As we know, there is a famous open problem that: whether the 6-dimensional sphere admits a complex structure? And this problem is related to the complex structure of projective space $\mathbb {CP}^3$. Let me explain it:
Let $X$ be a compact complex manifold diffeomorphic to $S^6$, and let $\tilde{X}$ be its blowup at one point $p\in X$, then $\tilde{X}$ is diffeomorphic to the complex projective space $\mathbb{CP}^3$. But $\tilde{X}$ has a different complex structure with $\mathbb {CP}^3$, so if $S^6$ admits a complex structure, we can show that the complex structure of projective space $\mathbb {CP}^3$ is not unique, which contradicts to our common belief.
And my question lies in: how can we show the blowing up a point at $S^6$ is diffeomorphic to $\mathbb{CP}^3$? Actually, I know the book 《complex geometry》 p102 by Huybrechts has provided an argument that blowing up a point at $S^6$ is diffeomorphic to the connected sum $S^6$#$\bar{\mathbb{CP}^3}$, but his method seems a little bit formal, did anyone have a more intuitive and simpler method?