Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was not even aware of the fact that there was another serious attack at FLT in the 80s - that of Y. Miyaoka.
Now, we usually don't pay much attention to purported proofs from cranks and obvious amateurs of famous open conjectures, but since this came from a serious mathematician, the issues with his attempted proof can only be instructive in terms of learning value. It is a good thing to learn not only from one's own mistakes but from the mistakes of others too.
So I started searching for more information, but to my surprise (and perhaps understandably) I have not been able to find any details for the past day (except that the author used techniques from Differential Geometry, which is still way too generic).
Hence my question:
What were the main ideas and the respective gaps in Miyaoka's attempted proof of FLT?
Addendum: As can be seen from Timothy Chow's answer, the statement in Stewert and Tall's book that "Miyaoka had used a technique parallel to that of Wiles, by translating the number-theoretic problem into a different mathematical theory — in this case, differential geometry" is actually a bit misleading in that regard. For, in fact, he did the opposite - he tried to transfer notions from differential geometry (and related alg. topology) to the arithmetic world rather than give a differential-geometric (in a strict sense) proof. Sorry to get the hopes of our differential geometers too high!