Talk:Function of several complex variables: Difference between revisions

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== I'm considering merge the ___domain of holomorphy into Several complex variables, but the ___domain of holomorphy (theory) may be the title of this page ==

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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)

== Is this right? ==

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

"''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>.''"

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.

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

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)

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

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

: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 ==

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

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

"''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:

"''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>

<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>"?

:"''<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>

''"

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

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.

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

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

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)

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

== Bad English and grammar ==

<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>

(IsSomeone thatplease right?)proofread Thisthis wouldarticle. read better if we could get rid of the "cases" curly bracket and just use normal English here.[[Special:Contributions/128.120.234.2372A02:6B6F:E98D:400:9BF:E37:5EAC:CB2C|128.120.234.2372A02:6B6F:E98D:400:9BF:E37:5EAC:CB2C]] ([[User talk:128.120.234.2372A02:6B6F:E98D:400:9BF:E37:5EAC:CB2C|talk]]) 0622:0916, 296 DecemberFebruary 20202025 (UTC)

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