Dimension of the minimum set for the real and complex Monge–Ampère equations in critical Sobolev spaces

Abstract

We prove that the zero set of a nonnegative plurisubharmonic function that solves $det\left(\partial \bar{\partial }u\right)\ge 1$ in ${ℂ}^n$ and is in $W^{2,n\left(n-k\right)∕k}$ contains no analytic subvariety of dimension $k$ or larger. Along the way we prove an analogous result for the real Monge–Ampère equation, which is also new. These results are sharp in view of well-known examples of Pogorelov and Błocki. As an application, in the real case we extend interior regularity results to the case that $u$ lies in a critical Sobolev space (or more generally, certain Sobolev–Orlicz spaces).