increasec21(X)while keeping the signature and the divisibility ofKX fixed. Note thatπ1(X\ΣX) = 1 sinceXis simply-connected and the sphereBX sewed together from exceptional spheres in both copies ofM intersectsΣX once. Hence the 4-manifold we obtain is again simply-connected (cf. [41, Section 3] for a related construction).
Remark 6.39. In principle it should also be possible to do the construction with triples of Lagrangian rim tori from Theorem 5.79 like in the previous sections to find inequivalent symplectic structures on simply-connected 4-manifolds, starting from an n-fold fibre sum M(n). Note that every fibre sum contributes 2g rim tori out of which we can form g Lagrangian triples. One can probably extend Example 5.73 to show that some of these rim tori are contained in nuclei N(2). In particular, this should work for the fibrationsX(m, n)in Section VI.5.3.
VI.3 Branched coverings
In the following sections we will describe another construction of simply-connected symplectic 4-manifolds with divisible canonical class. This construction uses branched coverings of algebraic sur-faces. We will first define the notion of branched coverings and give a criterion in Corollary 6.47 which ensures that the branched coverings we use are simply-connected if we start with a simply-connected manifold.
VI.3.1 Definition
LetMn be a closed, oriented manifold andFn−2 a closed, oriented submanifold of codimension 2.
Suppose that the fundamental class[F]∈Hn−2(M;Z)is divisible by some integerm >1. Choose a classB∈Hn−2(M;Z)such that[F] =mB. LetLF, LBdenote the complex line bundles with Chern classes
c1(LF) =P D[F], c1(LB) =P D(B).
Sincec1(LF) =mc1(LB), there exists an isomorphism L⊗Bm ∼=LF. We consider the following map
φ:LB→L⊗mB ,
x7→x⊗ · · · ⊗x (mfactors).
On each fibre, this map is given by
C→C⊗m, z7→z⊗ · · · ⊗z.
Letebe a basis vector of theC-vector spaceC. Thene⊗ · · · ⊗eis a basis ofC⊗m which induces an isomorphism
C⊗m →C
z1e⊗ · · · ⊗zme7→(z1·. . .·zm)e.
The composition
C→C⊗m→C
is then the mapz7→zm. On the unit circle, this is anm-fold covering. Hence we get
Lemma 6.40. LetLB, LF → M be complex line bundles withc1(LF) = mc1(LB)and denote the associated circle bundles byEF, EB. Then the map
φ:LB →L⊗mB ∼=LF
induces a fibrewisem-fold coveringEB→EF.
Lets:M →LFbe a section which vanishes alongF, is non-zero onM0 =M\Fand is transverse to the zero section.
Theorem 6.41. Consider the set
X=φ−1(s(M))⊂LB.
Then X is again a smooth manifold of dimension n. Letπ: X → M denote the restriction of the projectionLB →M.
• OverM0, the mapφ:φ−1(M0)→M0 is anm-fold cyclic covering.
• The intersection ofX with the zero section ofLB is a smooth submanifoldF ofXandπmaps F diffeomorphically ontoF.
• Letν(F)denote a tubular neighbourhood ofFinX. The projectionπmapsν(F)onto a tubular neighbourhoodν(F)ofF inM. Under the identificationF =F viaπ, there is a vector bundle isomorphismν(F) = ν(F)⊗mand the mapπcorresponds to the mapφabove. In other words, there are local coordinates of the formU×DX2 ⊂ν(F)andU×D2M ⊂ν(F), withU ⊂F ∼=F such thatπhas the form
U×DX2 →U ×DM2 ,(x, z)7→(x, zm).
For a proof, see [63].
Definition 6.42. Them-fold branched (or ramified) coveringM(F, B, m)ofM branched overFand determined byBis defined as
M(F, B, m) =φ−1(s(M))⊂LB.
SupposeMis a smooth complex algebraic surface andD⊂Ma smooth connected complex curve.
Ifm >0is an integer that divides[D]andB∈H2(M;Z)a homology class such that[D] =mB, then there exists a branched coveringM(D, B, m). Since the divisorDhas an associated holomorphic line bundle, one can show that the line bundleLB in the previous section can be chosen as a holomorphic line bundle as well (see [63]). This implies that the branched covering admits the structure of an algebraic surface. The invariants ofM can be calculated by the following proposition.
Proposition 6.43. Let D be a smooth connected complex curve in a complex surface M such that [D] = mB. Let φ:M(D, B, m) → M be the branched covering. Then the invariants of N :=
M(D, B, m)are given by:
(a) KN =φ∗(KM + (m−1)B) (b) c21(N) =m(KM + (m−1)B)2 (c) e(N) =me(M)−(m−1)e(D),
VI.3 Branched coverings 127 wheree(D) = 2−2g(D) =−(KM ·D+D2)by the adjunction formula.
Proof. The formula fore(N)can be calculated by the well-known formula for the Euler characteristic of a space decomposed into two pieces (which we used already in the proof of Corollary 5.14) and the formula for standard, unramified coverings:
e(N) =e(N \D) +e(ν(D))−e(∂ν(D))
=me(M\D) +e(D) =m(e(M)−e(D)) +e(D)
=me(M)−(m−1)e(D).
HereDdenotes the complex curve inNover the branching divisorDas in Theorem 6.41. The formula forc21(N)follows then by the signature formula of Hirzebruch [63]:
σ(N) =mσ(M)−m2−1 3m D2. The formula forKN can be found in [8, Chapter I, Lemma 17.1].
We will consider the particular case that the complex curveDis in the linear system|nKM|and hence represents in homology a multiplenKM of the canonical class ofM. Letm > 0be an integer dividingnand writen=ma.
Lemma 6.44. LetDbe a smooth connected complex curve in a complex surfaceM with[D] =nKM. Then the invariants of them-fold ramified coverφ:M(D, aKM, m)→Mbranched overDare given by:
(a) KN = (n+ 1−a)φ∗KM (b) c21(N) =m(n+ 1−a)2c21(M)
(c) e(N) =me(M) + (m−1)n(n+ 1)c21(M)
Proof. We have[D] =nKM andB =aKM. Hence we can calculate:
KM + (m−1)B = (1 +ma−a)KM = (n+ 1−a)KM e(D) =−(KM ·D+D2)
=−(n+n2)c21(M) =−n(n+ 1)c21(M)
This implies the formulas.
VI.3.2 The fundamental group of branched covers
LetMnbe a closed oriented manifold andFn−2 a closed oriented submanifold. Suppose that[F] = mBand consider the branched coveringM = M(F, B, m). Even if the base manifoldM is simply-connected the fundamental group ofMis in general non-trivial. The following theorem can be used to ensure that the branched covers are simply-connected. LetM0=M\Fdenote the complement ofF. Theorem 6.45. LetMnbe a closed oriented manifold andFn−2a closed oriented submanifold such that[F]is a non-torsion class inHn−2(M;Z). Suppose in addition that the fundamental group ofM0 is abelian. Then for allmandBwith[F] =mBthere exists an isomorphism
π1(M(F, B, m))∼=π1(M).
Proof. Letk >0denote the maximal integer dividing[F]. Thenmdivideskand we can writek=ma witha > 0. LetM0 =M(F, B, m)\F. Denote the meridian toF inM0, i.e. the class of a fibre in
∂ν(F), byσ. By Proposition A.3 we get
π1(M(F, B, m))∼=π1(M0)/N(σ).
We have an exact sequence
0→π1(M0)→π∗ π1(M0)→Zm→0,
sinceπ:M0 → M0is anm-fold cyclic covering. The assumption thatπ1(M0)is abelian implies that π1(M0)is also abelian. Therefore, the normal subgroups generated by the fibres in these groups are cyclic and we get an exact sequence of subgroups
0→Zaσ →m·Zmaσ →Zm→0,
where σ is the meridian of F in M0. The surjection Zmaσ → Zm implies that for each element α ∈π1(M0)there is an integerr ∈Zsuch thatα+rσmaps to zero inZm and hence is in the image ofπ∗. In other words, the induced map
π∗:π1(M0)−→π1(M0)/hσi is surjective. The kernel of this map ishσi, hence
π1(M0)/hσi−→∼= π1(M0)/hσi. Again by Proposition A.3, this impliesπ1(M(F, B, m))∼=π1(M).
We will use this theorem in the case whereM is a 4-manifold and F an embedded surface. In general, the complement of a 2-dimensional submanifold in a 4-manifold does not have abelian fun-damental group even if M is simply-connected. However, this is sometimes the case if we consider complex curves in complex manifolds. The following theorem is due to Nori ([105], Proposition 3.27).
Theorem 6.46. LetM be a smooth complex algebraic surface andD, E⊂Msmooth complex curves which intersect transversely. Assume thatD02 >0for every connected componentD0 ⊂D. Then the kernel ofπ1(M\(D∪E))→π1(M \E)is a finitely generated abelian group.
In particular, forE=∅, this implies that the kernel of π1(M0)→π1(M)
is a finitely generated abelian group ifDis connected andD2 >0, whereM0denotesM \D. IfM is simply-connected it follows thatπ1(M0)is abelian. Together with Theorem 6.45 we get the following corollary to Nori’s theorem.
Corollary 6.47. Let M be a simply-connected, smooth complex algebraic surface and D ⊂ M a smooth connected complex curve withD2 >0. LetM be a cyclic ramified cover ofMbranched over D. ThenM is also simply-connected.
If the divisor not only satisfiesD2 >0but is ample, there is a more general theorem by Cornalba [27]:
VI.3 Branched coverings 129 Theorem 6.48. LetMbe ann-dimensional smooth complex algebraic manifold andD⊂M a smooth ample divisor. LetM be a ramified cover ofM branched overD. Then
πk(M)∼=πk(M), 0≤k≤n−1, andπn(M)surjects ontoπn(M).
In particular, we get in the case of complex surfaces (n= 2):
Corollary 6.49. LetM be a smooth complex algebraic surface andD⊂M a smooth ample divisor.
LetM be a ramified cover ofM branched overD. Thenπ1(M)∼=π1(M).
In a different situation, Catanese [20] has also used restrictions on divisors to ensure that the com-plement of a curve in a surface and certain ramified coverings are simply-connected.
Example 6.50. LetM =CP2 andDa smooth complex curve of degreen > 0representingnH ∈ H2(M;Z), where H = [CP1] denotes the class of a hyperplane. The canonical class of CP2 is K =−3P D(H). By the adjunction formula,
g(D) = 1 +12(K·D+D2)
we can compute the genus ofD:g(D) = 1 +12n(n−3). SinceD2>0andCP2is simply-connected, the complementCP2\Dhas abelian fundamental group by Nori’s theorem. This implies that
π1(CP2\D)∼=H1(CP2\D;Z)∼=Zn,
which has been proved by Zariski in 1929 [147]. We can also consider then-fold cyclic branched covering
φ:M =M(D, H, n)→M.
By Corollary 6.47 the complex algebraic surfaceM is simply-connected. The invariants are given by the formulas in Proposition 6.43:
KM = (n−4)φ∗H c21(M) =n(n−4)2
c2(M) = 3n+ (n−1)n(n−3)
sincec2(CP2) = 3ande(D) =−n(n−3). The calculation
c21(M)−2c2(M) =n(n2−8n+ 16)−n(6 + 2n2−8n+ 6)
=n(−n2+ 4), implies
σ(M) =−13(n2−4)n
Note thatM is a simply-connected 4-manifold such that KM is divisible byd = n−4. However, c21(M)grows with the third power ofdand is rather larger. One can show thatM is diffeomorphic to a complex hypersurface inCP3of degreen(cf. [56, Exercise 7.1.6]).