Contracting exceptional divisors by the Kähler-Ricci flow

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.

39 pages, v2 minor corrections; v3 due to a gap in the previous argument of Section 3, the assertion that the G-H limit coincides with the metric completion has been removed

References in corpus (7)

• Notes on Perelman's papers
• The Kähler-Ricci flow on surfaces of positive Kodaira dimension
• On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow
• The Kähler-Ricci flow with positive bisectional curvature
• Kahler-Ricci flow on stable Fano manifolds
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Cited by in corpus (7)

• KAWA lecture notes on the Kähler-Ricci flow
• Kähler currents and null loci
• The Chern-Ricci flow on complex surfaces
• Collapsing of the Chern-Ricci flow on elliptic surfaces
• Regularizing properties of Complex Monge-Ampère flows II: Hermitian manifolds
• The Chern-Ricci flow and holomorphic bisectional curvature
• Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows