多変数関数論4

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958: １３２人目の素数さん [sage] 2025/06/04(水) 10:11:54.56 ID:Vo5laslH

https://rio2016.5ch.net/test/read.cgi/math/1745503590/432-433
＜AI Copilot さんが どれくらい賢いか？＞(そこそこ賢いようですね)

スタート
いま 数学の多変数複素関数論で 話題になっていることは 何ですか
　↓ （google英訳（を若干手直し））
What is the hot topic right now in the theory of functions of several complex variables ?

Copilotさんの回答1
One of the current hot topics in the theory of functions of several complex variables is the (L^2)-theory and its applications. Recent research has focused on the (L^2) existence theorem and (L^2) extension theorem, which play a crucial role in understanding positivity conditions in complex geometry. These results are being used to develop criteria for Griffiths positivity and Nakano positivity of Hermitian metrics on holomorphic vector bundles.
Another active area of research is the study of plurisubharmonic functions, which are essential in complex analysis and geometry. Researchers are investigating the strong openness property and stability of multiplier submodule sheaves associated with singular Nakano semipositive Hermitian metrics.

Would you like more details on any of these topics? I can help break them down further!

参考Link（殆ど 下記 Abstract通り？）
https://link.springer.com/article/10.1134/S0040577924010112
Home Theoretical and Mathematical Physics Article
Recent progress in the theory of functions of several complex variables and complex geometry
Research Articles
Published: 01 February 2024
Volume 218, pages 163–176, (2024) Xiangyu Zhou
Cite this article
Abstract
We give a survey on recent progress on converses of L^2 existence theorem and L^2
extension theorem which are two main parts in
L^2 -theory, and their applications in getting criteria of Griffiths positivity and characterizations of Nakano positivity of (singular) Hermitian metrics of holomorphic vector bundles, as well as the strong openness property and stability property of multiplier submodule sheaves associated to singular Nakano semipositive Hermitian metrics on holomorphic vector bundles.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1724146576/958

959: １３２人目の素数さん [sage] 2025/06/04(水) 10:12:18.08 ID:Vo5laslH

つづき

さて さらに
Yes, please more.

Copilotさんの回答2
Absolutely! Let's dive deeper.
(L^2)-Theory in Several Complex Variables
The (L^2)-method is a powerful tool in complex analysis, particularly in solving the (\bar{\partial})-equation.
This approach, pioneered by Lars Hörmander, provides existence theorems for holomorphic functions under certain curvature conditions. 1)
Recent research has refined the Oka–Cartan theory using (L2)-methods, leading to precise on hlomorphic extension. 2)
The Bergman kerenel, a fundamental object in complex analysis, has olso been extensively using L2-techniques.2)

Plurisubharmonic Functions
Plurisubharmonic functions are crucial in complex analysis and geometry.
They generalize subharmonic functions to multiple complex variables and play a key role in defining pseudoconvex domains.3)
These functions are used to study multiplier ideal sheaves, which have applications in algebraic geometry and singularity theory. 4)
The Levi problem, which characterizes domains of holomorphy, was historically solved using plurisubharmonic functions.3)
Would you like to explore specific applications or recent breakthroughs in these areas?
I can help break them down further!

参考
1)
https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/hormander_l2_estimates.pdf
Lars H¨ormander and the theory of L2 estimates for the ∂ operator Jean-Pierre Demailly

2)
https://link.springer.com/book/10.1007/978-4-431-56852-0
L² Approaches in Several Complex Variables
Towards the Oka–Cartan Theory with Precise Bounds
Authors: Takeo Ohsawa Book 2018

3)
https://en.wikipedia.org/wiki/Plurisubharmonic_function
Plurisubharmonic function

4)
https://www.math.ucdavis.edu/~hunter/m205b_18/projects/Plurisubharmonic_Functions.pdf
Plurisubharmonic Functions and Pseudoconvex Domains Thomas Jackson June 8, 2018

"Would you like to explore specific applications or recent breakthroughs in these areas?
I can help break them down further!"
とお誘いがありますが
まあ、ここらでお茶濁す　；ｐ）
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1724146576/959

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