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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 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).