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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 blow up $Bl_Z X$ of $X$ along $Z$?

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    $\begingroup$ See arxiv.org/pdf/1210.1687.pdf , The proof of Lemma 2.7 math.leidenuniv.nl/scripties/pasquotto.pdf , also page 7 and 8 of Ruan paper arxiv.org/pdf/math/0611592.pdf is interesting also $\endgroup$
    – user21574
    Commented Nov 27, 2017 at 23:50
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    $\begingroup$ The topological description of blow-ups is very well explained in McDuff's paper "Examples of simply-connected symplectic non-Kählerian manifolds"; see paragraph 2 and especially Definition 2.2. In that definition, $\tilde{V}$ is the neighborhood of the zero section of $\mathcal{O}_{\mathbf{P}(N_{Z/X})}(1)$, whose total space can be identified with $Y- S$ where $Y = \mathbf{P}(N_{Z/X} \oplus \mathcal{O}) $ and $S$ is the section of $Y \to Z$ corresponding to the quotient $N_{Z/X} \oplus \mathcal{O} \to \mathcal{O}$. So topologically $Bl_ZX$ is indeed a connected sum of $X$ with $Y$. $\endgroup$
    – HYL
    Commented Nov 30, 2017 at 14:59
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    $\begingroup$ This is also explained in Griffiths-Harris, Chapter 4, Section 6: "Blowing up a submanifold". In this section they compute the cohomology of a blow-up by a Mayer-Vietoris argument for the open cover given by $U = X \setminus Z$ and $V=$ a tubular neighborhood of the exceptional divisor in $\mathrm{Bl}_Z(X)$. $\endgroup$
    – Dan Petersen
    Commented Nov 30, 2017 at 18:03

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