Smooth structures on $\mathbb{CP}^3$

Question

How many distinct smooth structures does $\mathbb{CP}^3$ admit? How many almost-complex structures? How many complex structures? References would be great. Thanks.

Answer 1

This only answers the second question but is much to long for a comment so...

By Proposition 8 of "CUBIC FORMS AND COMPLEX 3-FOLDS" by Okonek and Van de Ven, the almost complex structures are in bijection with integral lifts of the second Steifel Whitney class $w_{2} \in H^{2}(\mathbb{CP}^{3},\mathbb{Z}_{2})$. In this case $w_{2} = 0$, so almost complex structures on $\mathbb{CP}^{3}$ are in bijection with cohomolgy classes of the form $$2 \alpha \in H^{2}(\mathbb{CP}^{3},\mathbb{Z}) \cong \mathbb{Z}.$$

In particular there are infinitely many distinct almost complex structures on $\mathbb{CP}^{3}$.

Such a cohomology class can be recovered from an almost complex structure by taking the first Chern class.