K\"ahler-Ricci Tangent Flows in the Analytic Minimal Model Program

Abstract

We describe certain finite-time singularities of the K\"ahler-Ricci flow arising in the analytic minimal model program. Assuming that convergence to an asymptotically conical K\"ahler-Ricci shrinker is realized by holomorphic maps, we prove that, in a fixed holomorphic gauge, the nearby flow is modeled on the shrinker at the level of K\"ahler potentials. Consequently, every noncollapsed K\"ahler-Ricci flow through singularities in complex dimension two is modeled on a shrinker-cone-expander transition, confirming a strong form of Song's conjectural picture. We also show analogous results in higher dimensions under the Calabi ansatz, and improve known results in the compact shrinker case. These give the first compact Ricci flows through conical singularities whose small-scale behavior is fully described.