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


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


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


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


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.

For clarification, I also recommend the discussion in §2.5 in Daniel Huybrechts' Complex Geometry; in particular, the proof of Prop. 2.5.8 seems relevant, and writes down the explicit map you do. Another reference is János Kollár's "The structure of algebraic threefolds," in §9.2 (although it is very concise).

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