Let $X\subset \mathbb{C} \mathbb{P}^{N}$ be a smooth (ie non singular) complex submanifold of $\mathbb{C} \mathbb{P}^N$ of complex dimension $k$. Let $\overline{X}$ be the closure of $X$ inside $\mathbb{C}\mathbb{P}^N$.

1. Is it true that $\overline{X}-X$ is an algebraic variety?

2. Is it true that the ``dimension'' of $\overline{X}-X$ is strictly less than the dimension of $X$?

3. In particular does $\overline{X}$ always define a homology class $$ [\overline{X}] \in H_{2k}(\mathbb{C} \mathbb{P}^N, \mathbb{Z}) $$
   the basic idea being that the singularities of $\overline{X}$ are of complex codimension one, hence real codimension two.

Everything is over complex numbers. Note that although $X$ may not be connected, I am assuming that every connected component of $X$ has the same dimension $k$.