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Examples of symplectic 4-manifolds with positive signature

We construct a new family {K_n} of simply connected symplectic 4-manifolds with the property c_1^2(K_n)/chi(K_n) ->9 (as n goes to infinity).


Introduction
The homeomorphism type of a compact, closed, simply connected symplectic 4-manifold is uniquely determined by its Euler characteristic e, signature σ, and the intersection form Q. If Q is indefinite, as it happens in most of the cases, it is determined by e,σ and its parity. It is natural to ask for which combinations (e, σ, spin) there exists such symplectic manifold, and how many distinct representatives there are. The first question is called the geography problem, the second is referred to as the botany. Both have been previously studied in the context of complex surfaces.
Symplectic 4-manifolds can be equipped with an almost complex structure, and this provides the possibility to use some notions coming from the complex geometry. In particular, we can define Chern classes and Chern numbers. Chern classes, in general, depend on the choice of an almost complex structure, but c 2 is the same as Euler characteristic e, and c 2 1 , which in the complex case denotes the self-intersection of a canonical divisor, can be extended uniquely by Hirzebruch signature formula c 2 1 = 3σ + 2e. Similarly, we can define the holomorphic Euler characteristic χ h by setting χ h = 1 4 (σ + e). The existence of the almost complex structure guarantees this number to be an integer.
The convention in 4-manifold geography is to use pairs (χ h , c 2 1 ) instead of (e, σ), and we will follow it in this article. Important fact about Chern numbers is the Bogomolov-Miyaoka-Yau inequality, asserting that c 2 1 ≤ 9χ h holds for complex surfaces of general type. It was conjectured by by Fintushel [10], [6]) in exploring the geography problem for simply connected symplectic 4-manifolds of positive signature, or equivalently those with c 2 1 ≥ 8χ h , provided families of examples close to the Bogomolov-Miyaoka-Yau line (B-M-Y line).
These constructions followed similar pattern -started with a complex surface with nontrivial fundamental group lying on, or close to the B-M-Y line, and performed an operation which preserved symplectic structure and reduced the topological complexity of the original surface. The operation in use was either symplectic connected normal sum used several times, fibered knot surgery or logarithmic transformation in a cusp neighborhood.
The botany question for complex surfaces has a partial answer -the number of non-diffeomorphic complex surfaces of general type in a given homeomorphism type is always finite, although there is no uniform bound on this number. The other large family, elliptic surfaces, is also well understood (see [4] for further references). However, for simply connected elliptic surfaces in a given homeomorphism type there are infinitely many different smooth structures. These two results reflects the nature of restrictions given by the complex (geometric) structure.
In symplectic case, the picture is less clear due to the availability of many gluing constructions, which are topological in nature. Fintushel and Stern in [2] provided another infinite family of symplectic 4-manifolds with different Seiberg-Witten (SW ) invariants, all of them homeomorphic to the standard K3 surface E(2). The method, fibered knot surgery, is different from the logarithmic transformation, and can be used with similar success for any symplectic 4-manifold containing cusp fiber and satisfying some additional properties (e.g. existence of a c-embedded torus fiber). In particular, it can be used also for symplectic manifolds coming from (some) complex surfaces of general type. This indicates that the botany problem for symplectic 4-manifolds comprises infinitely many diffeomorphism types for a much larger class of homeomorphism types labeled by triples (χ h , c 2 1 , spin). In the following we will construct a family K n of simply connected symplectic 4-manifolds close to the B-M-Y line. Moreover, for each K n there exist infinitely many mutually non-diffeomorphic symplectic 4-manifolds, all homeomorphic to K n . Theorem 1.1 There are simply connected symplectic 4-manifolds K n with Chern numbers c 2 (K n ) = n 7 + 12n 5 − 12n 4 + 6n 3 + 22, c 2 1 (K n ) = 3n 7 + 20n 5 − 24n 4 +6n 3 +2, holomorphic Euler characteristic χ h (K n ) = 1 3 (n 7 +8n 5 )−3n 4 +n 3 +2, and signature σ(K n ) = 1 3 (n 7 − 4n 5 ) − 2n 3 − 14. Acknowledgement: The author would like to thank Zoltán Szabó for many helpful discussions and suggestions.
2 Construction and homological properties of X n In [5], Hirzebruch gave an example of a sequence X n of minimal complex surfaces of general type with the following Chern numbers c 2 (X n ) = n 7 c 2 1 (X n ) = 3n 7 − 4n 5 . We briefly recall his construction, and also discuss some topological properties of smooth models of these algebraic surfaces.
Let ζ = e 2πi/6 and T be the elliptic curve Let T × T be a complex surface, and denote its points by (z, w). There is a well defined complex multiplication by algebraic integers α ∈ Q[ζ] on T . The elliptic curve in T × T given by w = αz is then isomorphic to T for every algebraic integer α. Consider the following curves on T × T All four curves pass through the origin and do not intersect each other anywhere else. It is, in fact, the consequence of the special choice of ζ as a generator of the elliptic curve T . Then 1 − ζ = −ζ 2 is also a unit in Q[ζ] and the curves T 1 , T ζ do not intersect outside of the origin. Other cases are trivial.
Let U n be the lattice in T × T consisting of n 4 points For each point of U n there are four curves passing through it, parallel to the curves T 0 , T ∞ , T 1 and T ζ . Denote the union of n 2 curves parallel to T i as D i for i ∈ {0, ∞, 1, ζ}. Altogether we have 4n 2 elliptic curves forming four parallel families. Except for the points in U n there are no other intersection points. After blowing up n 4 points of U n we get a smooth 4-manifold Y n = T 4 #n 4 CP 2 with Euler characteristic n 4 . There are n 4 exceptional curves L j (j ∈ U n ) resulting from blow-ups. Denote byD i the proper transforms of D i after the blow-up; these curves are smooth and do not intersect each other.
Since each of the homology classes [D i ] for i ∈ {0, ∞, 1, ζ} is divisible by n 2 , we can construct a complex algebraic surface X n which is a branched n 3 -fold cover of Y n branched overD i for each i ∈ {0, ∞, 1, ζ}, where eachD i will have the branching index n. Denote this covering map as π : X n → Y n .
Then a calculation of Euler characteristic gives For the Chern number c 2 1 we have For the actual computation, and for the further details we refer the reader to [5].
The map π : X n → Y n can be extended via the natural projections to the factors of the torus T × T . Then we have the following diagram: The map p 1 respects the natural fibration of T × T , singular fibers occur over the n 2 points in the set U n,0 = U n ∩ T 0 , and are formed by the curveD ∞ and exceptional curves L j . The situation for the map p 2 is analogous.
Composed maps p 1 •π, p 2 •π give two different (singular) fibrations of space X n over a 2-dimensional torus. In fact, these maps are holomorphic outside the blownup points and corresponding exceptional curves, which leads us to the observation that a generic fiber of p 1 • π is a smooth holomorphic curve.
For a regular point z in T 0 , i.e. point not belonging to U n,0 , the inverse image p −1 1 (z) is an elliptic curve parallel to T ∞ . This fiber is intersected 3n 2 times by the curvesD 0 ,D ζ andD 1 . Note, that these points are branching points of the map π restricted to the map of two-dimensional fibers in X n and Y n over the point z. By the following computation of Euler characteristic of a regular fiber Hence the genus of a regular fiber is g(F reg ) = 1 + 3 2 n 5 − 3 2 n 4 . For the singular value y of the composed map p 1 • π and the respective singular fiber F sing = (p 1 • π) −1 (y) the situation is a bit more complicated.
We know that y belongs to U n,0 , and since the construction is completely symmetric with respect to the points in U n,0 , it is enough to describe the singular fiber over one point. We can choose the point y = (0, 0). Then the inverse image p −1 1 (y) consists of n 2 exceptional spheres L j corresponding to the blown up points over y, connected with and intersected by the proper transformT ∞ of the torus T ∞ . Moreover, each of the exceptional spheres is intersected at exactly 3 different points byD 0 ,D ζ andD 1 , respectively. This gives us the complete information about the map π| y : F sing → p −1 1 (y). The singular fiber F sing consists of n 2 tori -copies ofT ∞ , connected with n 2 copies of branched coversL j of exceptional spheres L j (j ∈ U n ∩ F sing ). The Euler characteristic ofL j is e(L j ) = n 3 · e(L j \ 4 pts.) + n 2 · e(4 pts.) = −2n 3 + 4n 2 , and the Euler characteristic of the whole fiber F sing is e(F sing ) = n 2 · e(L j ) + n 2 · e(T ∞ ) − n 4 · e(pt.) = −2n 5 + 3n 4 .

Topology of X n
In this section we describe the topology of X n via the study of its fundamental group. Since our ultimate goal is to construct simply connected 4-manifold obtained from X n by a certain gluing, it would be convenient to find an embedded surface with small genus which generates as much of the fundamental group of X n as possible. We use the fact that the maps p 1 • π, p 2 • π map the space X n to tori T 0 and T ∞ .
Denote by F 1 = (p 1 • π) −1 (z) a regular fiber over some regular point z ∈ T 0 , and F 2 = (p 2 • π) −1 (w) a regular fiber over some regular point w ∈ T ∞ . Recall that the resulting surfaces F 1 , F 2 are smooth complex curves. In the previous section we computed the genera of these curves, they are the same, and equal g(F 1 ) = g(F 2 ) = 1 + 3 2 n 5 − 3 2 n 4 . The number of intersection points between F 1 and F 2 can be deduced from the following observation. Surface F 2 projects down by the map π to an embedded torus in Y n . The map π restricted to F 2 is then an n 3 -fold branched cover. When we project resulting torus further down by the map p 1 into T 0 we get, actually, one-to-one map. Therefore the restricted map p 1 • π| F2 : F 2 → T 0 is an n 3 -fold branched cover as well. As a consequence we get that the number of intersection points between F 1 and F 2 is exactly n 3 . Note that all intersections are transversal and positive.
The following lemma shows that the union F 1 ∪ F 2 contains all the topological information we need.

Lemma 3.1 The natural inclusion of fundamental groups
Proof We will show that any loop γ ∈ π 1 (X n ) can be deformed into a loop lying inside F 1 ∪F 2 . We can assume that the base point p of the fundamental group π 1 (X n ) is one of intersection points of F 1 and F 2 .
The map p 1 • π is singular over points in the set U n,0 . Since the singular locus (p 1 • π) −1 (U n,0 ) in X n has dimension 2, by using standard transversality argument we can assume that the representative of γ avoids singularities.
Consider the projection l = (p 1 • π)(γ). This gives a loop inside T 0 avoiding singular points in U n,0 . The inverse image of l, the space (p 1 • π) −1 (l) is therefore a surface bundle over l, and the fiber of this bundle over the point p is exactly F 1 . Using the fact that F 2 is a branched cover of T 0 , we can deduce that the intersection of the bundle and the surface F 2 is an n 3 -fold cover of the loop l, in particular F 2 intersects each fiber in n 3 points.
Since the loop γ lies inside the bundle whose fiber is connected, we can homotopically deform γ into γ′ which will lie inside F 2 except for the part lying above the point p and in the fiber F 1 . This proves the lemma. Lemma 3.2 There exists an embedded symplectic surface F in X n such that the inclusion map i : π 1 (F ) → π 1 (X n ) is surjective. The surface F has genus g(F ) = 3n 5 − 3n 4 + n 3 + 1 and self-intersection 2n 3 .
Proof We obtain F from F 1 ∪F 2 by standard resolution of singular intersection points in F 1 ∩ F 2 . It is easy to see that e(F ) = 2 · e(F reg ) − 2n 3 = −6n 5 + 6n 4 − 2n 3 . Since F 1 , F 2 are regular fibers, each of them have zero self-intersection number. The resolution of each positive intersection point increases the self-intersection by 2, hence Finally, the resolution of a singular point is only a local operation, but it might, and in fact does, change the fundamental group of respective surfaces. Nevertheless, the map π 1 (F ) → π 1 (F 1 ∪ F 2 ) is surjective, its kernel being generated by loops in F which can be contracted to singular points in F 1 ∪ F 2 . This proves the lemma.

Construction of K n
Surfaces F 1 and F 2 from the last section were non-degenerate holomorphic curves. After the resolution of their intersection points we got a smooth embedded surface F , which is embedded symplectically.
We will perform connected symplectic sum operation (as defined by Gompf in [3]) to the (symplectic) manifold X n along embedded symplectic surface F to obtain simply connected symplectic 4-manifold K n . The ingredient we still need to provide is the second summand -a simply connected symplectic manifold N n , preferably with small Chern numbers c 2 1 and c 2 , and with embedded symplectic surface of same genus and opposite self-intersection as F .
We begin with a K3 surface E(2), equipped with the standard elliptic fibration π : E(2) → CP 1 . In order to obtain a non-degenerate holomorphic curve (or symplectic surface) of relatively large negative self-intersection, we need to select a number of points on a curve, and blow them up. Resulting proper transform of the original curve will have the required negative self-intersection, decreasing by 1 after each blow-up. The adjunction formula for complex and symplectic manifolds implies that this is basically the only effective procedure how to obtain such surfaces.
Picking and blowing-up 2n 3 − 2 points on a selected section of E(2) gives a smoothly embedded rational complex curve C with self-intersection −2n 3 inside complex surface E(2)#(2n 3 − 2)CP 2 . Note that the elliptic fibration remained untouched outside blown-up points. Now we can perform fibered knot surgery (see [2]), in a small tubular neighborhood of a regular elliptic fiber T in a neighborhood of one of the cusp fibers in E(2). Let K ֒→ S 3 be a torus knot K (p,q) with genus Proof The fact that π 1 (N n \ F ′ ) = 1 follows from the triviality of π 1 (E(2) \ (C ∪ T )), since this space fibers over S 2 \ {pt.} with a (singular) simply connected fiber S 2 \ {pt.} coming from one of cusps. The rest of the statement follows directly from the construction of N n and straightforward computations. Now we can define K n as K n = X n # F N n , and prove Theorem 1.1.
Proof of Theorem 1. 1 We have done almost all the work in Lemmas 2.1, 3.1, 3.2 and 4.1. It remains to show that the manifold K n is simply connected. This can be obtained by combining facts that π 1 (F ) → π(X n ) is surjective, that π 1 (N n \ F ′ ) = 1, and by using Seifert-Van Kampen theorem for the union K n = X n # F N n . Computations of characteristics are routine.
Corollary 4.2 For each 4-manifold K n there exist infinitely many symplectic smooth manifolds of the same homeomorphism type which are not diffeomorphic to it. There exist infinitely many smooth manifolds of the same homeomorphism type not admitting any symplectic structure, as well.
Proof The K3 surface E(2) contains the nucleus N (2), and at least one symplectically c-embedded torus in the complement E(2) \ N (2), because this complement is a Milnor fiber M (2, 3, 11) (see [4], [8]). In our construction we used only the inside of the nucleus -we used a regular fiber in the cusp neighborhood for the fibered knot surgery N n = (E(2)#(2n 3 − 2)CP 2 ) K , and a section for the symplectic connected sum K n = X n # F N n . It means that the manifold K n could be produced in such a way that the c-embedded torus in the complement E(2) \ N (2) stayed intact during the surgeries. This gives us a possibility to perform another fibered knot surgery in the neighborhood of the remaining cusp.
By reasoning similar to that in [2] we see that the manifolds after the knot surgery will have different SW -invariants than those of K n , provided the Alexander polynomial of the knot is non-trivial. The infinite number of fibered knots with different Alexander polynomials provide the infinite number of symplectic manifolds homeomorphic but not diffeomorphic to K n . It can be shown (see [2], Cor. 1.3), that taking the fibered knot surgery along any knot, which is not fibered, and does not have a monic Alexander polynomial will produce a manifold not admitting a symplectic structure. Again, there is an infinite number of such knots with different non-monic Alexander polynomials. Remark 4.3 Similar results were obtained previously by J. Park, [6], and Stipsicz, [9]. Both constructed infinite families of simply connected 4-manifolds with the ratio c 2 1 /χ h very close to 9. However, their examples did not get above the line c 2 1 = α · χ h , with α ≈ 8.76, and α = 116/13 ≈ 8.923, respectively. The 4-manifolds constructed by J. Park are spin. In this case the constructions are more rare, since there are fewer building blocks -spin complex surfaces near B-M-Y line.
In [10], Stipsicz improved his result and constructed a family of simply connected symplectic 4-manifolds with (χ h , c 2 1 ) asymptotically approaching the line c 2 1 = 9χ h . In comparison with our result, this family converges more rapidly, meaning that for comparable values of χ h manifolds constructed by Stipsicz have higher value of c 2 1 , and are closer to the B-M-Y line. Reason for this is, basically, the starting manifold. Our construction starts with X n below the line, while the other construction uses complex surfaces lying on the B-M-Y line.
In [7], J. Park used Stipsicz's method and populated a large part of the region corresponding to the positive signature, however the Euler characteristics of these 4-manifolds have to be quite large.
Some desirable results going further in this direction would be the construction of infinite family of 4-manifolds in constant distance from the B-M-Y line (compare with the results of Chen, [1], for complex surfaces of general type), or the construction of an example above it, which would contradict the conjecture by Fintushel and Stern. At the present time, the available tools and techniques are insufficient for such constructions.
Finally, we have to add that even the smallest manifolds constructed here are still rather large. The two smallest K n 's with positive signature have the following invariants