In short, I'd like to know the following:

Is there an irreducible analytic variety that has to be defined by at least two distinct sets of holomorphic functions?

Are there two irreducible analytic varieties that overlap completely at some open set but are distinct?

Analytic variety of complex manifold $M$ is defined as $V\subset M$ such that each point $x\in V$ has a neighborhood $U$ such that $U \cap V$ is a zero locus of holomorphic functions $f_1 \cdots f_m$ defined at $U$.

This definition had me wondering about examples of analytic varieties that are not simply zero loci of a few functions. Do we truly need an analytic variety to be a patchwork of loci of different set of functions? Also, can one patch be "continued" in at least two different ways?

- Is there an irreducible analytic variety that can't be described as a zero locus of a single set of functions $\{ f_1 , \cdots f_m\}$? In other words, can we find two distinct sets of functions $\{f_j\}, \{g_j\}$ defined on $U_f,U_g$ which have common zero locus when restricted to $U_f \cap U_g$, but $f_j$ cannot be simultaneously extended to $U_g$ and vice versa.

As far as I know, this cannot happen for $M=\mathbb C$ since we can always find a meromorphic function that has a prescribed divisor.

- Are there irreducible analytic varieties $V_1 \neq V_2$ such that $\exists U$ such that $U \cap V_1 = U \cap V_2 \neq \emptyset$?

A simple version of this question would ask: are there functions $f, g$ on $\mathbb C^n$ such that $V(f) \neq V(g)$ but $\exists U $ such that $U \cap V(f) = U \cap V(g) \neq \emptyset$?