We apply a recent theorem of Li and the first author to give some criteria for the K-stability of Fano varieties in terms of anticanonical Q-divisors. First, we propose a condition in terms of certain anti-canonical Q-divisors of given Fano variety, which we conjecture to be equivalent to the K-stability. We prove that it is at least sufficient condition and also relate to the Berman-Gibbs stability. We also give another algebraic proof of the K-stability of Fano varieties which satisfy Tian's alpha invariants condition.

We prove that the Gromov-Hausdorff compactification of the moduli space of Kähler-Einstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tian's theorem on the existence of Kähler-Einstein metrics on smooth Del Pezzo surfaces and classifies all the degenerations of such metrics. The proof is based on a combination of both algebraic and differential geometric techniques.

We prove that various GIT semistabilities of polarized varieties imply semi-log-canonicity.

We algebraically prove K-stability of polarized Calabi-Yau varieties and canonically polarized varieties with mild singularities. In particular, the "stable varieties" introduced by Kollár-Shepherd-Barron [KSB88] and Alexeev [Ale94], which form compact moduli space, are proven to be K-stable although it is well known that they are not necessarily asymptotically (semi)stable. As a consequence, we have orbifold counterexamples, to the folklore conjecture "K-stability implies asymptotic stability". They have Kähler-Einstein (orbifold) metrics so the result of Donaldson [Don01] does not hold for orbifolds.

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.