Metric flips with Calabi ansatz

Abstract

We study the limiting behavior of the Kahler-Ricci flow on $\mathbb{P}(\mathcal{O}_{\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus (m+1)})$, assuming the initial metric satisfies the Calabi symmetry. We show that the flow either shrinks to a point, collapses to $\mathbb{P}^n$ or contracts a subvariety of codimension m+1 in Gromov-Hausdorff sense. We also show that the Kahler-Ricci flow resolves certain type of conical singularities in Gromov-Hausdorff sense.