I will present joint work in progress with Mattias Jonsson. Kontsevich and Soibelman have given a conjectural description of the Gromov-Hausdorff limit of a maximally degenerate family of polarized Calabi-Yau manifolds in terms of the Berkovich space attached to the degeneration. Motivated by this, Mustata, Nicaise and Xu recently studied the essential skeleton of this Berkovich space, which is a natural realization of the dual complex of a minimal model of the degeneration. We show that the volume form induced by a holomorphic form of top degree on a fiber converges, in some appropriate sense, to an explicit Lebesgue type measure on the essential skeleton.

Rational curves play a crucial role in the classification of projective manifolds via the minimal model program (MMP). The construction of these curves on projective manifolds or low-dimensional Kähler manifolds is based on deformation theory of curves and, in the projective case, reduction to positive characteristic. In this talk I will present a new strategy to construct rational curves via a "subadjunction formula for adjoint cohomology classes". This is joint work with Junyan Cao.

The abundance conjecture predicts that if the canonical bundle of a projective manifold is nef, then it is semi-ample. A special case is proved by Kawamata for big and nef canonical bundles. We give an analytic proof of Kawamata's theorem using the Ricci flow, L² theory and degeneration of Riemannian manifolds. We further construct unique Kähler-Einstein metrics with a global Riemannian structure on canonical models. We will also discuss a more general analytic base point free theorem.

In this talk I will discuss work in progress with Valentino Tosatti. I will introduce a transcendental definition of the restricted volume of a big (1,1) class, and discuss the conjectural relation between the retricted volume and the non-Kähler locus.