Let $\Omega\subset\mathbb{C}^n$ be open, let $F:\Omega\to\mathbb{C}^r$ be holomorphic, and let $a\in\mathbb{C}^r$. Suppose that for every $p\in F^{-1}(a)$, the [Jacobian matrix](/page/Jacobian%20Matrix) $JF_p$ has complex rank $r$. Then $F^{-1}(a)$ is a complex submanifold of $\Omega$ of complex dimension $n-r$.