Contracting exceptional divisors by the Kähler-Ricci flow

Details Contracting exceptional divisors by the Kähler-Ricci flow Contracting exceptional divisors by the Kähler-Ricci flow

2010-03-03

arxiv.org/abs/1003.0718v3

Mathematics

Differential Geometry

39 pages, v2 minor corrections; v3 due to a gap in the previous argument of Section 3, the assertion that the G-H limit coinci

Scientific paper

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.

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