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started a minimum at analytification, mainly interested for the moment in collecting the references now given there which discuss analytification of algebraic (etc.) stacks
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.
Thanks!
have added more of an actual definition to analytification and included some more pointers to the literature
How about this:
Let $\mathcal{C} \hookrightarrow Sch_{\mathbb{C}}$ be a subcategory of complex schemes on which analytification restricts to a fully faithful functor to complex manifolds $\mathcal{C} \to CplxMfd$. Let $F : CplxMfd \to SmthMfd$ 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 $\mathcal{C}$, hence over $CplxMfd$, and then left Kan extends that along $F$ to a presheaf on smooth manifolds.
Is it possible to choose $\mathcal{C}$ such that this left Kan extension operation exhibits analytification in that it sends the presehaf represented by some suitably nice $\mathbb{C}$-scheme $X$ to the presheaf represented by the smooth manifold underlying its analytification $X^{an}$
?
or how about this:
for $k \hookrightarrow \mathbb{C}$ a field, write
$analytify : SmthSchm_k \longrightarrow SmthMfd$for analytification $X \mapsto X(k)$. I suppose this preserves étale covers and hence produces via homotopy left Kan extension a left adjoint to pullback
$Sh_\infty(SmthSchm_k) \stackrel{L Lan_{analyt}}{\longrightarrow} Sh_{\infty}(SmthMfd) \,.$Composed with the shape modality this yields the analytic homotopy type functor
$Sh_\infty(SmthSchm_k) \stackrel{L Lan_{analyt}}{\longrightarrow} Sh_{\infty}(SmthMfd) \stackrel{\Pi}{\longrightarrow} \infty Grpd \,.$considered for instance by Dugger-Isaksen (here).
Now for
$Sh_\infty(SmthSchm_k) \stackrel{\stackrel{L Lan_{analyt}}{\longrightarrow} }{\underset{analytify^\ast}{\longleftarrow}} Sh_{\infty}(SmthMfd)$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 $Sh_\infty(SmthSchm)$ at these “smooth hypercovers pulled back under analytification”, would we still end up with a topos?
To get an essential geometric morphism, you need
$analytify: SmthSchm_k \to SmthMfd$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.
Thanks for joining in!
So we have $(L Lan_{analytify} \dashv analytify^\ast)$ 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 $L Lan_{analytify}$ 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 $\Pi$ or $L Lan_{analytify}$ as above, then we want to regard lifts through this left adjoint as “equipping a space with further geometric structure”. For $\Pi$ then lifts through it are choices of “smooth structure”, while lifts through $L Lan_{analytify}$ 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 $X \in Sh_\infty(SmthMfd)$. I suppose in a cohesive $\infty$-topos there is a canonical way of having a moduli stack of lifts of $X$ through $\Pi$, namely what in HoTT notation would be “$M(X) := \underset{\hat X \in Obj}{\sum} (\Pi(\hat X) \simeq X)$”. It would be nice if some such abstract nonsense would also produce moduli stacks for lifts through $L Lan_{analytify}$.
Why can’t you define this $M(X)$ with $analytify_!$ instead of $\Pi$?
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).
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 $analytify^*$ be fully faithful. Hm…
Or since Mike just mentioned it in another thread, what might be the localization of $Sh_\infty(SmthSchm)$ 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.)
If you fix a $\mathbb{C}$-variety $X$ and look only at the small étale site, then that is a left exact localization since $analytify$ preserves finite limits. This looks like the canonical factorization of $analytify^*$ as a conservative morphism followed by the inclusion of a sub-topos.
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