Complete Calabi-Yau metrics provide singularity models for limits of Kahler-Einstein metrics. We study complete Calabi-Yau metrics with Euclidean volume growth and quadratic curvature decay. It is known that under these assumptions the metric is always asymptotic to a unique cone at infinity. Previous work of Donaldson-S. gives a 2-step degeneration to the cone in the algebro-geometric sense, via a possible intermediate object (a K-semistable cone). We will show that such intermediate K-semistable cone does not occur. This is in sharp contrast to the case of local singularities. This result together with the work of Conlon-Hein also give a complete algebro-geometric classification of these metrics, which in particular confirms Yau’s compactification conjecture in this setting. I will explain the proof in this talk, and if time permits I will describe a conjectural picture in general when the curvature decay condition is removed. Based on joint work with Junsheng Zhang (UC Berkeley).

Toric symplectic manifolds contain an interesting and well-studied family of Lagrangian tori, called toric fibres. In this talk, we address the natural question of which toric fibres are equivalent under Hamiltonian diffeomorphisms of the ambient space. On one hand, we use a symmetric version of McDuff's probes to construct such equivalences and on the other hand, we give certain obstructions coming from Chekanov's classification of product tori in symplectic vector spaces combined with a lifting trick from toric geometry. We will discuss many four-dimensional examples in which a full classification can be achieved.