Northwestern-Notre Dame-UIC Complex Geometry Seminar

I will discuss the existence of a complete Calabi-Yau metric on the complement of a simple normal crossings anti-canonical divisor with two components, as well as possible extensions to more general situations.​ This is joint work with Yang Li.

Recent developments in complex geometry have highlighted the importance of real Monge-Ampère equations on singular affine manifolds, in particular for the SYZ conjecture concerning collapsing families of Calabi-Yau manifolds. We show that for symmetric data, the real Monge-Ampère equation on the unit simplex admits a unique Aleksandrov solution. This is concluded as a special case of a result giving necessary and sufficient conditions in terms of optimal transport for existence of solutions. I will outline the proof and explain a built in phenomena reminiscent of free boundary problems. Time permitting, I will discuss an application to the SYZ conjecture related to recent work by Y. Li.