Suppose $ B_{n}(0, 1) $ is the open ball of radius 1 in $n$-dim complex space $\mathbb{C}^{n}$ and $B_{m}(0,1)$ is the open ball of radius 1 in $m$-dim complex space $\mathbb{C}^{m}$. Let $V$ be a $n$-dim analytic variety (irreducible analytic set) of $B_{n}(0, 1)\times B_{m}(0,1) \subset \mathbb{C}^{m+n} $. Let $\pi_{1}$ be the projection map: $B_{n}(0, 1)\times B_{m}(0,1) \to B_{n}(0, 1)$.
Suppose $V $ is also contained in $ B_{n}(0, 1)\times B_{m}(0,\frac{1}{2}) $, where $B_{m}(0,\frac{1}{2})$ is the open ball of radius $\frac{1}{2}$ in $\mathbb{C}^{m}$. Is $\pi_{1}: V\to B_{n}(0, 1)$ a ramified map? (We require a ramified map shoud be a sujective, finite fibre map.)
I know that we have this property locally because of local property of analytic sets. But I am not sure whether it still hold globally. Could someone give an answer or some references? Thank you very much.
