When is a complex surface not symplectic

Question

Is there an invariant in terms of Euler characteristic $e$ and signature $\sigma$, that tells us a given complex surface is not symplectic. Some related results are- Complex surface is Kahler iff $b_{1}$ is even. A four manifold admits an almost complex structure then $e+\sigma=0\mod4$. Complex surfaces of general type are Kahler. If a four manifold admits a Lefschetz fibration then it is symplectic. But none of these help with my question.