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Lefschetz type pencils on contact manifolds. (English) Zbl 1101.53055

The author defines the concept of Lefschetz contact pencil and proves the following nice result. Given a contact closed manifold \((C,D)\) (respectively exact) and \(\alpha\in H_{2n-1}(C, \mathbb R)\) which is a reduction of an integer class, there exists a contact pencil on \(C\) (respectively oriented) whose fibers are contact submanifolds homologous to \(\alpha\). This result is analgous to the result of S. K. Donaldson [J. Differ. Geom. 53, No. 2, 205–236 (1999; Zbl 1040.53094)]. The main tool is the generalization of the local transversality theorem proved in the work of Donaldson. The results of the author are related to the results on existence of a convex structure in contact manifolds of E. Giroux and J. P. Mohsen. The results of the paper help to understand the possibility of finding isotopic constructions for contact manifolds constructed in A. Ibort, D. Martinez-Torres, F. Presas [J. Differ. Geom. 56, No. 2, 235–283 (2000; Zbl 1034.53088)] where the author partially translates Donaldson’s idea of working with \(J\)-holomorphic sections to the contact case, see S. K. Donaldson [J. Differ. Geom. 44, No. 4, 666–705 (1996; Zbl 0883.53032)]. In the end of the paper the author discusses applications of contact pencils to Gromov’s ideas of contact blow-ups.

MSC:

53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
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