Abstract. We study the collapsing behavior of the Kähler-Ricci flow on a compact Kähler manifold X admitting a holomorphic submersion X π − → Σ inherited from its canonical bundle, where Σ is a Kähler manifold with dim C Σ < dim C X. We show that the flow metric degenerates at exactly the rate of e −t as predicted by the cohomology information, and so the fibres π −1 (z), z ∈ Σ collapse at the optimal rate diamt(π −1 (z)) e −t/2 . Consequently, it leads to some analytic and geometric extensions to the regular case of Song-Tian's works [ST1,ST2]. Its applicability to general Calabi-Yau fibrations will also be discussed in local settings.
Abstract. We study the collapsing behavior of the Kähler-Ricci flow on a compact Kähler manifold X admitting a holomorphic submersion X π − → Σ where Σ is a Kähler manifold with dim C Σ < dim C X. We give cohomological and curvature conditions under which the fibers π −1 (z), z ∈ Σ collapse at the optimal rate diamt(π −1 (z)) ∼ (T − t) 1/2 .
We study the Kähler-Ricci flow on a class of projective bundles P(O Σ ⊕ 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 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 × P 1 )-type. This generalizes and extends part of Song-Weinkove's work [SW1] on Hirzebruch surfaces.
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