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Collapsing fibres under Kähler Ricci flow on Hirzebruch manifolds Dixon, Kael


In this article we study the Kähler Ricci flow on a class of ℂℙ¹ bundles over ℂℙⁿ⁻¹ known as Hirzebruch manifolds. These are defined by ℙ(Hⁱ⊕ℂ-1), where H is the canonical line bundle, ℂ is the trivial line bundle, and n,i∈ℕ. We follow the work by Song and Weinkove, who study solutions to the Kähler Ricci flow for a Calabi symmetric Kähler metrics on Hirzebruch manifolds. They were able to show that, depending on the initial Kähler class, the Ricci flow would reach a finite time singularity corresponding to the manifold either shrinking to a point, contracting the zero section to a point, or collapsing the fibres. In this paper, we investigate how the fibres collapse in the latter case with the further assumptions that the singularity is formed at a type I rate, and that the length of a generic vector does not decay too quickly in some sense. In this case we show that the fibres converge to round spheres after blowing up around a singular point on a fibre.

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