[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(1): 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(25/37) AAS
>>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
932(1): 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) 19:59:33.09 ID:ivt0JCXh(26/37) AAS
>>931
つづき
４）
ZFCG-modelを考えたけど、ギブアップしたのかな（＾＾；
P67
Section 3: Inter-universal Formalism: the Language of Species
(†G) Given any set x, there exists a universe V such that x ∈ V .
We shall refer to a ZFC-model that also satisfies this additional axiom of the
Grothendieck school as a ZFCG-model.
P68
Although we shall not discuss in detail here the quite difficult issue of whether
or not there actually exist ZFCG-models, we remark in passing that it may be
possible to justify the stance of ignoring such issues in the context of the present
series of papers ? at least from the point of view of establishing the validity of
various “final results” that may be formulated in ZFC-models ? by invoking the
work of Feferman [cf. [Ffmn]]. Precise statements concerning such issues, however,
lie beyond the scope of the present paper [as well as of the level of expertise of the
author!].
In the following discussion, we use the phrase “set-theoretic formula” as it is
conventionally used in discussions of axiomatic set theory [cf., e.g., [Drk], Chapter 1,
§2], with the following proviso: In the following discussion, it should be understood
that every set-theoretic formula that appears is “absolute” in the sense that its
validity for a collection of sets contained in some universe V relative to the model
of set theory determined by V is equivalent, for any universe W such that V ∈ W,
to its validity for the same collection of sets relative to the model of set theory
determined by W [cf., e.g., [Drk], Chapter 3, Definition 4.2].
Remark 3.1.2.
(ii) One interesting point of view that arose in discussions between the author
and F. Kato is the following. The relationship between the classical approach to
discussing mathematics relative to a fixed model of set theory ? an approach in
つづく
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