Lemma 3

Let X be a finite dimensional connected complex-analytic manifold and let A be a subset of X satisfying the following condition:

For any $x\in X$, there exists an analytic function germ $f_x$, not vanishing

at $x$, such that the germ of A at $x$ is contained in the germ at $x$ of the set

of zeros of $f_x$.

Then X -- A is dense in X and connected.

The density of X-- A follows from Differentiable and Analytic Manifolds, Results, 3.2.5. We can assume that A is closed (General Topology, Chap. I, §11, no. 1, Prop. 1). For any $x\in X$, there exists an open neighbourhood V of $x$ and an isomorphism $c$ from V to an open ball in $\mathbf{C}^n$ such that $c(A\cap V)$ is contained in the set of zeros of a holomorphic function not identically zero on $c(V)$. Then, by Lemma $2, V\cap (X$ -- A) is connected. In view of Lemma 1, this proves that X -- A is connected.