Kähler-Ricci Flow on Projective Bundles over Kähler-Einstein Manifolds

We study the Kähler-Ricci flow on a class of projective bundles $\mathbb{P}(\mathcal{O}_Σ\oplus L)$ over compact Kähler-Einstein manifold $Σ^n$. Assuming the initial Kähler metric $ω_0$ admits a U(1)-invariant momentum profile, we give a criterion, characterized by the triple $(Σ, L, [ω_0])$, under which the $\mathbb{P}^1$-fiber collapses along the Kähler-Ricci flow and the projective bundle converges to $Σ$ in Gromov-Hausdorff sense. Furthermore, the Kähler-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.

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