Type-I singularity conjecture for compact Kähler surfaces

Establish that every finite-time singularity of the Kähler–Ricci flow on a compact Kähler surface is a Type-I singularity, including the unresolved volume-collapsing, non-extinction case.

Background

The paper studies finite-time collapsing Kähler–Ricci flows on ruled surfaces and proves that, under the specified cohomological assumptions, the flow develops a Type-I singularity whose tangent flow is the standard product shrinker on $\mathbb{P}^1\times\mathbb{C}$. This addresses an important class of volume-collapsing Kähler–Ricci flows on compact complex surfaces.

The broader conjecture remains unresolved in the volume-collapsing case without extinction. Prior results cover the volume-noncollapsing case and certain explicit or symmetric collapsing examples, while the present work proves the conjecture for ruled surfaces and regular $\mathbb{P}^1$-fibrations under the stated class condition.