Minimality and irreducibility of symplectic four-manifolds

0.13 MB·English

by  M. J. D. Hamilton

MINIMALITY AND IRREDUCIBILITY OF SYMPLECTIC FOUR-MANIFOLDS
M.J.D.HAMILTONANDD.KOTSCHICK
6
0 ABSTRACT. We provethat all minimalsymplectic four-manifoldsare essentially irreducible. We
0 also clarify the relationship between holomorphic and symplectic minimality of Ka¨hler surfaces.
2 This leads to a new proof of the deformation-invarianceof holomorphic minimality for complex

n surfaceswithevenfirstBettinumberwhicharenotHirzebruchsurfaces.
a
J

2

1. INTRODUCTION AND STATEMENT OF RESULTS
]
G
Inthispaperwediscusscertaingeometricandtopologicalpropertiesofsymplecticfour-manifolds.
S
Our main concern is the notion of minimality, and its topological consequences. We shall extend
.
h to manifolds with b+ = 1 the irreducibility result proved in [7, 8] for the case that b+ > 1. We
t 2 2
a
also show that holomorphic and symplectic minimality are equivalent precisely for those Ka¨hler
m
surfaceswhicharenotHirzebruchsurfaces. TogetherwithworkofBuchdahl[2],thisyieldsanew
[
proof of the deformation-invariance of holomorphic minimality for complex surfaces with even

2
first Betti number,againwiththeexceptionofHirzebruch surfaces.
v
4
0 1.1. Minimality. A complex surface is said to be minimal if it contains no holomorphic sphere
2 of selfintersection −1, see for example [1]. A symplectic four-manifold is usually considered
9
to be minimal if it contains no symplectically embedded sphere of selfintersection −1, see for
0
5 example[13,4]. InthecaseofaKa¨hlersurfacebothnotionsofminimalitycanbeconsidered,butit
0
isnotatallobviouswhethertheyagree. Intherecentliteratureonsymplecticfour-manifoldsthere
/
h are frequent references to (symplectic) minimality, and often Ka¨hler surfaces are considered as
t
a examples,butwehavefoundnoexplicitdiscussionoftherelationshipbetweenthetwodefinitions
m
inprint,comparee. g. [13, 14, 15, 19, 4, 8, 5].
:
v An embedded holomorphic curve in a Ka¨hler manifold is a symplectic submanifold. There-
i fore, for Ka¨hler surfaces symplectic minimality implies holomorphic minimality. The following
X
counterexampletotheconverseshouldbewellknown:
r
a
Example 1. Let X = P(O ⊕ O(n)) be the nth Hirzebruch surface. If n is odd and n > 1, then
n
X isholomorphicallyminimalbutnotsymplecticallyminimal.
n
In Section 2 below we explain this example in detail, and then we prove that there are no other
counterexamples:
Theorem 1. A Ka¨hler surface that is not a Hirzebruch surface X with n odd and n > 1 is
n
holomorphicallyminimalifand onlyifit issymplecticallyminimal.
A proof can be given using the known calculations of Seiberg–Witten invariants of Ka¨hler sur-
faces. UsingSeiberg–Witten theory,it turnsout that fornon-ruledKa¨hler surfaces symplecticand
Date:September5,2005;MSC2000:primary57R17,57R57,32J15,secondary53D35,32G05,32J27.
ThesecondauthorwishestothankY.EliashbergforhishospitalityatStanfordUniversitywhilesomeofthiswork
wascarriedout.
1
2 M.J.D.HAMILTONANDD.KOTSCHICK
holomorphic minimality coincide because they are both equivalent to smooth minimality, that is,
the absence of smoothlyembedded (−1)-spheres. The case of irrational ruled surfaces is elemen-
tary.
Such a proof is not satisfying conceptually, because the basic notions of symplectic topology
should be well-defined without appeal to results in gauge theory. Therefore, in Section 2 we give
a proof of Theorem 1 within the framework of symplectic topology, using Gromov’s theory of
J-holomorphic curves. We shall use results of McDuff [13] for which Gromov’s compactness
theoremis crucial. Essentiallythesameargumentcan beused toshowthatsymplecticminimality
isadeformation-invariantproperty,seeTheorem3. Thisnaturalresultislurkingunderthesurface
of McDuff’s papers [13, 14, 15], and is made explicit in [16], compare also [18, 19]. Of course
this result is also a corollary of Taubes’s deep work in [21, 22, 23, 8], where he showed, among
otherthings,that if thereis a smoothlyembedded (−1)-sphere, then there is also a symplectically
embeddedone.
InSection 2weshallalsoprovethatforcompactcomplexsurfaces withevenfirst Betti number
whicharenotHirzebruchsurfacesholomorphicminimalityispreservedunderdeformationsofthe
complex structure. This result is known, and is traditionally proved using the Kodaira classifi-
cation, cf. [1]. The proof we give is intrinsic and independent of the classification. Instead, we
combinethe result of Buchdahl [2] with the deformation invariance of symplecticminimalityand
Theorem 1.
1.2. Irreducibility. Recall that an embedded (−1)-sphere in a four-manifold gives rise to a con-
nectedsumdecompositionwhereoneofthesummandsisacopyofCP2. Forsymplecticmanifolds
no other non-trivial decompositions are known. Gompf [4] conjectured that minimal symplectic
four-manifolds are irreducible, meaning that in any smooth connected sum decomposition one of
thesummandshastobeahomotopysphere. InSection3belowweshallprovethefollowingresult
inthisdirection:
Theorem 2. Let X be a minimal symplectic 4-manifold with b+ = 1. If X splits as a smooth
2
connectedsumX = X #X ,thenoneoftheX isanintegralhomologyspherewhosefundamental
1 2 i
grouphas nonon-trivialfinitequotient.
Formanifoldswithb+ > 1thecorrespondingresultwasfirst provedin[7]and publishedin[8].
2
AsanimmediateconsequenceoftheseresultsweverifyGompf’sirreducibilityconjectureinmany
cases:
Corollary 1. Minimal symplectic 4-manifolds with residually finite fundamental groups are irre-
ducible.
To proveTheorem 2 we shall follow the strategy of the proof for b+ > 1 in [7, 8]. In particular
2
we shallusethe deep work of Taubes [21, 22, 23], which produces symplecticsubmanifoldsfrom
information about Seiberg–Witten invariants. What is different in the case b+ = 1, is that the
2
Seiberg–Witten invariants depend on chambers, and one has to keep track of the chambers one is
workingin.
In addition to conjecturing the irreducibility of minimal symplectic four-manifolds, Gompf [4]
also raised the question whether minimal non-ruled symplectic four-manifolds satisfy K2 ≥ 0,
where K is the canonical class. For manifolds with b+ > 1 this was proved by Taubes [21, 22],
2
comparealso [8, 23]. Thecaseb+ = 1was then treated by Liu[12], who refers tothisquestionas
2
“Gompf’sconjecture”. Liu[12]alsoprovedthatminimalsymplecticfour-manifoldswhicharenot
MINIMALITYANDIRREDUCIBILITYOFSYMPLECTICFOUR-MANIFOLDS 3
rational or ruled satisfy K ·ω ≥ 0. We shall use Liu’s inequalities to keep track of the chambers
in ourargument. Althoughtheresultsof Liu[12], and also thoseofLi–Liu[10, 11], are related to
Theorem2,thistheoremdoesnotappearthere,oranywhereelseintheliteraturethatweareaware
of.
2. NOTIONS OF MINIMALITY
First we discuss the Hirzebruch surfaces X = P(O ⊕O(n)), with n odd and > 1, in order to
n
justifytheassertionsmadeinExample1 intheIntroduction.
If n = 2k +1, considerthe union of a holomorphicsection S of X of selfintersection −n and
n
of k disjoint parallel copies of the fiber F. This reducible holomorphic curve can be turned into
a symplecticallyembedded sphere E by replacing each of the transverse intersections of S and F
by asymplecticallyembeddedannulus. Then
E ·E = (S +kF)2 = S ·S +2k S ·F = −n+2k = −1 .
This shows that X is not symplectically minimal. To see that it is holomorphically minimal,
n
note that a homology class E containing a smooth holomorphic (−1)-sphere would satisfy E2 =
K·E = −1,andwouldthereforebeS+kF,asabove. However,thisclasshasintersectionnumber
E ·S = (S +kF)·S = −n+k = −k −1 < 0
with the smooth irreducible holomorphic curve S. Therefore, E can only contain a smooth irre-
ducibleholomorphiccurveifE = S, inwhichcase k = 0 and n = 1.
Next we prove that for all other Ka¨hler surfaces symplectic and holomorphic minimality are
equivalent.
Proofof Theorem 1. Inviewofthediscussionin1.1above,weonlyhavetoprovethatif(X,ω)is
a Ka¨hler surface which is not a Hirzebruch surface X with n odd and n > 1, then holomorphic
n
minimalityimpliessymplecticminimality.
We start by assuming that (X,ω) is not symplectically minimal, so that it contains a smoothly
embedded (−1)-sphere E ⊂ X with ω| 6= 0. Orient E so that ω| > 0, and denote by [E] ∈
E E
H (X;Z) the corresponding homology class. The almost complex structures J compatible with
2
ω are all homotopic to the given integrable J ; in particular their canonical classes agree with
∞
the canonical class K of the Ka¨hler structure. It is elementary to find a compatible J for which
the sphere E with the chosen orientation is J-holomorphic. Therefore E satisfies the adjunction
formula
1
g(E) = 1+ (E2 +K ·E) .
2
We conclude that K · E = −1. (Note that the orientation of E is essential here.) This implies
in particular that the expected dimension of the moduli space of J-holomorphic curves in the
homologyclass [E]vanishes.
Let J be the completion–with respect to a suitable Sobolov norm–of the space of C∞ almost
complex structures compatible with ω, cf. [17]. McDuff has proved that, for almost complex
structures J from an everywhere dense subset in J, there is a unique smooth J-holomorphic
sphereC inthehomologyclass[E], see Lemma3.1in [13].
The uniqueness implies that the curve C varies smoothly with J. One then uses Gromov’s
compactness theorem fora smoothfamily ofalmost complexstructures to concludethat for all J,
not necessarily generic, there is a uniqueJ-holomorphicrepresentativeof thehomologyclass [E]
which, if it is not a smooth curve, is a reducible curve C = C such that each C is a smooth
Pi i i
4 M.J.D.HAMILTONANDD.KOTSCHICK
J-holomorphicsphere. CompareagainLemma3.1in[13]and[17]. (Inthesereferences reducible
J-holomorphiccurves arecalled cusp curves.)
LetJ beasequenceofgenericalmostcomplexstructuresinJ whichconvergestotheintegrable
j
J∞ as j→∞. For each Jj there is a smooth Jj-holomorphicsphere Ej in the homologyclass [E].
As j→∞, the Ej converge weakly to a possibly reducible J∞-holomorphic curve E∞. If E∞ is
irreducible, then it is a holomorphic (−1)-sphere, showing that (X,J∞) is not holomorphically
minimal. IfE isreducible, let
∞
k
E∞ = XmiCi
i=1
be the decomposition into irreducible components. The multiplicities m are positive integers.
i
Each C isan embedded sphere,and therefore theadjunctionformulaimplies
i
C2 +K ·C = −2 .
i i
Multiplyingby m and summingoveri weobtain
i
k k k
XmiCi2 +K ·XmiCi = −2Xmi .
i=1 i=1 i=1
Nowthesecondterm ontheleft handsideequalsK ·E = −1, sothatwehave
k k
XmiCi2 = 1−2Xmi .
i=1 i=1
It follows that there is an index i such that C2 ≥ −1. If C2 = −1 for some i, then we again
i i
conclude that (X,J∞) is not holomorphicallyminimal. If Ci2 ≥ 0 for some i, then (X,J∞) is bi-
rationallyruledorisrational,cf.Proposition4.3inChapterVof[1]. Thus,ifitisholomorphically
minimal, it is either a minimal ruled surface or CP2, but the latter is excluded by our assumption
that(X,J∞)isnotsymplecticallyminimal. If(X,J∞)wereruledoverasurfaceofpositivegenus,
π
X −→ B, then theembedding of the(−1)-sphere E would be homotopicto a map with imagein
a fiber, because π| : E→B would be homotopicto a constant. But this would contradict the fact
E
thatE has non-zero selfintersection.
Thus we finally reach the conclusion that (X,J∞) is ruled over CP1. If it is holomorphically
minimal,thenit isaHirzebruch surface X withnoddand n > 1, becauseX is notholomorphi-
n 1
cally minimal,andX has evenintersectionformand istherefore symplecticallyminimal.
2k
ThisconcludestheproofofTheorem 1. (cid:3)
Remark1. Wehaveusedthefactthattheexistenceofarationalholomorphiccurveofnon-negative
selfintersection ina complexsurface impliesthatthesurface is rational orruled. Such astatement
alsoholdsin thesymplecticcategory,cf. [13], butwedo notneed thathere.
The exposition of the proof of Theorem 1 can be shortened considerably if one simply uses
McDuff’s Lemma 3.1 from [13] as a black box. We havechosen to includesomeof the details so
thatthereadercanseethatthedegenerationoftheJ -holomorphiccurvesE asj→∞istheexact
j j
inverseoftheregenerationused inthediscussionofExample1.
The following theorem, Proposition 2.3.A in [16], can be proved by essentially the same argu-
ment,allowingthesymplecticform tovary smoothly,comparealso[13, 19]:
MINIMALITYANDIRREDUCIBILITYOFSYMPLECTICFOUR-MANIFOLDS 5
Theorem3([16]). Symplecticminimalityisadeformation-invariantpropertyofcompactsymplec-
ticfour-manifolds.
Notethatholomorphicminimalityofcomplexsurfacesisnotinvariantunderdeformationsofthe
complexstructure. In the Ka¨hlercase theHirzebruch surfaces X withn odd are all deformation-
n
equivalent, but are non-minimal for n = 1 and minimal for n > 1. In the non-Ka¨hler case there
are otherexamplesamongtheso-calledclass VII surfaces.
Forcomplexsurfaces ofnon-negativeKodairadimensionitistruethatholomorphicminimality
is deformation-invariant, but the traditional proofs for this are exceedingly cumbersome, see for
example [1], section 7 of Chapter VI, where it is deduced from the Kodaira classification and a
whole array of additional results. For the case of even first Betti number we now give a direct
proof, whichdoes notusetheclassification.
Theorem 4. Let X be a holomorphically minimal compact complex surface with even first Betti
number, which is not a Hirzebruchsurface X with n odd. Then any surface deformationequiva-
n
lentto X is alsoholomorphicallyminimal.
Proof. Let X with t ∈ [0,1] be a smoothly varying family of complex surfaces such that X =
t 0
X. Buchdahl [2] has proved that every compact complex surface with even first Betti number is
Ka¨hlerian, without appealing to any classification results. Thus, each X is Ka¨hlerian, and we
t
wouldliketochooseKa¨hlerformsω andω onX andX respectively,whichcan bejoinedbya
0 1 0 1
smoothfamilyofsymplecticformsω . There aretwoways tosee thatthisispossible.
t
On the one hand, Buchdahl [2] characterizes the Ka¨hler classes, and one can check that one
can choose a smoothly varying family of Ka¨hler classes for X , which can then be realized by a
t
smoothly varying family of Ka¨hler metrics. On the other hand, we could just apply Buchdahl’s
resultforeach valueoftheparametertseparately,withoutworryingaboutsmoothvariationofthe
Ka¨hlerformwiththeparameter,andthenconstructasmoothfamilyω ofsymplecticnotnecessar-
t
ily Ka¨hler forms from this, cf. [19] Proposition 2.1. In detail, start with arbitrary Ka¨hler forms ω
t
on X . As the complex structure depends smoothly on t, there is an open neighbourhood of each
t
t ∈ [0,1] such that ω is a compatible symplectic form for all X with s in this neighbourhood
0 t0 s
of t . By compactness of [0,1], we only need finitely many such open sets to cover [0,1]. On
0
the overlaps we can deform these forms by linear interpolation, because the space of compatible
symplecticformsisconvex. Inthiswayweobtainasmoothlyvaryingfamilyofsymplecticforms.
Now X = X was assumed to be holomorphically minimal and not a Hirzebruch surface X
0 n
with odd n. Therefore, Theorem 1 shows that X is symplectically minimal, and Theorem 3 then
0
impliesthatX is alsosymplecticallyminimal. Theeasy directionof Theorem1 showsthat X is
1 1
holomorphicallyminimal. (cid:3)
Letusstressoncemorethat thisresultisnotnew,butitsproofis. Theaboveproofdoesnotuse
the Kodaira classification. The only result we have used from the traditional theory of complex
surfaces is that a surface containing a holomorphic sphere of positive square is rational, which
enteredintheproofofTheorem1. Wehavenotusedthegeneralizationofthisresulttosymplectic
manifolds, and we have not used any Seiberg–Witten theory either. Our proof does depend in an
essential way on the work of Buchdahl [2]. Until that work, the proof that complex surfaces with
evenfirst Betti numbersareKa¨hlerian depended ontheKodairaclassification.
6 M.J.D.HAMILTONANDD.KOTSCHICK
3. CONNECTED SUM DECOMPOSITIONS OF MINIMAL SYMPLECTIC FOUR-MANIFOLDS
Inthissectionweproverestrictionsonthepossibleconnectedsumdecompositionsofaminimal
symplectic four-manifold with b+ = 1, leading to a proof of Theorem 2. To do this we have to
2
leavetherealm ofsymplectictopologyanduseSeiberg–Witten gaugetheory.
Let X be a closed oriented smooth 4-manifold with b+(X) = 1. We fix a Spinc structure s and
2
a metric g on X and consider the Seiberg–Witten equations for a positive spinor φ and a Spinc
connectionA:
D+φ = 0
A
F+ = σ(φ,φ)+η ,
Aˆ
where the parameter η is an imaginary-valued g-self-dual 2-form. Here Aˆ denotes the U(1)-
connection on the determinant line bundle induced from A, so that F is an imaginary-valued
Aˆ
2-form. A reduciblesolutionoftheSeiberg–Witten equationsisasolutionwithφ = 0.
For every Riemannian metric g there exists a g-self-dual harmonic 2-form ω with [ω ]2 = 1.
g g
Because b+(X) = 1, this 2-form is determined by g up to a sign. We choose a forward cone,
2
i.e.oneofthetwoconnectedcomponentsof{α ∈ H2(X;R)|α2 > 0}. Thenwefixω bytaking
g
theformwhosecohomologyclasslies intheforward cone.
Let L be the determinant line bundle of the Spinc structure s. The curvature F represents
Aˆ
2πc (L)incohomology,andeveryformwhichrepresentsthisclasscanberealizedasthecurvature
1
i
of Aˆ for a Spinc connection A. For given (g,η) there exists a reducible solution of the Seiberg–
Witten equations if and only if there is a Spinc connection A such that F+ = η, equivalently
Aˆ
(c (L)− i η)·ω = 0. Definethediscriminantoftheparameters (g,η)by
1 2π g
i
∆ (g,η) = (c (L)− η)·ω .
L 1 g
2π
One dividesthespace of parameters (g,η) forwhich thereare no reduciblesolutionsinto theplus
andminuschambersaccordingtothesignofthediscriminant. Twopairsofparameters(g ,η )and
1 1
(g ,η ) can be connected by a path avoiding reducible solutions if and only if their discriminants
2 2
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 invariants SW+(X,s), SW−(X,s) which are constant on the corresponding
chambers.
Suppose now that X has a symplectic structure ω. Then ω 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
determinesacanonical class K and aSpinc structuresK−1 withdeterminantK−1. Onecan obtain
everyotherSpinc structurebytwistingsK−1 withalinebundleE,to obtainsK−1 ⊗E. ThisSpinc
structurehas determinantK−1 ⊗E2.
The Taubes chamber is the chamber determined by parameters (g,η)with g chosen such that it
isalmostKa¨hlerwithω = ω and
g
i
η = F+ − rω with r ≫ 0 ,
Aˆ0 4
whereAˆ isacanonical connectionon K−1. Wehavethefollowing:
0
Lemma 1. TheTaubes chamberistheminuschamber,forthechoiceofforwardconeas above.
MINIMALITYANDIRREDUCIBILITYOFSYMPLECTICFOUR-MANIFOLDS 7
Proof. We have
i i i 1
(c (−K)− η)·ω = ( F − F+ − rω)·ω
1 2π g 2π Aˆ0 2π Aˆ0 8π
i 1
−
= ( F − rω)·ω
2π Aˆ0 8π
1
= − rω2 < 0,
8π
becausethewedgeproductofaself-dual andan anti-self-dualtwo-form vanishes. (cid:3)
Thefollowingtheorem isduetoTaubes [20,21, 22], compare[10, 11]forthecaseb+ = 1.
2
Theorem 5. TheSeiberg–WitteninvariantintheminuschamberforthecanonicalSpinc structure
is non-zero. More precisely, SW−(X,sK−1) = ±1. Moreover, if SW−(X,sK−1 ⊗E) is non-zero
and E 6= 0, then for a generic ω-compatiblealmostcomplex structureJ, the Poincare´ dual of the
Chern classof E can berepresented bya smoothJ-holomorphiccurveΣ ⊂ X.
Wehavethe followingmoreprecise versionofthesecond part ofTheorem 5, which is also due
toTaubes.
Proposition 1. Suppose SW−(X,sK−1 ⊗E) is non-zero, and E 6= 0. Then for a generic almost
complex structureJ compatible with ω there exist disjointembedded J-holomorphiccurves C in
i
X suchthat
n
PD(c1(E)) = Xmi[Ci],
i=1
whereeachC satisfiesK ·C ≤ C ·C andeachmultiplicitym isequalto1, except possiblyfor
i i i i i
thoseifor whichC isa toruswithself-intersectionzero.
i
This depends on a transversality result for J-holomorphic curves, see Proposition 7.1 in [22]
and also[23, 8]. Proposition1 immediatelyimpliesthefollowing:
Corollary 2. If SW−(X,sK−1 ⊗E) 6= 0 with E2 < 0 then X contains an embedded symplectic
(−1)-sphereΣ.
Proof. Choose a generic compatiblealmost complex structureJ as in Proposition 1, and consider
E = m C . Then E2 = m2C2 because the C are disjoint, hence C2 < 0 for some j. We
Pi i i Pi i i i j
can computethegenusofC from theadjunctionformula:
j
1
g(C ) = 1+ (C ·C +K ·C ) ≤ 1+C ·C ≤ 0.
j j j j j j
2
HenceΣ = C is aspherewithself-intersectionnumber−1. (cid:3)
j
Afterthesepreparations wecan nowproveTheorem2.
Proofof Theorem 2. Let (X,ω) be a closed symplectic4-manifold with b+ = 1. We denoteby K
2
boththefirstChernclassofanycompatiblealmostcomplexstructure,andthecomplexlinebundle
withthisChern class.
First,supposethat(X,ω)is symplecticallyminimaland rationalorruled. Then,by theclassifi-
cationofruledsymplecticfour-manifolds,X isdiffeomorphiceithertoCP2,toanevenHirzebruch
surface, or to a geometrically ruled Ka¨hler surface over a complex curve of positive genus, com-
paree. g. [17]. Thesemanifoldsare allirreducibleforpurely topologicalreasons. Thisisclear for
8 M.J.D.HAMILTONANDD.KOTSCHICK
CP2 and 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, inanyconnected sumdecompositiononeofthesummandsissimplyconnected. Ifthis
summandwerenotahomotopysphere,thentheothersummandwouldbeasmoothfour-manifold
with the same fundamental group but with strictly smaller Euler characteristic than the ruled sur-
face. This is impossible, because the irrational ruled surfaces realize the smallest possible Euler
characteristicfortheirfundamentalgroups,compare[6].
Thus, we may assume that (X,ω) is not only symplectically minimal, but also not rational or
ruled. ThenLiu’sresultsin[12]tellus thatK2 ≥ 0and K ·ω ≥ 0.
If X decomposes as a connected sum X = M#N then one of the summands, say N, has
negativedefiniteintersectionform. Moreover,thefundamentalgroupofN hasnonon-trivialfinite
quotients, by Proposition 1 of [9]. In particular H (N;Z) = 0, and hence the homology and
1
cohomologyofN aretorsion-free. IfN isanintegralhomologysphere,thenthereisnothingmore
toprove.
Suppose N is not an integral homology sphere. By Donaldson’s theorem [3], the intersection
form of N is diagonalizable over Z. Thus there is a basis e ,...,e of H2(N;Z) consisting of
1 n
elementswithsquare−1 which arepairwiseorthogonal. Write
n
K = KM +Xaiei ,
i=1
with K ∈ H2(M;Z). The a ∈ Z are odd, because K is a characteristic vector. This shows
M i
in particular that K is not a torsion class. Its orthogonal complement K⊥ in H2(X;R) is then a
hyperplane. AsK2 ≥ 0andb+(X) = 1,thehyperplaneK⊥ doesnotmeetthepositivecone. Thus
2
Liu’sinequalityK ·ω ≥ 0 mustbestrict: K ·ω > 0.
Now we know SW−(X,sK−1) = ±1 from Taubes’s result, where sK−1 is the Spinc structure
with determinant K−1 induced by the symplectic form ω. The inequality (−K) · ω < 0 shows
that a pair (g,0) is in the negative, i. e. the Taubes chamber, whenever g is almost Ka¨hler with
fundamentaltwo-formω. AsK⊥ doesnotmeetthepositivecone,allpairs(g,0)areinthenegative
chamber,forallRiemannianmetricsg. WechooseafamilyofRiemannianmetricsg onX which
r
pinchestheneckconnectingM andN downtoapointasr → ∞. Forr largewemayassumethat
g convergesto metricsonthe(punctured)M and N, which wedenoteby g and g .
r M N
Lemma 2. If we choose the forward cone for M to be such that it induces on X the forward
conedetermined bythesymplecticstructure, thenfor every Riemannianmetricg′ on M, thepoint
(g′,0) is in the negative chamber of M with respect to the Spinc structure s on M obtained by
M
restrictionof s .
K−1
Proof. Thechamberisdetermined bythesignofc (s)·ω . Wehave
1 g
0 > (−K)·ωgr = c1(sK−1)·ωgr = c1(sM)·ωgr +c1(sN)·ωgr
−→ c (s )·ω +c (s )·ω , asr → ∞ .
1 M gM 1 N gN
Weknowthat ω is self-dual harmonicwithrespect tog , and hence vanishesbecauseb+(N) =
gN N 2
0. Thisimpliesthatc1(sK−1)·ωgr convergestoc1(sM)·ωgM forr → ∞. Thus
c (s )·ω ≤ 0 .
1 M gM
MINIMALITYANDIRREDUCIBILITYOFSYMPLECTICFOUR-MANIFOLDS 9
However,we havec (s ) = K−1, and
1 M M
n
K2 = K2 +Xa2 ≥ K2 +n ≥ n ≥ 1 ,
M i
i=1
showing that K⊥ does not meet the positivecone of M. Thus c (s )·ω < 0. Again because
M 1 M gM
K⊥ doesnot meetthepositiveconeofM, thisinequalityholdsforallmetrics g′ on M. (cid:3)
M
The degeneration of the g as r goes to infinity takes place in the negative chamber for s ,
r K−1
where the Seiberg–Witten invariant is ±1, and, by Lemma 2, g is in the negative chamber for
M
sM. It followsthatSW−(M,sM) = ±1.
We now reverse the metric degeneration, but use a different Spinc structure on N. Instead
of using s with c (s ) = − n a e , we use the unique Spinc structure s′ with c (s′ ) =
N 1 N Pi=1 i i N 1 N
a e − n a e . For every metric on N there is a unique reducible solution of the Seiberg–
1 1 Pi=2 i i
Witten equations for this Spinc structure with η = 0. Gluing this solution to the solutions on M
given by the invariant SW−(M,sM), we find SW−(X,s′) = ±1, where s′ is the Spinc structure
on X obtained from sM and s′N, compare Proposition 2 of [9]. We have s′ = sK−1 ⊗ E, with
E = a e . Therefore E2 = −a2 ≤ −1, and Corollary 2 shows that X is not minimal. This
1 1 1
completestheproofofTheorem 2. (cid:3)
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MATHEMATISCHESINSTITUT,LUDWIG-MAXIMILIANS-UNIVERSITA¨TMU¨NCHEN,THERESIENSTR.39,80333
MU¨NCHEN, GERMANY
E-mailaddress:[email protected]
MATHEMATISCHESINSTITUT,LUDWIG-MAXIMILIANS-UNIVERSITA¨TMU¨NCHEN,THERESIENSTR.39,80333
MU¨NCHEN, GERMANY
E-mailaddress:[email protected]