The coarse moduli space of a flat analytic groupoid

Abstract

Let $\mathcal X$ be a flat analytic groupoid $R_X\stackrel{s}{\underset{t}{\rightrightarrows }}X$ such that the holomorphic map $j=(s,t):R_X\rightarrow X\times X$ is finite. In this paper, we prove that there exist a (unique up to isomorphism) complex space $Q(\mathcal X )$ and a holomorphic map $q:X\rightarrow Q(\mathcal X )$ which is a GC quotient (see Definition 3.1). This extends to analytic groupoids the Main Theorem proved by Keel and Mori in the algebraic context (Keel and Mori in Ann Math 145(1):193–213, 1997, 1.1 Theorem).

1 Introduction

Objects like groupoids have been created in attempt to generalize algebraic varieties, analytic spaces, differentiable manifolds to entities carrying higher-order information or to manage such kind of objects when some operations performed on them no longer yield something endowed of the same structures, for instance, quotients of proper actions of Lie groups on complex analytic spaces. A groupoid simultaneously contains data on objects of a moduli problem and on the isomorphisms between them.

One can see an analytic groupoid as a pair of complex spaces $R\stackrel{s}{\underset{t}{\rightrightarrows }} X$ (over a fixed complex space $S$) along with five holomorphic maps $s,t, e, m, i$

$$\begin{aligned} R_X\stackrel{s}{\underset{t}{\rightrightarrows }} X\stackrel{e}{\rightarrow } R_X\>\>\>;\>\>\>\>R_X\times _{s,X,t} R_X\stackrel{m}{\rightarrow }R_X\>\>\>;\>\>\>\>R_X\stackrel{i}{\rightarrow }R_X \end{aligned}$$

(1)

satisfying certain properties (see Sect. 2). Isomorphisms are determined by the space $R$ through the maps $s$ and $t$, and the whole groupoid structure endows $X$ with an equivalence relation. A coarse moduli space may be thought of as one space to which the groupoid maps in a “universal” way. It does not need to exist, if the groupoid is general. In the case of an analytic groupoid, if the coarse moduli space exists, it must be a complex analytic space and its underlying set of points must be the quotient set of the isomorphism classes of the groupoid or equivalently must be the quotient of $X$ by the mentioned equivalence relation. The question about the existence of such a complex space for a class of special étale groupoids arose in the paper [2]. In that manuscript, we used the notion of hyperbolicity given in [1] in order to prove a Brody theorem for such a class of étale groupoids. The coarse moduli space plays a crucial role in the proof of that theorem.

The main result of this manuscript is the following:

Theorem 1.1

Let $\mathcal X$ be a flat analytic groupoid with $j$ finite. Then there exists a complex space $Q(\mathcal X )$ which is a GC quotient.

The proof begins along the same lines as in [6], sharing Proposition 3.2 as reduction step to the étalecase, after which it takes an entirely different path. The idea of reducing to the étalecase goes back to Mumford and Fogarty [8], Appendix to Chapter 5, A], where a particular moduli problem of algebraic origin over $\mathbb C$ is considered. In our paper, the key result of this step is Lemma 3.3 which corresponds to [6, Lemma 3.3]. The proof in our context uses complex variables tools, in particular the crucial Theorem of Frisch (cfr. [5]) on the generic flatness of a morphism of complex spaces. The étaleanalytic case is settled in Sect. 4 by means of the Cartan’s Theorem on quotients of complex spaces (cfr. [3]), which demands the construction of a rich enough sheaf of functions on the quotient, made possible by using the groupoid structure of $\mathcal X$ (see Lemma 6).

2 Preliminaries

2.1 Analytic groupoids

An analytic groupoid $\mathcal X$ is a pair $(X,R_X)$ of (reduced) complex spaces with five holomorphic maps $s,t,e,m,i$

$$\begin{aligned} R_X\stackrel{s}{\underset{t}{\rightrightarrows }} X\stackrel{e}{\rightarrow } R_X\>\>\>;\>\>\>\>R_X\times _{s,X,t}R_X\stackrel{m}{\rightarrow } R_X\>\>\>;\>\>\>\>R_X\stackrel{i}{\rightarrow }R_X \end{aligned}$$

(2)

which satisfy the following properties

• (a) $s\circ e=t\circ e=\mathsf{id}_R$,$\>s\circ i=t$, $\>t\circ i=s$,$\>s\circ m=s\circ p_2$, $\>t\circ m=t\circ p_1$ (where $p_1$, $p_2$ are the natural projections);

• (b) associativity: $m\circ (m\times \mathsf{id}_{R_X}) =m\circ (\mathsf{id}_{R_X}\times m)$;

• (c) both compositions

  $$\begin{aligned} R_X=R_X\times _{s,X}X=X\times _{X,t}R_X\stackrel{\mathsf{id}_{R_X}\times e}{\underset{e\times \mathsf{id}_{R_X}}{\rightrightarrows }}R_X\times _{s,X,t}R_X\stackrel{m}{\rightarrow }R_X \end{aligned}$$

  are the identity map on $R_X$;

• (d) inverse: $m\circ (i\times \mathsf{id}_{R_X})=e\circ s$,$\>m\circ (\mathsf{id}_{R_X}\times i)=e\circ t$.

For an analytic groupoid $\mathcal X$ will also use the notation $R_X\stackrel{s}{\underset{t}{\rightrightarrows }}X$ to make explicit the structure maps.

An analytic groupoid $R_X\stackrel{s}{\underset{t}{\rightrightarrows }} X$ is said to be flat, respectively, étale, smooth if $s, t$ are. Similarly, we define finite and quasi-finite groupoids. A smooth analytic groupoid is called an analytic Artin groupoid.

$R_X$ is the set of the isomorphisms of $R_X\stackrel{s}{\underset{t}{\rightrightarrows }} X$: if $\phi \in R_X$, then $x=s(\phi )$, $y=t(\phi )$ are the source and the target of $\phi$, respectively, and we write $\phi :x\rightarrow y$. The composition $m(\phi ,\psi ):x\rightarrow z$ of two isomorphisms $\phi :y\rightarrow z$, $\psi :x\rightarrow y$ is also denoted by $\phi \circ \psi$ or $\phi \psi$. By $\{x,y\}$, we denote the analytic subset $\{\phi \in R_X:s(\phi )=x, t(\phi )=y\}$ of all isomorphisms $x\rightarrow y$.

Given an analytic groupoid $R_X\stackrel{s}{\underset{t}{\rightrightarrows }} X$, let $j:R_X\rightarrow X\times X$ be the map $\phi \mapsto \big (s(\phi ),t(\phi )\big )$. The analytic subset $S=j^{-1}\left( \Delta _{X\times X}\right)$ is called the stabilizer of $R_X\stackrel{s}{\underset{t}{\rightrightarrows }} X$. $S$ is said to be quasi-finite if the canonical map $S\rightarrow X$ is quasi-finite. We remark that this is equivalent to require $j$ to be a quasi-finite map, because of the underlying groupoid structure. If the stabilizer $S$ is quasi-finite and $j:R_X\rightarrow X\times X$ is a closed map, then $j$ is proper with finite fibers; equivalently, it is a finite morphism.

It can be proved that $j$ is a closed map if and only if $S\rightarrow X$ is. In particular, $j$ is finite if and only if $S\rightarrow X$ is.

2.2 Homomorphisms and equivalences

A homomorphism $F$ of analytic groupoids $\mathcal X$, $\mathcal X '$ defined by $R_X\stackrel{s}{\underset{t}{\rightrightarrows }}X$ and $R_{X'}\stackrel{s'}{\underset{t'}{\rightrightarrows }}X'$, respectively, is a pair $(f_0,f_1)$ of holomorphic maps $f_0:X\rightarrow X'$, $f_1: R_X\rightarrow R_{X'}$ commuting with all the structure maps. Thus if $\phi :x\rightarrow y$ is an isomorphism of $(X,R_X)$, then $f_1(\phi )$ is an isomorphism $f_0(x)\rightarrow f_0(y)$.

If $F=(f_0,f_1)$ and $G=(g_0,g_1)$ are homomorphisms, a natural transformation $\mathsf T$ from $F$ to $G$ is a holomorphic map $\mathsf{T}:X\rightarrow R_{X'}$ such that

• (i) T $(x)$ is an isomorphism $f_0(x)\rightarrow g_0(x)$ for every $x\in X$;

• (ii) if $\phi :x\rightarrow y$ is an isomorphism of $(X,R_X)$ then $g_1(\phi )\mathsf{T}(x)=\mathsf{T}(y)f_1(\phi ).$

A homomorphism $F: \mathcal X \rightarrow \mathcal X '$ is called an equivalence if

• (1) the map $t'p_1:R_{X'}\times _{s',X',f_0}X\rightarrow X'$ where $p_1$ is the projection on the first factor is a surjective submersion, i.e., $U\subset X'$ is open iff $(t'p_1)^{-1}(U)$ is open and

• (2) the square

  

  (3)

  is a fibered product.

In particular,

• (1$^\prime$) condition (1) says that every $x'\in X'$ is connected to some point $f_0(y)$ in the image of $f_0$ by an isomorphism $\phi ':f_0(y)\rightarrow x'$, $\phi '\in R_{X'}$;

• (2$^\prime$) condition (2) says that $F$ induces a biholomorphism between $\{x,y\}$ and $\{f_0(x),f_0(y)\}$ for every $(x,y)\in X\times X$.

Remark 2.1

Let $R_X\stackrel{s}{\underset{t}{\rightrightarrows }}X$ and $R_{X'}\stackrel{s'}{\underset{t'}{\rightrightarrows }}X'$ equivalent analytic groupoids. Then $j=(s,t)$ is a finite morphism if and only if $j'=(s',t')$ is.

2.3 Restriction

(cfr. [6, Remark 2.6]) Let $R_X\stackrel{s}{\underset{t}{\rightrightarrows }} X$ be an analytic groupoid. Every holomorphic map $f:W\rightarrow X$ of complex spaces induces an analytic groupoid $R_{|W}\stackrel{s_W}{\underset{t_W}{\rightrightarrows }}W$, called restriction and defined by the following diagram, where all squares are fiber squares:

(4)

the structure maps $s_W$, $t_W$ are defined by the first horizontal and the left vertical compositions in the diagram (4), respectively.

Remark 2.2

In particular, if $f$ is an embedding, we have

$$\begin{aligned} R_{|W}=\big \{(\phi ,w)\times (w',\phi ')\in \big (R_X\times W\big )\times \big (W\times R_X\big ):\phi =\phi ',w=t(\phi ),w'=s(\phi )\big \} \end{aligned}$$

and $s_W\big ((\phi ,t(\phi ))\times (s(\phi ),\phi )\big )=s(\phi )$, $t_W\big ((\phi ,t(\phi ))\times (s(\phi ),\phi )\big )=t(\phi ).$ It follows that $j_W$ is a closed map if $j$ is.

3 The coarse moduli space

3.1 Basic definitions

Given an analytic groupoid $R_X\stackrel{s}{\underset{t}{\rightrightarrows }}X$, we are interested in finding a complex space $X'$ and a holomorphic map $q$ making the diagram

$$\begin{aligned} R_X\stackrel{s}{\underset{t}{\rightrightarrows }} X\stackrel{q}{\rightarrow } X' \end{aligned}$$

commutative and preserving as much information about the diagram as possible. Let $X$ and $U$ be complex spaces; by $X(U)$, we will denote the set of holomorphic maps $\mathsf{Hom}(U,X)$. A relation is a holomorphic map $j:R\rightarrow X\times X$ from a complex space $R$; this is a pre-equivalence relation if the image of $j(U):R(U)\rightarrow X(U)\times X(U)$ is an equivalence relation of sets for all complex spaces $U$. When $j(U)$ is injective for all complex spaces $U$, we will say that $j$ is an equivalence relation. The associated sheaf to the presheaf $U\rightarrow X(U)/R(U)$ is called the quotient sheaf of a pre-equivalence relation and will be denoted as $X/R_X$. Notice that, if $\mathcal X$ is a groupoid $R_X\stackrel{s}{\underset{t}{\rightrightarrows }} X$, the pair $j=(s,t):R_X\rightarrow X\times X$ is a pre-equivalence relation. Moreover, setting $U=\mathsf {pt}$, we get a set-theoretic equivalence relation $R=R(\mathcal X )$ on the set $X(\mathsf{pt})=X$ associated with $\mathcal X$ by considering $j(\mathsf {pt})$. Two points $x,y\in X$ are $R$-equivalent, $x\sim y$, if there exists $\phi \in R_X$ such that $s(\phi )=x, t(\phi )=y$. We denote by $[x]$ the set $t\big (s^{-1}(x)\big )$ of all the points equivalent to $x$ and by $Q(\mathcal X )$ the quotient set of this set-theoretic equivalence.

Following [6, Definition 1.8], we state the

Definition 3.1

Let $j:R_X\rightarrow X\times X$ be a pre-equivalence relation and $q:X\rightarrow X/R_X\rightarrow Q$ be a map of complex spaces, which factorizes through the sheaf $X/R_X$. Consider the following properties:

• (G) $q$ gives a bijection between the sets $\big (X/R_X\big )(\mathsf{pt}) \rightarrow Q(\mathsf{pt})$;

• (C) $q$ is universal among holomorphic maps to complex spaces;

• (UC) $(X\times _Q Q')/(R_X\times _Q Q')\rightarrow Q'$ satisfies $(C)$ for any flat map $Q'\rightarrow Q$;

• (US) $q$ is a universal submersion, i.e., $U\subset Q$ is open iff $q^{-1}(U)$ is open, and this property is preserved by base change;

• (F) for every domain $U\subset Q$, the homomorphism

  $$\begin{aligned} q^*:\mathcal O _Q(U)\rightarrow \big \{f\in \mathcal O _X\big (q^{-1}(U)\big ):f\circ s=f\circ t\big \} \end{aligned}$$

  defined by $h\mapsto h\circ q$ is an isomorphism. (i.e., the local holomorphic functions on $Q$ are the $R_X$-invariant (local) holomorphic functions on $X$).

If $q$ satisfies (C) (respectively (UC)), it is called a (respectively uniform) categorical quotient. When it satisfies (G) and (C), it is a coarse moduli space; (G), (US) and (F) a geometric quotient and all of the above a GC quotient.

Remark 3.1

A homomorphism $F:\mathcal X \rightarrow \mathcal X '$ of analytic groupoids with respective GC quotients $Q(\mathcal X )$ and $Q(\mathcal X ')$ determines a holomorphic map $Q(\mathcal X )\rightarrow Q(\mathcal X ')$. If $F$ is an equivalence and $\mathcal X$ admits a GC quotient $Q(\mathcal X )$, then $\mathcal X '$ admits a GC quotient $Q(\mathcal X ')$ and the induced map by $F$ is a biholomorphism.

3.2 Reduction

The reduction step consists in proving the following

Proposition 3.2

If any analytic groupoid with quasi-finite flat structure maps and $j$ finite admits a GC quotient, then so does any flat groupoid with $j$ finite.

The proof of Proposition 3.2 is consequence of the following lemmas.

Lemma 3.3

Let $R_X\stackrel{s}{\underset{t}{\rightrightarrows }} X$ be a flat analytic groupoid. Assume that $j:R\rightarrow X\times X$ is finite. Then for every point $y\in X$, there exist a point $x\sim y$, a neighborhood $U(x)\Subset X$ of $x$ and a closed analytic subspace $g:W\hookrightarrow U(x)\subset X$ such that

• (i) $x\in W$ and $W\cap [x]=\{x\}$;

• (ii) the morphism $p:W\times _{(g,t)} R\rightarrow R\stackrel{s}{\rightarrow }X$ is quasi-finite flat and surjective over $U(x)$ (hence surjective for the qff topology over $U(x)$);

• (iii) the holomorphic maps $s_W, t_W:R_{|W}\rightarrow W$ defined by the diagram (4) are quasi-finite flat;

• (iv) $R_{|W}\stackrel{s_W}{\underset{t_W}{\rightrightarrows }} W$ is an analytic groupoid with $j_W:R_{|W}\rightarrow W\times W$ finite.

Proof

We first observe the following

• (a) since $j$ is a closed map, $j(R_X)$ is a closed analytic subset of $X\times X$ (cfr. [9, Theorem 3, Ch. VII]);

• (b) if $s^{-1}(x)$ is the geometric fiber of $s$ over $x\in X$, the holomorphic map $t_{|s^{-1}(x)}$ is closed, for if $C\subset s^{-1}(x)$ we have

  $$\begin{aligned} \{x\}\times t_{|s^{-1}(x)}(C)=\{x\}\times t(C)=j(C) \end{aligned}$$

  which is closed if $C$ is;

• (c) in particular, for every $x\in X$ the subset $[x]$ is closed and analytic.

Step 1.

There exists a point $x\sim y$ such that $s^{-1}(x)$ is regular at each point of $s^{-1}(x)\cap t^{-1}(x)= j^{-1}(x,x)$.

Let $\tau =t_{|s^{-1}(y)}$ and $\Sigma _1= \tau ^{-1}\big (\mathsf{Sing}\,[y]\big )$, $\Sigma _2=\tau ^{-1}\big (\tau (\mathsf{Sing}\,s^{-1}(y))\big )$. Then $\tau$ gives a holomorphic map $s^{-1}(y)\,\!\backslash \!\,\Sigma _1\cup \Sigma _2\rightarrow [y]\,\!\backslash \!\,\mathsf{Sing}\,[y]$ and we take for $x$ a regular value for this map.

Now let $x=t(v)$, $v\in s^{-1}(y)\,\!\backslash \!\,\Sigma _1\cup \Sigma _2$, and consider the isomorphism $\Phi :s^{-1}(y)\rightarrow s^{-1}(x)$ given by $w\mapsto w\circ i(v)$ where $i:R\rightarrow R$ is the inverse. Then, we directly check that $\Phi \big (s^{-1}(y)\cap t^{-1}(x)\big ) =s^{-1}(x)\cap t^{-1}(x)$; hence, $s^{-1}(x)$ is regular at the points of $s^{-1}(x)\cap t^{-1}(x)$.

Step 2.

Let $V\Subset X$ be a neighborhood of $x$ and $W'\ni x$ an analytic subset of a neighborhood of $\overline{V}$ such that $W'\cap [x]$ is 0-dimensional at $x$ (we can take $W'=\{f_1=\cdots =f_k=0\}$, $k=\mathrm{dim}_{_{\mathbb{C }}}s^{-1}(x)$, where $f_1,\ldots ,f_k,\ldots ,f_m$ are minimal generators of the maximal ideal of $\mathcal O _{X,x}$). Denote by $g:W'\rightarrow X$ the canonical immersion. Since $s^{-1}(x)$ is regular, in particular Cohen–Macauley, at each point of $j^{-1}(x,x)$, and $s$ is flat, [7], (20.E)] applies (see also [4], Ch. $\mathbf{0}_\mathrm{III}$, (10.2.4)]) showing that the map $p:W\times _{(g,t)} R\rightarrow R\stackrel{s}{\rightarrow }X$ is flat over $x$ and $s_{W'}:R_{|W'}\rightarrow W'$ is flat and quasi-finite over $x$. The same holds for $t_W$ since $t=s\circ i$. Since $j$ is finite, $j_W$ is finite (see Remark 2.2).

We wish now to show that the flatness property of $p$ can be extended on an open neighborhood $U(x)\subset X$ of $x$. Let $A\subset W'\times _{(g,t)} R_X$ be the subset on which $p$ is not flat. By a theorem of Frisch, $A$ is a closed analytic subset (cfr. [5]).

We prove that, by the properness of $j$, the point $x$ cannot be of accumulation for $p(A)$ (notice that $p(A)$ depends on $x$ since $W'$ does), implying the existence of an open neighborhood $U(x)$ on which $p$ is flat. Let $\{(y_n, \phi _n)\}$ be a sequence of points in $A$ whose image through $p$ converge to a point $x$. Two cases may occur: $\{t(y_n)\}$ converges to a point $y\in W'$; then $j(\phi _n)=(s(\phi _n), t(\phi _n))\rightarrow (x,y)$ and, since $j$ is proper, there exists a subsequence $\{\phi _{n'}\}$ converging to a $\phi \in R$. By continuity of $j$, we have $s(\phi )=x$, $t(\phi )=y$, i.e., $x=p(y,\phi )$ contradicting the properties of the point $x$. The other possible case happening is that $\{t(y_n)\}$ converges to a $y'\in \overline{W'}$, by possibly shrinking $W'$ a bit. We are assuming that $y'\not \in W'$, thus $y'\ne x$. Since $j(R)$ is closed, we have that $(x,y')\in j(R)$ because $j(s(\phi _n), y_n)$ converges to it. But this is absurd, because it implies that $y'\sim x$, hence $y'\in [x]\cap \overline{W'}$, but we may assume $[x]\cap \overline{W'}=[x]\cap W'=\{x\}$, by possibly further shrinking $W'$, and take $W=W'\cap U(x)$. $\square$

To use the GC quotient of $R_{|W}\stackrel{s_W}{\underset{t_W}{\rightrightarrows }}W$, whose existence we may assume by hypothesis, on the groupoid $R_X\stackrel{s}{\underset{t}{\rightrightarrows }}X$, we will repeatedly appeal to the following result

Lemma 3.4

With the notations of Lemma 3.3 the following statements hold:

• (1) if $U(x)\rightarrow Q$ is a GC quotient, then the image $V$ of the composition

  $$\begin{aligned} W\hookrightarrow U(x)\rightarrow Q \end{aligned}$$

  along with the induced map $W\rightarrow V$ is a GC quotient;

• (2) if $W\rightarrow W/(R_{|W})\rightarrow Z$ is a GC quotient, then

  $$\begin{aligned} U(x)\rightarrow U(x)/R_{|U(x)}\cong W/(R_{|W})\rightarrow Z \end{aligned}$$

  is a GC quotient.

Proof

Statements (1) and (2) correspond to parts (2) and (3) in [6, Lemma 3.1] where $X$ is replaced by $U(x)$. Their proof just uses the definition of GC quotient and general nonsense; hence, it applies in our context, as well, observing that the condition on the map

$$\begin{aligned} p:W\times _{(g,t)}R_{|U(x)}\rightarrow R_{|U(x)}\stackrel{s_{|U(x)}}{\rightarrow }U(x) \end{aligned}$$

in our case is satisfied by the Lemma 3.3. $\square$

Proof of Proposition 3.2

Because of Lemma 3.4 part (2), the existence of a GC quotient $Q(U(x))$ for $R_{|U(x)}\stackrel{s_{|U(x)}}{\underset{t_{|U(x)}}{\rightrightarrows }}U(x)$ is reduced to the existence of a GC quotient for $R_{|W}\stackrel{s_W}{\underset{t_W}{\rightrightarrows }} W$, which we have by assumption in the Proposition 3.2 and the properties of $W$ proved in Lemma 3.3. In order to end the proof of Proposition 3.2, it suffices to build a GC quotient for $R\rightrightarrows X$ out of the $Q(U(x))$’s for $U(x)$ covering $X$. That is possible by invoking the following

Lemma 3.5

Let $R_X\stackrel{s}{\underset{t}{\rightrightarrows }}X$ be a flat groupoid with $j$ finite and $X$ being open covered by subspaces $V_i$, each part of a groupoid. If all the groupoids on $V_i$ admit a GC quotient $Q(V_i)$ for all $i$, then so does $X$.

Proof

By the unicity of the GC quotient, it suffices to show the existence of GC quotients for the induced groupoids $R_{V_i\cap V_j}\rightrightarrows V_i\cap V_j$. This assures the GC quotients $Q(V_i)$ can be glued together to form one for the whole $X$. Lemma 3.4 part (1) applies to the present situation by letting $g:W=V_i\cap V_j\hookrightarrow V_i=X$. $\square$

4 Proof of the main theorem

The Proposition 3.2 reduces the proof of the main theorem to the case of an analytic groupoid $\mathcal X$ represented by $R_X\stackrel{s}{\underset{t}{\rightrightarrows }}X$ with quasi-finite flat structure maps, hence quasi-finite étale, and $j$ finite.

The key point in the proof of the existence of the GC quotient for such groupoids is Cartan’s theorem on quotients of complex spaces (cfr. [3]).

Let $S$ be the stabilizer of $\mathcal X$ and $A=s(S)=t(S)$. Since $j$ is proper, $A$ is a closed analytic subset.

Consider the equivalence relation on $X$ introduced in 3.1:

$$\begin{aligned} x\sim y \; \text {if and only if there exists} \; \phi \in R_X \;\text {such that}\; s(\phi )=x \;\text {and}\; t(\phi )=y. \end{aligned}$$

(5)

For every $x\in X$, the set $[x]=t(s^{-1}(x))$ is finite. Moreover, because of the underlying groupoid structure, if $x\sim y$ and $x\in A$ then $y\in A$.

For every subset $N\subset X$, we define $\widehat{N}:=t(s^{-1}(N))$ and we call it the saturated of $N$ by $R$.

Let $Q(\mathcal X )$ be the set $X/\!\!\sim$ equipped with the quotient topology and $q:X\rightarrow Q(\mathcal X )$ the natural projection.

• $Q(\mathcal X )$ is locally compact and Hausdorff. Let $z=q(x)\ne \zeta =q(y)\in Q(\mathcal X )$ and $\overline{U}_x\Subset X$, $\overline{U}_y\Subset X$ neighborhoods of $x$, $y$, respectively, such that $\widehat{\overline{U}_x}$ and $\widehat{\overline{U}_y}$ are disjoint. Then $q(\overline{U}_x)$ and $q(\overline{U}_y)$ are disjoint compact neighborhoods of $z$ and $\zeta$, respectively.

• $q:X\rightarrow Q(\mathcal Y )$ is a continuous open map with finite fibers. Let $O\subset X$ be an open set. Then $q^{-1}\big (q(O))=\widehat{O}$ which is open in $X$.

• If $X_0$ is a connected component of $X$, then $q_{|X_0}$ is proper from $X_0$ to $q(X_0)$. Indeed a sequence $\{x_n\}$ is convergent in $X$ if and only if $[x_n]$ is, so $q$ is closed (the topology of $Q(\mathcal X )$ being countable basis).

• $B=q(A)$ is a closed subset of $Q(\mathcal X ).$ We first observe that if $X_k$ is a connected component of $X$, and $A_k=A\cap X$, then $B_k:=q(A_k)$ is an analytic subset. If $q(X_k)\cap q(X_l)\ne \varnothing$, then $B_k=B_l$ on $q(X_k)\cap q(X_l)\ne \varnothing .$

• $q^{-1}(B)=A$ and $q:X\!\backslash \!A\rightarrow Q(\mathcal X )\!\backslash \!B$ is a finite topological covering.

The first statement follows at once since isomorphic objects have isomorphic automorphism groups. As for the second, let $z=q(x)\in Q(\mathcal X )\!\backslash \!B$ with $x\in X\!\backslash \!A$. Then it is possible to find an open neighborhood $U_x\subset X\!\backslash \!\,A$ of $x$ such that $t(V_i)\cap t(V_j)=\varnothing$ for $i\ne j$, where as usual $s^{-1}(U_x)=\amalg _iV_i$, which means that $q(U_x)$ is uniformly covered by $q$ since $q^{-1}(q(U_x))=\amalg _i t(V_i)$.

We are now able to prove the following theorem:

Theorem 4.1

Assume that $\mathcal X =\{R_X\stackrel{s}{\underset{t}{\rightrightarrows }}X\}$ is an étalegroupoid with $j: R_X\rightarrow X\times X$ finite. Then

• (1) there exists a complex structure on $Q(\mathcal X )$ such that $q$ is holomorphic;

• (2) $B=q(A)$ is a closed analytic subspace, $q^{-1}(B)=A$ and $q:X\!\backslash \!A\rightarrow Q(\mathcal X )\!\backslash \!B$ is étale;

• (3) $Q(\mathcal X )$ is a $GC$ quotient.

Proof

(1) On $Q(\mathcal Y )\!\backslash \!B$, the complex structure is the one descended from $X\!\backslash \!A$ through the covering map $q_{|X\!\backslash \!A}$.

In order to prove the theorem, we apply the quoted Cartan’s Theorem componentwise to $X$. Namely, we prove that for every connected component $X_k$ of $X$, there exists a complex structure on $B_k=q(X_k)$ such that $q_{|X_k}$ is holomorphic, thus we may assume that $X=X_k$.

According to Cartan’s Theorem, this involves showing that for any point $\bar{x}\in B_k$, there exists an open neighborhood $N=N(\bar{x})$ such that the algebra

$$\begin{aligned} \mathcal A (N)=\{h\in C^0(N,\mathbb{C }): h\circ q\in \mathcal O (q^{-1}(N))\} \end{aligned}$$

separates points in $N$, where $\mathcal O$ is the sheaf of germs of holomorphic functions on $X$. Given $x\in X$, we let

$$\begin{aligned} s^{-1}(x)=\big \{x_1,\dots ,x_m\}\>\>\mathrm{e}\>\> J=\{j\in \{1,\cdots ,m\}:x_j\in s^{-1}(x)\cap t^{-1}(x)\big \}. \end{aligned}$$

Consider an $s$-uniformly covered Stein neighborhood $U=U(x)$ of $x$ and denote $s^{-1}(U)= \amalg _{i=}^m V_i$ such that

• (i) $t(V_i)\cap U=\varnothing$ if $i \not \in J$ and

• (ii) $t(V_j)=U$ for all $j\in J$.

For any $j\in J$, we let $\lambda _j$ be the corresponding holomorphic section $U\rightarrow V_j$ of $s$. If $f:U\rightarrow \mathbb{C }$ is a holomorphic function, we define a new holomorphic function $g_f:U\rightarrow \mathbb{C }$ as

$$\begin{aligned} g_f(z)=\prod \limits _{j\in J}f\left( t(\lambda _j(z))\right) . \end{aligned}$$

(6)

for $z\in U$. $\square$

Lemma 4.2

If $z,z'\in U$ and $z\sim z'$ then $g_f(z)=g_f(z')$.

Proof

By assumption, $z'=t(\lambda _k(z))$ for some $k\in J$, hence

$$\begin{aligned} g_f(z')=\prod \limits _{j\in J}f\left( t(\lambda _j(t(\lambda _k(z)))\right) . \end{aligned}$$

(7)

By the groupoid structure in $\mathcal X$, we have

$$\begin{aligned} t\Bigl (m\,\bigl ( \;\lambda _k(z),\lambda _j(t(\lambda _k(z)))\;\bigr ) \Bigr )=t\Bigl (\lambda _j\bigl (t(\lambda _k(z))\bigr )\Bigr ) \end{aligned}$$

(8)

and

$$\begin{aligned} s\Bigl (\;m\,\bigl ( \;\lambda _k(z),\lambda _j(t(\lambda _k(z)))\;\bigr )\; \Bigr )=z. \end{aligned}$$

(9)

Let $h(j)$ be a positive integer such that $m\,\bigl ( \,\lambda _k(z),\lambda _j(t(\lambda _k(z))\,) \,\bigr )\in V_{h(j)}$. We have that $h(j)\in J$, because $j\in J$ so the element represented by the equality (8) lies in $U$ and $t(V_{h(j)})=U$. Moreover, the points $\{x_j: j\in J\}$ form a group under the operation $m$, hence the correspondence $h:J\rightarrow J$ is a permutation. On the other hand, all the $\lambda _i$ are sections of $s$, therefore, by the Eq. (9) we have

$$\begin{aligned} \lambda _{h(j)}(z)=m\,\bigl ( \; \lambda _k(z)\;,\;\lambda _j(t(\lambda _k(z)))\; \bigr ). \end{aligned}$$

Summing up, the three sets of points

• $\{t(\lambda _j(z)):j\in J\}$

• $\{t(\lambda _{h(j)}(z)): j\in J\}$

• $\{t(\lambda _j(t(\lambda _k(z)))): j\in J\}$

all coincide. This shows that the map in (6) is the same as the map in (7).

Let $N=q(U)$ where $U$ is a Stein domain in $X$; $N$ is open and the holomorphic function $g_f$ factors through $Q(\mathcal X )$ for all $f\in \mathcal O (U)$ and it defines a continuous function $h_f$ on $N$. For any $f\in \mathcal O (U)$, the function $h_f\circ q$ holomorphically extends to a holomorphic function $l_f$ defined on the saturation $\widehat{U}=U\amalg _{i\not \in J} t(V_i)$ of $U$: let $h_f\circ q_{|t(V_i)}=g_f\circ s\circ \mu _i$, where $\mu _i:t(V_i)\rightarrow V_i$ is the $i$-th section of $t$ (making $U$ a little smaller, we may assume $t$ is a biholomorphism on its images when restricted to $V_i$ for all $i$). By construction, $l_f$ is holomorphic on $q^{-1}(N)$. We show that the function algebra $\mathcal A (N)$ separates points. It suffices to show that the set of functions $\{l_f\}_{f\in \mathcal O (U)}$ separates points. Take $\bar{x}\ne \bar{y}\in N$, with $x,y\in U$; then, since $U$ is Stein, there exists $f\in \mathcal O (U)$ such that $f(x)=0$ and $f(y)=1.$ It follows that $h_f(\bar{x})=0$ and $h_f(y)=1$.

Clearly, on $N\!\backslash \!B_k$, the complex structure just introduced coincides with the one descended from $X_k\!\backslash \!A_k$ thus, in order to conclude the proof of part i) of the theorem, we have to show that if $X_k$ and $X_l$ are two connected components of $X$ such that $q(X_k)\cap q(X_l)\ne \varnothing$ and $q_k$, $q_l$ are the restrictions of $q$ to them, then the respective complex structures patch together on $q_k(X_k)\cap q_l(X_l)\ne \varnothing$. Assuming that $U(a)\subset q(X_k)\cap q(X_l)$ is an open neighborhood of $a$ and that $f\circ q_k$ is holomorphic for a continuous function $f:U(a)\rightarrow \mathbb C$, we want to prove that $f\circ q_l$ is holomorphic, as well. Let $y\in q_l^{-1}(a)$, which is a discrete-set, and assume $q(V)=U(a)$ where $V\subset X_k$ is a connected open set uniformly covered by $s$. Let furthermore $W=\sigma (V)$, for a local section $\sigma$ di $s$, be the connected component of $s^{-1}(V)$ such that $s:W\stackrel{\cong }{\rightarrow } V$ and $t:W\stackrel{\cong }{\rightarrow } t(W)$, $t(W)\cap q_l^{-1}(a)=y$. Then $f\circ q_k=f\circ q_l\circ t\circ \sigma$ is holomorphic as function restricted to $V$; therefore, $f\circ q_l$ is holomorphic since $t$ and $\sigma$ are biholomorphisms when restricted to $W$ and $V$, respectively.

(2) It is a consequence of the fact that $q$ is quasi-finite and

$$\begin{aligned} q_{|X\!\backslash \!A}: X\!\backslash \!A\rightarrow Q(\mathcal X )\!\backslash \!B \end{aligned}$$

is a covering map.

(3) $Q(\mathcal X )$ satisfies the condition (G) of the Definition 3.1 because, in the analytic site, the sections of the quotient presheaf of $X$ modulo the relation coming from $R_X$ on the point are the same as the sections on the point of its associated sheaf $X/R_X(\mathsf{pt})$ and the former are in bijective correspondence with the points in $Q(\mathcal X )$.

Condition (C): let $Z$ be any complex space and $f:X\rightarrow X/R_X\rightarrow Z$ be any holomorphic map factoring through $X/R_X$; to construct a holomorphic map $\rho =\rho _f:Q(\mathcal X )\rightarrow Z$ making the diagram commute, we first define it set-theoretically, using the property (G) above. Clearly, we may assume that $X$ is connected. Such $\rho$ is continuous since $q$ is proper: if $\{[y_n]\}$ is a sequence in $Q(\mathcal X )$ converging to $[y]$, it is possible to find a sequence $\{x_m\}$ in $X$ converging to $x$ such that $q(x)=[y]$ and $q(x_m)=[y_m]$. It follows that $\{\rho (y_m)\}=\{f(x_m)\}$ converges to $f(x)=\rho (y)$. To prove that $\rho$ is holomorphic fix a domain $U\subset Q(\mathcal X )$ a domain $V\subset Z$ isomorphic to an analytic subset of a domain in $\mathbb{C }^N$ with $\rho (U)\subset V$; thus, up to a biholomorphism, $\rho _{|U}$ is given by $N$ holomorphic functions $\rho _1,\ldots ,\rho _N$ on $U$. For $i=1,2,\ldots ,N$, we have $(z_i\circ \rho )\circ q=z_i\circ f$ which is holomorphic and $R_X$-invariant. By the very construction of the complex space, $Q(\mathcal X )$ $z_i\circ \rho _i$ is holomorphic whence $\rho _i=z_i\circ \rho _i$ is holomorphic.

This proves that condition (C) is satisfied by $Q(\mathcal X )$.

Condition (UC) refers to the diagram

where $Q$ states for $Q(\mathcal X )$ and $f$ is a flat holomorphic map. To be shown that the canonical sheaf morphism

$$\begin{aligned} (X\times _Q Q')/(R_X\times _Q Q')\rightarrow Q' \end{aligned}$$

satisfies the property (C) above. The above argument implies that it suffices to show that $Q'$ is the set-theoretic quotient of $X\times _Q Q'$ by the relation coming from $R_X\times _Q Q'$, being all the maps of the previous diagram holomorphic, $q'$ in particular. It is known that colimits are stable under base change in the category of sets, and this translates in the canonical map

$$\begin{aligned} (X\times _Q Q')/(R_X\times _Q Q')\rightarrow (X/R_X)\times _Q Q' \end{aligned}$$

being bijection (set-theoretic isomorphism), but the latter is $Q\times _Q Q'=Q'$.

Condition (F) states that holomorphic functions on $Q(\mathcal X )$ are set-theoretic functions $f:Q(\mathcal X )\rightarrow \mathbb C$ such that $f\circ q$ are holomorphic functions with the property that $f\circ q\circ s=f\circ q\circ t$.

Finally, condition (US) is satisfied because $q$ is surjective, and for every connected component, $X_k$ of $X$ the holomorphic map $q_{|X_k}$ is finite. $\square$