On the tangent flow to the collapsing Kähler-Ricci flow on Hirzebruch surfaces

TL;DR

The paper addresses the problem of identifying tangent-flow models for volume-collapsing finite-time singularities of the Kähler-Ricci flow on Hirzebruch surfaces. Using Bamler’s tangent-flow framework, it shows that every tangent flow at the singular time is a non-flat gradient shrinking Kähler-Ricci soliton with finitely many orbifold singularities, rather than a flat constant flow, and proves finiteness of these singularities through a symplectic-filling/topological argument. The approach places the result within the broader program of classifying singularity models for 4-dimensional Ricci flows and clarifies the structure of collapsing behavior on complex surfaces. The findings provide a precise description of the local models governing volume-collapsing flows and advance understanding of singularity formation in Kähler geometry on Hirzebruch surfaces.

Abstract

In this paper, we study the collpasing Kähler-Ricci flow on Hirzebruch surfaces, which develops finite time singularities. We show that any tangent flow based at a point in the singular time slice is the Kähler-Ricci flow associated with a nonflat gradient Kähler-Ricci shrinker with finitely many orbifold singularities .

Theorems & Definitions (21)

• Conjecture 1.1
• Theorem 1.2: Main Theorem
• Theorem 2.1: Tangent flow, Bam21b
• Theorem 2.2: Structure of the tangent flow
• Remark 2.3
• Theorem 2.4
• proof
• Theorem 3.1
• Corollary 3.2
• Lemma 3.3