Lefschetz fibrations and the Hodge bundle

Smith, Ivan¹

¹ New College, University of Oxford, Oxford OX1 3BN, United Kingdom

Geometry & topology, Tome 3 (1999) no. 1, pp. 211-233.

Résumé

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.

Détail

DOI : 10.2140/gt.1999.3.211

Keywords: symplectic geometry, Lefschetz fibration, stable curves, signature

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