In this section we prove restrictions on the possible connected sum decompositions of a minimal sym-plectic four-manifold withb+2 = 1, leading to a proof of Theorem 3.3. To do this we have to leave the realm of symplectic topology and use Seiberg–Witten gauge theory.
LetXbe a closed oriented smooth4-manifold withb+2(X) = 1. We fix aSpinc structuresand a metricgonXand consider the Seiberg–Witten equations for a positive spinorφand aSpincconnection A:
D+Aφ = 0 F+ˆ
A = σ(φ, φ) +η ,
where the parameterηis an imaginary-valuedg-self-dual2-form. HereAˆdenotes theU(1)-connection on the determinant line bundle induced fromA, so thatFAˆis an imaginary-valued2-form. A reducible solution of the Seiberg–Witten equations is a solution withφ= 0.
For every Riemannian metric g there exists a g-self-dual harmonic2-form ωg with [ωg]2 = 1.
Becauseb+2(X) = 1, this2-form is determined bygup to a sign. We choose a forward cone, i. e. one of the two connected components of{α ∈H2(X;R)|α2 >0}. Then we fixωg by taking the form whose cohomology class lies in the forward cone.
LetLbe the determinant line bundle of theSpincstructures. The curvatureFArepresents 2πi c1(L) in cohomology, and every form which represents this class can be realized as the curvature ofAˆfor a Spinc connectionA. For given(g, η)there exists a reducible solution of the Seiberg–Witten equations if and only if there is aSpinc connectionAsuch thatF+ˆ
A = η, equivalently(c1(L)− 2πi η)·ωg = 0.
Define the discriminant of the parameters(g, η)by
∆L(g, η) = (c1(L)−2πi η)·ωg.
One divides the space of parameters (g, η) for which there are no reducible solutions into the plus and minus chambers according to the sign of the discriminant. Two pairs of parameters(g1, η1)and (g2, η2)can be connected by a path avoiding reducible solutions if and only if their discriminants have the same sign, i. e. if and only if they lie in the same chamber. A cobordism argument then shows that the Seiberg–Witten invariant is the same for all parameters in the same chamber. In this way we get the invariantsSW+(X,s),SW−(X,s)which are constant on the corresponding chambers.
Suppose now thatX has a symplectic structure ω. Thenω determines an orientation of X and a forward cone in H2(X;R). We will take the chambers with respect to this choice. Moreover, ω determines a canonical class K and a Spinc structure sK−1 with determinant K−1. One can obtain every other Spinc structure by twistingsK−1 with a line bundleE, to obtainsK−1 ⊗E. This Spinc structure has determinantK−1⊗E2.
III.3 Connected sum decompositions of minimal symplectic4-manifolds 25 The Taubes chamber is the chamber determined by parameters(g, η)withgchosen such that it is almost K¨ahler withωg=ωand
η=FA+
0 −4irω with r0, whereA0is a canonical connection onK−1. We have the following:
Lemma 3.8. The Taubes chamber is the minus chamber, for the choice of forward cone as above.
Proof. We have
(c1(−K)−2πi η)·ωg = (2πi FAˆ
0 −2πi F+ˆ
A0− 8π1 rω)·ω
= (2πi F−ˆ
A0 −8π1 rω)·ω
= −8π1 rω2 <0,
because the wedge product of a self-dual and an anti-self-dual two-form vanishes.
The following theorem is due to Taubes [131, 132, 134], compare [88, 89] for the caseb+2 = 1.
Theorem 3.9. The Seiberg–Witten invariant in the minus chamber for the canonicalSpincstructure is non-zero. More precisely,SW−(X,sK−1) = ±1. Moreover, ifSW−(X,sK−1 ⊗E)is non-zero and E 6= 0, then for a genericω-compatible almost complex structureJ, the Poincar´e dual of the Chern class ofEcan be represented by a smoothJ-holomorphic curveΣ⊂X.
We have the following more precise version of the second part of Theorem 3.9, which is also due to Taubes.
Proposition 3.10. SupposeSW−(X,sK−1 ⊗E) is non-zero, andE 6= 0. Then for a generic almost complex structureJ compatible withω there exist disjoint embeddedJ-holomorphic curvesCi inX such that
P D(c1(E)) =
n
X
i=1
mi[Ci],
where eachCi satisfiesK ·Ci ≤ Ci·Ci and each multiplicitymi is equal to1, except possibly for thoseifor whichCiis a torus with self-intersection zero.
This depends on a transversality result forJ-holomorphic curves, see Proposition 7.1 in [134] and also [135, 80]. Proposition 3.10 immediately implies the following:
Corollary 3.11. IfSW−(X,sK−1 ⊗E) 6= 0withE2 <0thenX contains an embedded symplectic (−1)-sphereΣ.
Proof. Choose a generic compatible almost complex structureJ as in Proposition 3.10, and consider E =P
imiCi. ThenE2 =P
im2iCi2because theCi are disjoint, henceCj2 <0for somej. We can compute the genus ofCj from the adjunction formula:
g(Cj) = 1 +12(Cj·Cj+K·Cj)≤1 +Cj·Cj ≤0.
HenceΣ =Cj is a sphere with self-intersection number−1.
After these preparations we can now prove Theorem 3.3.
Proof of Theorem 3.3. Let(X, ω) be a closed symplectic4-manifold withb+2 = 1. We denote byK both the first Chern class of any compatible almost complex structure, and the complex line bundle with this Chern class.
First, suppose that(X, ω) is symplectically minimal and rational or ruled. Then, by the classifi-cation of ruled symplectic four-manifolds,X is diffeomorphic either toCP2, to an even Hirzebruch surface, or to a geometrically ruled K¨ahler surface over a complex curve of positive genus, compare e. g. [95]. These manifolds are all irreducible for purely topological reasons. This is clear forCP2and for the even Hirzebruch surfaces, because the latter are diffeomorphic to S2×S2. For the irrational ruled surfaces note that the fundamental group is indecomposable as a free product. Therefore, in any connected sum decomposition one of the summands is simply connected. If this summand were not a homotopy sphere, then the other summand would be a smooth four-manifold with the same funda-mental group but with strictly smaller Euler characteristic than the ruled surface. This is impossible, because the irrational ruled surfaces realize the smallest possible Euler characteristic for their funda-mental groups, compare [78].
Thus, we may assume that(X, ω)is not only symplectically minimal, but also not rational or ruled.
Then Liu’s results in [90] tell us thatK2 ≥0andK·ω≥0.
IfXdecomposes as a connected sumX =M#N then one of the summands, sayN, has negative definite intersection form. Moreover, the fundamental group of N has no non-trivial finite quotients, by Proposition 1 of [81]. In particularH1(N;Z) = 0, and hence the homology and cohomology ofN are torsion-free. IfN is an integral homology sphere, then there is nothing more to prove.
SupposeNis not an integral homology sphere. By Donaldson’s theorem [31], the intersection form ofN is diagonalizable overZ. Thus there is a basise1, ..., enofH2(N;Z)consisting of elements with square−1which are pairwise orthogonal. Write
K =KM +
n
X
i=1
aiei,
with KM ∈ H2(M;Z). The ai ∈ Z are odd, because K is a characteristic vector. This shows in particular that K is not a torsion class. Its orthogonal complement K⊥ in H2(X;R) is then a hyperplane. As K2 ≥ 0andb+2(X) = 1, the hyperplaneK⊥does not meet the positive cone. Thus Liu’s inequalityK·ω ≥0must be strict:K·ω >0.
Now we knowSW−(X,sK−1) =±1from Taubes’s result, wheresK−1 is theSpincstructure with determinantK−1 induced by the symplectic formω. The inequality(−K)·ω < 0shows that a pair (g,0)is in the negative, i. e. the Taubes chamber, whenevergis almost K¨ahler with fundamental two-formω. AsK⊥ does not meet the positive cone, all pairs(g,0)are in the negative chamber, for all Riemannian metrics g. We choose a family of Riemannian metricsgr on X which pinches the neck connectingM andN down to a point asr → ∞. For r large we may assume thatgr converges to metrics on the (punctured)MandN, which we denote bygM andgN.
Lemma 3.12. If we choose the forward cone forM to be such that it induces onXthe forward cone determined by the symplectic structure, then for every Riemannian metricg0 onM, the point(g0,0)is in the negative chamber ofM with respect to theSpinc structuresM onM obtained by restriction of sK−1.
Proof. The chamber is determined by the sign ofc1(s)·ωg. We have
0>(−K)·ωgr =c1(sK−1)·ωgr = c1(sM)·ωgr+c1(sN)·ωgr
−→ c1(sM)·ωgM +c1(sN)·ωgN,asr→ ∞.
III.3 Connected sum decompositions of minimal symplectic4-manifolds 27 We know thatωgN is self-dual harmonic with respect togN, and hence vanishes becauseb+2(N) = 0.
This implies thatc1(sK−1)·ωgr converges toc1(sM)·ωgM forr→ ∞. Thus c1(sM)·ωgM ≤0.
However, we havec1(sM) =KM−1, and KM2 =K2+
n
X
i=1
a2i ≥K2+n≥n≥1,
showing thatKM⊥ does not meet the positive cone ofM. Thusc1(sM)·ωgM <0. Again becauseKM⊥ does not meet the positive cone ofM, this inequality holds for all metricsg0onM.
The degeneration of thegrasrgoes to infinity takes place in the negative chamber forsK−1, where the Seiberg–Witten invariant is±1, and by the LemmagM is in the negative chamber forsM. It follows thatSW−(M,sM) =±1.
We now reverse the metric degeneration, but use a different Spinc structure on N. Instead of usingsN withc1(sN) = −Pn
i=1aiei, we use the uniqueSpinc structure s0N withc1(s0N) = a1e1− Pn
i=2aiei. For every metric onNthere is a unique reducible solution of the Seiberg–Witten equations for thisSpinc structure withη = 0. Gluing this solution to the solutions onM given by the invariant SW−(M,sM), we findSW−(X,s0) =±1, wheres0is theSpincstructure onXobtained fromsM and s0N, compare Proposition 2 of [81]. We haves0 =sK−1 ⊗E, withE=a1e1. ThereforeE2 =−a21 ≤
−1, and Corollary 3.11 shows thatXis not minimal. This completes the proof of Theorem 3.3.
Chapter IV
On the conformal systoles of 4-manifolds
Contents
IV.1 Introduction . . . . 29 IV.2 Definitions . . . . 31 IV.3 Proofs of the theorems on conformal systoles . . . . 32 IV.4 Symplectic manifolds . . . . 33 IV.5 The 54-conjecture and some examples . . . . 33
We extend a result of M. Katz on the conformal systoles for blow-ups of the projective plane to all four-manifolds withb+2 = 1and odd intersection form of type(+1)⊕n(−1). The same result holds for all four-manifolds withb+2 = 1with even intersection form of type−nE8⊕Hforn≥0and which are symplectic or satisfy the so-called 54-conjecture.1
IV.1 Introduction
There are several notions of systolic invariants for Riemannian manifolds, which were introduced by M. Berger and M. Gromov (see [59] and [9, 28] for an overview). The most basic concept is the k-systole sysk(X, g)of a Riemannian manifoldX, defined as the infimum over the volumes of all cycles representing non-zero classes in Hk(X;Z). In this note we discuss a different systole, namely the conformal systole, which depends only on the conformal class of the Riemannian metric. We briefly review its definition (see Section IV.2 for details).
Let(X2n, g)be a closed oriented even dimensional Riemannian manifold. The Riemannian metric defines anL2-norm on the space of harmonicn-forms onXand hence induces a norm on the middle-dimensional cohomologyHn(X;R). The conformaln-systole confsysn(X, g)is the smallest norm of a non-zero element in the integer latticeHn(X;Z)RinHn(X;R). It is known that for a fixed manifold X the conformaln-systoles are bounded from above asgvaries over all Riemannian metrics. Hence the supremumCS(X) = supgconfsysn(X, g)of the conformal systoles over all metricsgis a finite number, which isa prioria diffeomorphism invariant ofX.
The interest in the literature has been to find bounds forCS(X)that depend only on the topology ofX, e.g. the Euler characteristic ofX, whereXruns over some class of manifolds. In [18] P. Buser
1This chapter has been published under the same title in Manuscripta math. 121, 417-424 (2006).
and P. Sarnak proved the following inequalities for the closed orientable surfacesΣsof genuss: there exists a constantC >0independent ofssuch that
C−1logs < CS(Σs)2 < Clogs, ∀s≥2. (4.1) In dimension 4, M. Katz [70] proved a similar inequality for the conformal 2-systole of blow-ups of the complex projective planeCP2: there exists a constantC >0independent ofnsuch that
C−1√
n < CS(CP2#nCP2)2 < Cn, ∀n >0. (4.2) In his proof, M. Katz used a conjecture on the period map of 4-manifoldsXwithb+2 = 1. The period map is defined as the map taking a Riemannian metric g to the point in the projectivization of the positive cone inH2(X;R)given by theg-selfdual direction (see Section IV.2). The conjecture, which is still open, claims that this map is surjective. However, an inspection of the proof of M. Katz shows that this surjectivity conjecture in full strength is not needed and that in fact his theorem holds in much greater generality.
In Section IV.3, we first remark that the following proposition holds as a consequence of recent work of D. T. Gay and R. Kirby [50].
Proposition 4.1. The period map for all closed 4-manifolds withb+2 = 1has dense image.
Using the argument of M. Katz, this implies the following theorem.
Theorem 4.2. There exists a universal constantCindependent ofXandn=b2(X)such that C−1√
n < CS(X)2 < Cn, (4.3) for all closed 4-manifoldsXwithb+2 = 1which have odd intersection form.
Another consequence of Proposition 4.1 is the following theorem.
Theorem 4.3. LetX, X0be closed 4-manifolds withb+2 = 1which have isomorphic intersection forms.
ThenCS(X) =CS(X0).
This shows that in dimension 4 the invariantCSis much coarser than a diffeomorphism invariant.
Theorem 4.3 can be compared to a result of I. K. Babenko ([5], Theorem 8.1.), who showed that a certain 1-dimensional systolic invariant for manifolds of arbitrary dimension is a homotopy-invariant.
Theorem 4.3 enables us to deal with even intersection forms. SupposeX is a closed 4-manifold withb+2 = 1and even intersection form. By the classification of indefinite even quadratic forms, the intersection form of Xis isomorphic toH⊕(−k)E8 for somek ≥ 0. In particular, for eachr ∈ N there are only finitely many possible even intersection forms of rank less or equal thanr. Hence by Theorem 4.3, the invariantCStakes only finitely many values on all 4-manifolds with even intersection form, b+2 = 1andb2 ≤ r. We will show that symplectic 4-manifoldsX with b+2(X) = 1 and even intersection form necessarily haveb2(X)≤10(see Section IV.4). The same bound holds ifXsatisfies the so-called 54-conjecture (see Section IV.5). Hence together with Theorem 4.2, we get the following corollary, which possibly covers all 4-manifolds withb+2 = 1.
Corollary 4.4. There exists a universal constantCindependent ofXandn=b2(X)such that C−1√
n < CS(X)2 < Cn, (4.4) for all closed 4-manifolds X withb+2 = 1which are symplectic or have odd intersection formQor satisfy the 54-conjecture ifQis even.