Chern scalar curvature and finite-time singularities of the conformal Chern--Calabi flow

Abstract

On every unit-volume flat complex three-torus, we construct a solution of the volume-preserving conformal Chern--Calabi flow with a finite-time Chern Type-I singularity at any prescribed point. The full spatial supremum has an exact asymptotic coefficient, and prescribed rescalings converge to a complete nonflat radial shrinking solution. The positive part of Chern scalar curvature can remain arbitrarily small throughout the flow, as can each prescribed subcritical spatial curvature norm. We determine the punctured terminal metric, its one-point completion and unique nonflat tangent cone, and the terminal energy asymptotics. We also exclude Chern Type-II under a logarithmically improved critical weak $L^3$ bound on Chern scalar curvature, and prove a separate scale-invariant integral criterion involving first Chern--Ricci curvature. The constructed singular solutions satisfy both criteria.