The Kähler-Ricci flow on Fano bundles

Abstract

We study the behavior of the Kähler-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 kähler class, then the fibers collapse in finite time and the metrics converge sub-sequentially in Gromov-Hausdorff sense to a metric on the base.

Rewrite some part of the introduction, add more references, and correct the computation in the proof of Lemma 2.2. Welcome any Comments. arXiv admin note: text overlap with arXiv:1003.0718 by other authors