Lefschetz pencils on symplectic manifolds. (English) Zbl 1040.53094

Let \((V,\omega)\) be a compact symplectic manifold of dimension \(2n\). A topological Lefschetz pencil \((V,A,f)\) on \(V\) consists of a codimension 4 submanifold \(A\) of \(V\), a finite set of points \(\{b_\lambda\}\subset V\setminus A\) and a smooth map \(f: V\setminus A \to S^2\) whose restriction to \(V\setminus (A\cup\{b_\lambda\})\) is a submersion such that \(f(b_\lambda)\neq f(b_\mu)\) for \(\lambda\neq\mu\). The main results of this paper is the following existence theorem for symplectic submanifolds of compact symplectic manifolds: Suppose that \([\omega]\in H^2(V;R)\) is the reduction of an integral class \(h\). For a sufficiently large integer \(k\) there is a topological Lefschetz pencil on \(V\) whose fibers are symplectic subvarieties, homologous to \(k\) times the Poincaré dual of \(h\). Moreover, the author proves that the topological Lefschetz pencils obtained above are in some sense asymptotically unique. For the proof the author uses the techniques from complex geometry in his previous paper [ibid. 44, 666–705 (1996; Zbl 0883.53032)].


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


Zbl 0883.53032
Full Text: DOI Link