Complex surfaces which are fibre bundles

Jonathan A. Hillman

School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, AUSTRALIA e-mail: jonh@ maths.usyd.edu.au

ABSTRACT

We characterize complex surfaces admitting holomorphic submersions to complex curves and quotients of such surfaces by free actions of finite groups in terms of their Euler characteristics and fundamental groups.

AMS Subject Classification (1991): Primary 32J15. Secondary 32L05, 57N13. Key words and phrases: complex surface, elliptic surface, fibre bundle.

It is an easy consequence of the classification of surfaces that a minimal compact complex surface S is ruled over a curve C of genus ≥ 2 if and only if π_{1}(S) ≅ π_{1}(C) and χ(S) = 2χ(C). (See Chapter VI of [BPV]). We shall give a similar characterization of the complex surfaces which admit holomorphic submersions to complex curves of genus ≥ 2, and more generally of quotients of such surfaces by free actions of finite groups. However we shall use the classification only to handle the cases of non-Kähler surfaces.

In order to avoid a possible ambiguity in the use of the word “surface” we shall use the term “P D_{2}^{+}-group” for the fundamental group of an aspherical closed orientable 2-manifold.

Theorem 1. Let S be a complex surface. Then S has a finite covering space which admits a holomorphic submersion onto a complex curve, with base and fibre of genus ≥ 2, if and only if π = π_{1}(S) has normal subgroups K < π̂

Typeset by AMS-TEX

such that K and π̂/K are P D_{2}^{+}-groups, [π : π̂] < ∞ and [π : π̂]χ(S) = χ(K)χ(π̂/K) > 0.

Proof. The conditions are clearly necessary. Suppose that they hold. Then S is aspherical, by Theorem IV.1 of [Hi]. In particular, π is torsion free and π_{2}(S) = 0, so S is minimal. After enlarging K if necessary we may assume that π/K has no nontrivial finite normal subgroup. Let Ŝ be the finite covering space corresponding to π̂. Then β_{1}(Ŝ) ≥ 4. If β_{1}(Ŝ) were odd then Ŝ would be minimal properly elliptic, by the classification of surfaces. But then either χ(S) = 0 or Ŝ would have a singular fibre and the projection of Ŝ to the base curve would induce an isomorphism on fundamental groups [CS]. Hence β_{1}(Ŝ) is even and so Ŝ and S are Kähler (see Theorem 4.3 of [Wl]). Since π/K is not virtually Z^{2} it is isomorphic to a discrete group of isometries of the upper half plane ℍ^{2} and β_{1}^{(2)}(π/K) ≠ 0. Hence there is a properly discontinuous holomorphic action of π/K on ℍ^{2} and a π/K-equivariant holomorphic map from the covering space S_{K} to ℍ^{2}, with connected fibres, by Theorems 4.1 and 4.2 of [ABR]. Let B and B̂ be the complex curves ℍ^{2}/(π/K) and ℍ^{2}/(π̂/K), respectively, and let h : S → B and ĥ : Ŝ → B̂ be the induced maps. The quotient map from ℍ^{2} to B̂ is a covering projection, since π̂/K is torsion free, and so π_{1}(ĥ) is an epimorphism with kernel K.

The map h is a submersion away from the preimage of a finite subset D ⊂ B. Let F be the general fibre and F_{d} the fibre over d ∈ D. Fix small disjoint discs Δ_{d} ⊂ B about each point of D, and let B^{∗} = B − ∪_{d∈D}Δ_{d}, S^{∗} = h^{−1}(B^{∗}) and S_{d} = h^{−1}(Δ_{d}). Since h|_{S^{∗}} is a submersion π_{1}(S^{∗}) is an extension of π_{1}(B^{∗}) by π_{1}(F). The inclusion of ∂S_{d} into S_{d}−F_{d} is a homotopy equivalence. Since F_{d} has real codimension 2 in S_{d} the inclusion of S_{d} − F_{d} into S_{d} is 2-connected. Hence π_{1}(∂S_{d}) maps onto π_{1}(S_{d}).

Let m_{d} = [π_{1}(F_{d})] : Im(π_{1}(F))]. After blowing up S at singular points of F_{d} we may assume that it has only normal crossings. We may then pull h|_{S_{d}} back over a suitable branched covering of Δ_{d} to obtain a singular fibre F̃_{d} with no multiple components and only normal crossing singularities. In that case F̃_{d} is obtained from F by shrinking vanishing cycles, and so π_{1}(F) maps onto π_{1}(F̃_{d}). Since blowing up a point on a curve does not change the fundamental group it follows from §9 of Chapter III of [BPV] that in general m_{d} is finite.

We may regard B as an orbifold with cone singularities of order m_{d} at d ∈ D. By the Van Kampen theorem (applied to the space S and the orbifold B) the image of π_{1}(F) in π is a normal subgroup and h induces an isomorphism from π/π_{1}(F) to π_{1}^{orb}(B). Therefore the kernel of the canonical map from π_{1}^{orb}(B) to π_{1}(B) is isomorphic to K/Im(π_{1}(F)). But this is a finitely generated normal subgroup of infinite index in π_{1}^{orb}(B), and so must be trivial. Hence π_{1}(F) maps onto K, and so χ(F) ≤ χ(K).

Let D̂ be the preimage of D in B̂. The general fibre of ĥ is again F. Let F̂_{d} denote the fibre over d ∈ D̂. Then χ(Ŝ) = χ(F)χ(B) + Σ_{d∈D̂}(χ(F̂_{d}) − χ(F)) and χ(F̂_{d}) ≥ χ(F), by Proposition III.11.4 of [BPV]. Moreover χ(F̂_{d}) > χ(F) unless χ(F̂_{d}) = χ(F) = 0, by Remark III.11.5 of [BPV]. Since χ(B̂) = χ(π̂/K) < 0, χ(Ŝ) = χ(K)χ(π̂/K) and χ(F) ≤ χ(K) it follows that χ(F) = χ(K) < 0 and χ(F̂_{d}) = χ(F) for all d ∈ D̂. Therefore F̂_{d} ≅ F for all d ∈ D̂ and so ĥ is a holomorphic submersion. □

We may construct examples of such surfaces as follows. Let n > 1 and C_{1} and C_{2} be two curves such that Z/nZ acts freely on C_{1} and with isolated fixed points on C_{2}. Then the quotient S of C_{1} × C_{2} under the induced action is a complex surface and the projection from C_{1} × C_{2} to C_{2} induces a surjective holomorphic mappping from S to C_{2}/(Z/nZ) with critical values corresponding to the fixed points.

Corollary. The surface S admits such a holomorphic submersion onto a complex curve if and only if π/K is torsion free. □

Similar arguments may be used to characterize surfaces ruled over curves of genus ≥ 2 and to show that a Kähler surface S is a minimal properly elliptic surface with no singular fibres if and only if χ(S) = 0 and π = π_{1}(S) has a normal subgroup A ≅ Z^{2} such that π/A is virtually torsion free and indicable, but is not virtually abelian. Moreover, if S is not a ruled surface then it is a complex torus, a hyperelliptic surface, an Inoue surface, a Kodaira surface or a minimal elliptic surface if and only if χ(S) = 0 and π_{1}(S) has a normal subgroup A which is poly-Z and not cyclic, and such that π/A is infinite and virtually torsion free indicable. (See Chapter X.§5 of [Hi]).

We may combine Theorem 1 with some observations deriving from the classification of surfaces for our second result.

Theorem 2. Let S be a complex surface such that π = π_{1}(S) ≠ 1. If S is homotopy equivalent to the total space E of a bundle over a closed orientable 2-manifold then S is diffeomorphic to E. Proof. Let B and F be the base and fibre of the bundle, respectively. Suppose first that χ(F) = 2. Then χ(B) ≤ 0, for otherwise S would be simplyconnected. Hence π_{2}(S) is generated by an embedded S^{2} with self-intersection 0, and so S is minimal. Therefore S is ruled over a curve diffeomorphic to B, by the classification of surfaces.

Suppose next that χ(B) = 2. If χ(F) = 0 and π ≅ Z̸^{2} then π ≅ Z ⊕ (Z/nZ) for some n > 0. Then S is a Hopf surface and so is determined up to diffeomorphism by its homotopy type, by Theorem 12 of [Kt]. If χ(F) = 0 and π ≅ Z^{2} or if χ(F) < 0 then S is homotopy equivalent to S^{2} × F, so χ(S) < 0, w_{1}(S) = w_{2}(S) = 0 and S is ruled over a curve diffeomorphic to F. Hence E and S are diffeomorphic to S^{2} × F.

In the remaining cases E and F are both aspherical. If χ(F) = 0 and χ(B) ≤ 0 then χ(S) = 0 and π has one end. Therefore S is a complex torus, a hyperelliptic surface, an Inoue surface, a Kodaira surface or a minimal properly elliptic surface. (This uses Bogomolov’s theorem on class V II_{0} surfaces [Te]). The Inoue surfaces are mapping tori of self-diffeomorphisms of S^{1} × S^{1} × S^{1}, and their fundamental groups are not extensions of Z^{2} by Z^{2}, so S cannot be an Inoue surface. As the other surfaces are Seifert fibred 4-manifolds E and S are diffeomorphic, by [Ue].

If χ(F) < 0 and χ(B) = 0 then S is a minimal properly elliptic surface. Let A be the normal subgroup of the general fibre in an elliptic fibration. Then A ∩ π_{1}(F) = 1 (since π_{1}(F) has no nontrivial abelian normal subgroup) and so [π : A.π_{1}(F)] < ∞. Therefore E is finitely covered by a cartesian product T × F, and so is Seifert fibred. Hence E and S are diffeomorphic, by [Ue].

The remaining case (χ(B) < 0 and χ(F) < 0) is an immediate consequence of Theorem 1, since such bundles are determined by the corresponding extensions of fundamental groups (see Theorem IV.1 of [Hi]). □

A simply-connected smooth 4-manifold which fibres over a 2-manifold must be homeomorphic to CP^{1} × CP^{1} or CP^{2}♯\overline{CP^{2}}. (See [FQ], or Chapter IX of [Hi]). Is there such a surface of general type? (Qin has shown that no surface of general type is diffeomorphic to CP^{1} × CP^{1} or CP^{2}♯\overline{CP^{2}} [Qi]).

Corollary. If moreover the base has genus 0 or 1 or the fibre has genus 2 then S is finitely covered by a cartesian product.

Proof. A holomorphic submersion with fibre of genus 2 is the projection of a holomorphic fibre bundle and hence S is virtually a product, by [Ks]. □

Up to deformation there are only finitely many algebraic surfaces with given Euler characteristic > 0 which admit holomorphic submersions onto curves [Pa]. By the argument of the first part of Theorem 1 this remains true without the hypothesis of algebraicity, for any such complex surface must be Kähler, and Kähler surfaces are deformations of algebraic surfaces (see Theorem 4.3 of [Wl]). Thus the class of bundles realized by complex surfaces is very restricted. Are there any others which admit symplectic structures? Which extensions of P D_{2}^{+}-groups by P D_{2}^{+}-groups are realized by complex surfaces (i.e., not necessarily aspherical)?

References

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