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I am looking for an example of a compact complex surface with $h^{1,0} < h^{0,1}$. The bound that $h^{1,0} \leq h^{0,1}$ is known. In the Kähler case, $h^{p,q}=h^{q,p}$, so the example cannot be (for example) a projective variety or a complex torus. Does anyone know of such an example? Thanks.

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up vote 5 down vote accepted

The standard example of such a thing is the Hopf surface.

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