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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 22nd 2014

    started a minimum at analytification, mainly interested for the moment in collecting the references now given there which discuss analytification of algebraic (etc.) stacks

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeMay 22nd 2014
    • (edited May 22nd 2014)

    The account of analytification and GAGA (listed in books in algebraic geometry) which is accessible to undergraduates, fully accurate and modern is in Neeman’s book! Other references are less closed and also less accessible to non-experts.

    • Amnon Neeman, Algebraic and analytic geometry, London Math. Soc. Lec. Note Series 345
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 22nd 2014

    Thanks!

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2014

    have added more of an actual definition to analytification and included some more pointers to the literature

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2014
    • (edited May 23rd 2014)

    How about this:

    Let 𝒞Sch \mathcal{C} \hookrightarrow Sch_{\mathbb{C}} be a subcategory of complex schemes on which analytification restricts to a fully faithful functor to complex manifolds 𝒞CplxMfd\mathcal{C} \to CplxMfd. Let F:CplxMfdSmthMfdF : CplxMfd \to SmthMfd be the forgetful functor to smooth manifolds. Now consider the operation that takes a suitably well behaved \mathbb{C}-scheme, first regards it as a presheaf over 𝒞\mathcal{C}, hence over CplxMfdCplxMfd, and then left Kan extends that along FF to a presheaf on smooth manifolds.

    Is it possible to choose 𝒞\mathcal{C} such that this left Kan extension operation exhibits analytification in that it sends the presehaf represented by some suitably nice \mathbb{C}-scheme XX to the presheaf represented by the smooth manifold underlying its analytification X anX^{an}

    ?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2014
    • (edited Jun 21st 2014)

    or how about this:

    for kk \hookrightarrow \mathbb{C} a field, write

    analytify:SmthSchm kSmthMfd analytify : SmthSchm_k \longrightarrow SmthMfd

    for analytification XX(k)X \mapsto X(k). I suppose this preserves étale covers and hence produces via homotopy left Kan extension a left adjoint to pullback

    Sh (SmthSchm k)LLan analytSh (SmthMfd). Sh_\infty(SmthSchm_k) \stackrel{L Lan_{analyt}}{\longrightarrow} Sh_{\infty}(SmthMfd) \,.

    Composed with the shape modality this yields the analytic homotopy type functor

    Sh (SmthSchm k)LLan analytSh (SmthMfd)ΠGrpd. Sh_\infty(SmthSchm_k) \stackrel{L Lan_{analyt}}{\longrightarrow} Sh_{\infty}(SmthMfd) \stackrel{\Pi}{\longrightarrow} \infty Grpd \,.

    considered for instance by Dugger-Isaksen (here).

    Now for

    Sh (SmthSchm k)analytify *LLan analytSh (SmthMfd) Sh_\infty(SmthSchm_k) \stackrel{\stackrel{L Lan_{analyt}}{\longrightarrow} }{\underset{analytify^\ast}{\longleftarrow}} Sh_{\infty}(SmthMfd)

    to have a further right adjoint and hence to qualify as an (essential) geometric moprhism, by adjunction it would have to be true that given a smooth hypercover and regarding it as a simplicial presheaf on schemes by probing it on analytifications of test schemes, then it remains a hypercover.

    How unlikely is that? Or else, if we forced this by further localizing Sh (SmthSchm)Sh_\infty(SmthSchm) at these “smooth hypercovers pulled back under analytification”, would we still end up with a topos?

    • CommentRowNumber7.
    • CommentAuthorMarc Hoyois
    • CommentTimeJun 23rd 2014
    • (edited Jun 23rd 2014)

    To get an essential geometric morphism, you need

    analytify:SmthSchm kSmthMfd analytify: SmthSchm_k \to SmthMfd

    to be both continuous (in the ∞-sense) and cocontinuous. It seems to me that only the discrete topology satisfies the latter requirement, since there are analytic covering sieves that contain no algebraic maps. The discrete topology fails the first requirement, of course.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2014

    Thanks for joining in!

    So we have (LLan analytifyanalytify *)(L Lan_{analytify} \dashv analytify^\ast) but not a priori a geometric morphism, since not in general a further right adjoint. Nevertheless, it looks like a good idea to think of LLan analytifyL Lan_{analytify} as akin to the extra left adjoint that produces homotopy types. So what might be a good set up in which this lives most generally?

    Here I am after something like this: given a left adjoint of the sort such as Π\Pi or LLan analytifyL Lan_{analytify} as above, then we want to regard lifts through this left adjoint as “equipping a space with further geometric structure”. For Π\Pi then lifts through it are choices of “smooth structure”, while lifts through LLan analytifyL Lan_{analytify} are like choices of complex structure, “of polarization” etc.

    I want to find a neat conceptual way to speak abstractly about the moduli stack of complex/polarization structures on a given XSh (SmthMfd)X \in Sh_\infty(SmthMfd). I suppose in a cohesive \infty-topos there is a canonical way of having a moduli stack of lifts of XX through Π\Pi, namely what in HoTT notation would be “M(X):=X^Obj(Π(X^)X)M(X) := \underset{\hat X \in Obj}{\sum} (\Pi(\hat X) \simeq X)”. It would be nice if some such abstract nonsense would also produce moduli stacks for lifts through LLan analytifyL Lan_{analytify}.

    • CommentRowNumber9.
    • CommentAuthorMarc Hoyois
    • CommentTimeJun 23rd 2014

    Why can’t you define this M(X)M(X) with analytify !analytify_! instead of Π\Pi?

    I think one of the key issues is that there are not enough topological or analytic coverings that are algebraically defined. If we believe that there is a sensible way to assigne an ∞Grpd to a scheme, in such a way that for a smooth complex variety you get the homotopy type of the underlying manifold, then we have no choice but to enlarge the category of schemes so that we have more coverings.

    Joseph Ayoub was recently giving a series of talks in Essen about the foliated topology, a topology on the category of foliated schemes (in char 0) which is in some sense finer than the étale topology. For instance, one can realize the universal cover of ×\mathbb {C}^\times as a foliated scheme. However, the theory turns out to be quite hairy (and although this point was mentioned in the introduction, I think the real motivation behind it is somewhat different).

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2014

    Thanks. It’ good to just talk about it, helps me organize my thoughts.

    So a passage to a finer topology I was vaguely wondering about in the last line of #6.

    Let’s see, what we would actually need to make that moduli-stack formula work is that analytify *analytify^* be fully faithful. Hm…

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2014

    Or since Mike just mentioned it in another thread, what might be the localization of Sh (SmthSchm)Sh_\infty(SmthSchm) at those morphisms which become equivalences under analytification?

    (Am just firing off this question in case anyway happens to know right away or else to come back to it later. Will have to disappear now to do something else for the moment.)

    • CommentRowNumber12.
    • CommentAuthorMarc Hoyois
    • CommentTimeJun 24th 2014

    If you fix a \mathbb{C}-variety XX and look only at the small étale site, then that is a left exact localization since analytifyanalytify preserves finite limits. This looks like the canonical factorization of analytify *analytify^* as a conservative morphism followed by the inclusion of a sub-topos.

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