Volume 9, issue 2 (2005)

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ISSN (electronic): 1364-0380
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Singular Lefschetz pencils

Denis Auroux, Simon K Donaldson and Ludmil Katzarkov

Geometry & Topology 9 (2005) 1043–1114

arXiv: math.DG/0410332


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,ω) can be decomposed into (a) two symplectic Lefschetz fibrations over discs, and (b) a fibre bundle over S1 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.

near-symplectic manifolds, singular Lefschetz pencils
Mathematical Subject Classification 2000
Primary: 53D35
Secondary: 57M50, 57R17
Forward citations
Received: 1 November 2004
Accepted: 30 May 2005
Published: 1 June 2005
Proposed: Robion Kirby
Seconded: Dieter Kotschick, Ronald Stern
Denis Auroux
Department of Mathematics
Massachusetts Institute of Technology
Massachusetts 02139
Simon K Donaldson
Department of Mathematics
Imperial College
London SW7 2BZ
United Kingdom
Ludmil Katzarkov
Department of Mathematics
University of Miami
Coral Gables
Florida 33124
Department of Mathematics
UC Irvine
California 92612