The Teichm\"uller and Riemann Moduli Stacks

TL;DR

This paper investigates the structure of higher-dimensional Teichmüller and Riemann moduli stacks, introducing a foliation concept, constructing a holonomy groupoid, and characterizing these stacks as Artin analytic stacks under automorphism bounds.

Contribution

It introduces a foliation model and explicit holonomy groupoid construction for moduli stacks, providing a new analytic stack characterization.

Findings

Holonomy groupoid objects form a finite-dimensional analytic space

Source and target maps are smooth morphisms

Moduli stacks are characterized as Artin analytic stacks

Abstract

The aim of this paper is to study the structure of the higher-dimensional Teichm\"uller and Riemann moduli spaces, viewed as stacks over the category of complex manifolds. We first show that the space of complex operators on a smooth manifold admits a foliation transversely modeled on a translation groupoid, a concept that we define here. We then show how to construct explicitly a holonomy groupoid for such a structure and show that in this case its objects and morphisms form a finite-dimensional analytic space and its source and target maps are smooth morphisms. This holonomy data encodes how to glue the local Kuranishi spaces to obtain a groupoid presentation of the Teichm\"uller and Riemann moduli stacks, which can thus be characterized as Artin analytic stacks. This is achieved under the sole condition that the dimension of the automorphism group of each structure is bounded by a fixed integer.