Abstract:

In recent work with V. Apostolov, X. Fu, and Y. Ustinovskiy, we gave an extension of the Calabi-Yau theorem to the setting of generalized K\”ahler (GK) geometry, a relatively new subject with roots in complex geometry, mathematical physics, and Poisson geometry.  In this talk I will describe a complete solution to this problem in a restricted case, which reduces to studying a fully nonlinear, mixed convex/concave, scalar PDE: the twisted Monge-Ampere equation.