  We prove that the zero set of a nonnegative plurisubharmonic function that
solves $\det (\partial \overline{\partial} u) \geq 1$ 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).
