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

Fong, Frederick Tsz-Ho

doi:10.1090/S0002-9947-2013-05726-1

Kähler-Ricci flow on projective bundles over Kähler-Einstein manifolds
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by Frederick Tsz-Ho Fong PDF

Trans. Amer. Math. Soc. 366 (2014), 563-589 Request permission

Abstract:

We study the Kähler-Ricci flow on a class of projective bundles $\mathbb {P}(\mathcal {O}_\Sigma \oplus L)$ over the 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 the Gromov-Hausdorff sense. Furthermore, the Kähler-Ricci flow must have Type I singularity and is of $(\mathbb {C}^n \times \mathbb {P}^1)$-type. This generalizes and extends part of Song-Weinkove’s work on Hirzebruch surfaces.

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