= Hopf surface
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A primary Hopf surface is a quotient of $\mathbb C^2\setminus\{0\}$ by an infinite cyclic group generated by a suitable contracting holomorphic map; reversing the generator gives the expanding dilation description. The scalar dilation $z\mapsto2z$ gives a compact surface diffeomorphic to $S^3\times S^1$ via polar coordinates and log radius. Hence $b_1=1$. The <Betti and Hodge bounds for compact complex surfaces> force $h^{1,0}=0$ and $h^{0,1}\geq1$, furnishing a strict example of $h^{1,0}<h^{0,1}$.
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