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Singular Lefschetz pencils. (English) Zbl 1077.53069

In the paper under review the authors consider structures analogous to symplectic Lefschetz pencils in the context of a closed \(4\)-manifold equipped with a “near-symplectic” structure. Their main result asserts that, up to blowups, every near-symplectic \(4\)-manifold \((X,\omega)\) can be decomposed into: two symplectic Lefschetz fibrations over discs; and a fibre bundle over \(S^1\) which relates the boundaries of the Lefschetz fibrations to each other via a sequence of fiberwise bundle additions taking place in a neighborhood of the zero set of the \(2\)-form \(\omega\). Conversely, from such a decomposition one can recover a near-symplectic structure.

MSC:

53D35 Global theory of symplectic and contact manifolds
57M50 General geometric structures on low-dimensional manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
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References:

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