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 $$Z(M, [[\omega ]])$$ Z ( M , [ [ ω ] ] ) depending on symplectic deformation equivalence class $$[[\omega ]]$$ [ [ ω ] ] . We first prove that there exists a 6-dimensional smooth manifold M with more than one deformation equivalence class with different signs of $$Z(M, [[\omega ]] )$$ 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 $$(M, \omega )$$ ( M , ω ) of dimension $$\ge \!\!4$$ ≥ 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 $$\omega $$ ω .