2025/12/26 by Alexander Bednarek, Bednarek, Alexander
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · doi:10.48550/arxiv.2512.21910
This is the second of two papers studying both the geometric structure of Fano fibrations and the application to Kähler-Ricci flows developing a singularity in finite time. We assume that the Kähler-Ricci flow on a compact Kähler manifold has a rational initial metric and develops a singularity in finite time such that the manifold admits a Fano fibration structure. Moreover, it is assumed that the volume form of the flow collapses uniformly at the rate of C-1(T-t)n-m Ω≤ ω(t)n≤ C(T-t)n-mΩ. Under this setting, a diameter bound is obtained in any compact set away from singular fibres and the diameter of the fibres is proven to collapse at the optimal rate √(T-t). Furthermore, several precise C0-estimates are proven for the potential of the complex Monge-Ampere flow which involve the potentials of singular twisted Kähler-Einstein metrics on the base variety from part I. Finally, in the case of Kähler-Einstein Fano fibres, we deduce Type I scalar curvature in any compact set away from singular fibres and globally for a submersion.