Singular Lefschetz pencils

Auroux, Denis¹ ; Donaldson, Simon K² ; Katzarkov, Ludmil³

¹ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
² Department of Mathematics, Imperial College, London SW7 2BZ, United Kingdom
³ Department of Mathematics, University of Miami, Coral Gables, Florida 33124, USA, Department of Mathematics, UC Irvine, Irvine, California 92612, USA

Geometry & topology, Tome 9 (2005) no. 2, pp. 1043-1114.

Résumé

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.

Détail

DOI : 10.2140/gt.2005.9.1043

Keywords: near-symplectic manifolds, singular Lefschetz pencils

Bibliographie

[1] D Auroux, Symplectic 4–manifolds as branched coverings of $\bold C\bold P^2$, Invent. Math. 139 (2000) 551

[2] D Auroux, A remark about Donaldson's construction of symplectic submanifolds, J. Symplectic Geom. 1 (2002) 647

[3] S K Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996) 666

[4] S K Donaldson, Lefschetz fibrations in symplectic geometry, from: "Proceedings of the International Congress of Mathematicians, Vol II (Berlin, 1998)" (1998) 309

[5] S K Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999) 205

[6] S Donaldson, I Smith, Lefschetz pencils and the canonical class for symplectic four-manifolds, Topology 42 (2003) 743

[7] D T Gay, R Kirby, Constructing symplectic forms on 4–manifolds which vanish on circles, Geom. Topol. 8 (2004) 743

[8] R E Gompf, Toward a topological characterization of symplectic manifolds, J. Symplectic Geom. 2 (2004) 177

[9] K Honda, Transversality theorems for harmonic forms, Rocky Mountain J. Math. 34 (2004) 629

[10] K Honda, Local properties of self-dual harmonic 2–forms on a 4–manifold, J. Reine Angew. Math. 577 (2004) 105

[11] C Lebrun, Yamabe constants and the perturbed Seiberg–Witten equations, Comm. Anal. Geom. 5 (1997) 535

[12] F Presas, Submanifolds of symplectic manifolds with contact border

[13] C H Taubes, The geometry of the Seiberg–Witten invariants, from: "Surveys in differential geometry, Vol III (Cambridge, MA, 1996)", Int. Press, Boston (1998) 299

[14] C H Taubes, Seiberg–Witten invariants and pseudo-holomorphic subvarieties for self-dual, harmonic 2–forms, Geom. Topol. 3 (1999) 167

[15] W P Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976) 467