K\"ahler-Ricci Flow on Projective Bundles over K\"ahler-Einstein Manifolds

TL;DR

This paper investigates the behavior of the K"ahler-Ricci flow on certain projective bundles over K"ahler-Einstein manifolds, identifying conditions for fiber collapse and singularity types, extending previous work on specific surfaces.

Contribution

It provides a criterion for fiber collapse and singularity type of the K"ahler-Ricci flow on projective bundles, generalizing earlier results to broader classes of manifolds.

Findings

Fiber collapse occurs under specific geometric conditions.

Flow develops Type I singularities of a particular type.

Convergence to the base manifold in Gromov-Hausdorff sense.

Abstract

We study the K\"ahler-Ricci flow on a class of projective bundles $\mathbb{P}(\mathcal{O}_\Sigma \oplus L)$ over compact K\"ahler-Einstein manifold $\Sigma^n$. Assuming the initial K\"ahler metric $\omega_0$ admits a U(1)-invariant momentum profile, we give a criterion, characterized by the triple $(\Sigma, L, [\omega_0])$, under which the $\mathbb{P}^1$-fiber collapses along the K\"ahler-Ricci flow and the projective bundle converges to $\Sigma$ in Gromov-Hausdorff sense. Furthermore, the K\"ahler-Ricci flow must have Type I singularity and is of $(\C^n \times \mathbb{P}^1)$-type. This generalizes and extends part of Song-Weinkove's work \cite{SgWk09} on Hirzebruch surfaces.