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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2014

    have added a tad more content to Stein manifold and cross-linked a bit more

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2014

    Added the statement of Cartan’s theorem B and added to the Idea-section a remark that therefore Stein manifolds play the role in complex geometry of Cartesian spaces in smooth manifold theory, for purposes of abelian (Cech)-sheaf cohomology.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeMay 28th 2014
    • (edited May 28th 2014)

    Affine schemes of algebraic geometry.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2014

    Or close at least. I have added something here.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeMay 28th 2014
    • (edited May 28th 2014)

    I am not saying that (about the analytification), but precisely what you say above: affine are cohomologically trivial in the sense as proved in chapter 3 of Hartshorne’s book and this is a usually given statement when algebraic geometers look at analytic spaces. Many other deep properties are also analogous.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2019

    somebody just alerted me:

    this page here has been and still is referring to

    • Zachary Maddock, Dolbeault cohomology (pdf)

    for proof of some of its statements (existence of “good” Stein covers). However, the link to that pdf

      http://www.math.columbia.edu/~maddockz/notes/dolbeault.pdf
    

    no longer works, and Google seems to see no other trace of it either.

    (?)

    diff, v12, current

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeJan 17th 2019

    Zachary Maddock is on Linkedin so you might be able to contact him and put a copy of the document on the Lab if it seems worth it.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2019
    • (edited Jan 17th 2019)

    For what it’s worth, I have found and uploaded an old copy of the file (here)

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2021

    adjusted the wording of the example of open Riemann surfaces (here) and added pointer to the classical reference:

    diff, v15, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2021

    adjusted the wording of the example of open Riemann surfaces (here) and added pointer to the classical reference:

    diff, v15, current

    • CommentRowNumber11.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 15th 2021

    Just a sanity check: are disjoint unions of Stein manifolds again Stein manifolds? Including countably infinite disjoint unions? The answer seems to be yes, simply by staring at the definition, but perhaps I am missing something?

    The reason I am asking about this is that Lárusson in his paper https://arxiv.org/abs/math/0101103v3, at the beginning of Section 5 asks “It is natural to ask whether a finite homotopy sheaf on S satisfies descent.”

    Here S is the site of Stein manifolds and holomorphic maps, and a finite homotopy sheaf is a presheaf that satisfies the homotopy descent condition with respect to finite covers.

    It would seem to me that the answer to Lárusson’s question should be negative as stated, e.g., because we can take the presheaf P that assigns to a Stein manifold M the abelian group of holomorphic functions on M that vanish on all but finitely many connected components of M.

    Then P satisfies descent with respect to finite covers, i.e., is a finite homotopy sheaf in Lárusson’s terminology. However, it does not satisfy descent with respect to (say) countable disjoint covers.

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 15th 2021
    • (edited Sep 15th 2021)

    I don’t see why not. Use a characterisation of n-dimensional Stein manifolds that involves proper embeddings in C^m for some m. In fact there is a uniform bound on the needed m in terms of n, so for countable disjoint unions there is a common m. Then properly embed embed the countable copies of C^m in C^m+1 as parallel affine hypersurfaces.

    Or else I can ask Finnur today, as his office is next to mine, if that’s not convincing enough. Why do you ask?

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 15th 2021

    Re #12: Basically, I am wondering why the example in the last two paragraphs of #11 is not a (trivial) counterexample to what Lárusson is suggesting in his paper https://arxiv.org/abs/math/0101103v3, at the beginning of Section 5, where he asks “It is natural to ask whether a finite homotopy sheaf on S satisfies descent.”

    • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 16th 2021

    I should add that one doesn’t need all the dimensions of the disjoint pieces to be equal, but you do need a bound on the dimensions. I spoke with Finnur today and he seemed to indicate that if the dimensions of the pieces are unbounded, then the disjoint union isn’t Stein. Maybe this is it.

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 16th 2021
    • (edited Sep 16th 2021)

    if the dimensions of the pieces are unbounded, then the disjoint union isn’t Stein

    That’s not the example I had in mind, though.

    Disjoint unions of points (finite or countable) are 0-dimensional Stein manifolds.

    Finitely supported complex-valued functions on such manifolds form a presheaf that satisfies finite homotopy descent, but does not satisfy descent with respect to covers of arbitrary cardinality. This would seem to constitute a simple counterexample to the question stated in Section 5 of his paper, but probably I am just misreading something.

    • CommentRowNumber16.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 17th 2021

    Oh, sorry, I was kinda ignoring the homotopy sheaf material :-)

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeSep 17th 2021

    Dmitri’s point in #11 (repeated in #13) isn’t even to do with homotopy, it’s a rather basic observation about sites and sheaves.

    Dmitri brought up a possible subtlety in the definition of Stein manifolds only as an attempt to find a technical loophole clause that would explain why Lárusson doesn’t consider what seems to be the immediate conclusion.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeFeb 21st 2022
    • (edited Feb 21st 2022)

    For when the editing functionality is back, to add the example that complements of hyperplane arrangements in Stein manifolds are again Stein (e.g. the configuration spaces of points in a punctured Riemann surface are Stein).

    This is:

    • Ferdinand Docquier, Hans Grauert, Satz 1 in: Levisches Problem und Rungescher Satz fuer Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann. 140, 94–123 (1960) (doi:10.1007/BF01360084)

    recalled, e.g., in

    • Graham C. Denham, Alexander I. Suciu, Rem. 2.8 in: Local systems on complements of arrangements of smooth, complex algebraic hypersurfaces, Forum of Mathematics, Sigma 6 (2018) (arXiv:1706.00956, doi:10.1017/fms.2018.5)
    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeFeb 21st 2022

    Also to add pointers for this example: Universal covers of Stein manifolds are again Stein:

    • Karl Stein, Überlagerungen holomorph-vollständiger komplexer Räume, Archiv der Mathematik 7 354–361 (1956) (doi:10.1007/BF01900686)

    recalled, e.g., in:

    • Lei Ni, Luen-Fai Tam, p. 39 of: Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature (arXiv:math/0304096)
    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeFeb 21st 2022

    This monograph cares to spell out the proof that homolorphic de Rham cohomology of Stein manifolds computes ordinary cohomology:

    • Lars Hörmander, Chapter VII of: An introduction to complex analysis in several variables, North-Holland Mathematical Library 7 (ISBN:9780444884466)

    review and generalization in:

    • Xiaojun Huang, Hing Sun Luk, Stephen S.-T. Yau, Punctured local holomorphic de Rham cohomology, J. Math. Soc. Japan 55(3): 633-640 (2003) (doi:10.2969/jmsj/1191418993)
    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeFeb 21st 2022

    Also good to mention: The intersection of any finite number of open Stein subsets inside some complex manifold (which itself need not be Stein) is again Stein:

    From this and the usual facts about the holomorphic de Rham complex (e.g. as usefully reviewed by M. Stevenson here: pdf), it is immediate that the the holomorphic De Rham cmplex satisfies descent when regarded as an \infty-stack over the site of Stein manifolds (i.e. any shift of it, seen under the Dold-Kan construction).

    I wonder if any author already says this, in a citable way?

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeFeb 22nd 2022

    Is there a citable textbook that would make explicit the fact that over Stein domains the global sections of a twisted holomorphic de Rham complex computes the cohomology in the given local system?

    several authors who discuss related issues don’t mention this particular combination of the statement (i.e. holomorphic + twists + Stein).

    For instance Voisin’s book (translated by Schneps), where this is part of Cor. 5.4 (here) the issue seems to be a little lost in translation (“varieties” on p. 131 remains ambiguous).

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeOct 22nd 2023

    made more explicit (here) that on Stein manifolds holomorphic de Rham cohomology coincides with ordinary de Rham cohomology.

    I gather that this statement may originate with

    though I haven’t actually seen this article, since the DOI (which I get from SemanticScholar here) seems to be broken.

    But the same statement appears then also in

    which I have added, too.

    diff, v18, current