現代数学の系譜工学物理雑談古典ガロア理論も読む79

[過去ﾛｸﾞ] 現代数学の系譜工学物理雑談古典ガロア理論も読む79 (1002ﾚｽ)
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931: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2020/01/03(金) 19:57:31.73 ID:ivt0JCXh

>>930
つづき
３）
(PB1) の、PはParshin、BはBogomolovか

P64
In the following
discussion, we shall refer to this geometry as the Schwarz-theoretic geometry of
D. Perhaps the most fundamental difference between the proofs of Parshin and
Bogomolov lies in the fact that
(PB1) Whereas Parshin’s proof revolves around estimates of displacements arising from actions of elements of the fundamental group on a certain two dimensional complete [Kobayashi] hyperbolic complex manifold by means
of the holomorphic geometry of the Kobayashi distance, i.e., in effect,
the Schwarz-theoretic geometry of D, Bogomolov’s proof [cf. the review of Bogomolov’s proof given in [BogIUT]] revolves around estimates of
displacements arising from actions of elements of the fundamental group
on a one-dimensional real analytic manifold [i.e., a universal covering of
a copy of the unit circle S1] by means of the real analytic symplectic
geometry of the upper half-plane.
Here, it is already interesting to note that this fundamental gap, in the case of
results over complex function fields, between the holomorphic geometry applied in
Parshin’s proof of the Mordell Conjecture and the real analytic symplectic geometry
applied in Bogomolov’s proof of the Szpiro Conjecture is highly reminiscent of the
fundamental gap discussed in Remark 2.3.3, (iii), in the case of results over number
fields, between the arithmetically holomorphic nature of the proof of the Mordell
Conjecture given in [Falt] and the “arithmetically quasi-conformal” nature of
the proof of the Szpiro Conjecture [cf. Corollary 2.3] via inter-universal Teichm¨uller
theory given in the present series of papers. That is to say,

つづく

http://rio2016.5ch.net/test/read.cgi/math/1573769803/931

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