Singular Lefschetz pencils

By D. Auroux, Simon Donaldson, L. Katzarkov. 2004

108 citations recorded by the source corpus.

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 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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/383211b4cf894a0e64b153c253b531224f811a4c

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Citations in the corpus

Wrinkled fibrations on near-symplectic manifolds (2007, methodology): “This is an adaptation due to Auroux, Donaldson and Katzarkov of Gompf’s generalization of Thurston’s argument used in finding a symplectic form on a Lefschetz fibration.”
Lagrangian matching invariants for fibred four-manifolds: I (2006, background): “The construction of Lagrangian matching invariants was guided by Taubes’ programme [29] to obtain the Seiberg–Witten invariants of a near-symplectic four–manifold as generalised Gromov invariants.”
Round handles, logarithmic transforms and smooth 4‐manifolds (2010, methodology): “(These examples of broken Lefschetz fibrations are due to Auroux, Donaldson and Katzarkov [4], and the description via handlebodies we use here as given in [6].)”
Six Lectures on Four 4-Manifolds (2006, methodology): “χ h : A FAKE PROJECTIVE PLANE 45 (8n−8,4n−1) are, for n>1, diﬀeomorphic or even symplectomorphic. However, since both these deformation types are branched covers of S2 ×S2 we can use techniques of [2,14] to braid the branched set to show that indeed H(4n− 1) can be obtained from H′(4n−1) via a ±1”
Existence of broken Lefschetz fibrations (2008, unknown): “G T ] 1 2 Fe b 20 08”