Volume 9, issue 2 (2005)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Singular Lefschetz pencils

Denis Auroux, Simon K Donaldson and Ludmil Katzarkov

Geometry & Topology 9 (2005) 1043–1114

arXiv: math.DG/0410332

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,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.

Keywords
near-symplectic manifolds, singular Lefschetz pencils
Mathematical Subject Classification 2000
Primary: 53D35
Secondary: 57M50, 57R17
References
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Publication
Received: 1 November 2004
Accepted: 30 May 2005
Published: 1 June 2005
Proposed: Robion Kirby
Seconded: Dieter Kotschick, Ronald Stern
Authors
Denis Auroux
Department of Mathematics
Massachusetts Institute of Technology
Cambridge
Massachusetts 02139
USA
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
USA
Department of Mathematics
UC Irvine
Irvine
California 92612
USA