# explicit (holomorphic) map revealing blow-up as a connected sum with $\overline{\mathbb{CP}}^n$

I am trying to understand the statement that a blow-up of a complex manifold $M$ at a point $p$ is equivalent to the connected sum of $M$ with $\overline{\mathbb{CP^n}}$ and, being a physicist, I need the explicit map between charts. I have looked at a number of sources, but found Broens' thesis particularly useful. Using this one may construct a transition function from the blow-up of $M$, $\tilde{M}$, to $\overline{\mathbb{CP^n}}$. The problem I have is that it doesn't look very holomorphic.

the set-up

If $M$ has co-ordinates $z'$ with $z'(p)=0$, then the blow-up $\tilde{M}$ is given by the set of points

$\{ (z', [w]) \in \mathbb{C}^n \times \mathbb{CP^{n-1}} \;|\; z'_iw_j=z'_jw_i\}$

Now define a map $\psi:\overline{\mathbb{CP^n}}\to\tilde{M}$ by

$\psi(\overline{Z}_0:Z_1:Z_2…:Z_n)=(Z_0Z/|Z|^2,\;[Z])$

where $Z=(Z_1,Z_2,…Z_n)\neq0$. The inverse of this map is, I think,

$\psi^{-1}(z',[z'])=(|z'|^2:z'_1:z'_2:…z'_n)$

A ball in $\mathbb{CP^n}$ about the point $q$ with co-ordinates $(1:0:0…:0)$ is defined by the set

$B_\epsilon=\{ (\overline {Z}_0:Z)\;|\;\;|Z|/|Z_0|\;\leq\;1/\epsilon \}$

which is then removed to give the set $V_\epsilon$ that will be glued on

$V_\epsilon=\{ (\overline {Z}_0:Z)\;|\;\;|Z|/|Z_0|\;>\;1/\epsilon \}$

the image of which is

$\psi(V_\epsilon)=\{ (Z_0Z/|Z|^2,\;[Z])\;|\;\;|Z|/|Z_0|\;>\;1/\epsilon \}$

so if we define the co-ordinates on $M$ to be $z'=Z_0Z/|Z|^2$ we find the set of points

$U_\epsilon=\psi(V_\epsilon)=\{ (z',[z'])\;|\;\;|z'|<\epsilon \}$

now make the annulus for the connected sum by constructing $U_\delta$, $\delta<\epsilon$, and the corresponding annulus on $V_\epsilon$.

So, given a point on the annulus in $M$, $z'\in U_\epsilon-U_\delta$, we get a point

$(|z'|^2:z'_1:z'_2:…z'_n)=(1:z'_1/|z'|^2:z'_2/|z'|^2:…z'_n/|z'|^2)$.

on $\overline{\mathbb{CP^n}}$ with inhomogeneous co-ordinates $z'_i/|z|^2$

the problem There's a good chance that I have totally misunderstood something, but if we take $\zeta_i$ to be the inhomogeneous co-ordinates on $\overline{\mathbb{CP^n}}$, then the above relates the charts of $M$ and $\overline{\mathbb{CP^n}}$ via

$\zeta_i=z'_i/|z|^2$

which is not holomorphic. Given that, for example, the del Pezzo surfaces are blow-ups, and also complex manifolds, I would have expected a holomorphic map between charts.

What is important about the differential-topological construction you ask about is that the blowup $\operatorname{Bl}_pM$ of a complex $n$-manifold $M$ is only (orientation-preservingly) diffeomorphic to $M \# \overline{\mathbf{CP}^n}$, and not in fact holomorphic to it, since in particular $\overline{\mathbf{CP}^n}$ isn't a complex manifold, and so the connected sum construction is not performed in the holomorphic category.