We give a criterion under which a solution of the Kähler–Ricci flow contracts exceptional divisors on a compact manifold and can be uniquely continued on a new manifold. As tends to the singular time from each direction, we prove the convergence of in the sense of Gromov–Hausdorff and smooth convergence away from the exceptional divisors. We call this behavior for the Kähler–Ricci flow a canonical surgical contraction. In particular, our results show that the Kähler–Ricci flow on a projective algebraic surface will perform a sequence of canonical surgical contractions until, in finite time, either the minimal model is obtained, or the volume of the manifold tends to zero.
"Contracting exceptional divisors by the Kähler–Ricci flow." Duke Math. J. 162 (2) 367 - 415, 1 February 2013. https://doi.org/10.1215/00127094-1962881