Lefschetz pencils on symplectic manifolds

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 We will leave the discussion of these further topics for future papers, and concentrate here on the proofs of the main existence theorems.