Title: Symplectic structure on ruled surfaces and a generalized adjunction formula
Abstract: In this section, we state the theorems needed in our paper.Recently, Seiberg and Witten ([SW], [Wi]) have introduced a new set of 4-manifold invariants.These invariants are in similar spirit to Donaldson invariants but much easier to handle.Various longstanding conjectures including the Thom conjecture are proved using Seiberg-Witten invariants.An important ingredient in the proof of the Thom conjecture by Kronheimer and Mrowka is the wall crossing formula for manifolds with b 1 = 0.Seiberg-Witten invariants take on a very simple form for Kahler surfaces ([Wi], [B], [FM1]).All the basic classes are explicitly known and in particular, the anticanonical bundle is always a basic class.A large part of this story is generalized to symplectic manifolds by Taubes who ([T1],[T2], [T3], [T4]) proved several remarkable theorems on Seiberg-Witten invariants of symplectic four-manifolds.Recall that every symplectic manifold has a complex line bundle, K (called the canonical bundle), which is canonical up to isomorphism.The first theorem of Taubes is Theorem 1. ([T1]) Let M be an oriented symplectic four-manifold with b + 2 ≥ 2. Let ω be a symplectic form compatible with the orientation.Then c 1 (K -1 ) on M has Seiberg-Witten invariant ±1.This result clearly shows that the Seiberg-Witten invariant is an important tool to study the differential topology of symplectic four-manifolds; we will see in this paper that indeed this has many applications.The next two theorems of Taubes give very strong constraints on symplectic forms and almost complex structures supporting symplectic structures. Theorem 2. ([T2]) Let M , ω, K be as in the above theorem and let E ∈ H 2 (M ; Z) have nonzero Seiberg-Witten invariant.Then