On Metric tangent cones at Klt singularities

Abstract: Kawamata log terminal (Klt) singularities appear on the Gromov-Hausdorff limits of Kahler-Einstein Fano manifolds. Donaldson-Sun conjectured that the metric tangent cones at such metric Klt singularities depend only on the algebraic structures of the singularities. We will discuss our recent work (joint with Chenyang Xu) toward proving this conjecture. An ingredient in this work is a generalization of Martelli-Sparks-Yau's volume minimization principle for Sasakian-Einstein metrics.