Higher order estimates for collapsing Calabi-Yau metrics

Hans-Joachim Hein, Fordham University
Fine Hall 314

Consider a compact Calabi-Yau manifold X with a holomorphic fibration F: X to B over some base B, together with a "collapsing" path of Kahler classes of the form [F*omega_B] +  t * [omega_X] for t in (0,1]. Understanding the limiting behavior as t to 0 of the Ricci-flat Kahler forms representing these classes is a basic problem in geometric analysis that has attracted a lot of attention since the celebrated work of Gross-Wilson (2000) on elliptically fibered K3 surfaces. The limiting behavior of these Ricci-flat metrics is still not well-understood in general even away from the singular fibers of F. A key difficulty arises from the fact that Yau's higher order estimates for the complex Monge-Ampere equation depend heavily on bounds on the curvature tensor of a suitable background metric, but such bounds are simply not available in this collapsing situation. I will explain recent joint work with Valentino Tosatti where we manage to bypass Yau's method in some cases, proving higher order estimates even though the background curvature blows up.