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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 3rd 2012
   • (edited Oct 3rd 2012)
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   Author: Urs
   Format: MarkdownItexstarted _[[complex analytic space]]_ but I really have some basic questions on this topic, at the time of posting this I am really a layperson: is it right that every complex analytic space is locally isomorphic to a [[polydisk]]? So then they are all locally contractible as topological spaces. Are they also locally contractible as seen by étale homotopy? (So: do they admit covers by polydsisks such that if in the Cech-nerves of these covers all disks are sent to points, the resulting simplicial set is contractible?)

   started complex analytic space

   but I really have some basic questions on this topic, at the time of posting this I am really a layperson:

   is it right that every complex analytic space is locally isomorphic to a polydisk?

   So then they are all locally contractible as topological spaces. Are they also locally contractible as seen by étale homotopy? (So: do they admit covers by polydsisks such that if in the Cech-nerves of these covers all disks are sent to points, the resulting simplicial set is contractible?)

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 3rd 2012
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   Author: Todd_Trimble
   Format: MarkdownItexI'm a layperson too, but based on the definitions I've just read, it seems the answer is no, not locally isomorphic to a polydisk. Those that are are called complex manifolds. A complex analytic space is locally modeled on analytic varieties. So any algebraic variety over $\mathbb{C}$ (viewed as a scheme) would be an analytic variety. For example, the subvariety of $\mathbb{C}^2$ defined by the locus of $x y$, the union of two intersecting lines. Not locally a polydisk at the origin.

   I’m a layperson too, but based on the definitions I’ve just read, it seems the answer is no, not locally isomorphic to a polydisk. Those that are are called complex manifolds.

   A complex analytic space is locally modeled on analytic varieties. So any algebraic variety over $\mathbb{C}$ (viewed as a scheme) would be an analytic variety. For example, the subvariety of $\mathbb{C}^2$ defined by the locus of $x y$, the union of two intersecting lines. Not locally a polydisk at the origin.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 3rd 2012
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   Author: Urs
   Format: MarkdownItexThanks. I got myself mixed up here.

   Thanks. I got myself mixed up here.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 3rd 2012
   • (edited Oct 3rd 2012)
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   Author: Urs
   Format: MarkdownItexAh, now I see what happened to me: So a _smooth_ complex analytic space is locally isomorphic to a polydisk. For instance [p. 2 here](https://www.wisdom.weizmann.ac.il/~vova/Inven_1999_137_contra.pdf#page=2). (This is probably dead basic, but I feel I am lacking some basic experience here.)

   Ah, now I see what happened to me:

   So a smooth complex analytic space is locally isomorphic to a polydisk. For instance p. 2 here.

   (This is probably dead basic, but I feel I am lacking some basic experience here.)

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 3rd 2012
   • (edited Oct 3rd 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSo my question then is (and should have been before): are _smooth_ complex analytic spaces locally étale-contractible? (as before, this is probably a most basic question, but anyway).

   So my question then is (and should have been before):

   are smooth complex analytic spaces locally étale-contractible?

   (as before, this is probably a most basic question, but anyway).