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

الأصل · EN

We study the Kähler-Ricci flow on a class of projective bundles P(OΣ⊕ L) over compact Kähler-Einstein manifold Σⁿ. Assuming the initial Kähler metric ω₀ admits a U(1)-invariant momentum profile, we give a criterion, characterized by the triple (Σ, L, [ω₀]), under which the P¹-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 (ⁿ × P¹)-type. This generalizes and extends part of Song-Weinkove's work SgWk09 on Hirzebruch surfaces.