# How to imagine zeros of an analytic function of several variables

Let $f(z_1,\cdots, z_n)$ be a holomorphic function of several variables in an open subset of $\mathcal C^n$. Let $Z(f)=\{ (z_1,\cdots, z_n) \: | \: f=0\}$ be the zero set of $f$.

If $n=1$, the zeros set consists of isolated points. How to generalize this to $n$ dimensions?

I know that $Z(f)$ has zero Lebesgue measure in $2n$ dimensional space. But this does not help: for $n=1$ it includes both isolated points and line segments.

Can the concept of limit/isolated/accumulation point be usefully generalized in this context? What's the best way to characterize $Z(f)$?

Informally, the zero set of a holomorphic function that does not vanish identically (let's suppose it is defined on a domain, i.e. a connected open set) is a locally finite union of complex $(n-1)$-dimensional surfaces; after a change of variables, these are the graphs of holomorphic functions of $n-1$ variables. The simple example $f(z_1,z_2) = z_1\cdot z_2$ shows that it is in general not a submanifold, but most points of the zero set have a neighbourhood $U$ (in $\mathbb{C}^n$) such that $U\cap Z(f)$ is a submanifold of $U$, the set of non-regular points has lower dimension (than $n-1$).
The zero set of a non-constant holomorphic function does not locally disconnect $\mathbb{C}^n$: If $D\subset \mathbb{C}^n$ is a connected open subset, and $f \in \mathcal{O}(D)$, then $D\setminus Z(f)$ is connected [trivially, if $f\equiv 0$].