Is there a complex manifold diffeomorphic to $\mathrm{SU}(3)$?


This question arises in a StackExchange discussion by HK Lee, Ted Shifrin and Jason DeVito:
<https://math.stackexchange.com/questions/488959/way-distinguishing-whether-or-not-complex-manifold> 


I believe it is also related to Etesi's work on a complex structure for $S^6$ where he makes use of the fibration $\mathrm{SU}(3) \to G\_2 \to S^6$. So as an aside question, I would like to know whether Etesi's work implies the existence of a complex manifold dffeomorphic to $\mathrm{SU}(3)$?



Is there an integrable complex structure on $\mathrm{SU}(3)$?
It is an old theorem of Samelson that any compact Lie group $G$ of even rank has an integrable complex structure, which, in particular applies to the case of $\mathrm{SU}(3)$. Basically, one chooses a Cartan subalgebra $\frak{t}\subset\frak{g}$, which gives a splitting of the Lie algebra into (complex) root spaces, choose a base for the root system and then consider the positive root spaces, use those to define the $(1,0)$-vector fields and then define the complex structure however one likes on the Cartan sub-algebra $\frak{t}$ (which one can do because it has even dimension, by assumption), and then one shows that the left-invariant almost complex structure that this defines on $G$ is always integrable. (It follows almost immediately from the structure equations.)


Here is an explicit construction in the case of $\mathrm{SU}(3)$: Let $g:\mathrm{SU}(3)\to \mathrm{GL}(3,\mathbb{C})\subset M\_{3,3}(\mathbb{C})$ be the standard inclusion homomorphism of $\mathrm{SU}(3)$ into the group of invertible $3$-by-$3$ complex matrices. Then $g$ satisfies $g^\*g = I\_3$ and $\det(g)=1$ (where $g^\*$ means the conjugate transpose of $g$), so the canonical left invariant form $\gamma = g^{-1}\,\mathrm{d}g$ satisfies $\gamma^\*+\gamma=0$ and $\mathrm{tr}(\gamma)=0$, so it can be written in the form
$$
\gamma = \begin{pmatrix}
i\theta\_1 & \omega\_3 & \omega\_2\\
-\overline{\omega\_3} & i\theta\_2 & \omega\_1\\
-\overline{\omega\_2} & -\overline{\omega\_1} & i\theta\_3\end{pmatrix},
$$
where the $\omega\_i$ are $\mathbb{C}$-valued $1$-forms and the $\theta\_i$ are $\mathbb{R}$-valued $1$-forms satisfying $\theta\_1+\theta\_2+\theta\_3 = 0$. By the definition of $\gamma$, we have the structure equation $\mathrm{d}\gamma = - \gamma\wedge\gamma$. 


Now let $\lambda\in \mathbb{C}\setminus\mathbb{R}$ be any constant that has nonzero imaginary part and set $\omega\_0 = \theta\_1 + \lambda\,\theta\_2$. Then the $\mathbb{C}^4$-valued $1$-form $\omega = (\omega\_0,\omega\_1,\omega\_2,\omega\_3)$ defines a left-invariant $\mathbb{C}^4$-valued coframing on $\mathrm{SU}(3)$, i.e., it provides a linear identification of each tangent space of $\mathrm{SU}(3)$ with $\mathbb{C}^4$, and hence defines a unique almost complex structure $J\_\lambda$ on $\mathrm{SU}(3)$ for which the $1$-forms $\omega\_i$ are of $J\_\lambda$-type $(1,0)$. (This $J\_\lambda$ really does depend on the choice of $\lambda$ above.)


Finally, the structure equation $\mathrm{d}\gamma = -\gamma\wedge\gamma$ implies
$$
\mathrm{d}\omega\_i \equiv 0 \mod \omega\_0,\omega\_1,\omega\_2,\omega\_3
$$
for $0\le i\le 3$. (For example, 
$$
\mathrm{d}\omega\_1 = i(\theta\_3-\theta\_2)\wedge\omega\_1 +\overline{\omega\_3}\wedge\omega\_2
\equiv 0 \mod \omega\_0,\omega\_1,\omega\_2,\omega\_3\,,
$$
the other cases are similar.) Thus, the Nijnhuis tensor of $J\_\lambda$ vanishes, and so there is a unique complex structure on $\mathrm{SU}(3)$ for which the forms $\omega\_i$ are of type $(1,0)$. (Because this almost complex structure $J\_\lambda$ is left-invariant, it is *a fortiori* real-analytic, so the Newlander-Nirenberg theorem is not needed, the integrability follows from earlier results.)


