Let $X\subset \mathbb{C} \mathbb{P}^{N}$ be a smooth, 
algebraic (locally closed)  complex 
submanifold of $\mathbb{C} \mathbb{P}^N$ 
of complex dimension $k$. More concretely, $X$ is of the 
following type 
$$ X := \{ p \in \mathbb{C} \mathbb{P}^N: \phi_1(p) =0, \phi_2(p) \neq 0  \} $$
where $\phi_1$ and $\phi_2$ are sections of some holomorphic 
vector bundle and whenever $\phi_2(p) \neq 0$, $\phi_1$ is transverse to the zero set. 
  

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 $k$.