The Kähler–Ricci flow on Fano bundles

Abstract

We study the behavior of the Kahler–Ricci flow on some Fano bundles 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 Kahler class, then the fibers collapse in finite time and the metrics converge sub-sequentially in Gromov–Hausdorff sense to a metric on the base.