Talk:Function of several complex variables: Difference between revisions

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

"''Since a non-compact (open) Riemann surface<ref name =Weyl1913>{{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=The concept of a Riemann surface | orig-year=1913 | url=https://archive.org/details/dieideederrieman00weyluoft | publisher=[[Dover Publications]] | ___location=New York | edition=3rd | isbn=978-0-486-47004-7 | year=2009 | mr=0069903}}</ref> always has a non-constant single-valued holomorphic function,<ref name =Behnke–Stein1948>{{Citation | author = Heinrich Behnke & Karl Stein | title=Entwicklung analytischer Funktionen auf Riemannschen Flächen | year=1948 | journal=Mathematische Annalen| volume=120 | pages=430–461|doi=10.1007/BF01447838|zbl =0038.23502 | s2cid=122535410 }}</ref> 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>.''" [[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]]) 17:46, 28 December 2022 (UTC)

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:

"''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)