Description:

Title: Complex Monge-Ampere equation for positive (p,p) forms

Abstract: A complex Monge-Ampère equation for differential (p,p) forms is introduced on compact Kähler manifolds. For any 1≤p<n, we show the existence of smooth solutions unique up to adding constants. For p=1, this corresponds to the Calabi-Yau theorem proved by S. T. Yau, and for p=n−1, this gives the Monge-Ampère equation for (n−1) plurisubharmonic functions solved by Tosatti-Weinkove. For other p values, this defines a non-linear PDE that falls outside of the general framework of Caffarelli-Nirenberg-Spruck. In this talk, we will give an overview of this theory and discuss the main ideas involved in the proof of existence of solutions.