Does there exist a holomorphic structure on $S^6$? Does the six-sphere $S^6$ admits any holomorphic structure? 
Can someone tell me if there is any development in research of holomorphic structures on $S^6$ as we know $S^6$ has an almost complex structure which cannot be induced by any holomorphic structure?
 A: In March $2015$, Gábor Etesi submitted the paper Complex structure on the six dimensional sphere from a spontaneous symmetry breaking to the arXiv. A revised version was posted in April and was published in the Journal of Mathematical Physics, see here. 
Due to the importance of the problem that the paper claimed to solve, Misha Verbitsky, an expert in the area, began a discussion of the paper on MathOverflow, a discussion which included the author's input (see the answers). 
There has since been an erratum to the paper which has been published, but the main claim apparently still holds.
There is also a short note ($8$ pages) which suppresses the physical background and motivation, and instead focuses on the mathematical structure of the proof.
It is unclear whether the proof by Etesi has been accepted by the mathematical community yet. I don't think it has. That isn't to say it is incorrect though.

Putting the above aside, there have been some surprising results about complex structures on $S^6$, if they were to exist. For example, Claude LeBrun showed in Orthogonal Complex Structures on $S^6$ that there is no complex structure on $S^6$ such that the round metric is hermitian. Another result by Campana, Demailly and Peternell is that if $S^6$ admits a complex structure, the only meromorphic functions are constants; see The Algebraic Dimension of
Compact Complex Threefolds
with vanishing second Betti Number.
