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

Abstract

We study the Kähler-Ricci flow on a class of projective bundles $\mathbb{P}({\mathcal{O}}_{\Sigma }\oplus L)$ over compact Kähler-Einstein manifold ${\Sigma }^n$. Assuming the initial Kähler 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 Kähler-Ricci flow and the projective bundle converges to $\Sigma$ 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.

arXiv commentrevised version for publication, to appear in Trans. Amer. Math. Soc