Song Sun, University of California, Berkeley
Complete Calabi-Yau metrics asymptotic to cones
Complete Calabi-Yau metrics provide singularity models for limits of Kahler-Einstein metrics. We study complete Calabi-Yau metrics with Euclidean volume growth and quadratic curvature decay. It is known that under these assumptions the metric is always asymptotic to a unique cone at infinity. Previous work of Donaldson-S. gives a 2-step degeneration to the cone in the algebro-geometric sense, via a possible intermediate object (a K-semistable cone). We will show that such intermediate K-semistable cone does not occur. This is in sharp contrast to the case of local singularities. This result together with the work of Conlon-Hein also give a complete algebro-geometric classification of these metrics, which in particular confirms Yau’s compactification conjecture in this setting. I will explain the proof in this talk, and if time permits I will describe a conjectural picture in general when the curvature decay condition is removed. Based on joint work with Junsheng Zhang (UC Berkeley).