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Abstract
The thesis is divided into two parts. In Part I, after some background material, we give a signature formula for Lefschetz fibrations in terms of the cohomology ring of the moduli space of curves. Using this we are able to prove a conjecture of Bogomolov et al: there are no symplectic Lefschetz fibrations with monodromy group contained in the Torelli group. We then study the geometry of Lefschetz fibrations via the "quantisation" representations of mapping class groups in conformal field theory. In the abelian theory we give a necessary condition on these quantisation representations to arise from Kähler surfaces with positive first Betti number. This condition is a consequence of the Hard Lefschetz theorem; we find counter-examples to some possible translations of the statement to non-abelian structure groups. We then compute the explicit monodromy of the quantisation bundles arising from simple algebraic singularities ("cusp fibres") as a natural precursor to more detailed future applications.
In Part II of the thesis we turn to explicit topological questions. Lefschetz pencils are stratified by the genus of the generic fibre, and if this genus is less than two there is a complete classification. We prove that (up to blowing up and down if there are reducible fibres) genus two fibrations all arise as branched covers of rational ruled surfaces. Rich families of examples show the restrictive nature of the fibre-irreducible assumption. If we also assume a Kähler structure we obtain a complete classification, giving a new and simpler proof of an unpublished result of Chakiris. Modulo a (difficult) conjecture on isotopy classes of symplectic submanifolds, we observe that the Kähler assumption is unnecessary and that the techniques should give a new and geometric proof of the classification of elliptic Lefschetz fibrations.
