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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2014

    am starting complex analytic infinity-groupoid (in line with “smooth infinity-groupoid” etc.) and higher complex analytic geometry. Currently there is mainly a pointer to Larusson. To be expanded.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2014

    added the actual statement of Larusson’s \infty-presheaf-formulation of the Oka condition here (or at least the first part of it)

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 30th 2014

    Cohesive over Grpd\infty Grpd?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2014
    • (edited May 30th 2014)

    No, that’s the thing. Instead we should take this to be the base topos and have cohesion over it, as at smooth E-infinity-groupoid. I would tend to say “smooth analytic \infty-groupoid” for objects in

    Sh (CartSp,Sh (SteinSp))Sh (SteinSp) Sh_\infty(CartSp, Sh_\infty(SteinSp)) \to Sh_\infty(SteinSp)

    but maybe that’s too ambiguous and I should instead say “analytically-fibered smooth infinity-groupoid” or something like this. Not sure yet.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2014
    • (edited Jun 7th 2014)

    I think I have been really confused about this. It is cohesive, isn’t it.

    Take the site of complex polydiscs with its canonical coverage. That’s a dense subsite of that of complex manifolds and of that of Stein spaces, so the hypercomplete \infty-toposes will all agree. Now since the coverage by polydiscs is completed by the standard Grothendieck topology, a simplicial presheaf satisfies descent with respect to hypercovers in the complex analytic site precisely if it does so already for hypercovers which are degreewise coproducts of polydiscs. (Unless I am mixed up.) The would-be Π\Pi-functor sends such to the simplicial set obtained by replacing each polydisc by a point, hence to the “complex analytic étale homotopy type” of the polydisc. But that is contractible (because forgetting the complex structure it is in particular the homotopy type of the underlying manifold, hypercovered by open balls). So from this point on the argument for cohesion proceeds as that for smooth manifolds.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 10th 2014

    I have promoted the argument that AnalyticGrpd\mathbb{C}Analytic \infty Grpd is cohesive to a numbered proposition (here) and added a pointer to the Examples section in the entry on cohesive \infty-toposes.

    One naturally feels a certain urge to use the forgetful functor CplxMfdSmthMfdCplxMfd \to SmthMfd to induce some geoemtric morphism between AnalyticGrpd\mathbb{C}Analytic \infty Grpd and SmoothGrpdSmooth\infty Grpd, but I am not sure if it works…

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 10th 2014
    • (edited Jun 10th 2014)

    Now I see that just this statement is also Hopkins-Quick 12, lemma 2.3 + prop. 2.4 + lemma 2.5 + prop. 2.6. I have added corresponding pointers in the entry.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 10th 2014

    Had many of the consequences of AnalyticGrpd\mathbb{C}Analytic \infty Grpd being cohesive already been observed? If you find so many of the indications of cohesive (infinity,1)-topos – structures in a candidate for cohesion, can this ever tell you it is cohesive?

    • CommentRowNumber9.
    • CommentAuthorColin Tan
    • CommentTimeJul 16th 2014
    In my apprenticeship with my PhD supervisor, it seems that complex manifolds are actually smooth manifolds equipped with a holomorphic structure. It's rare to be able to stay in the holomorphic category. Usually one is given a smooth map and has to verify partial differential equations to know that this map is holomorphic. In other words, for actual complex geometers on the ground, it would be very helpful to actual practice to be able to speak of both complex manifolds and smooth manifolds in a single topos. For instance, complex geometers use hermitian metrics and plurisubharmonic functions, which are not holomorphic.
    • CommentRowNumber10.
    • CommentAuthorColin Tan
    • CommentTimeJul 16th 2014
    As a concrete example of the need to speak of complex manifolds and smooth manifolds in the same topos, the traditional articulation of the Cauchy integral formula requires to be able to speak of real curves in the complex affine line.
    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 16th 2014

    So I was trying to understand the following, but so far with little success:

    the forgetful functor

    ComplexManifoldsSmoothManifolds ComplexManifolds \longrightarrow SmoothManifolds

    induces an adjunction (not a geometric morphism, though) between the toposes

    ComplexAnalyticGrpdSmoothGrpd ComplexAnalytic \infty Grpd \longrightarrow Smooth \infty Grpd

    in order to “include” smooth geoemtry into complex geometry, one might look at the factorization of this adjunction through its induced reflection, as discussed at idempotent monad – The associated idempotent monad .

    This gives a reflective subcategory

    SomethingComplexAnalyticGrpd Something \hookrightarrow ComplexAnalytic \infty Grpd

    being the localization of complex analytic \infty-groupoids at those maps which become equivalences when forgetting the complex analytic structure and remembering just the smooth structure.

    Is there anything useful to say about this SomethingSomething? I am not sure. Also, I doubt that it is a good thing to consider.

    One should probably try something else. Not sure yet what.

    • CommentRowNumber12.
    • CommentAuthorZhen Lin
    • CommentTimeJul 16th 2014

    I don’t see why you wouldn’t get a geometric morphism. The “inclusion” of complex manifolds into smooth manifolds is cover-reflecting (i.e. has the cover lifting property in the sense of Mac Lane and Moerdijk) so you should get a covariantly-induced geometric morphism. This is certainly true for 1-toposes.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJul 16th 2014

    Thanks for mentioning this!

    (I have now created covering lifting property, which used to be missing on the nLab, and pointed to it from morphism of sites).

    So in the case at hand we have an essential geometric morphism, since the functor also preserves covers.

    • CommentRowNumber14.
    • CommentAuthorColin Tan
    • CommentTimeJul 17th 2014

    For certain complex analytic spaces, the action of the Galois group of the complex field over the real field acts by conjugation on each plot can be patched up to a conjugation involution on the said complex analytic space. Examples are complex affine spaces, complex projective spaces, some complex tori and most (all?) complex algebraic groups.

    In this way, real spaces provide a common context for complex analytic spaces equipped with a conjugation involution and smooth spaces equipped with the trivial involution.

    There is something particularly interesting about Something{\mathrm{Something}}. Namely, I wish to look at the complex affine line {\mathbb{C}}. The conjugation involution of {\mathbb{C}} is not holomorphic, but is a smooth equivalence (diffeomorphism). More explicitly, for each complex analytic space XX, there is its conjugate complex analytic space X¯\bar{X}, of which it is anti-biholomorphic to. In particular, there are two distinct (i.e. not biholomorphic) complex analytic spaces {\mathbb{C}} and ¯\bar{{\mathbb{C}}}. However, these two complex analytic spaces are indistinguishable in Something{\mathrm{Something}}. One reason to “include” smooth spaces is in other to define, say (1,1)(1,1)-forms. Is there already an internal definition of (1,1)(1,1)-forms and/or anti-holomorphic derivatives?

    • CommentRowNumber15.
    • CommentAuthorColin Tan
    • CommentTimeOct 2nd 2015
    Is there currently a page or terminology to refer to a sheaf of sets over the site of Stein manifolds? Just 1-geometry, not higher geometry.
    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2015

    Is there currently a page or terminology to refer to a sheaf of sets over the site of Stein manifolds? Just 1-geometry, not higher geometry.

    I don’t think so. Please create it!

    • CommentRowNumber17.
    • CommentAuthorColin Tan
    • CommentTimeOct 4th 2015
    • (edited Oct 4th 2015)
    Would every complex analytic space be a sheaf of sets over the Stein site?
    • CommentRowNumber18.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 5th 2015

    @Colin I don’t see why not…

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