Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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 2\mathbb{C}^2 defined by the locus of xyx 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).

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)