The Space of Symplectic Structures on Closed 4-manifolds

Let X be a 2n−dimensional smooth manifold. A 2−form ω on X is said to be non-degenerate if, for each q ∈ X and for each nonzero vector v in the tangent space TqX, there is a tangent vector v ∈ TqX such that ω(u, v) 6= 0. A symplectic structure onX is a non-degenerate closed 2−form. The fundamental example of a symplectic structure is ω0 = ∑ i dxi ∧ dyi on R = {(x1, y1, ..., xn, yn)}. In fact, by the Darboux Theorem, every symplectic structure is locally like (R2n, ω0). Symplectic structures first appeared in Hamiltonian mechanics. A Kähler form on a complex manifold is symplectic, thus we also find a rich source of symplectic manifolds in algebraic geometry. Thirty years ago people even wondered whether there are closed non-Kähler symplectic manifolds. We have now gradually realized that the world of Kähler manifolds only occupies a tiny part of the symplectic world (see [27], [64] and the references therein). Two of the basic questions about symplectic structures are (see [58]): 1. Which smooth manifolds support symplectic structures? 2. How many symplectic structures, up to appropriate equivalence, are there on a given smooth manifold? In this survey we focus on the second question for closed smooth 4−manifolds (for the first question in dimension 4 see [40]). In section 2 we review some fundamental facts about symplectic structures. In section 3 we survey what is known about the space of symplectic structures in dimension 4. In the case of b+ = 1 we have a rather good understanding. Especially, for a rational or ruled manifold, there is the deep uniqueness result that a symplectic form is determined by its cohomology class up to diffeomorphisms. We further give a simple description of the moduli spaces for such a manifold. We also point out various possible extensions to the case b+ > 1. In section 4 we compare the space of symplectic forms and the space of Kähler manifolds on a manifold admitting a Kähler structure. As a by product we describe an example of non-holomorphic Lefschetz fibration on a Kähler surface. We would like to thank P. Biran, J. Dorfmeister, R. Friedman and M. Usher for useful discussions. We are very grateful to the referee for reading it carefully and making many useful suggestions.