m at h . SG ] 1 9 Ju n 20 01 Uniqueness of symplectic canonical class , surface cone and symplectic cone of 4 − manifolds with b + = 1

Let M be a closed oriented smooth 4−manifold admitting symplectic structures. If M is minimal and has b + = 1, we prove that there is a unique symplectic canonical class up to sign, and any real second cohomology class of positive square is represented by symplectic forms. Similar results hold when M is not minimal. §1. Introduction Let M be a smooth, closed oriented 4−manifold. An orientation-compatible symplectic structure on M is a closed 2-form ω such that ω ∧ω is nowhere vanishing and agrees with the orientation. Let Ω M be the moduli space of such 2-forms. In the first part of this paper, we devote ourselves to the understanding of the topology of this moduli space, which can be studied in three ways. First of all, on Ω M , there is a natural equivalence relation, the deformation equivalence. ω 1 and ω 2 in Ω 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 group of orientation-preserving diffeomor-phisms on the set of connected components of Ω M , and the number of deformation classes of symplectic structures is just the number of the orbits of this action. Secondly, there is a map of canonical class K : Ω M −→ H 2 (M ; Z). Any symplectic structure determines a homotopy class of compatible almost complex structures on the cotangent bundle, whose first Chern class is called the symplec-tic canonical class. For each symplectic canonical class K, if we let Ω M,K be the subset of Ω M , whose elements have K as the symplectic canonical class, then Ω M is the disjoint union of the Ω M,K. There is also a natural equivalence relation on the set of symplectic canonical classes. We say two symplectic canonical classes K 1 and K 2 are equivalent if there is an orientation-preserving diffeomorphism φ such that φ * K 1 = ±K 2. Symplectic structures in a connected component have the same symplectic canonical class. Moreover if two symplectic structures are related by an orientation-preserving diffeomorphism, so are their symplectic canonical classes. Therefore the set of deformation equivalence classes of orientation-compatible sym-plectic structures maps onto the set of equivalent classes of symplectic canonical classes, and can be understood via the latter.