Geometry & Topology, Vol. 9 (2005) Paper no. 24, pages 1043--1114.

Singular Lefschetz pencils

Denis Auroux, Simon K Donaldson, Ludmil Katzarkov

Abstract. We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4-manifold equipped with a `near-symplectic' structure (ie, a closed 2-form which is symplectic outside a union of circles where it vanishes transversely). Our main result asserts that, up to blowups, every near-symplectic 4-manifold (X,omega) can be decomposed into (a) two symplectic Lefschetz fibrations over discs, and (b) a fibre bundle over S^1 which relates the boundaries of the Lefschetz fibrations to each other via a sequence of fibrewise handle additions taking place in a neighbourhood of the zero set of the 2-form. Conversely, from such a decomposition one can recover a near-symplectic structure.

Keywords. Near-symplectic manifolds, singular Lefschetz pencils

AMS subject classification. Primary: 53D35. Secondary: 57M50, 57R17.

DOI: 10.2140/gt.2005.9.1043

E-print: arXiv:math.DG/0410332

Submitted to GT on 1 November 2004. Paper accepted 30 May 2005. Paper published 1 June 2005.

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Denis Auroux, Simon K Donaldson, Ludmil Katzarkov
Department of Mathematics, Massachusetts Institute of Technology
Cambridge, MA 02139, USA
Department of Mathematics, Imperial College
London SW7 2BZ, United Kingdom
Department of Mathematics, University of Miami, Coral Gables, FL 33124, USA
and Department of Mathematics, UC Irvine, Irvine, CA 92612, USA
Email: auroux@math.mit.edu, s.donaldson@imperial.ac.uk, lkatzark@math.uci.edu

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