The K\"ahler-Ricci flow on Fano bundles

TL;DR

This paper investigates the K"ahler-Ricci flow on specific Fano bundles, demonstrating fiber collapse and convergence to a metric on the base in certain cases.

Contribution

It provides new results on fiber collapse and metric convergence for the K"ahler-Ricci flow on particular Fano bundles with specific initial conditions.

Findings

Fibers collapse in finite time under the flow.

Metrics converge to a base metric in Gromov-Hausdorff sense.

Results apply to bundles with fibers like blown-up projective spaces.

Abstract

We study the behavior of the K\"ahler-Ricci flow on some Fano bundle which is a trivial bundle on one Zariski open set. We show that if the fiber is $\mathbb{P}^{m}$ blown up at one point or some weighted projective space blown up at the orbifold point and the initial metric is in a suitable k\"ahler class, then the fibers collapse in finite time and the metrics converge sub-sequentially in Gromov-Hausdorff sense to a metric on the base.