Lefschetz fibrations and the Hodge bundle

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

Volume 3, issue 1 (1999)

Ivan Smith