Is the union of 2 complex analytic sets still a complex analytic set?

Question

Is the union of 2 complex analytic sets still a complex analytic set? The notion of the analytic set is the usual one in complex analysis and comeplex geometry.

Answer 1

As long as they are closed analytic subsets of some common domain (or manifold), then yes. Locally, as the comment to the question makes clear, you could just multiply the defining functions to get functions that vanish on both.

However, if you are talking about local analytic subvarieties (Whitney's terminology), that is, if they are not closed, then no, a union need not be a subvariety.

The distinction is as follows. Let $U$ be a domain or a manifold. $X \subset U$ is an analytic subset (a subvariety, sometimes emphasized as "closed subvariety of $U$") if for every $p \in U$ there is a neighborhood $V$ of $p$, and holomorphic functions $f_1,\ldots,f_k$ defined in $V$, such that $X \cap V = \{ z \in V : f_1(z) = 0 , \ldots , f_k(z) = 0 \}$.

A definition of a local subvariety is the same except it starts different, it says "for every $p \in X$" instead of "for every $p \in U$". The distinction may seem trivial but it is important. For example in two dimensions, the set given by $z_1 = 0$, $\text{Re}\ z_2 > 0$ is a local analytic subvariety, but not a closed subvariety of ${\mathbb C}^2$. Its union with the subvariety (actual, honest, closed subvariety of ${\mathbb C}^2$) given by $z_2 = 0$ is not a subvariety in either sense (the origin is a problem!)

A local subvariety is a (closed) subvariety of some open subset. A union of subvarieties is a subvariety if they are subvarieties of the same open subset. Otherwise not necessarily.

So the upshot is: it depends on the definition of "complex analytic set". I'd bet 90% of time people mean a closed subvariety of some set or other, especially if they say "subvariety of $U$" or "analytic subset of $U$". But if considering subvarieties as generalizations of submanifolds, then it makes sense to talk about local subvarieties as that's the way we generally think of submanifolds. Of course, the definition of "submanifold" is yet another can of worms for different reasons.

The upshot of the upshot? Be careful about what definition the source that you are reading is using. If you are using a book, check what the definition says carefully and then compare above.