Non-existence of certain Einstein metrics on some symplectic manifolds

We present a necessary and sufficient condition for an Einstein metric on a symplectic manifold (M,ω) to be ω-compatible. An immediate consequence of the solution of the Goldberg conjecture by Sekigawa is that there exists no Einstein metric with non-negative Ricci curvature satisfying our condition on compact symplectic manifolds that admit no Kähler structure.