Ricci flow and birational surgery

Abstract

We study the formation of finite time singularities of the Kahler-Ricci flow in relation to high codimensional birational surgery in algebraic geometry. We show that the Kahler-Ricci flow on an n-dimensionl Kahler manifold contracts a complex submanifold ${\mathbb{P}}^m$ with normal bundle ${\oplus }_{j=1}^{n-m}{\mathcal{O}}_{{\mathbb{P}}^m}(-a_j)$ for $a_j\in {\mathbb{Z}}^{+}$ and ${\sum }_{j=1}^{n-m}{a}_j\le m$ in Gromov-Hausdorff topology with suitable initial Kahler class. We also show that the Kahler-Ricci flow resolves a family of isolated singularities uniquely in Gromov-Hausdorff topology. In particular, we construct global and local examples of metric flips by the Kahler-Ricci flow as a continuous path in Gromov-Hausdorff topology.

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