I am reading about complex and symplectic blow-ups from McDuff-Salamon's Introduction to Symplectic Topology and have some troubling extracting the main point from all the details.
If I understand correctly, when considering Kähler manifolds, both blow-ups perform the same topological procedure but the symplectic blow-up gives different Kähler forms depending on the ball embedding (actually, can even given nonKähler forms). As far as I can tell, the main thing that happens as we change the ball embeddings is that we get different volumes for the blow-up. And if we carefully choose a particular embedding, we can in fact, obtain the complex blow-up. Is this an accurate summary?
There is also this fact: in real dim other than 2 and 6, if you have two symplectic manifolds and you connect sum them in the usual topological way, there is no symplectic form on the connect sum which restricts to the original forms. This follows from the fact that the only spheres with almost complex structure are $S^2$ and $S^6$.
Applying this to blow-ups, if we blow-up $\mathbb{CP}^2$ at a point, topologically, what we have is $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$. Then, the fact above tells us that this blow-up does not have a symplectic form which restricts to the original symplectic forms on the two manifolds. So neither blow-up procedure gives a symplectic form which restricts to the symplectic form on $\overline{\mathbb{CP}^2}$. On the other hand, if we take the connect sum $\mathbb{CP}^2 \# \mathbb{CP}^2$, this is not even a symplectic manifold by Seiberg-Witten theory. What is it about this reverse of orientation that makes such a difference?
I think some of my trouble might be that I want to intuitively think of blow-ups as this topological process of connect sum but it does not serve me well geometrically, either in the complex or symplectic setting.