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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 3rd 2012
• (edited Oct 3rd 2012)

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?)

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeOct 3rd 2012

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 3rd 2012

Thanks. I got myself mixed up here.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 3rd 2012
• (edited Oct 3rd 2012)

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.)

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 3rd 2012
• (edited Oct 3rd 2012)

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).