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.
Notes on file formats
Ivan Smith
New College, Oxford OX1 3BN, England
Email: smithi@maths.ox.ac.uk
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