Talk:Function of several complex variables: Difference between revisions

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::Withdraw from merge the ___domain of holomorphy. I might suggest merging conditions that are equivalent to the ___domain of holomorphy, but I thought that should be considered on the ___domain of holomorphy. Writing the ___domain of holomorphy on this page has no effect on withdrawal. thanks!--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 13:53, 26 December 2020 (UTC)

== A memo about the structure of the section ==

== Unclear sentence ==

The [https://math.berkeley.edu/courses/choosing/course-descriptions#math212 course catalog]. --[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 03:19, 21 January 2022 (UTC)

In the section '''Radius of convergence of power series''', this sentence:

== Is this right? ==

"''In the power series <math>\sum_{k_1,\dots,k_n=0}^\infty c_{k_1,\dots,k_n}(z_1-a_1)^{k_1}\cdots(z_n-a_n)^{k_n}\ </math>, it is possible to define ''n'' combination of <math>r_\nu</math><ref group=note>This combination may not be unique.</ref>

The section '''Stein manifold''' begins as follows:

:"''<math>\begin{cases}

\text{Absolutely converge on}\ \{ z=(z_1, z_2, \dots, z_n) \in {\Complex}^n \mid | z_\nu - a_\nu | < r_\nu, \text{ for all } \nu = 1,\dots,n \}\\

\text{Does not absolutely converge on}\ \{ z=(z_1, z_2, \dots, z_n) \in {\Complex}^n \mid | z_\nu - a_\nu | > r_\nu, \text{ for all } \nu = 1,\dots,n \}

\end{cases}</math>

''"

"''Since a non-compact (open) Riemann surface always has a non-constant single-valued holomorphic function, and satisfies the [[second axiom of countability]], the open Riemann surface can be thought of ''1''-dimensional complex manifold to have a holomorphic embedding into a complex plane <math>\Complex</math>.''"

is very poorly worded and makes no sense in normal English. I hope someone knowledgeable about this subject who is also familiar with English can rewrite this so that it is readable and accurate.

But wait. A holomorphic embedding is certainly also a topological embedding. But most non-compact surfaces do not have any topological embeddings into the complex plane.

I'm '''guessing''' that what is '''meant''' is this:

For instance, the torus with one point removed (the "punctured torus") has no topological embedding into the plane.

... it is possible to define ''n'' positive real numbers <math>r_\nu</math> such that the power series

Question: Is it possible that the phrase "holomorphic embedding" should be replaced with '''holomorphic immersion'''? Or just '''holomorphic mapping'''? [[Special:Contributions/2601:200:C000:1A0:9AE:98DB:C7E0:3910|2601:200:C000:1A0:9AE:98DB:C7E0:3910]] ([[User talk:2601:200:C000:1A0:9AE:98DB:C7E0:3910|talk]]) 17:57, 28 December 2022 (UTC)

<math>\begin{cases}

\text{is absolutely convergent on}\ \{ z=(z_1, z_2, \dots, z_n) \in {\Complex}^n \mid | z_\nu - a_\nu | < r_\nu, \text{ for all } \nu = 1,\dots,n \}\\

\text{and is not absolutely convergent on}\ \{ z=(z_1, z_2, \dots, z_n) \in {\Complex}^n \mid | z_\nu - a_\nu | > r_\nu, \text{ for all } \nu = 1,\dots,n \}

\end{cases}</math>

:Apparently Gunning and Narasimhan proved that every non-compact Riemann surface '''does''' in fact '''immerse''' in the complex plane.

(Is that right?) This would read better if we could get rid of the "cases" curly bracket and just use normal English here.[[Special:Contributions/128.120.234.237|128.120.234.237]] ([[User talk:128.120.234.237|talk]]) 06:09, 29 December 2020 (UTC)

:(R. C. Gunning and Raghavan Narasimhan, Immersion of open Riemann surfaces, Math. Annalen 174 (1967), 103–108.)

:Thank you for the advice. I tried to fix it.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 02:04, 31 December 2020 (UTC)

:Therefore I will correct the text to reflect this fact. [[Special:Contributions/2601:200:C000:1A0:9AE:98DB:C7E0:3910|2601:200:C000:1A0:9AE:98DB:C7E0:3910]] ([[User talk:2601:200:C000:1A0:9AE:98DB:C7E0:3910|talk]]) 18:58, 28 December 2022 (UTC)

== Please say what you mean by superscript n ==

== Proposed merge of [[Several complex variables]] into [[Complex analysis]] ==

The definition of a '''coherent sheaf''' is given as follows:

Properly belongs as one article, functions of one complex variable are a special case of functions of several complex variables and both should be discussed in the same article with complex analysis. Furthermore, [[complex variables]] is an unacceptable and confusing DAB page which should also redirect here. If those articles need a DAB, it can be accomplished with a hat note and a link to a new [[complex variables (disambiguation)]]. Size is not an issue. [[User:Footlessmouse|Footlessmouse]] ([[User talk:Footlessmouse|talk]]) 07:03, 31 October 2020 (UTC)

:{{Ping|Footlessmouse}} Nice to me you. I'm in support of inserting Several complex variables into the complex analysis page, but please wait a bit for page consolidation.In complex analysis, holomorphic is a characteristic property, which is different from several real variables.In other words, we need to write about the differences from several real variables.In order to clarify the difference from complex analysis, it may be possible to integrate several pages included in [[:Category:Several complex variables]], or to have duplicate contents.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 13:11, 1 November 2020 (UTC)

"''A '''coherent sheaf''' on a [[ringed space]] <math>(X, \mathcal O_X)</math> is a sheaf <math>\mathcal F</math> satisfying the following two properties:

<ol type="1">

<li> <math>\mathcal F</math> is of ''finite type'' over <math>\mathcal O_X</math>, that is, every point in <math>X</math> has an [[open neighborhood]] <math>U</math> in <math>X</math> such that there is a surjective morphism <math>\mathcal{O}_X^n|_{U} \to \mathcal{F}|_{U} </math> for some natural number <math>n</math>;</li>

<li> for arbitrary open set <math>U\subseteq X</math>, arbitrary natural number <math>n</math>, and arbitrary morphism <math>\varphi: \mathcal{O}_X^n|_{U} \to \mathcal{F}|_{U} </math> of <math>\mathcal O_X</math>-modules, the kernel of <math>\varphi</math> is of finite type.''"</li></ol>

But what does the superscript n mean in the symbol "<math>\mathcal{O}_X^n|_{U}</math>"?

The article does not say, and nothing in the linked article [[ringed space]] uses this notation.

I can guess two distinct possibilities for what "<math>\mathcal{O}_X^n|_{U}</math>" means.

I hope someone knowledgeable about this subject can explain the notation and avoid having thousands of future readers of Wikipedia also have to guess what it means. [[Special:Contributions/2601:200:C082:2EA0:E1A8:CCAE:61A1:827C|2601:200:C082:2EA0:E1A8:CCAE:61A1:827C]] ([[User talk:2601:200:C082:2EA0:E1A8:CCAE:61A1:827C|talk]]) 02:25, 9 February 2023 (UTC)

== Bad English and grammar ==

Someone please proofread this article. [[Special:Contributions/2A02:6B6F:E98D:400:9BF:E37:5EAC:CB2C|2A02:6B6F:E98D:400:9BF:E37:5EAC:CB2C]] ([[User talk:2A02:6B6F:E98D:400:9BF:E37:5EAC:CB2C|talk]]) 22:16, 6 February 2025 (UTC)