Ricci flow and birational surgery

TL;DR

This paper investigates how the Kahler-Ricci flow develops finite-time singularities related to high codimensional birational surgeries, providing new insights into metric flips and singularity resolution in algebraic geometry.

Contribution

It demonstrates the contraction of complex submanifolds and the resolution of isolated singularities by the Kahler-Ricci flow, introducing new examples of metric flips in Gromov-Hausdorff topology.

Findings

Contraction of complex submanifolds with specific normal bundles under Kahler-Ricci flow.

Resolution of isolated singularities in Gromov-Hausdorff topology.

Construction of metric flips as continuous paths in Gromov-Hausdorff topology.

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 $\sum_{j=1}^{n-m} a_j \leq m$ 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.