# Metric contraction of the cone divisor by the conical K\"ahler-Ricci flow

## TL;DR

This paper studies the conical K"ahler-Ricci flow on Hirzebruch surfaces, showing it either converges to a sphere or point, or contracts the cone divisor, providing new insights into singularity formation and flow behavior.

## Contribution

It introduces the first example of the conical K"ahler-Ricci flow contracting the cone divisor to a point and proposes a conjectural framework for finite time singularities on K"ahler surfaces.

## Key findings

- Flow converges to Riemann sphere or a point in finite time.
- Flow contracts the cone divisor to a point, leading to a projective orbifold.
- First example of cone divisor contraction in conical K"ahler-Ricci flow.

## Abstract

We use the momentum construction of Calabi to study the conical K\"ahler-Ricci flow on Hirzebruch surfaces with cone angle along the exceptional curve, and show that either the flow Gromov-Hausdorff converges to the Riemann sphere or a single point in finite time, or the flow contracts the cone divisor to a single point and Gromov-Hausdorff converges to a two dimensional projective orbifold. This gives the first example of the conical K\"ahler-Ricci flow contracting the cone divisor to a single point. At the end, we introduce a conjectural picture of the geometry of finite time non-collapsing singularities of the flow on K\"ahler surfaces in general.