On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces

2022/03/08 by Cifarelli, Charles, Conlon, Ronan J., Deruelle, Alix · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2203.04380

Abstract

We show that the underlying complex manifold of a complete non-compact two-\linebreak dimensional shrinking gradient Kähler-Ricci soliton (M, g, X) with soliton metric g with bounded scalar curvature R_g whose soliton vector field X has an integral curve along which R_g\not→0 is biholomorphic to either ℂ×ℙ¹ or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension.

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• Linear stability and instability of Kähler Ricci solitons
• On Sun-Zhang's theory of Fano fibrations -- weighted volumes, moduli and bubbling Fano fibrations

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