Complex surface
= Complex surface
{title2=$\dim_{\mathbb C}M=2$}
A complex surface is a <complex manifold> of complex dimension two, hence real dimension four. Compact complex surfaces need not admit a <Kähler metric>; a <Hopf surface> is an example. Every <holomorphic one-form> on a compact complex surface is closed, and the <Betti and Hodge bounds for compact complex surfaces> still apply without Kähler assumptions.