There's a very nice topological description of blow ups of complex manifolds at a point as connected sum with projective space. The following is an attmept to understand whether there's a higher dimensional generalization to this:


Let $X$ be a complex manifold and $Z \subset X$ a complex submanifold. Let $N_Z$ be the normal bundle and let $1_Z$ denote the trivial (complex) rank 1 bundle on $Z$.

The projectivized completion $Y = \mathbb{P}(N_Z \oplus 1_Z)$ can be written as the union $N_Z \cup\mathbb{P}N_Z$.

We can now do the following differential topological construction:

Let $U \subset N_Z$ be a tubular neighborhood of the zero section  isomorphic to a tubular neighborhood $U^{'} \subset X$ of $Z$ in $X$ (exists by tubular neighborhood theorem). Let $\tilde X$ be the space obtained by gluing $Y$ and $X$ along $U$ and $U^{'}$. 

> Is $\tilde X$ diffeomorphic to the the blow up $Bl_Z X$ of $X$ along
> $Z$?