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On the Monodromy of 4-Dimensional Lagrangian Fibrations On the Monodromy of 4-dimensional Lagrangian Fibrations Dissertation zur Erlangung des Doktorgrades der Fakult¨at f¨ur Mathematik und Physik der Albert-Ludwigs-Universit¨at Freiburg im Breisgau vorgelegt von Christian Thier Oktober 2008 Dekan: Prof. Dr. Kay K¨onigsmann 1. Gutachter: Prof. Dr. Bernd Siebert 2. Gutachter: Prof. Dr. Manfred Lehn Datum der Promotion: 05.12.2008 F¨ur Doris, Karl, Lisa und Heike Acknowledgements First of all I would like to thank my supervisor Prof. Dr. Bernd Siebert for introducing me to this subject and for his support. Especially for the opportunity to visit him when he was on a sabbatical leave at the University of California San Diego. I also thank him and Prof. Dr. Miles Reid for inviting me to give a talk on this work in a Seminar at the University of Warwick. I am grateful to PD. Dr. Vsevolod V. Shevchishin for many useful discussions on my thesis and thank him for giving me the opportunity to give a talk on this thesis in a workshop at the MPI for Mathematics in Bonn. I would also like to thank Prof. Dr. Manfred Lehn for his interest in my work and for inviting me to give a talk at the Johannes Gutenberg-University Mainz. As to the non-mathematical acknowledgements, I express at this point my grat- itude towards my parents on whose support and understanding I could always count. Last but not least I thank very warmly Heike Kaprolat for her support and encouragement and for enduring me. ON THE MONODROMY OF 4-DIMENSIONAL LAGRANGIAN FIBRATIONS CHRISTIAN THIER Contents 1. Introduction 9 1.1. Introduction and Results 9 1.2. Compact hyperk¨ahler manifolds 13 1.3. Lagrangian fibrations and affine structures 15 1.4. Elliptic K3-surfaces 20 1.5. Monodromy 24 2. Group theory 30 2.1. Mapg and Sp(2g, Z) 30 2.2. Central extensions of Sp(4, Z) 38 2.3. The universal cover of the symplectic group Sp(4, Z) 42 2 2.4. The class γ∗ H (Sp(4, Z), Z) 52 ∈ 2.5. Generators and relations in H2(Sp(4, Z), Z) 53 2.6. The central extension Sp(4, Z) 56 3. Geometry 61 c 3.1. Evaluation of cohomology classes 61 3.2. The geometrical interpretation of the generator κ∗ + γ∗ 62 3.3. The geometrical interpretation of the generator γ∗ 80 3.4. A formula for deg(∆) and an upper bound for deg(∆) 90 3.5. Further restrictions on the values of deg(∆) and γ∗ 92 3.6. Classification of surfaces, fibred by genus two curves 97 3.7. Example of a surface with (s =2) 101 3.8. Construction of an abelian fibration 112 3.9. Sawons results and a dichotomy 124 3.10. Construction again 127 3.11. Intersection theory on 127 M2 3.12. Two examples with non-unipotent monodromy 128 7 8 CHRISTIAN THIER 3.13. Discussion 131 References 132 ON THE MONODROMY OF 4-DIMENSIONAL LAGRANGIAN FIBRATIONS 9 1. Introduction 1.1. Introduction and Results. Compact hyperk¨ahler manifolds are a very spe- cial class of complex manifolds. This is manifest in the fact that up to deformation there are only very few examples known. As compact hyperk¨ahler manifolds ad- mit a holomorphic-symplectic form, they are even dimensional. In dimension two compact hyperk¨ahler manifolds are K3-surfaces and in dimension four the known examples are up to deformation either Hilbert schemes of points on K3-surfaces or generalised Kummer varieties. Therefore one would either like to construct new examples of compact hyperk¨ahler manifolds or else understand why there exist so few of them. The restrictive nature of compact hyperk¨ahler manifolds also governs the type of fibration that such manifolds admit. By a theorem, due to Matsushita [45], we know that a fibration on a compact hyperk¨ahler manifold is a fibration in complex-Lagrangian tori. Furthermore we know that the base of such a fibration is a Fano variety with the same Hodge numbers as the complex projective space [46]. Sawon explained in [65] that and how Lagrangian fibrations could be used to classify compact hyperk¨ahler manifolds up to deformation. In fact all of the known examples of compact hyperk¨ahler manifolds can be deformed into compact hyperk¨ahler manifolds that admit a Lagrangian fibration. In this thesis we study Lagrangian fibrations on four-dimensional hyperk¨ahler manifolds. The discriminant locus ∆ of such a fibration is a curve in P2. Around components of ∆ the fibration exhibits monodromy. Outside the discriminant locus the Lagrangian fibration endows the base P2 with the structure of an inte- gral affine manifold. We are mainly interested in the discriminant locus and in the monodromy. Control over the degree of ∆ might lead to a classification of Lagrangian fibrations, see [64] and [72]. For fibrations whose fibres are polarised abelian varieties, the monodromy of the Lagrangian fibration (or equivalently that of the affine structure) lies in an appropriate integral symplectic group. We study the case of principal polarisations. In this case the monodromy transformations lie in Sp(4, Z). Studying group theoretical properties of Sp(4, Z) we are able to prove a monodromy theoretical formula for the degree of the discriminant locus and an upper bound for deg(∆), (Theorem 3.15 and Corollary 3.25). Further we prove under additional assumptions a dichotomy for such fibrations. (Theorem 3.47). In chapter one we first review the definition of a compact hyperk¨ahler manifold, 10 CHRISTIAN THIER the known examples and some results on Lagrangian fibrations. Then we look at elliptic fibrations on K3-surfaces from the point of view of monodromy and finally we discuss types of unipotent monodromy transformations, in particular we focus on transformations that are “unipotent of rank one”. This means that the trans- formation is unipotent and that its logarithm has rank one. In chapter two we describe the relation between the integral symplectic group Sp(4, Z) and the mapping class group Map2 of a surface of genus two. In order to study the monodromy of principally polarised Lagrangian fibrations we study central extensions 0 Z E Sp(4, Z) 1 −→ −→ −→ −→ of Sp(4, Z) by Z. The latter are classified by the group cohomology group 2 H (Sp(4, Z), Z). We use a presentation of Map2 due to Wajnryb [75] and Mat- sumoto [49] to describe natural Z-extensions of Sp(4, Z). This leads to two natural 2 extensions that we denote by κ∗ + γ∗ and γ∗ that generate H (Sp(4, Z), Z) (Sec- tions 2.3 and 2.4). Further we describe a central extension Sp(4, Z) of Sp(4, Z) by Z2 and show that unipotent transformations of rank one have a distinguished lift c in this central extension. By a distinguished lift of a symplectic transformation in a central extension E we mean a distinguished inverse image under the homomor- phism E Sp(4, Z). The distinguished lift in Sp(4, Z) leads to distinguished −→ lifts in all central Z-extensions of Sp(4, Z). c In chapter three we study what we call monodromy factorisations of Lagrangian fibrations. Restricting a Lagrangian fibration f : X P2 to a general line l P2, −→ ⊂ 1 one obtains an abelian fibration f X : Xl l with base P . This fibration has | l −→ singular fibres over the points where l intersects ∆. Choosing an appropriate pre- sentation of the fundamental group π (l ∆) yields a factorisation of the identity 1 \ into monodromy transformations. We call such a factorisation µl a monodromy factorisation of f. We make the assumption that the monodromy transformation T around a point of l ∆ is unipotent of rank one. This assumption puts restric- ∩ tions on the singularities of the general singular fibre of f : X P2. We show −→ that monodromy factorisations that satisfy this assumption admit a distinguished lift in central Z-extensions of Sp(4, Z). The distinguished lift of a monodromy factorisation is an integer and this allows us to evaluate classes in H2(Sp(4, Z), Z) on monodromy factorisations. Next we give geometrical interpretations of the two natural generators κ∗ + γ∗ ON THE MONODROMY OF 4-DIMENSIONAL LAGRANGIAN FIBRATIONS 11 2 and γ∗ of H (Sp(4, Z), Z). We prove that the evaluation of the class κ∗ + γ∗ on a monodromy factorisation µl gives the first Chern class of the canonical extension of the Hodge bundle of f X , (Theorem 3.7). This interpretation together with | l 1 Matsushitas result that R f X is isomorphic to the cotangentbundle of the base ∗O allows us to prove that for a Lagrangian fibration with principally polarised fibres and unipotent monodromy of rank one deg(∆) = 30+2γ∗(µl) where µl is a monodromy factorisation of f X (Theorem 3.15). The interpretation | l of the class γ∗ is as follows. The fibration f X : Xl l gives rise to a moduli | l −→ map ϕ : l −→ A2 into the compactification of the moduli space of principally polarised abelian sur- faces . Assume that the general fibre of the Lagrangian fibration is not reducible A2 as a principally polarised abelian variety. Denote by D the divisor on that 1 A2 parametrises reducible principally polarised abelian varieties. Then deg(ϕ∗D1) is given by γ∗(µ ), where γ∗(µ ) denotes the evaluation of γ∗ on a monodromy fac- − l l torisation µl (Theorem 3.20). In this sense γ∗(µl) counts the fibres of f X : Xl l | l −→ that are reducible as p.p.a.s.. Next we check the formula of Theorem 3.15 in an example of a Lagrangian fi- bration with degree of the discriminant locus equal to 30.
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