Theorem
(Kodaira-Enriques-et.al) If \((M,J,g)\) is a Kahler surface, then you can perform a finite sequence of blowdowns (contract a holomorphic sphere to a point, “finite” because this operation lowers the Betti number by 1 each time)
\begin{align*} M\to M_1\to M_2\to ... \to M_N \end{align*}such that any further blowdown of \(M_N\) results in a singularity, hence not a Kahler manifold. We call this a “Kahler surface of minimal type”.
Note that this doesn’t say anything about what happens to the form \(g\) over this process – which confuses me because I’m not sure what category these blowdowns take place in. But it leads to a question: is there a corresponding process for \(g\)? This leads to Ricci flows.