Scalar Curvature Functions of Almost-Kahler Metrics

For a closed smooth manifold M admitting a symplectic structure, we define a smooth topological invariant Z(M) using almost-Kahler metrics, i.e., Riemannian metrics compatible with symplectic structures. We also introduce Z(M, [[omega]]) depending on symplectic deformation equivalence class [[omega]]. We first prove that there exists a 6-dimensional smooth manifold M with more than one deformation equivalence class with different signs of Z(M, [[omega]]). Using Z invariants, we set up a Kazdan-Warner type problem of classifying symplectic manifolds into three categories. We finally prove that on every closed symplectic manifold (M, omega) of dimension >= 4, any smooth function which is somewhere negative and somewhere zero can be the scalar curvature of an almost-Kahler metric compatible with a symplectic form which is deformation equivalent to omega.