Ricci flow and birational surgery

2013/04/09 by Jian Song, Song, Jian
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1304.2607

openalex publication_date 2013/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ℙ^m with normal bundle ⊕_j=1^n-mO_ℙ^m(-a_j) for a_j∈ℤ⁺ and ∑_j=1^n-m a_j ≤ 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.

Citations

• On rotationally symmetric Kahler-Ricci solitons
• Convergence of Kähler-Ricci flow on Fano manifolds, II
• Supremum of Perelman's entropy and Kähler-Ricci flow on a Fano manifold
• Birational cobordisms and factorization of birational maps
• Ricci flow with surgery on three-manifolds
• Degenerations of Ricci-flat Calabi-Yau manifolds

Related

• Kahler-Ricci flow on blowups along submanifolds
• Finite time singularities of the Kähler-Ricci flow
• The Kahler-Ricci flow through singularities
• Metric flips with Calabi ansatz
• The Ricci Flow on Complete Noncompact Kähler Manifolds