#### 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