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«MICHAEL USHER Abstract. Let X1, X2 be symplectic 4-manifolds containing symplectic sur- faces F1, F2 of identical positive genus and opposite ...»

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Abstract. Let X1, X2 be symplectic 4-manifolds containing symplectic sur-

faces F1, F2 of identical positive genus and opposite squares. Let Z denote the

symplectic sum of X1 and X2 along the Fi. Using relative Gromov–Witten

theory, we determine precisely when the symplectic 4-manifold Z is minimal

(i.e., cannot be blown down); in particular, we prove that Z is minimal unless either: one of the Xi contains a (−1)-sphere disjoint from Fi ; or one of the Xi admits a ruling with Fi as a section. As special cases, this proves a conjecture of Stipsicz asserting the minimality of fiber sums of Lefschetz fibrations, and implies that the non-spin examples constructed by Gompf in his study of the geography problem are minimal.

1. Introduction Let (X1, ω1 ), (X2, ω2 ) be symplectic 4-manifolds, and let F1 ⊂ X1, F2 ⊂ X2 be two-dimensional symplectic submanifolds with the same genus whose homology classes satisfy [F1 ]2 + [F2 ]2 = 0, with the ωi normalized to give equal area to the surfaces Fi. For i = 1, 2, a neighborhood of Fi is symplectically identified by Weinstein’s symplectic neighborhood theorem [19] with the disc normal bundle νi of Fi in Xi. Choose a smooth isomorphism φ of the normal bundle to F1 in X1 (which is a complex line bundle) with the dual of the normal bundle to F2 in X2.

According to [2] (and independently [11]), the symplectic sum Z = X1 #F1 =F2 X2 = (X1 \ ν1 ) ∪∂ν1 ∼φ ∂ν2 (X2 \ ν2 ) carries a natural deformation class of symplectic structures (note that the diffeo- morphism type of Z may depend on the identification φ, as seen for instance in Example 3.2 of [2], a feature which is mostly suppressed from the notation here- inafter).

Symplectic sums along S 2 are well-understood; according to pp. 563–566 of [2] such a symplectic sum amounts to either blowing down a sphere of square −1 or −4 in one of the summands, taking the fiber sum of two ruled surfaces, or leaving the diffeomorphism type of one of the summands unchanged, and then possibly blowing up the result. Accordingly, we shall restrict our attention to symplectic sums along surfaces of positive genus.

Recall that a symplectic 4-manifold M is called minimal if it contains no symplec- tically embedded spheres of square −1, and hence cannot be expressed as a blowup of another symplectic 4-manifold. According to results arising from Seiberg–Witten theory ([18],[8]), this is equivalent to the condition that M not contain any smoothly embedded spheres of square −1.

In this note, we resolve completely the question of under what circumstances the

above symplectic sum Z is minimal. Our result may be summarized as follows:

1991 Mathematics Subject Classification. 53D35, 53D45.

–  –  –

Theorem 1.1.

Let the symplectic sum Z = X1 #F1 =F2 X2 be formed as above, and

assume that the Fi have positive genus g. Then:

(i) If either X1 \ F1 or X2 \ F2 contains an embedded symplectic sphere of square −1, then Z is not minimal.

(ii) If one of the summands Xi (for definiteness, say X1 ) admits the structure of an S 2 -bundle over a surface of genus g such that F1 is a section of this fiber bundle, then Z is minimal if and only if X2 is minimal.

(iii) In all other cases, Z is minimal.

Case (i) above should be obvious: if X1 admits a sphere of square −1 which misses F1, then cutting out a small neighborhood of F1 and replacing the neighborhood with something else will not change that fact. In the situation of Case (ii), note that since F1 and F2 have opposite squares, the complement of a neighborhood of F1 in the ruled surface X1 is diffeomorphic to a neighborhood of F2, and so effectively the symplectic sum operation cuts out a neighborhood of F2 and then glues it back in via a map which (since φ : ∂ν1 → ∂ν2 is a bundle isomorphism, not just a diffeomorphism) takes a meridian of F2 to itself; certainly for some choices of the gluing map the result is just diffeomorphic to X2 and so of course will have the same minimality properties, and indeed the author is unaware of any cases in which the diffeomorphism type of X2 is changed by summing with a ruled surface along a section with a nonstandard choice of gluing map.

Case (iii) is thus quite broad, and in particular confirms the belief expressed by the authors of [10] that the main theorem of that paper could be generalized. For some prior results asserting the minimality of symplectic sums in various special cases, see [17], Theorem 1.5 of [16], Theorem 2.5 of [14], [10], and [7].

We now turn to some corollaries of Theorem 1.1. Recall that a map f : X → Σ from a symplectic 4-manifold to a 2-manifold is called a symplectic Lefschetz fibration provided that its smooth fibers are symplectic submanifolds and f has just finitely many critical points, near each of which it is given in orientation preserving complex coordinates by (z1, z2 ) → z1 z2. f is called relatively minimal if none of its singular fibers (each of which is a nodal curve) contains a (−1)-sphere as a reducible component. Note that in general a Lefschetz fibration admits compatible symplectic structures provided that its fiber is homologically essential; this condition is automatic if the fiber genus is at least two or if the genus is one and the map has at least one critical point. The fiber sum of two symplectic Lefschetz fibrations whose fibers have the same genus is just the symplectic sum along a smooth fiber.

The following was conjectured by A. Stipsicz in [17]:

Corollary 1.2. Let fi : Xi → S 2 (i = 1, 2) be relatively minimal symplectic Lefschetz fibrations on 4-manifolds X1, X2 whose fibers F1, F2 have the same positive genus g, and assume that neither fi is the projection Σg × S 2 → S 2. Then the fiber sum X1 #F1 =F2 X2 is minimal.

Another consequence of Theorem 1.1 relates to the geography of minimal symplectic 4-manifolds, that is, the question of which pairs of integers (a, b) have the property that there is a minimal symplectic 4-manifold Ma,b such that c2 (Ma,b ) = a 1 and c2 (Ma,b ) = b. Any such pair (a, b) necessarily satisfies a+b ≡ 0 (mod 12) by the Noether formula. By performing symplectic sums on certain manifolds covered by Case (iii) of Theorem 1.1, R. Gompf in [2] realized a great many such pairs as the Chern numbers of symplectic manifolds with prescribed fundamental group; in the


case that the Chern numbers are consistent with Rohlin’s theorem (so that (a, b) has form (8k, 4k + 24l)) Gompf was able to arrange that the resulting manifold be spin and hence minimal, but in other cases minimality appeared likely but could not be proven. However, Theorem 1.1 in conjunction with Remark 2 after Theorem

6.2 of [2] allow us to deduce:

Corollary 1.3. Let G be any finitely presentable group. There is a constant r(G) with the property that if (a, b) ∈ Z2 satisfies a + b ≡ 0 (mod 12) and 0 ≤ a ≤ 2(b − r(G)) then there is a minimal symplectic 4-manifold Ma,b,G such that π1 (Ma,b,G ) ∼ = G, c2 (Ma,b,G ) = a, and c2 (Ma,b,G ) = b.

1 Note that most of Gompf’s examples involve taking symplectic sums in which at least one of the summands is rational, so these examples are not covered by previous results on the minimality of symplectic sums. In the simply connected case, different constructions have been used to obtain symplectic manifolds occupying large parts of the c2 c2 -plane and then to show that they are minimal using gauge theory; see, 1 e.g., Theorem 10.2.14 of [3].

The next two sections are occupied with the proof of Theorem 1.1. This proof splits into two parts: first, we use relative Gromov–Witten theory to give a condition on the pairs (X1, F1 ), (X2, F2 ) in terms of the intersection numbers of the Fi with holomorphic spheres which is sufficient to guarantee the minimality of Z (namely, the Fi should be “rationally K-nef,” defined below). We then see that surfaces of positive genus in symplectic 4-manifolds are always rationally K-nef except in the cases (i) and (ii) in Theorem 1.1; this follows from results of Seiberg–Witten theory concerning the canonical class ([18],[9]) when the ambient manifold is not a blowup of a rational or ruled surface, while the ruled case can be handled fairly directly and the rational case depends in part on the analysis of the chamber structure in the cohomology of rational surfaces that was carried out in [1]. Finally, in the last section we prove Corollaries 1.2 and 1.3.

Acknowledgements. I am indebted to T.J. Li for finding an oversight in an earlier version of this paper. This work was partially supported by an NSF Postdoctoral Fellowship.

2. Symplectic sums along rationally K-nef surfaces are minimal If (X, ω) is a symplectic 4-manifold, g ≥ 0, A ∈ H2 (X; Z), and α ∈ H ∗ (X; Z), X we let GWg,A (α) denote the Gromov–Witten invariant ([15]) counting perturbedpseudoholomorphic maps from a surface of genus g into X, representing the homology class A, and passing through a cycle Poincar´ dual to α.

e Suppose now that the symplectic sum Z = X1 #F1 =F2 X2 is not minimal, so that Z contains a symplectic sphere E of square −1. Using an almost complex structure J generic among those making E pseudoholomorphic, we immediately see that Z GW0,[E] (1) = ±1, since the operator ∂ J will (for generic J) have nondegenerate linearization at the embedding of E, and positivity of intersections (see, e.g., Theorem E.1.5 of [13]) prevents the existence of any other J-holomorphic spheres homologous to E. We now review how to use this nonvanishing to deduce the existence of nonzero relative Gromov–Witten invariants in certain homology classes in X1, X2 by means of the gluing results in [5].

The main theorem of [5] (or, similarly, that of [6]) provides a somewhat complicated formula expressing the Gromov–Witten invariants of Z in terms of the 4 MICHAEL USHER Gromov–Witten invariants of X1 and X2 relative to the fibers F1 and F2, the latter invariants having been defined in [4]. Without reproducing the formula, we recall the essential points, referring readers to [5] for details: in forming the symplectic sum, letting x1 and x2 be complex coordinates in the normal bundles ν1, ν2, one performs the identification by taking x1 x2 = λ for some small λ ∈ C∗ ; this results in a symplectic 6-manifold Z equipped with a projection to the disc whose fiber (Zλ, ωλ ) over λ ∈ D2 \ {0} is isotopic to Z but whose fiber over 0 is Z0 = X1 ∪F1 =F2 X2. As the parameter λ approaches zero, pseudoholomorphic curves in Zλ limit to trees of curves in Z0 consisting of: curves in X1 meeting F1 in isolated points; curves contained in the identified surfaces F = F1 = F2 ; and curves in X2 meeting F2 in isolated points. In fact there is a quantifiable finite-to-one correspondence between these trees (which are counted by a combination of (relative) Gromov–Witten invariants in X1, F, and X2 ) and the curves counted by the Gromov–Witten invariants of Z. This leads to a gluing formula (Theorem 12.4 of [5]) expressing the latter invariants in terms of the former.

Let us return to our case, where Z admits a sphere E of square −1, and consider what the gluing result of [5] allows us to deduce about the Gromov–Witten invariants of X1 and X2. First, notice that since E is a sphere the limiting tree discussed in the previous paragraph will consist only of genus-zero curves, and so will not have any components mapped into F, since π2 (F ) = 0. So the gluing Z formula will express the nonzero Gromov–Witten invariant GW0,[E] (1) in terms of invariants which count pairs (C1, C2 ) of possibly-disconnected curves (each of whose components are spheres) representing elements of the intersection-homology F F spaces HX1, HX2 defined in Section 5 of [4], and which have matching intersections 1 2 with the fiber F = F1 = F2. So we obtain nonzero relative invariants in both X1 and X2, which in particular count (generally disconnected) holomorphic curves in certain classes Ai ∈ H2 (Xi ; Z) (i = 1, 2) (subject to some additional constraints on their intersections with Fi ). We note also that while in the general situation considered in [5] the Gromov–Witten invariants are counts of maps satisfying a perturbed Cauchy-Riemann equation ∂ J u = ν, in our case we can take ν = 0 by virtue of the fact that [E] is a primitive class in H2 (Z; Z) (so that the relevant moduli space has no strata consisting of multiple covers). Thus the curves C1 and C2 will be genuinely pseudoholomorphic curves for almost complex structures J1, J2 on X1, X2, which may be chosen generically among those pairs of almost complex structures on the Xi which preserve T Fi and agree on the identified neighborhoods νi of Fi. Incidentally, in our proof of Theorem 1.1 we will in fact only need the Z fact that a nonvanishing invariant GW0,[E] (1) gives rise rise via a Gromov-type compactness theorem to (generally reducible, non-reduced) Ji -holomorphic curves Ci for some almost complex structures Ji which make the Fi pseudoholomorphic.

The full strength of the Ionel-Parker theorem, which shows that the Ci are in fact enumerated by nonvanishing relative invariants, is not needed for this conclusion.

Let d = A1 ∩ [F1 ] = A2 ∩ [F2 ]. Then according to Lemma 2.2 of [5], we have, where for a symplectic manifold M we denote the canonical class of M by κM, κZ, [E] = κX1, A1 + κX2, A2 + 2d.

But E is an embedded (−1) sphere and so satisfies κZ, [E] = −1 by the adjunction formula; thus we may suggestively rewrite the above equation as κX1 + P D[F1 ], A1 + κX2 + P D[F2 ], A2 = −1.



Accordingly, we make the following definition (recall from [13] that a pseudoholomorphic curve u : Σ → X is called simple if no two disjoint open sets in its

domain Σ have the same image; in this case the map u is generically injective):

Definition 2.1. Let (X, ω) be a symplectic four-manifold. An embedded symplectic surface F ⊂ X is called rationally K-nef if, whenever J is an almost complex structure preserving T F and A ∈ H2 (X; Z) is represented by a simple J-holomorphic sphere, we have κX + P D[F ], A ≥ 0, (2) where κX is the canonical class of X.

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