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I have added to Teichmüller theory a mini-paragraph Complex structure on Teichmüller space with a minimum of pointers to the issue of constructing a complex structure on Teichmüller space itself.
Maybe somebody has an idea on the following: The Teichmüller orbifold itself should have a neat general abstract construction as the full subobject on the étale maps of the mapping stack formed in smooth $\infty$-groupoids/smooth $\infty$-stacks into the Haefliger stack for complex manifolds : via Carchedi 12, pages 37-38.
Might we have a refinement of this kind of construction that would produce the Teichmüller orbifold directly as on objects in $\infty$-stacks over the complex-analytic site?
Maybe the idea that one should build a moduli space of ways of lifting a smooth manifold to a complex manifold is misleading. It may be better to think of lifting all the way from bare homotopy types (i.e. $K(\pi,1)$-s in the case of Riemann surfaces) to complex manifolds.
In that case we have the following natural construction of a complex analytic moduli space:
Let $\mathbf{H}$ be the homotopy theory of complex analytic ∞-groupoids and write, as usual, $\Pi$ for its shape modality and $\sharp$ for its sharp modality.
Then given a bare homotopy type $\Pi(S) \in \mathbf{H}$, to be thought of as the homotopy type of a 2-dimensional manifold, then the natural complex analytic homotopy type of all ways of equipping this with complex analytic structure is
$\underset{\Sigma \in Type}{\sum} (\Pi (dec_{Type} (\Sigma)) \simeq \Pi(S) ) \;\;\;\in \mathbf{H} \,,$hence the homotopy pullback in the diagram
$\array{ \underset{\Sigma \in Type}{\sum} (\Pi (dec_{Type} (\Sigma)) \simeq \Pi(S) ) &\longrightarrow& &\longrightarrow& \ast \\ \downarrow && \swArrow_{\simeq} && \downarrow^{\mathrlap{\vdash \Pi(S)}} \\ Type &\stackrel{decohese_{Type}}{\longrightarrow}& \sharp Type &\stackrel{\Pi}{\longrightarrow}& \sharp Type } \,,$where $Type \in \mathbf{H}$ is the “small object classifier” for small bundles of complex analytic homotopy types.
Over $\mathbb{C}^n$ a collection classified by $\underset{\Sigma \in Type}{\sum} (\Pi (dec_{Type} (\Sigma)) \simeq \Pi(S) )$ is a complex analytic bundle $\mathbf{\Sigma}\to \mathbb{C}^n$ of complex analytic higher stacks/groupoids over $\mathbb{C}^n$ equipped for each point $z$ of the base of an equivalence $\Pi(\Sigma_z) \simeq \Pi(S)$, where these choices of equivalences are not required to be complex analytic or even continuous. They are chosen for each point independently.
That seems to make good sense, though I need to think about this more.
Of course one will want to add the condition that $\Sigma_z$ is indeed a complex manifold. This can be done by invoking also the infinitesimal shape modality $\Pi_{inf}$ in terms of which one may express that $\Sigma_z$ is an étale infinity-groupoid over $\mathbb{C}$ and so the right complex analytic moduli space would be
$\underset{\Sigma \in Type}{\sum} \left( \left(\Pi (dec_{Type} (\Sigma)) \simeq \Pi(S) \right) and \left( isEtale\left(\Sigma\right) \right) and \left( is0Truncated\left(\Sigma\right) \right) \right) \in \mathbf{H} \,,$which is given by a more complicated but straightforward homotopy pullback.
Is there a sense in which you can use the space (preorder/category?) of kinds of cohesion, and have lifts from bare homotopy types factoring through intermediate cohesions? Maybe what’s happening here is related.
That’s exactly what I am banging my head against.
The above construction universally gives the moduli object for “cohesive structures” on any bare homotopy type in any context of cohesion.
I kept thinking that to speak of complex analytic moduli stacks of complex structures properly in this way I ought to factor the shape modality through “intermediate cohesion”
$\Pi_{complexAnalytic} \colon ComplexAnalytic \infty Grpd \stackrel{ForgetComplexStructure}{\longrightarrow} Smooth \infty Grpd \stackrel{\Pi_{smooth}}{\longrightarrow} \infty Grpd$and consider moduli stacks of lifts through the left arrow here. But I am not making progress with realizing this idea. So therefore here the change of perspective: maybe it is actually more natural to lift through all of $\Pi_{complexAnalytic}$.
Compared to the suggestion explored in parallel:
here the advantage is that the moduli stack is itself manifestly a complex analytic object, which is what is needed to proceed with the story of geoemtric quantization (because next we want to build the Hitchin connection on this). The disadvantage currently is that I have only a rough idea of how this construction relates in fine detail to the traditional moduli stacks.
That other construction has the advantage that it is clear that it gives the traditional moduli stack, the disadvantage is that it doesn’t automatically yield a complex structure on it and that it only works for complex structures on 2-dimensional manifolds (which is good for a start, but the general abstract theory should give the general case).
How would one gather all cohesions together? Presumably there’s a $(\infty, 2)$-category of $(\infty, 1)$-toposes and geometric morphisms. Some of these morphisms exhibit the domain topos as cohesive over the codomain topos. Composites of cohesive morphisms are cohesive.
Is there anything to say in reverse about when a cohesive morphism factors as the composite of two geometric morphisms, and the later are cohesive? E.g., what’s known about $ForgetComplexStructure$?
Can a general geometric morphism be factorised into a largest cohesive component, and maybe an infinitesimal component?
Sorry, one addendum to #4 first, maybe mainly to remind myself:
one reason why one should indeed want to have a concept of moduli space of “complex structures” which contains higher geometric complex analytic structures more general than just complex manifolds is that by the story of equivariant elliptic cohomology, we indeed want to be able to lift for instance a family of tori not just to an family of elliptic curve, but possibly to derived elliptic curves.
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