On the regular-convexity of Ricci shrinker limit spaces

In this paper we study the structure of the pointed-Gromov–Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger–Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any non-collapsing Ricci shrinker limit space is strongly convex, inspired by Colding–Naber’s original idea of parabolic smoothing of the distance functions.

Tropical Geometric Compactification of Moduli, II: A g Case and Holomorphic Limits

We compactify the classical moduli variety Ag of principally polarized abelian varieties of complex dimension g, by attaching the moduli of flat tori of real dimensions at most g in an explicit manner. Equivalently, we explicitly determine the Gromov–Hausdorff limits of principally polarized abelian varieties. This work is analogous to [50], where we compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov–Hausdorff limits along holomorphic families of abelian varieties and show that these form special nontrivial subsets of the whole boundary. We also do the same for algebraic curves case and observe a crucial difference with the case of abelian varieties.

Ricci curvature in Kähler geometry

These are the notes for lectures given at the Sanya winter school in complex analysis and geometry in January 2016. In Sec. 1, we review the meaning of Ricci curvature of Kähler metrics and introduce the problem of finding Kähler–Einstein metrics. In Sec. 2, we describe the formal picture that leads to the notion of K-stability of Fano manifolds, which is an algebro-geometric criterion for the existence of a Kähler–Einstein metric, by the recent result of Chen–Donaldson–Sun. In Sec. 3, we discuss algebraic structure on Gromov–Hausdorff limits, which is a key ingredient in the proof of the Kähler–Einstein result. In Sec. 4, we give a brief survey of the more recent work on tangent cones of singular Kähler–Einstein metrics arising from Gromov–Hausdorff limits, and the connections with algebraic geometry.