We consider degenerations of Calabi-Yau manifolds over higher dimensional bases in general. We then shall present a result on the equivalence of a uniform diameter bound as Ricci-flat Kähler-Einstein manifolds and that the limit varieties have canonical singularities at worst.

I will define the notion of multiplier subsheaves for generically defined variations of Hodge structures on smooth complex varieties (and more precisely for Hodge modules). I will present both algebraic and analytic constructions, inspired by those for multiplier ideals. Using Kodaira-Saito vanishing, I will prove a Nadel-type vanishing theorem for multiplier subsheaves, generalizing a number of vanishing theorems in algebraic geometry. If time permits, I will present an application to a Fujita-type freeness result for the lowest term in the Hodge filtration.

Chen-Sun-Wang showed that the Kähler-Ricci flow on a Fano manifold gives rise to a certain algebraic degeneration of the manifold. I will discuss in what sense this degeneration is optimal, or "most-destabilizing". An application of this result is a general convergence result for the Kähler-Ricci flow on Fano manifolds admitting a Kähler-Ricci soliton, generalizing works of Tian-Zhu and Tian-Zhang-Zhang-Zhu. This work is joint with Ruadhai Dervan.

The Yau-Tian-Donaldson conjecture, recently proved by Chen-Donaldson-Sun, and Tian, asserts that a Fano manifold X admits a Kähler-Einstein metric if and only if X is K-(poly)stable. I will present joint work with Robert Berman and Sebastien Boucksom, on a new, variational, proof of this conjecture in the case X has no vector fields. Our proof uses pluripotential theory and ideas from non-Archimedean geometry, but not use the continuity method nor Cheeger-Colding-Tian theory.