ガロア第一論文と乗数イデアル他関連資料スレ18

ガロア第一論文と乗数イデアル他関連資料スレ18 (447ﾚｽ)
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ﾘﾛｰﾄﾞ規制です｡10分ほどで解除するので､他のﾌﾞﾗｳｻﾞへ避難してください｡

299(2): 07/06(日)08:08 ID:+k1m9OFg(1/9) AAS
>>298
>武漢では開性定理の応用について
>サーベイがあった

"開性定理"は、初耳です
google検索のAI回答は下記ですが、合ってますか？
（日本語だと情報が少ないが、英語などでは情報があるかもです）

google検索
多変数関数論開性定理とは

＜AI による概要＞（AI の回答には間違いが含まれている場合があります）
多変数関数論における開性定理とは、多変数複素関数が局所的に正則であれば、その関数は開集合上で正則であるという定理です。より具体的に言うと、ある開集合内の点において、その点の近傍で正則な多変数複素関数は、その開集合全体で正則になるという定理です。
開性定理の概要:
多変数複素関数:複数の複素変数を入力とする関数です。例えば、f(z1, z2) のように表されます。
正則性:複素関数が微分可能であること、つまり、複素微分が定義できることを意味します。
局所正則性:ある点の近傍で正則であることです。
開集合:任意の点に対して、その点を含む十分小さな開円盤（近傍）がその集合に完全に含まれるような集合です。
大域正則性:開集合全体で正則であることです。
開性定理の重要性:
開性定理は、多変数複素関数論において、局所的な性質が、より広い範囲の集合における性質にどのように影響するかを示す重要な定理です。この定理によって、局所的に正則な関数を調べることで、その関数全体の性質を把握できる可能性が開かれます。
例:
例えば、ある多変数複素関数が、ある開集合Dのすべての点zで、その点の近傍で正則だとします。この時、開性定理により、その関数は開集合D全体で正則であることが保証されます。
関連する概念:
正則関数:複素微分可能な関数。
多変数複素関数論:複数の複素変数を扱う関数論。
岡潔:多変数複素関数論に大きな貢献をした数学者。
補足:
開性定理は、多変数複素関数論における基本的な定理の一つであり、様々な応用を持っています。例えば、複素解析の手法は、応用数学や物理学、工学など、幅広い分野で利用されています。

300(1): 07/06(日)08:19 ID:+k1m9OFg(2/9) AAS
>>299 補足

前記の
”日経サイエンス　 2025年8月号
量子論に虚数は避けられないのか”>>296-297

これで、『実数量子論』（実2変数 (x,y)の関数論を使う）のと
1変数複素関数論 z=x+iy
との違いは、"開性定理"にありってことですかね？　（＾＾

304(1): 07/06(日)17:49 ID:+k1m9OFg(3/9) AAS
>>301
google検索
Several Complex Variables openness conjecture openness theorem

検索結果（要約メモ：openness conjecture＝Demailly's Strong Openness Conjecture らしい）
1）
外部ﾘﾝｸ[pdf]:english.cas.cn
2 ページ
BCAS vol.34 No.3 2020 （Bulletin of the Chinese Academy of Sciences）
TKK Young Scientist Award in Mathematics and Physics
Solutions of Demailly’s Strong Openness Conjecture and Related Problems in Several Complex Variables
By YAN Fusheng (Staff Reporter)
The 2020 TKK Young Scientist Award in Mathematics & Physics went to Prof. GUAN Qi’an, a young talented mathematician from Peking University, for his solutions (joint with ZHOU Xiangyu) of a series of problems in several complex variables, particularly for his proof (joint with ZHOU Xiangyu) of the Demailly’s strong openness conjecture.
GUAN has mainly engaged himself in the study of several complex variables, which explores the properties and structures of holomorphic functions of several variables, and is also called complex analysis of several variables. Because the properties of holomorphic functions are largely affected by the geometric and topological properties of their domains of definition,the research involves not only the studying of local properties, but also of global properties.
In the research of several complex variables, various methods from partial differential equations, algebraic
geometry, complex geometry, topology, Lie groups and other areas are widely applied. The research of function theory of several complex variables has also driven the development of the above-mentioned research fields. For example, LU Qikeng proved “LU Qikeng Theorem” named after him; SIU Yum-Tong proved the deformational invariance of plurigenera of projective algebraic manifolds; ZHOU Xiangyu proved the extended future tube conjecture which was listed as an
unsolved problem in the Encyclopedia of Mathematics.
In cooperation with Prof. ZHOU Xiangyu, a CAS Member at the Academy of Mathematics and Systems Science (AMSS) under the Chinese Academy of Sciences,GUAN solved the optimal L2 extension problem,
proposing new ideas and methods. They established the optimal L2 extension theorem by unifying the previous
以下略
注）TKK：the Tan Kah Kee Young Scientist Award 陈嘉庚科学奖基金会外部ﾘﾝｸ:tsaf.cas.cn

つづく

305(1): 07/06(日)17:49 ID:+k1m9OFg(4/9) AAS
つづき

2）
外部ﾘﾝｸ[pdf]:annals.math.princeton.edu
A proof of Demailly's strong openness conjecture
Annals of Mathematics
2015/01/14 — In the present article, we discuss a more general conjecture — the strong openness conjecture about multiplier ideal sheaves for ...
12 ページ

3）
外部ﾘﾝｸ:arxiv.org
arXiv:2109.00353v1 [math.CV] 1 Sep 2021
Q Guan 著 · 2021 · 被引用数: 10 — The strong openness property is an important feature of multiplier ideal sheaves and used in the study of several complex variables, algebraic ...

4）
外部ﾘﾝｸ:arxiv.org
arXiv:2203.01648v4 [math.CV] 1 Apr 2024
S Bao 著 · 2022 · 被引用数: 13 — of several complex variables, complex algebraic geometry and complex differential geometry (see e.g. [48, 42, 44, 12, 13, 11, 14, 39, 40, 45 ...

BOUNDARY POINTS, MINIMAL L2 INTEGRALS AND CONCAVITY PROPERTY
SHIJIE BAO, QI’AN GUAN, AND ZHENG YUAN
Abstract. For the purpose of proving the strong openness conjecture of
multiplier ideal sheaves, Jonsson-Mustat¸˘a posed an enhanced conjecture and
proved the two-dimensional case, which says that: the Lebesgue measure of
the set {cFo(ψ)ψ − log |F| < log r} divided by r2 has a uniform positive lower
bound independent of r, for a plurisubharmonic function ψ and a holomorphic
function F near the origin o. Jonsson-Mustat¸˘a’s conjecture was proved by
Guan-Zhou depending on the truth of the strong openness conjecture. However, it is still a question whether one can prove Jonsson-Mustat¸˘a’s conjecture
without using the strong openness property, and obtain a sharp effectiveness
result for this conjecture.
In this article, we use an L2 method with the weight functions ψ − log |F|
and firstly consider a module at at a boundary point of the sublevel sets of
a plurisubharmonic function. By studying the minimal L2
integrals on the sublevel sets of a plurisubharmonic function with respect to the module at the
boundary point, we establish a concavity property of the minimal L2 integrals.
As applications, we obtain a sharp effectiveness result related to JonssonMustat¸˘a’s conjecture, which completes the approach from the conjecture to the strong openness property. We also obtain a strong openness property of the module and a lower semi-continuity property with respect to the module.

つづく

306(1): 07/06(日)17:50 ID:+k1m9OFg(5/9) AAS
つづき

5）
外部ﾘﾝｸ[pdf]:www2.math.kyushu-u.ac.jp
九大数理学研究院
The proof is based on a density formula of Zelditch, the Abel–Jacobi theory, Fekete points theory, and a new perturbation method. This is joint work (arXiv: ...
10 ページ
＜関連箇所＞
Young Mathematicians Workshop on Several Complex Variables 2024
Osaka Metropolitan University∗
13:40–14:20 Wang Xu (Sun Yat-sen University)
Optimal L2 extension of openness type and related topics
Abstracts
L2 extension theorems and optimal L2
extension theorems are important and powerful tools in several complex variables and complex geometry. There is a closely
related problem called the optimal L2
extension problem of openness type: given
a holomorphic section f defined on a neighbourhood U of a subvariety S, find a
holomorphic extension of f|S to the ambient manifold, whose L2 norm is optimally
controlled by the L2 norm of f on U. In this talk, I will present a solution on
weakly pseudoconvex K¨ahler manifolds, which generalizes a couple of known results. I will also discuss some connections and applications to related topics, such as sharper L2
extensions and generalized Suita conjectures. This talk is mainly
based on joint work with Prof. Xiangyu Zhou.

6）動画がある
外部ﾘﾝｸ:www.mathnet.ru
International Conference Dedicated to the 100th Anniversary of the Birthday of V. S. Vladimirov (Vladimirov-100)
January 10, 2023 14:00–14:30, Moscow, Steklov Mathematical Institute, room 430 (Gubkina 8) + Zoom
X. Zhou
Academy of Mathematics and Systems Science, Chinese Academy of Sciences
＜動画アドレス＞
外部ﾘﾝｸ[mp4]:www.mathnet.ru
Abstract: We'll talk about some recent results in several complex variables and complex geometry, e.g., the solution of Demailly's strong openness conjecture on multiplier ideal sheaves, converse of L^2 existence theorem due to Hoermander and Demailly, and their applications in some fundamental problems in complex geometry.

7）余録(openness conjectureは出てこないが、それ以前の整理には役立つだろう)
外部ﾘﾝｸ[pdf]:www.ms.u-tokyo.ac.jp
Some remarks on basic materials in several complex variables
J. Noguchi
30th Symposium of Complex Geometry, Kanazawa 2024
(引用終り)
以上

308(1): 現代数学の系譜雑談 ◆yH25M02vWFhP 07/06(日)18:05 ID:+k1m9OFg(6/9) AAS
Copilot：
What is Demailly's Strong Openness Conjecture
回答：
Demailly's Strong Openness Conjecture is a deep result in complex geometry and analysis, particularly in the theory of multiplier ideal sheaves associated with plurisubharmonic functions.

The Conjecture (Now a Theorem)
Let \varphi be a plurisubharmonic function on a complex manifold X. The conjecture asserts that:
\mathcal{I}_+(\varphi) = \mathcal{I}(\varphi)
where:
- \mathcal{I}(\varphi) is the multiplier ideal sheaf: the sheaf of holomorphic functions f such that |f|^2 e^{-\varphi} is locally integrable.
- \mathcal{I}_+(\varphi) := \bigcup_{\varepsilon > 0} \mathcal{I}((1+\varepsilon)\varphi)
This means that the ideal sheaf stabilizes under small perturbations of the weight function \varphi, reflecting a kind of openness in the space of integrable functions

Historical Context
Jean-Pierre Demailly proposed this conjecture as a strengthening of the earlier Openness Conjecture (by Demailly and Kollár), which was concerned with the integrability of exponentials of plurisubharmonic functions.

✅ Resolution
The conjecture was proven by Qi'an Guan and Xiangyu Zhou in 2015. Their work not only settled the conjecture but also led to further insights into:
- The behavior of singularities of plurisubharmonic functions
- The structure of multiplier ideal sheaves
- Applications to algebraic geometry and complex differential geometry
You can explore the original proof in the Annals of Mathematics article1) or the expanded version on arXiv2).
Would you like to dive into the techniques used in the proof, such as the L^2 extension theorem or curve selection lemma? I think you'd find the analytic machinery quite elegant.

注：
1)Annals of Mathematics article
外部ﾘﾝｸ:annals.math.princeton.edu
A proof of Demailly’s strong openness conjecture
Pages 605-616 from Volume 182 (2015), Issue 2 by Qi'an Guan, Xiangyu Zhou
Abstract
In this article, we solve the strong openness conjecture on the multiplier ideal sheaf associated to any plurisubharmonic function, which was posed by Demailly.

2)arXiv
外部ﾘﾝｸ[7158]:arxiv.org
[Submitted on 28 Jan 2014]
Strong openness conjecture and related problems for plurisubharmonic functions
Qi'an Guan, Xiangyu Zhou

314(1): 現代数学の系譜雑談 ◆yH25M02vWFhP 07/06(日)21:27 ID:+k1m9OFg(7/9) AAS
>>309
ご苦労さま

1）AIの解説は、私もすでに2カ所で投稿している
　一つは、>>308 "Copilot：What is Demailly's Strong Openness Conjecture"
　もう一つは、>>299 "google検索多変数関数論開性定理とは＜AI による概要＞・・・"
2）あんたのは、>>299のgoogle の＜AI による概要＞とほぼ同じ
　だが、あんたの問題は i)どのAIをつかったのか？ ii)どういう質問をしたのか？
　この2点の明示がないこと
　特に、”ii)どういう質問をしたのか？”は、大きな問題だな
　つまり、これを見た人が、自分の手持ちのAIに同じ質問をしようとしたときに
　それができない。あるいは、将来半年とか1年後に AIの進化やネット情報の更新があったとして
　もう一度同じ質問をしたいとき、それが出来ないってことだ

まあ、採点は御大がしてくれるだろうさ；ｐ）

さてしかし、>>299のgoogle の＜AI による概要＞のあとに
御大の>>301
"openness conjectureが解決された結果
開性定理が生まれた
そのeffective versionsが
複素幾何に応用されている"
が投稿されているだろ？

つまり、
1）openness conjecture とは？
2）それをいつだれがどのように解決したのか？
3）”effective versions”は、どんなものか？（複数形だよ）
4）”複素幾何に応用されている" の部分は、どうか？

1）と2）については、>>304-306にある
3）の”effective versions”は、まだ不十分だが
　(>>308 で ”You can explore the original proof in the Annals of Mathematics article1) or the expanded version on arXiv2).”とあるから、the expanded version on arXiv2)が該当の一つかも)
4）”複素幾何に応用されている"は、>>308 の”multiplier ideal sheaves”がキモらしい（以前御大がそう述べていたから）

結論として、>>309-312は、
上記の>>299のgoogle の＜AI による概要＞とほぼ同じじゃね？；ｐ）

315: 現代数学の系譜雑談 ◆yH25M02vWFhP 07/06(日)21:29 ID:+k1m9OFg(8/9) AAS
>>313
ID:a2BajG8e は、御大か
巡回ありがとうございます

いやー、>>314とかぶった（＾＾

316: 現代数学の系譜雑談 ◆yH25M02vWFhP 07/06(日)21:45 ID:+k1m9OFg(9/9) AAS
google検索
Demailly's Strong Openness Conjecture wiki
（これで、”Demailly's Strong Openness Conjecture”のwikipedia をさがしたら）
下記”周向宇”を嫁めと出る

外部ﾘﾝｸ:en.wikipedia.org
Xiangyu Zhou (Zhou Xiangyu, Chinese: 周向宇; pinyin: Zhōu Xiàngyǔ, born March 1965) is a Chinese mathematician, specializing in several complex variables and complex geometry. He is known for his 1998 proof of the "extended future tube conjecture", which was an unsolved problem for almost forty years.[1]

Selected publications
・Guan, Qi'an; Zhou, Xiangyu (2013). "Strong openness conjecture for plurisubharmonic functions". arXiv:1311.3781 [math.CV].
・Guan, Qi'an; Zhou, Xiangyu (2015). "A proof of Demailly's strong openness conjecture". Annals of Mathematics. 182 (2): 605–616. doi:10.4007/annals.2015.182.2.5. JSTOR 24523344.
・Guan, Qi'an; Zhou, Xiangyu (2015). "Effectiveness of Demailly's strong openness conjecture and related problems". Inventiones Mathematicae. 202 (2): 635–676. arXiv:1403.7247. Bibcode:2015InMat.202..635G. doi:10.1007/s00222-014-0575-3. S2CID 119317767.
・Guan, Qi'An; Zhou, Xiangyu (2017). "Strong openness of multiplier ideal sheaves and optimal
L2 extension". Science China Mathematics. 60 (6): 967–976. arXiv:1703.08387. doi:10.1007/s11425-017-9055-5. S2CID 119150408.

ついでに”Ohsawa"関連3点
・Guan, Qiʼan; Zhou, Xiangyu; Zhu, Langfeng (2011). "On the Ohsawa–Takegoshi extension theorem and the twisted Bochner–Kodaira identity". Comptes Rendus Mathematique. 349 (13–14): 797–800. doi:10.1016/j.crma.2011.06.001.
・Zhu, Langfeng; Guan, Qiʼan; Zhou, Xiangyu (2012). "On the Ohsawa–Takegoshi
L2 extension theorem and the Bochner–Kodaira identity with non-smooth twist factor". Journal de Mathématiques Pures et Appliquées. 97 (6): 579–601. doi:10.1016/j.matpur.2011.09.010
・Guan, Qi'An; Zhou, Xiangyu (2015). "Optimal constant in an
L2 extension problem and a proof of a conjecture of Ohsawa". Science China Mathematics. 58 (1): 35–59. arXiv:1412.0054. Bibcode:2015ScChA..58...35G. doi:10.1007/s11425-014-4946-4. S2CID 119139395

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