In October 2016, Atiyah famously posted a preprint to the arXiv, "The Non-Existent Complex 6-Sphere" containing a very brief proof $S^6$ admits no complex structure, which I immediately read and realized I lacked the background to understand. The preprint inspired interest in KR-theory and a few postings on MathOverflow (Atiyah's paper on complex structures on $S^6$; https://mathoverflow.net/a/253490/5792), but a Google search does not seem to turn up any exposition or commentary later than November. A paper so brief on such an important topic has, however, no doubt been thoroughly read and reread by competent experts since that time.
- Has an exposition with more detail been published anywhere?
- Is the argument of the paper currently thorougly understood? If not,
- What parts are currently understood?
- What remains to be more closely explained?