We study symplectic structures on Kähler surfaces with $p_g = 0$. We give an example of a projective surface which admits a symplectic structure which is not compatible with any Kähler metric.
"Symplectic Genric Complex Structures on Four-Manifolds with $b_+ = 1$." J. Symplectic Geom. 10 (4) 493 - 502, December 2012.