Volume 3, issue 1 (1999)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Lefschetz fibrations and the Hodge bundle

Ivan Smith

Geometry & Topology 3 (1999) 211–233

arXiv: math.SG/9907200

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 Kähler 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
Mathematical Subject Classification
Primary: 53C15
Secondary: 53C55, 58F99
References
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Publication
Received: 4 May 1999
Revised: 10 June 1999
Accepted: 8 July 1999
Published: 14 July 1999
Proposed: Ronald Stern
Seconded: Frances Kirwan, Walter Neumann
Authors
Ivan Smith
New College
University of Oxford
Oxford OX1 3BN
United Kingdom