold C contains no exceptional spheres.

(In this situation (M, C) is sometimes called a minimal pair.) Observe that given any such C in M one can obtain a minimal pair by blowing down a maximal collection of disjoint exceptional spheres in M − C.

Here are the main results.

Theorem 2.1 (Gromov–McDuff ) Let (M, ω) be a closed symplectic 4-manifold which contains an embedded symplectic 2-sphere C of self-intersection ≥ 0.

Then (M, ω) is symplectomorphic to a blow-up either of CP2 with its standard Kahler structure (which is unique up to scaling by a constant) or of a ruled symplectic manifold. In particular, if M − C is minimal then (M, ω) is either symplectomorphic to CP2 or is ruled.

Remarks. (1) In the above theorem, the natural hypothesis would be to assume M minimal. But then one does not recover the case of the topologically nontrivial S 2 -bundle over S 2, which coincides with the blow-up of CP2 at one point.

In order to include this case, we must assume only the minimality of M − C.

(2) The symplectomorphism of the theorem can be chosen so that the curve C corresponds either to a line or quadric in CP2 or to a fiber of the S 2 -bundle, or to a section of the bundle when the base is the sphere.

1 To say that a 2-dimensional submanifold S of M is symplectically embedded is equivalent to saying that the restriction ω|S never vanishes.

–  –  –

3 Preliminaries

3.1 Summary of basic facts on J-holomorphic curves All manifolds and maps will be assumed to be C ∞ -smooth unless specific mention is made to the contrary.

An almost complex structure J on a manifold M is an automorphism J : T M → T M of the tangent bundle such that J 2 = −Id. J is said to be tamed by ω if

–  –  –

As usual we write J = J (M, ω) for the set of all ω-compatible J (or the set of all ω-tame structures J, depending on the context). These spaces are both nonempty and contractible (because Sp(2n; R) retracts onto its maximal compact subgroup U (n)). (See [27, Chapter 2.5].) Given any J ∈ J (M, ω), we write c1 ∈ H 2 (M ; Z) for the first Chern class of the complex vector bundle (T M, J). This is independent of the choice of J ∈ J (M, ω) since the latter space is connected.

–  –  –

The corresponding unparametrized curve Im u will often be denoted C. Thus C has real dimension 2 and complex dimension 1. When Σ is the Riemann sphere, the curve is often said to be rational. It is important to note that a Jholomorphic map u : Σ → M is either a multiple cover, i.e. it factors through a holomorphic map Σ → Σ of degree 1, or it is somewhere injective in the sense that there is a point z ∈ Σ such that

–  –  –

This is proved in McDuff [13] with more details given in [26, 2.3].

3.2 Local properties of J-curves 3.2.1 Singularities We begin with a theorem essentially due to McDuff ([19, 22]) and improved by Sikorav in [36]. In particular, his argument works under much weaker smoothness assumptions on J.

Theorem 3.2 (Isolated singularities) Let (M, J) be an almost complex manifold and u, u two J-holomorphic maps to M defined on closed Riemann surfaces Σ, Σ.

Then the points where u(z) coincides with u (z) are isolated. Further, the points where du(z) vanishes are also isolated.

–  –  –

3.2.2 Positivity of intersections and the adjunction formula The positivity of intersections of two algebraic curves (which states that each point of intersection contributes positively) is a cornerstone of the algebraic geometry of surfaces. The previous theorem gives one hope that this principle extends to the almost complex case. And indeed it does, although the proof is much harder than in the integrable case. It is obvious that the sign of a point p of intersection of two J-curves is positive (and counts for +1) when the two curves are regular at p, and positivity is still quite easy to prove when at least one of the two curves is regular at p. It is more difficult when both curves are singular at p.

There is also a notion of positivity of intersection of a single curve: it says that each singular point of a complex curve gives a positive contribution to the self-intersection number of the curve. Again, this still holds in the almost complex case. The proof is quite elementary in the cases where the 1-jet of the singularity has the form (z k, z ) with k, relatively prime, but is harder in the general case. In any case, part (i) of the following theorem was conjectured by Gromov in [2]. The full result (except with a C 0 rather than C 1 φ) was proved by McDuff in [16, 19, 22] by a topological argument based on perturbations. An improved and more analytical proof of everything except the last statement was given by Micallef and White in [29], by a method which works under considerably weaker smoothness assumptions on J.

Theorem 3.4 (Positivity) (i) Let C, C be two closed J-holomorphic curves.

Then the contribution kp of each point of intersection of C and C to the intersection number C · C is a strictly positive integer. It is equal to 1 if and only if C and C are both regular at p and meet transversally at that point.

(ii) For each singularity p of a J-holomorphic curve C, there is a neighbourhood U of p in M and a C 1 -diffeomorphism φ from (U, p) to (B, 0) ⊂ (C2, 0) which sends C ∩ U to an isolated singularity at 0 of a complex curve in the ball B.

Moreover, there is a C 0 -perturbation of J to J with compact support inside U and a C 0 -small isotopy of C to C (or more precisely of its parametrisation) with compact support inside U such that the curve C is immersed and J holomorphic.

Exercise 3.5 Prove the adjunction formula. This says that, if (M 4, J) is an almost complex manifold and u : Σ → M is a J-holomorphic curve which does not factorize through another Riemann surface by a multiple branched covering (that is: u is not a multiple covering), then the virtual genus of C = Im(u) defined by gv (C) = 1 + (C · C − c1 (C)) is always greater or equal to the genus of Σ, and it is equal to the genus of Σ if and only if u is an embedding. Note that this means that the total weight of singularities of a J-curve in (M 4, J) is a topological invariant. In particular, any J-curve homologous to an embedded J-curve of the same genus is also embedded.

Hint: suppose first that the curve is immersed and decompose the first Chern class of the ambient tangent bundle along the curve in the tangent and normal directions. Compute the self-intersection of the curve in terms of the self-intersection of the curve inside its normal bundle and of the number of self-intersection points of the curve (remember that these self-intersections are all positive!) Then deduce the general case from the immersed case using the last assertion of the theorem. For more details see [16].

3.3 Global geometry: moduli spaces and Gromov’s compactness theorem We now describe the global behaviour of the space of all J-holomorphic curves in a given homology class. It turns out that this space, as any solution space of an elliptic system of PDEs, is finite dimensional with dimension given by the Atiyah-Singer index theorem, at least when J is generic. We will see that, when this space is not empty, it is either compact or can be compactified by addition of what Gromov calls cusp-curves, which are the analogue of reducible curves in algebraic geometry. Actually, the picture is again very similar to the one in the integrable case. But the proofs are more delicate and rely on Riemannian estimates like the isoperimetric inequality and on properties of elliptic operators.

3.3.1 Fredholm framework Let (M, ω) be a smooth compact symplectic manifold. As above, let J (M ) be the Fr´chet manifold of all almost complex structures which are tamed by e ω. For each class A ∈ H2 (M, Z) and each non-negative integer g, consider the space Mp (A, g, J ) of all triples (u, j, J) such that (i) j belongs to the Teichmuller space Tg of the closed real oriented surface Σg of genus g,

–  –  –



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