Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 15th 2014
   • (edited Jul 15th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to _[[Teichmüller theory]]_ a mini-paragraph [Complex structure on Teichmüller space](http://ncatlab.org/nlab/show/Teichmüller+theory#ComplexStructureOnTeichmüllerSpace) 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](http://ncatlab.org/nlab/show/Haefliger%20groupoid#Carchedi12). 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?

   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?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 16th 2014
   • (edited Jul 16th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexMaybe 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 16th 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIs 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](http://nforum.mathforge.org/discussion/5829/moduli-space-of-curvesof-riemann-surfaces/?Focus=48425#Comment_48425) is related.

   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.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 16th 2014
   • (edited Jul 16th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThat'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](http://nforum.mathforge.org/discussion/5829/moduli-space-of-curvesof-riemann-surfaces/?Focus=48425#Comment_48425): 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](http://nforum.mathforge.org/discussion/5829/moduli-space-of-curvesof-riemann-surfaces/?Focus=48425#Comment_48425) 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).

   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).

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 16th 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexHow 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?

   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?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 16th 2014
   • (edited Jul 16th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSorry, 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]].

   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.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMay 1st 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's * Francesco Bonsante, Andrea Seppi, _Anti-de Sitter geometry and Teichmüller theory_ ([arXiv:2004.14414](https://arxiv.org/abs/2004.14414)) <a href="https://ncatlab.org/nlab/revision/diff/Teichm%C3%BCller+theory/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/Teichm%C3%BCller+theory/12">v12</a>, <a href="https://ncatlab.org/nlab/show/Teichm%C3%BCller+theory">current</a>

   added pointer to today’s

   • Francesco Bonsante, Andrea Seppi, Anti-de Sitter geometry and Teichmüller theory (arXiv:2004.14414)

   diff, v12, current

8. • CommentRowNumber8.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 24th 2020
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexAdded details and doi link to English translation of Teichmüller's article, >_Variable Riemann surfaces_, In: Athanase Papadopoulos. Handbook of Teichmüller Theory, Volume IV, 19, European Mathematical Society, pp 787–803, 2014, IRMA Lectures in Mathematics and Theoretical Physics , doi:[10.4171/117-1/19](https://doi.org/10.4171/117-1/19). as well as the companion commentary article >* Annette A'Campo-Neuen, Norbert A'Campo, Lizhen Ji, Athanase Papadopoulos. _A commentary on Teichmüller's paper "Veränderliche Riemannsche Flächen" (Variable Riemann Surfaces)_, In: Athanase Papadopoulos. Handbook of Teichmüller Theory, Volume IV, 19, European Mathematical Society, pp.804-814, 2014, IRMA Lectures in Mathematics and Theoretical Physics, doi:[10.4171/117-1/20](https://doi.org/10.4171/117-1/20), [hal-00730238](https://hal.archives-ouvertes.fr/hal-00730238), arXiv:[1209.1882](https://arxiv.org/abs/1209.1882). <a href="https://ncatlab.org/nlab/revision/diff/Teichm%C3%BCller+theory/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/Teichm%C3%BCller+theory/13">v13</a>, <a href="https://ncatlab.org/nlab/show/Teichm%C3%BCller+theory">current</a>

   Added details and doi link to English translation of Teichmüller’s article,

   Variable Riemann surfaces, In: Athanase Papadopoulos. Handbook of Teichmüller Theory, Volume IV, 19, European Mathematical Society, pp 787–803, 2014, IRMA Lectures in Mathematics and Theoretical Physics , doi:10.4171/117-1/19.

   as well as the companion commentary article

   • Annette A’Campo-Neuen, Norbert A’Campo, Lizhen Ji, Athanase Papadopoulos. A commentary on Teichmüller’s paper “Veränderliche Riemannsche Flächen” (Variable Riemann Surfaces), In: Athanase Papadopoulos. Handbook of Teichmüller Theory, Volume IV, 19, European Mathematical Society, pp.804-814, 2014, IRMA Lectures in Mathematics and Theoretical Physics, doi:10.4171/117-1/20, hal-00730238, arXiv:1209.1882.

   diff, v13, current

9. • CommentRowNumber9.
   • CommentAuthorGuest
   • CommentTimeJan 14th 2022
   • PermaLink
   Author: Guest
   Format: TextNot sure if it deserves a separate article but this preprint about p-adic Teichmuller theory came out last year: https://arxiv.org/abs/2106.11452

   Not sure if it deserves a separate article but this preprint about p-adic Teichmuller theory came out last year: https://arxiv.org/abs/2106.11452
10. • CommentRowNumber10.
    • CommentAuthorGuest
    • CommentTimeJan 14th 2022
    • PermaLink
    Author: Guest
    Format: TextThere is also this preprint by the same author: https://arxiv.org/pdf/2111.06771.pdf

    There is also this preprint by the same author: https://arxiv.org/pdf/2111.06771.pdf
11. • CommentRowNumber11.
    • CommentAuthorGuest
    • CommentTimeJan 14th 2022
    • PermaLink
    Author: Guest
    Format: TextAnd this preprint: https://arxiv.org/pdf/2111.04890.pdf

    And this preprint: https://arxiv.org/pdf/2111.04890.pdf
12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2022
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks for collecting relevant references. Once editing functionality is back in a few days it would be great if you could add them. By the way, to make hyperlinks here on the $n$Forum, type [link text](url) and best to point not to pdf-s directly but to their abstract page: e.g. typing * [[Kirti Joshi]], *Comments on Arithmetic Teichmüller Spaces* ([arXiv:2111.06771](https://arxiv.org/abs/2111.06771)) produces: * [[Kirti Joshi]], *Comments on Arithmetic Teichmuller Spaces* ([arXiv:2111.06771](https://arxiv.org/abs/2111.06771)) (The link to the author name will work once editing is back and we create that page.)

    Thanks for collecting relevant references. Once editing functionality is back in a few days it would be great if you could add them.

    By the way, to make hyperlinks here on the $n$Forum, type

      [link text](url)
    

    and best to point not to pdf-s directly but to their abstract page: e.g. typing

      * [[Kirti Joshi]], *Comments on Arithmetic Teichmüller Spaces* ([arXiv:2111.06771](https://arxiv.org/abs/2111.06771))
    

    produces:

    • Kirti Joshi, Comments on Arithmetic Teichmuller Spaces (arXiv:2111.06771)

    (The link to the author name will work once editing is back and we create that page.)