Contracting exceptional divisors by the K\"ahler-Ricci flow

TL;DR

This paper establishes conditions under which the Kahler-Ricci flow contracts exceptional divisors on compact manifolds, leading to a canonical surgical contraction process that simplifies the manifold's structure.

Contribution

It introduces a criterion for contracting exceptional divisors via the Kahler-Ricci flow and proves convergence properties, enabling a sequence of canonical contractions towards minimal models.

Findings

Flow contracts exceptional divisors under certain conditions.

Convergence to the limit in Gromov-Hausdorff sense.

Finite-time termination with minimal model or volume collapse.

Abstract

We give a criterion under which a solution g(t) of the Kahler-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 convergence of g(t) in the sense of Gromov-Hausdorff and smooth convergence away from the exceptional divisors. We call this behavior for the Kahler-Ricci flow a canonical surgical contraction. In particular, our results show that the Kahler-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.