The Teichmüller and Riemann moduli stacks

Meersseman, Laurent

Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 879-945.

The aim of this paper is to study the structure of the Teichmüller and Riemann moduli spaces, viewed as stacks over the category of complex analytic spaces, for higher-dimensional manifolds. We show that both stacks are analytic in the sense that they are isomorphic to the stackification of a smooth analytic groupoid. We then show how to construct explicitly such an atlas as a sort of generalized holonomy groupoid. This is achieved under the sole condition that the dimension of the automorphism group of each structure is bounded by a fixed integer. All this can be seen as an answer to Question 1.8 of [48].

Le but de cet article est d’étudier la structure des espaces de modules de Teichmüller et de Riemann de variétés de dimension plus grande que $1$, considérés comme des champs sur la catégorie des espaces analytiques complexes. Nous montrons que ces deux champs sont analytiques, c’est-à-dire isomorphes à la champification d’un groupoïde analytique lisse. Nous donnons ensuite une construction explicite d’atlas comme groupoïde d’holonomie généralisé. Ces résultats sont valables dès que la dimension du groupe d’automorphismes de chaque structure est bornée par un entier fixé. On peut voir ce travail comme une réponse à la question 1.8 de [48].

Classification: 32G05, 58H05, 14D23
Keywords: Teichmüller space, deformations of complex structures, analytic groupoids, stacks and moduli problems
Mot clés : Espace de Teichmüller, déformations de structures complexes, groupoïdes analytiques, champs et problèmes de modules

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