Let $X$ be a complex compact Kähler minimal surface with zero algebraic dimension and $H^2(X,\mathcal{O}_X) \ne 0$. We know that according to Enriques–Kodaira classification, $X$ is either a torus or a K3 surface. In particular, its canonical bundle is trivial.
My question is, are there are any direct ways to show that the canonical bundle $K_X$ is trivial without using the classification of complex compact surface? Thank you!