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 can be decomposed into (a) two symplectic Lefschetz fibrations over discs, and (b) a fibre bundle over 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.
Citation
Denis Auroux. Simon K Donaldson. Ludmil Katzarkov. "Singular Lefschetz pencils." Geom. Topol. 9 (2) 1043 - 1114, 2005. https://doi.org/10.2140/gt.2005.9.1043
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