PomeLemma 9.2.8.4D71178D595A Let $A$ be a proper analytic subset of a connected complex manifold $M$. Then $A$ is nowhere dense in $M$. Moreover, if $U\subset M$ is a nonempty connected open set and $g_1,\ldots ,g_ k$ are holomorphic functions on $U$ with $A\cap U=\{ g_1=\cdots =g_ k=0\} $, then some $g_ i$ is not identically zero on $U$. In particular, every point of $A$ has a connected open neighborhood $U$ with a holomorphic function $g\not\equiv 0$ on $U$ such that $A\cap U\subset \{ g=0\} $.
Lemma 9.2.84D71178D595A
Proof. The interior of $A$ is open, and we show that it is closed in $M$. Let $a\in M$ be a limit of interior points of $A$, and choose a connected coordinate neighborhood $U$ of $a$ and functions $g_1,\ldots ,g_ k$ with $A\cap U=\{ g_1=\cdots =g_ k=0\} $. Each $g_ i$ vanishes on a nonempty open subset of $U$, hence on $U$ by the identity theorem; so $U\subset A$, and $a$ is an interior point. Since $M$ is connected and $A\neq M$, the interior of $A$ is empty. In the second assertion, if all $g_ i$ vanished identically on $U$, the nonempty open set $U$ would lie in $A$, contradicting the first. For the third, take for $U$ a connected coordinate ball on which $A$ is the common zero set of finitely many holomorphic functions, and for $g$ one of them that is not identically zero, which exists by the second assertion. $\square$
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