We identify $\mathbb{C}^n$ with $\mathbb{R}^{2n}$, and let $C\subset \mathbb{C}^n$ be a semi-algebraic curve, then can we always find a complex algebraic curve $V'\subset \mathbb{C}^n$ containing $C$?
In general, let $V\subset \mathbb{C}^n$ be a semi-algebraic set of dimension $m$, then is it true that the dimension of the Zariski closure of $V$ always has complex dimension $m$?
Thanks very much!