I'm trying to understand the proof (in Complex Geometry-Daniel Huybrechts ) that the Blow-up at a point $p$ of a complex manifold $X$ of dimension $n$, denote it by $B\ell_p(X)$ is diffeomorphic to $X\#\mathbb C\mathbb P^n$.
When $M$ and $M'$ are oriented, in the definition of connected sum we assume that $\eta$ is orientation-preserving and $\eta'$ is orientation-reversing. The proof in the above-cited book is the following.
Now, there are a pair of things I don't understand.
First, if I'm right, the connected sum $D\#\overline{\mathbb{CP}}^n$ is just $\overline{\mathbb{CP}}^n\setminus\eta'\left(\frac{1}{2}\overline{D}\right)$. Why don't we simpy find a orientation-preserving diffeomorphism $\overline{\mathbb{CP}}^n\setminus\eta'\left(\frac{1}{2}\overline{D}\right)\longrightarrow \hat{D}$?
Second, I don't understand why we need to take $D\#\overline{\mathbb{CP}}^n$ instead of $D\#\mathbb{CP}^n$.
I stumbled on this constructing reading McDuff, Salamon book on Sympletic geometry, section 7.1. I also need to understand what's happening and though the construction of Huybrechts is explicit, I can't get it too.
Connected sum of a disc $D$ with $\overline{\mathbb{CP}}^n$ is not a $\overline{\mathbb{CP}}^n$ without $\eta'\left(\frac{1}{2}\overline{D}\right)$. What is happening, is that we take a connected sum of two manifolds.
Connected sum can be defined the following way: I cut out open balls $B^n$ from two manifolds (let us think that they don't have border), get the border equal to $S^{n-1}$ and identify two spheres as I wish. This gives me connected sum topologically. But in order for the smooth structure be unambiguously defined, I can throw away an interiour of the ball (Huybrechts throws away the ball of diameter $\frac{1}{2}$ and identify the points of the rest)
Any local diffeomorphisms can be either orientation preserving or orientation reversing (it's because the determinant of Jacobian is a continuous function, and the sign can't change without passing through $0$, in which case the map is not diffeo anymore.) If Jacobian is positive, the map is o-preserving, otherwise o-reversing.
Also composition of local diffeos are reversing orientation as soon there were odd orientation-reversing diffeos. In our case $\nu$ and $\xi$ are orientation-preserving and $\nu'$ is orientation-reversing. I didn't check.
The conjugation of the complex structure is making $\mathbb{C}P^n$ to have opposite orientation as soon as $n$ is odd. Indeed, we had tangent vector $x_1, y_1, x_2, y_2$ where $y_i = I x_i$ and now we have tangent vector $x_1 -y_1, x_2, -y_2$ and the Jacobian has sign $(-1)^n$. I may be wrong, since I was always believing that conjugation is changing orientation for all $n$.
What I have heard is that taking different orientation during gluing of your manifold gives the same differential structure, but different complex and symplectic structure, which is happening here.
I hope that someone will clarify things and this answer will pull the question to the newsfeed. If you, Vincenzo, are still interested, please ask questions and I will try to fill the gaps later.