#### Volume 9, issue 2 (2005)

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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
##### 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 $\left(X,\omega \right)$ 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
##### Mathematical Subject Classification 2000
Primary: 53D35
Secondary: 57M50, 57R17
##### 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