Ricci flow and birational surgery

Original · EN

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 Pᵐ with normal bundle ⊕ⱼ₌₁ⁿ⁻ᵐOₚₘ(-aⱼ) for aⱼ+ and ∑ⱼ₌₁ⁿ⁻ᵐ aⱼ ≤ 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.