Using the chern character, it can be shown that there is no complex structure on $S^n$ if $n > 6$: See May's book: if $S^{2n}$ has a complex structure, let $\tau$ be the tangent bundle. $c\_n(\tau) = \chi(S^{2n}) = 2$ must be divisible by $(n-1)!$ by Husemöller, Fibre bundles, chapter 20, Theorem 9.8. So the only case left is $S^4$ and $S^6$.


Is there a complex structure on $S^4$ or $S^6$?



complex structure on S^n
It is known that $S^4$ doesn't even have an almost complex structure. (The sphere $S^6$ does have some, but whether any of them is the underlying almost complex structure of a complex manifold is open.) The lack of almost complex structure can be proved a number of ways, one way is by showing that an almost complex, compact, four manifold with $\dim\_{\mathbb{Q}}H^2(X,\mathbb{Q})=0$ has $\chi(X)=0$, but the four sphere doesn't. (It follows from the index theorem, here's a quick reference, first result: [Srinivasacharyulu, "Topology of complex manifolds", Canad. Math. Bull. 1966](https://doi.org/10.4153/CMB-1966-003-9).)


