Viktor Majewski (University of Waterloo)
Degenerations of exotic Calabi-Yau metrics through Atiyahs flop
The small resolution of the three-dimensional ordinary double point carries the classical Candelas–de la Ossafamily of asymptotically conical Calabi–Yau metrics. In this talk, I will describe the construction of a new one-parameter family of complete Calabi–Yau metrics on the same complex manifold. The construction begins withan explicit family of U(2)-invariant Kähler metrics adapted to the geometry of the exceptional curve and theasymptotic conical end. After establishing uniform Sobolev, curvature, and volume-growth estimates, one solvesthe noncompact complex Monge–Ampère equation to obtain Ricci-flat metrics in the corresponding Kählerclasses. A central issue is to understand the behaviour of this family as the area of the exceptional curve tends tozero. I will explain how a combination of weighted pluripotential estimates, Chern–Lu inequalities, symmetry,and analysis on the conifold yields uniform control away from the exceptional set and produces a singularCalabi–Yau metric on the conifold. The resulting degeneration has a different local geometric profile from theclassical Candelas–de la Ossa degeneration, providing an exotic family of Calabi–Yau metrics on the resolvedconifold
MC 5417