1999/01/01 by Simon Donaldson · 7 citations
Mathematics · #Advanced Combinatorial Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematics #Pure mathematics #Symplectic geometry
paper · pdf · doi:10.4310/jdg/1214425535
openalex publication_date 1999/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
This paper is a sequel to [3], in which techniques from complex geometry were adapted to prove a general existence theorem for symplectic submanifolds of compact symplectic manifolds.These submanifolds were obtained as the zero-sets of suitable sections of complex line bundles.In the present paper we take the ideas further, developing the symplectic analogue of "pencils", generated by a pair of sections of a line bundle.Our main results are a general existence theorem for topological Lefschetz pencils (Theorem 2 below), together with an asymptotic uniqueness statement (Theorem 20).These results, along with recent work of R. Gompf ([4, Theorem 10.2.18]), go some way towards giving a topological characterisation of symplectic manifolds; more generally they, along with various further extensions, give a means of translating many questions in symplectic topology into questions about the "monodromy" of the pencil.We will leave the discussion of these further topics for future papers, and concentrate here on the proofs of the main existence theorems.