View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Expo. Math. 23 (2005) 1–22 www.elsevier.de/exmath Fundamental groups of elliptic fibrations and the invariance of the plurigenera for surfaces with odd first Betti number Markus Dürr1 Institut für Mathematik, Universität Zürich, Switzerland Received 18 December 2003 Abstract We study the fundamental groups of elliptic fibrations and the close relationship between the Ko- daira dimension of an elliptic surface with odd first Betti number and the geometry of its base orbifold. We show that the plurigenera and hence the Kodaira dimension of a complex surface with odd first Betti number are determined by the isomorphism type of the fundamental group. ᭧ 2005 Elsevier GmbH. All rights reserved. MSC 2000: primary 14J27; secondary 14J80 Keywords: Elliptic fibrations; Fundamental groups; Plurigenera; Two-dimensional geometries 1. Introduction The goal of this article is twofold: we study elliptic surfaces and their fundamental groups, and we present a short and self-contained proof of the fact that the plurigenera of complex surfaces with odd first Betti number are topological invariants. By an elliptic surface we mean a closed complex analytic surface, which admits an elliptic fibration, i.e. a mapping onto a smooth curve whose general fibre is a smooth 1 Partially supported by EAGER— European Algebraic Geometry Research Training Network, Contract No. HPRN-CT-2000-00099 (BBW 99.0030), and by SNF, No. 2000-055290.98/1. E-mail address: [email protected]. 0723-0869/$ - see front matter ᭧ 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.exmath.2005.01.008 2 M. Dürr / Expo. Math. 23 (2005) 1–22 elliptic curve. Elliptic surfaces play a central role in the Kodaira–Enriques classification of complex surfaces. Of particular interest to us are the elliptic fibrations with odd first Betti number. By a result of Kodaira, these are exactly the elliptic surfaces which do not admit a Kähler metric. We completely determine their structure, and are thus able to give a presentation of their fundamental groups. Using only these results, we will give a conceptually simple and concise proof of the fact that the plurigenera and hence the Kodaira dimension of non-Kählerian surfaces are determined by the isomorphism type of the fundamental group. This result has first been obtained by Friedman and Morgan [3], but their proof is scattered over some 100 pages. The Kodaira dimension is the most fundamental invariant of a closed complex manifold X. It measures the growth of the plurigenera 0 ⊗n Pn(X) := dim H (KX ) 1 for large n. Here KX := det X denotes the canonical line bundle of X. In the case of curves the Kodaira dimension takes the values −∞, 0, 1, and these three values correspond to the three two-dimensional geometries in the sense of Thurston: the sphere S2, the Euclidean plane E2 and the hyperbolic plane H 2. We find an analogous situation in the case of elliptic fibrations with odd first Betti number, but here the Kodaira dimension of the surface is linked to the geometry of its base orbifold. The base orbifold is the base curve together with an additional structure defined by the elliptic fibration. Since the geometry of the orbifold is encoded in the fundamental group of the surface, this group determines the Kodaira dimension and moreover the plurigenera of the surface. This paper is organized as follows: Section 2 reviews the general theory of closed two-dimensional orbifolds and their fun- damental groups, and states a theorem relating the fundamental group of an elliptic fibration with the fundamental group of its base orbifold. In Section 3, we show that every elliptic fibration X → C with trivial first direct image sheaf can be obtained from a product C × E, where E is an elliptic curve, by means of logarithmic transformations. Moreover, we compute the fundamental groups of such surfaces. In Section 4, we study elliptic fibrations with odd first Betti number and their fundamental groups more closely. We compute the plurigenera of non-Kählerian elliptic surfaces, and show that their Kodaira is reflected by the geometry of the base orbifold. We deduce that the plurigenera of elliptic fibrations with odd first Betti number are topological invariants. Finally, in Section 4, we show, using an elegant argument by Friedman and Morgan, that the plurigenera of non-Kählerian surfaces are determined by the isomorphism type of their fundamental groups. 2. Orbifolds and the fundamental groups of elliptic fibrations 2.1. Orbifolds and their fundamental groups In the study of elliptic surfaces and their fundamental groups one is lead in a natural way to the notion of orbifold. M. Dürr / Expo. Math. 23 (2005) 1–22 3 Whereas an n-manifold is a Hausdorff, paracompact space, which is locally homeomor- n phic to R ,ann-orbifold is a Hausdorff, paracompact space, which is locally homeomorphic n to the quotient space of R by a finite group action. For a precise definition we refer the reader to Scott [7, p. 422]. One can introduce the notion of an orbifold covering and of a universal covering orbifold. Such a universal covering orbifold exists for any orbifold and is unique. A proof of this fact in the relevant case n = 2 can be found in [7, p. 423]. The existence of a universal covering leads then in a natural way to the notion of fundamental group of an orbifold. In dimension 2, any orbifold is homeomorphic to a manifold (possibly with boundary), because the only singular points are cone points, reflector lines and corner reflectors (see [7] for a definition). Note, however, that an orbifold with cone points is homeomorphic to a manifold as a topological space, but not isomorphic to this one as an orbifold. So one must distinguish carefully between a two-dimensional orbifold and its underlying topological surface. Recall that there are three two-dimensional geometries in the sense of Thurston — namely the unit sphere S2, the Euclidian plane E2 and the hyperbolic plane H 2. We say that an orbifold admits a geometric structure modelled on S2, E2 or H 2, if it is isomorphic as orbifold to the quotient of S2, E2 or H 2 by some discrete group of isometries. Following [7], we call an orbifold good, if it is covered by some manifold, or equivalently, if its universal covering orbifold is a manifold. Theorem 2.1. Everygood two-dimensional orbifold without boundaryadmits a geometric structure modelled on S2, E2 or H 2. Proof. This is Theorem 2.4 of Scott [7]. For a proof, Scott refers the reader to Thurston [8]. Proposition 2.2. Let C be a 2-orbifold with underlying surface a closed oriented surface of genus g and having as singular points r cone points of multiplicity mi,1i r. Then C is good unless • g = 0 and r = 1, or • g = 0, r = 2 and m1 = m2. Proof. This is a special case of Scott [7, Theorem 2.3]. If C is a 2-orbifold with underlying surface a closed oriented surface of genus g and r cone points of multiplicity mi,1i r, we define r 1 (C) := 2g − 2 + 1 − . mi i=1 and call −(C) the orbifold Euler number of C. Proposition 2.3. Let C be a good 2-orbifold with underlying surface a closed oriented surface of genus g and having as singular points r cone points of multiplicity mi,1i r. 4 M. Dürr / Expo. Math. 23 (2005) 1–22 Then C admits a geometric structure modelled on • S2 if and onlyif (C) < 0; • E2 if and onlyif (C) = 0; • H 2 if and onlyif (C) > 0. Proof. See [7, p. 427f]. C orb(C) C When is an orbifold, we denote by 1 the fundamental group of the orbifold . Proposition 2.4. Let C be a 2-orbifold with underlying surface a closed oriented surface of genus g and having as singular points r cone points of multiplicity mi,1i r. A orb(C) presentation of the fundamental group 1 is g r mi a1,b1,...,ag,bg,s1,...,sr |si = 1, [aj ,bj ] si = 1 . j=1 i=1 Proof. See [7, p. 424]. We call such a group a group associated to the signature (g; m1,...,mr ). As permut- ing the multiplicities does not affect the isomorphy type of the associated group, we will henceforth assume that mi mi+1 for 1i r − 1. Proposition 2.5. Let G1 and G2 be two groups associated to the signatures (g; m1,...,mr ) (g; m ,...,m ) G G and 1 r . If 1 and 2 are isomorphic as abstract groups, then the signatures agree unless the groups are finite cyclic. If G1 is finite cyclic, then g = 0 and r = 0, 1 or 2. Proof. For a proof of the above proposition, see for instance [5, p. 132]. Proposition 2.6. Fix a signature (g; m1,...,mr ) and let G denote an associated group. Then the centre of G is trivial unless • G is finite cyclic, or • g = 1 and r = 0, or • (g; m1,...,mr ) = (0; 2, 2, 2n) with n a positive integer. In the last case we have s ,s ,s |s2 = s2 = s2n = ,ss s = ={ ,sn} Centre 1 2 3 1 2 3 1 1 2 3 1 1 3 for n2. Proof. Suppose that G is not finite cyclic. Therefore, by Theorem 2.1, G has a faithful representation as a discrete and cocompact subgroup of the group of orientation preserving isometries of the sphere S2, the Euclidean plane E2 or the hyperbolic plane H 2.