Kähler-Ricci flow on projective bundles over Kähler-Einstein manifolds

Abstract

We study the Kähler-Ricci flow on a class of projective bundles P ( O Σ ⊕ L ) \mathbb {P}(\mathcal {O}_\Sigma \oplus L) over the compact Kähler-Einstein manifold Σ n \Sigma ^n . Assuming the initial Kähler metric ω 0 \omega _0 admits a U ( 1 ) U(1) -invariant momentum profile, we give a criterion, characterized by the triple ( Σ , L , [ ω 0 ] ) (\Sigma , L, [\omega _0]) , under which the P 1 \mathbb {P}^1 -fiber collapses along the Kähler-Ricci flow and the projective bundle converges to Σ \Sigma in the Gromov-Hausdorff sense. Furthermore, the Kähler-Ricci flow must have Type I singularity and is of ( C n × P 1 ) (\mathbb {C}^n \times \mathbb {P}^1) -type. This generalizes and extends part of Song-Weinkove’s work on Hirzebruch surfaces.