Geometry & Topology, Vol. 3 (1999) Paper no. 9, pages 211--233.

Lefschetz fibrations and the Hodge bundle

Ivan Smith

Abstract. Integral symplectic 4-manifolds may be described in terms of Lefschetz fibrations. In this note we give a formula for the signature of any Lefschetz fibration in terms of the second cohomology of the moduli space of stable curves. As a consequence we see that the sphere in moduli space defined by any (not necessarily holomorphic) Lefschetz fibration has positive "symplectic volume"; it evaluates positively with the Kahler class. Some other applications of the signature formula and some more general results for genus two fibrations are discussed.

Keywords. Symplectic geometry, Lefschetz fibration, stable curves, signature

AMS subject classification. Primary: 53C15. Secondary: 53C55, 58F99.

DOI: 10.2140/gt.1999.3.211

E-print: arXiv:math.SG/9907200

Submitted to GT on 4 May 1999. (Revised 10 June 1999.) Paper accepted 8 July 1999. Paper published 14 July 1999.

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Ivan Smith
New College, Oxford OX1 3BN, England

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