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 (462ﾚｽ)
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304: １３２人目の素数さん [] 2025/07/06(日) 17:49:17.04 ID:+k1m9OFg

>>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）
https://english.cas.cn/bcas/2020_3/202102/P020210210712420301203.pdf
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 https://tsaf.cas.cn/en/

つづく

http://rio2016.5ch.net/test/read.cgi/math/1748354585/304

305: １３２人目の素数さん [] 2025/07/06(日) 17:49:40.26 ID:+k1m9OFg

つづき

2）
https://annals.math.princeton.edu/wp-content/uploads/annals-v182-n2-p05-p.pdf
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）
https://arxiv.org/pdf/2109.00353
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）
https://arxiv.org/pdf/2203.01648
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

http://rio2016.5ch.net/test/read.cgi/math/1748354585/305

306: １３２人目の素数さん [] 2025/07/06(日) 17:50:04.08 ID:+k1m9OFg

つづき

5）
https://www2.math.kyushu-u.ac.jp/~kusakabe/YMWSCV2024/abstracts.pdf
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
https://www.mathnet.ru/eng/present37745
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＞
https://www.mathnet.ru/PresentFiles/37745/37745.mp4
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)
https://www.ms.u-tokyo.ac.jp/~noguchi/Nog_remark_SCV_Kanazawa2024.pdf
Some remarks on basic materials in several complex variables
J. Noguchi
30th Symposium of Complex Geometry, Kanazawa 2024
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1748354585/306

308: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/07/06(日) 18:05:08.73 ID:+k1m9OFg

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
https://annals.math.princeton.edu/2015/182-2/p05
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
https://arxiv.org/abs/1401.7158
[Submitted on 28 Jan 2014]
Strong openness conjecture and related problems for plurisubharmonic functions
Qi'an Guan, Xiangyu Zhou

http://rio2016.5ch.net/test/read.cgi/math/1748354585/308

314: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/07/06(日) 21:27:10.86 ID:+k1m9OFg

>>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）

http://rio2016.5ch.net/test/read.cgi/math/1748354585/314

316: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/07/06(日) 21:45:05.15 ID:+k1m9OFg

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

https://en.wikipedia.org/wiki/Xiangyu_Zhou
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

http://rio2016.5ch.net/test/read.cgi/math/1748354585/316

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