It is well known that complex projective three space $\mathbb{C}\mathbf{P}^3$ is a complex manifold. However it also possess a non-integrable almost-complex structure (as discussed in this [article](https://www.sciencedirect.com/science/article/pii/S0393044020301728#b12) for instance). Can somebody give an explicit constructuion of this almost-complex structure. I also wonder how, of if, it interacts with the homogeneous action of the unitary group $U(3)$ (or indeed $SU(3)$) on $\mathbb{C}\mathbf{P}^3$.



Non-integrable almost complex structure for complex projective $3$-space
There are lots of non-integrable almost-complex structures on $\mathbb{CP}^3$, but the one you are looking for is, I suspect, the following.


First I will explain the twistor fibration, which is a submersion $t \colon \mathbb{CP}^3 \to S^4$ with fibres $S^2$. To do this, fix an identification $\mathbb{C}^4 \cong \mathbb{H}^2$ with the quaternionic plane. Now each complex line in $\mathbb{C}^4$ generates a quaternionic line in $\mathbb{H}^2$. This gives a map $t \colon \mathbb{CP}^3 \to \mathbb{HP}^1$. The fibre of $t$ over a point $p \in \mathbb{HP}^1$ is the set of all complex lines in $\mathbb{C}^4$ that lie in the quaternionic line corresponding to $p$. This quaternionic line is a copy of $\mathbb{C}^2$ inside $\mathbb{C}^4$ and so $t^{-1}(p) \cong \mathbb{CP}^1 \cong S^2$. Meanwhile, $\mathbb{HP}^1$ is the 1-point compactification of $\mathbb{H}$ (just as $\mathbb{CP}^1$ is the 1-point compactification $\mathbb{C}$) and so the range of $t$ is diffeomorphic to $S^4$. This gives the fibration $t \colon \mathbb{CP}^3 \to S^4$.


Now I will explain how to use this to give a *non*-integrable almost-complex structure on $\mathbb{CP}^3$. The tangent spaces to the fibres of $t$ determine the so-called vertical tangent bundle, which is a complex line-subbundle $V \leq T\mathbb{CP}^3$. Meanwhile, the Fubini-Study metric in $\mathbb{CP}^3$ gives a complement $H$ to $V$ and so we have a decomposition into complex summands $T\mathbb{CP}^3 = V \oplus H$. With this in hand, we can define $J \colon T\mathbb{CP}^3 \to T\mathbb{CP}^3$ by $J = - i \oplus i$. In other words, $J$ agrees with the standard complex structure on $H$ and *minus* the standard complex structure on $V$.


In the language of twistor geometry, $\mathbb{CP}^3$ is the twistor space of $S^4$. The Aityah-Hitchin-Singer twistor almost complex structure is the standard one on $\mathbb{CP}^3$ (which is integrable when the base Riemannian 4-manifold is self-dual) whilst the one I described here is the Eells-Salamon twistor almost complex structure and this is *never* integrable.


You also asked about how to describe this by thinking of $\mathbb{CP}^3$ as a homogeneous space, so now I'll try and explain something in that direction. The isometries of $S^4$, namely $\mathop{SO}(5)$, lift to $\mathbb{CP}^3$ preserving the twistor fibration. It follows that they also preserve both the integrable and non-integrable complex structures. The action of $\mathop{SO}(5)$ on $\mathbb{CP}^3$ is transitive with stabiliser of a point isomorphic to $\mathop{U}(2)$. From this perspective, $\mathbb{CP}^3 \cong \mathop{SO}(5)/\mathop{U}(2)$. Write
$$
\mathfrak{so}(5) = \mathfrak{u}(2) \oplus \mathfrak{p}
$$
(orthongonal with respect to the Killing form). Then an $\mathop{SO}(5)$-invariant almost-complex structure on $\mathbb{CP}^3$ corresponds to a choice of almost complex structure on $\mathfrak{p}$ which is invariant under the action of $\mathop{U}(2)$. As a $\mathop{U}(2)$-representation,
$$
\mathfrak{p} \cong \mathbb{C}^2 \oplus \Lambda^2 \mathbb{C}^2
$$
(where $\mathbb{C}^2$ is the fundamental representation of $\mathop{U}(2)$). This corresponds to the splitting $T\mathbb{CP}^3 = H \oplus V$ desribed above. It follows that there are precisely two $\mathop{SO}(5)$-invariant almost-complex structures on $\mathbb{CP}^3$ (up to overall sign), and they are the standard one and the one in which $J$ has been reversed on $\Lambda^2\mathbb{C}^2 \cong V$.


Finally, as a remark, you can check that the non-integrable almost-complex structure given here actually has $c\_1=0$ (e.g. by using the $\mathop{SO}(5)$-invariant description above). This means it is not even isotopic to the integrable complex structure.


