Contracting exceptional divisors by the Kähler–Ricci flow

Abstract

We give a criterion under which a solution $g\left(t\right)$ of the Kähler–Ricci flow contracts exceptional divisors on a compact manifold and can be uniquely continued on a new manifold. As $t$ tends to the singular time $T$ from each direction, we prove the convergence of $g\left(t\right)$ 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.