Scalar curvature bounds and tangent cones for Kähler--Ricci shrinkers

Abstract

We prove a uniform scalar curvature bound, depending only on the dimension, for Kähler--Ricci shrinkers. We also prove that every noncompact Kähler--Ricci shrinker has a unique tangent cone at infinity, naturally homeomorphic to the base of its polarized Fano fibration. For shrinkers with maximal volume growth, this cone is isometric to the metric completion of the regular part equipped with the intrinsic distance induced by the limiting Kähler metric.

arXiv commentv2: Added metric-completion results for Kähler-Ricci shrinkers with maximal volume growth and finite-time singularities of volume-noncollapsed Kähler-Ricci flows