Abstract
The aim of this paper is to put a nice analytic structure on the higher-dimensional Teichm\uller and Riemann spaces, which are only defined as topological spaces. We show that they can be described as Artin analytic stacks of families of complex manifolds, that is admitting a presentation as a groupoid whose objects and morphisms form a finite-dimensional analytic space and whose source and target maps are smooth morphisms. This is achieved under the sole condition that the dimension of the automorphism group of each structure is bounded by a fixed integer. The main point here is the existence of a global (multi)-foliated structure on the space of complex operators whose holonomy data encodes how to glue the local Kuranishi spaces to obtain a groupoid presentation of the Teichm\uller space.
Highlights
Let X be a smooth oriented compact surface
We insist on the fact that we work exclusively in the C-analytic context, since we deal with arbitrary compact complex manifolds
We show that each 1-parameter subgroup of the automorphism group Aut0(X0) of X0 acts on the Kuranishi space K0 of X0
Summary
Let X be a smooth oriented compact surface. The Riemann moduli space M s(X) is defined as the set of Riemann surfaces C∞-diffeomorphic to X up to biholomorphism. The Teichmüller space T s(X) is roughly defined as the set of Riemann surfaces diffeomorphic to X up to biholomorphism C∞-isotopic to the identity (see Section 2.1). Examples of Teichmüller stacks are an unending source of examples of analytic stacks, showing all the complexity and richness of their structure, far from finite dimensional group actions and leaf spaces. Let X be a smooth (i.e., C∞) oriented compact connected manifold of even dimension The unoriented Teichmüller space of a compact surface has two connected components, corresponding to the two possible orientations. We insist on the fact that we work exclusively in the C-analytic context, since we deal with arbitrary compact complex manifolds This forces us to adapt and sometimes to transform the definitions of stacks coming from algebraic geometry. It is interesting to have this flexibility, for example we will often take for V a connected component of I , even if it is not saturated
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