Dimension of the minimum set for the real and complex Monge-Amp\`{e}re equations in critical Sobolev spaces

We prove that the zero set of a nonnegative plurisubharmonic function that solves $\det (\mathrm{\partial }\bar{\mathrm{\partial }}u)\ge 1$ in ${\mathbb{C}}^n$ and is in $W^{2,\frac{n(n-k)}{k}}$ contains no analytic sub-variety of dimension $k$ or larger. Along the way we prove an analogous result for the real Monge-Amp`ere equation, which is also new. These results are sharp in view of well-known examples of Pogorelov and B{\l}ocki. 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).