Let $(M,\omega, J, g)$ be a $4$ dimensional Kahler manifold. Put $\omega '=\star \omega$ where $\star$ is the hodge operator associated the metric $g$.

Is $(M,\omega ')$ a symplectic manifold? Is it necessarilly symplectic equivalent to the original structure $(M,\omega)$?Namely, is thete a diffeomorphism $f$ which carries $\omega ' $ to $\omega$?