ABSTRACT
A symplectic structure on a manifold X is a closed non-degenerate 2-form ω on X. Typical examples of closed symplectic manifolds are Kahler manifolds. In the two-dimensional case, a symplectic structure is a nowhere-vanishing area form on a surface and such structures are determined by the underlying topological structure and total area. In higher dimensions, the problem is more subtle. There are some constructions of symplectic manifolds which do not admit Kahler structures. Gromov introduced the technique of pseudo-holomorphic curves and showed many interesting results. Among them, he showed that if a closed symplectic four-manifold X is diffeomorphic to the complex projective plane and contains a symplectically embedded two-sphere representing the generator of the second homology of X, then X is actually symplectomorphic to the complex projective plane with the Fubini-Study form. This chapter reviews Taubes’ theorems. It gives some numerical consequences of Taubes’ non-triviality result. The chapter discusses symplectic four-manifolds with “positive first Chern class”.
