Collapsing Ricci-flat metrics: estimates or no estimates?

Valentino Tosatti, Courant Institute, NYU
Fine Hall 110

Compact Calabi-Yau manifolds admit Ricci-flat Kahler metrics thanks to a celebrated theorem of Yau, a unique such metric in each Kahler cohomology class. If we degenerate the class, the corresponding Ricci-flat metrics will also degenerate, and one can ask whether they admit uniform bounds (in all C^k norms) away from a closed analytic subvariety. This is known when the metrics are not collapsing, but the collapsing case is much more challenging. I will discuss how the answer to this question is positive when the limiting class comes from the base of a fibration (joint with Hein), and negative in general when there is no such fibration (joint with Filip), and raise some related open questions.