Contracting exceptional divisors by the K\"ahler-Ricci flow II

TL;DR

This paper studies the behavior of the K"ahler-Ricci flow on complex surfaces, showing convergence to orbifold limits and extending previous results, with applications to higher-dimensional analogues and minimal surfaces.

Contribution

It proves smooth and Gromov-Hausdorff convergence of the flow in new settings, including orbifold points and higher-dimensional P^1-bundles, extending prior work.

Findings

Flow blows down exceptional divisors to orbifold points

Gromov-Hausdorff limit matches metric completion of the flow

Solutions converge to K"ahler-Einstein orbifolds in specific cases

Abstract

We investigate the case of the Kahler-Ricci flow blowing down disjoint exceptional divisors with normal bundle O(-k) to orbifold points. We prove smooth convergence outside the exceptional divisors and global Gromov-Hausdorff convergence. In addition, we establish the result that the Gromov-Hausdorff limit coincides with the metric completion of the limiting metric under the flow. This improves and extends the previous work of the authors. We apply this to P^1-bundles which are higher-dimensional analogues of the Hirzebruch surfaces. In addition, we consider the case of a minimal surface of general type with only distinct irreducible (-2)-curves and show that solutions to the normalized Kahler-Ricci flow converge in the Gromov-Hausdorff sense to a Kahler-Einstein orbifold.