Singular Lefschetz pencils. (English) Zbl 1077.53069

Auroux, Denis; Donaldson, Simon K.; Katzarkov, Ludmil

Geom. Topol. 9, 1043-1114 (2005).

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.

Reviewer: Mircea Puta (Timişoara)

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

Keywords:

near-symplectic manifolds; singular Lefschetz pencils

References:

[1]	D Auroux, Symplectic 4-manifolds as branched coverings of \(\mathbf C\mathbf P^2\), Invent. Math. 139 (2000) 551 · Zbl 1080.53084 · doi:10.1007/s002220050019
[2]	D Auroux, A remark about Donaldson’s construction of symplectic submanifolds, J. Symplectic Geom. 1 (2002) 647 · Zbl 1171.53343 · doi:10.4310/JSG.2001.v1.n3.a4
[3]	S K Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996) 666 · Zbl 0883.53032
[4]	S K Donaldson, Lefschetz fibrations in symplectic geometry (1998) 309 · Zbl 0909.53018
[5]	S K Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999) 205 · Zbl 1040.53094
[6]	S Donaldson, I Smith, Lefschetz pencils and the canonical class for symplectic four-manifolds, Topology 42 (2003) 743 · Zbl 1012.57040 · doi:10.1016/S0040-9383(02)00024-1
[7]	D T Gay, R Kirby, Constructing symplectic forms on 4-manifolds which vanish on circles, Geom. Topol. 8 (2004) 743 · Zbl 1054.57027 · doi:10.2140/gt.2004.8.743
[8]	R E Gompf, Toward a topological characterization of symplectic manifolds, J. Symplectic Geom. 2 (2004) 177 · Zbl 1084.53072 · doi:10.4310/JSG.2004.v2.n2.a1
[9]	K Honda, Transversality theorems for harmonic forms, Rocky Mountain J. Math. 34 (2004) 629 · Zbl 1071.58002 · doi:10.1216/rmjm/1181069872
[10]	K Honda, Local properties of self-dual harmonic 2-forms on a 4-manifold, J. Reine Angew. Math. 577 (2004) 105 · Zbl 1065.53066 · doi:10.1515/crll.2004.2004.577.105
[11]	C LeBrun, Yamabe constants and the perturbed Seiberg-Witten equations, Comm. Anal. Geom. 5 (1997) 535 · Zbl 0901.53028
[12]	F Presas, Submanifolds of symplectic manifolds with contact border · Zbl 1101.53055
[13]	C H Taubes, The geometry of the Seiberg-Witten invariants, Int. Press, Boston (1998) 299 · Zbl 0956.57019
[14]	C H Taubes, Seiberg-Witten invariants and pseudo-holomorphic subvarieties for self-dual, harmonic 2-forms, Geom. Topol. 3 (1999) 167 · Zbl 1027.53111 · doi:10.2140/gt.1999.3.167
[15]	W P Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976) 467 · Zbl 0324.53031 · doi:10.2307/2041749