  For a closed smooth manifold $M$ admitting a symplectic structure, we define
a smooth topological invariant $Z(M)$ using almost-K\"ahler 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 classes 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 $\geq 4$, any smooth function which is somewhere negative and
somewhere zero can be the scalar curvature of an almost-K\"ahler metric
compatible with a symplectic form which is deformation equivalent to $\omega$.
