 Hopf surface

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 $z\mapsto 2z$ 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 $h^{0,1}\ge 1$, furnishing a strict example of $h^{1,0}<h^{0,1}$.