Paper detail

Scalar Curvature Functions of Almost-Kähler Metrics

For a closed smooth manifold $M$ admitting a symplectic structure, we define a smooth topological invariant $Z(M)$ using almost-Kähler metrics, i.e. Riemannian metrics compatible with symplectic structures. We also introduce $Z(M, [[ω]])$ depending on symplectic deformation equivalence class $[[ω]]$. We first prove that there exists a 6-dimensional smooth manifold $M$ with more than one deformation equivalence classes with different signs of $Z(M, [[ω]] )$. 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 $(M, ω)$ of dimension $\geq 4$, any smooth function which is somewhere negative and somewhere zero can be the scalar curvature of an almost-Kähler metric compatible with a symplectic form which is deformation equivalent to $ω$.