Let $X$ be a complex manifold of dimension $d$ and assume that $f:X\to \mathbb{C}$ is holomorphic and not identically equal to zero.
In this case the zero-set $X_1$ of $f$ is "almost" a submanifold of co-dimension $1$. Let $X_2$ be the subset of $X_1$ where $\nabla f =0$. Then $X_1\backslash X_2$ is an embedded submanifold of co-dimesnion $1$.
Can we find an embedded submanifolds $Y$, of co-dimension at least $1$, which cover $X_2$?
If the answer is negative, here is what I hope to achieve. I am looking for a finite sequence $X_m\subset...\subset X_2\subset X_1\subset X_0=X$, such that $X_{k+1}$ is the intersection of $X_k$ and zero-sets of some functions (like $X_2$ is the intersection of $X_1$ with zero-sets of all partial derivatives of $f$), $X_k\backslash X_{k+1}$ is an embedded submanifold of co-dimension $k$, and $X_m$ can be covered by an embedded submanifold.
From my understanding, something like this is called stratification of analytic sets, but I know approximately nothing on the topic.
PS in my specific case, $f$ is a Jacobian of a holomorphic map, but this is probably irrelevant.