Uniqueness of symplectic canonical class , surface cone and symplectic cone of 4 − manifolds with b + = 1

Let M be a smooth, closed oriented 4−manifold. An orientation-compatible symplectic structure on M is a closed two form ω such that ω ∧ ω is nowhere vanishing and gives the orientation. Two such symplectic structures ω 1 and ω 2 on M are said to be deformation equivalent if there is an orientation-preserving diffeomorphism φ such that φ * ω 1 and ω 2 are connected by a path of symplectic forms. Clearly, the number of deformation class of symplectic structures is just the number of the orbit of the group of orientation-preserving diffeomorphisms on the set of connected components of orientation compatible symplectic structures. The determination of the number of deformation class of symplectic structures is a central problem in symplectic topology. It turns out to be a very difficult problem. In higher dimension, the first result was due to Ruan. In [R1] he discovered that on a series of 6−manifolds there are more than one deformation class of symplectic structures. In dimension four, the first breakthrough was made by Taubes [T3], who showed that there is one deformation class on CP 2. We [LL2] later showed that the uniqueness holds for rational and ruled surfaces. In this paper we will first study a simpler problem. Any symplectic structure determines a homotopy class of compatible almost complex structures on the cotangent bundle, whose first Chern class is called the symplectic canonical class. Symplectic structures in a connected component has the same symplectic canonical class. And if two symplectic structures are related by a diffeomorphism, so are their symplectic canonical classes. Two symplectic canonical classes K 1 and K 2 are equivalent if there is a diffeomorphism φ such that φ * K 1 = ±K 2. All Kahler surfaces have only one equivalent class of symplectic canonical classes (see [B] and [FM]). In fact, any minimal symplectic 4−manifold with b + ≥ 2 and two basic Seiberg-Witten classes have this property as well. However, Mc-Mullen and Taubes [MT], and later LeBrun [Le] and Simth [Sm], have constructed manifolds with b + > 1 and Inequivalent classes of symplectic canonical classes.