Fermat’s Last Theorem – Main Ideas and Gaps in Yoichi Miyaoka’s Attempted Proof (1988)

Question

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!

Answer 1

Here's some information from Barry Cipra's June 1988 article "Fermat's Theorem remains unproved" in Science magazine.

Parshin showed that the arithmetical version of a certain inequality involving geometric invariants of surfaces—an inequality that Miyaoka proved for the geometric case in 1974—would lead by a series of steps to a bound on the size of possible exponents for which Fermat's Last Theorem could be false. $\ldots$

Miyaoka's work is directed at proving the arithmetical inequality. Miyaoka, who is an expert in algebraic geometry but a relative newcomer to the arithmetical theory, proceeded by analogy with the geometric case. But according to Enrico Bombieri, a professor of mathematics at the Institute for Advanced Studies [sic] in Princeton, the translation is not straightforward. "Things go over, but with some qualifications," Bombieri says. "The naïve extension doesn't go through."

The problem, according to Barry Mazur of Harvard University, is the lack of a good arithmetical analog of a crucial geometric object known as the tangent bundle. Mazur, who helped Miyaoka analyze the proof, explains that Miyaoka had "a very interesting idea" to replace the tangent bundle with a "generic" bundle, with the assumption that the generic bundle can be chosen so as to have suitably nice properties. This seems not to be the case.

The effort is not wasted, however, Mazur says that Miyaoka has carried the idea of substituting generic bundles for the tangent bundle back to the original geometric case. "Given any choice of a bundle, you'll get some inequalities," Mazur says. "It's a perfectly reasonable and interesting geometric question to ask what's the structure of this whole complex set of inequalities." Answering such questions will very possibly lead to a deeper understanding of Miyaoka's original geometric proof.

More information about how the inequality in question (known as the Bogomolov–Miyaoka–Yau inequality) relates to Fermat's Last Theorem can be found in the appendix (by Paul Vojta) to Serge Lang's book Introduction to Arakelov Theory.