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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!

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    I still don't believe the dimension thing, but you're right about the Kahler form. On the other hand, what I should have said is that since $h^{0,2}\neq 0$ by assumption by Hodge symmetry $h^{2,0}\neq 0$. Thus there is at least a global section of $K_X$. Not sure how to proceed from here.2012-08-22

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