The Teichmüller and Riemann moduli stacks

Abstract

To Alberto Verjovsky on his 70th birthday.Abstract. 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 admit a smooth analytic groupoid as atlas. We then show how to construct explicitly such an atlas as a sort of generalized holonomy groupoid for such a structure. 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 [40].

The Teichmüller Stack

“…Let M be a smooth oriented compact manifold of even dimension 2n. We define as in [6] the Teichmüller stack as the quotient stack and Diff 0 (M ) is the group of diffeomorphisms of M which are C ∞ -isotopic to the identity. For M being a hyperbolic surface of genus g, this is exactly the classical Teichmüller space of M , that is a complex manifold which embeds as an open set in C 3g−3 .…”

“…In the higher-dimensional case, this is in general not even locally an analytic space (as shown by Hirzebruch or Hopf surfaces, see [6,Ex. 11.3 and 11.6]).…”

“…This explains why we define it as a stack. In [6] (see also [7] for a comprehensive introduction), we study thoroughly the structure of this stack and gives an explicit atlas of it.…”

“…Theorem 7.1 gives the most general picture. Building on [6], we define in Section 3 the analytic Artin stack [K 0 /Aut 0 (X 0 )] and we show that, at a Kähler point, the Teichmüller stack is locally the quotient of the stack [K 0 /Aut 0 (X 0 )] by the finite group Γ (see (4.1) for its definition). In particular, Corollary 7.3 asserts that at a Kähler point with no non-trivial global holomorphic vector Date: July 28, 2018.…”

“…The statements are easy consequences of [6]. They are obtained by applying a classical property of the cycle space of a Kähler manifold due to Lieberman [4] to show the finiteness of the set of maximal elements of Hol and its identification with Γ.…”