A characterization of complex quasi-projective manifolds uniformized by unit balls

Abstract

In 1988 Simpson extended the Donaldson–Uhlenbeck–Yau theorem to the context of Higgs bundles, and as an application he proved a uniformization theorem which characterizes complex projective manifolds and quasi-projective curves whose universal coverings are complex unit balls. In this paper we give a necessary and sufficient condition for quasi-projective manifolds to be uniformized by complex unit balls. This generalizes the uniformization theorem by Simpson. Several byproducts are also obtained in this paper.

Appendix A: Metric rigidity for toroidal compactification of non-compact ball quotients

The main motivation of this appendix is to provide one building block for Theorem 5.7.(ii). Our main result, Theorem A.7, says that there is no other smooth compactification for non-compact ball quotient than the toroidal one, so that the Bergman metric grows “mildly” near the boundary. Besides its own interests, this result is applied to show that the smoothness of D in Theorem A is necessary if one would like to characterize non-compact ball quotients.

1.1 A.1: Toroidal compactifications of quotients by non-neat lattices

In this section, we recall a well known way of constructing the toroidal compactifications of ball quotients in the case where the lattice has torsion at infinity. The reader will find more details about the natural orbifold structure on these compactifications in [21]. For our purposes, the basic result given in Proposition A.1 will be sufficient.

Recall that we say that a lattice $\Gamma \subset PU(n,1)$ is neat (cf. [7]) if for any $g \in \Gamma$, the subgroup of ${\mathbb {C}}^*$ generated by the eigenvalues of g is torsion free. This implies that $\Gamma$ is torsion free and that all parabolic elements of $\Gamma$ are unipotent, so that the toroidal compactifications of $\frac{{\mathbb {B}}^n}{\Gamma }$ provided by [2, 44] are smooth (there is no "torsion at infinity"). Note that by [7, Proposition 17.4] in the arithmetic case, and [6], or [50, Theorem 6.11] in general, any lattice in PU(n, 1) admits a finite index neat normal sublattice.

Proposition A.1

Let $\Gamma \subset PU(n, 1)$ be a torsion free lattice, and let $\Gamma ' \subset \Gamma$ be a finite index normal neat sublattice. Let $U = \frac{{\mathbb {B}}^n}{\Gamma }$, $U' = \frac{{\mathbb {B}}^n}{\Gamma '}$, and denote by $X'$ the smooth toroidal compactification of $U' = \frac{{\mathbb {B}}^n}{\Gamma '}$ as constructed in [2, 44].

Then the natural action of the finite group $G = \frac{\Gamma }{\Gamma '}$ on $U'$ extends to $X'$, and the quotient $X = \frac{X'}{G}$ is a normal projective space, with boundary $X - U$ made of quotient of abelian varieties by finite groups. Moreover, when $\Gamma$ is arithmetic, X coincides with the toroidal compactification of U constructed in [2].

Before explaining how to prove Proposition A.1, let us recall the construction of $X'$ as it is defined in [44] (see also [11] for a similar discussion).

Each component D of $X' - U'$ is associated to a certain $\Gamma '$-orbit of points of $\partial {\mathbb {B}}^n$, whose points are called the $\Gamma '$-rational boundary components of $\partial B^n$ (cf. [2, Chapter 3] or [44, §1.3]). Let $b \in \partial {\mathbb {B}}^n$ be such a point, and let $N_b \subset PU(n,1)$ be its stabilizer. This is a maximal parabolic real subgroup of PU(n, 1); let us denote by $W_b$ its unipotent radical. This group is an extension $1 \rightarrow U_b \rightarrow W_b \overset{\pi }{\rightarrow } A_b \rightarrow 1$, where $A_b \cong {\mathbb {C}}^{n-1}$, and $U_b \cong {\mathbb {R}}$ is the center of $W_b$. Let $L_b = \frac{N_b}{W_b}$. This reductive group can be embedded as a Levi subgroup in $N_b$, so that $N_b = W_b \cdot L_b$. Moreover, we have a further decomposition $L_b = U(n-1) \times {\mathbb {R}}$. (all this description can be obtained e.g. by specializing the discussion of [3, Section 1.3] or [2, Section 4.2] to the case of the ball).

This Lie theoretic description of $N_b$ can be understood more easily by expressing the action of the previous groups on the horoballs tangent to b. Let $(S_b^{(N)})_{N \ge 0}$ be the family of these horoballs. Each $S_b^{(N)} \subset {\mathbb {B}}^n$ can be described as an open subset in a Siegel domain of the third kind, as follows:

$$\begin{aligned} S_b^{(N)} \simeq \{ (z', z_n) \in {\mathbb {C}}^{n-1} \times {\mathbb {C}} \; | \; \mathrm{Im}\, z_n > || z ' ||^2 + N \}. \end{aligned}$$

(A.1.1)

We have $S_b^{(0)} \cong {\mathbb {B}}^n$, and when $b = (0, ..., 0, 1)$, the change of coordinates between the two descriptions of the ball is given by the Cayley transform

$$\begin{aligned}&(w_1,\ldots , w_{n-1}, w_n) \in {\mathbb {B}}^n \mapsto (z', z_n)\\&\quad = \left( \frac{w_1}{1 - w_n}, \ldots , \frac{w_{n-1}}{1 - w_n}, i \frac{1 + w_n}{1 - w_n}\right) \in S^{(0)}_{(0, \dots , 0, 1)}. \end{aligned}$$

Now, if $g \in W_b$, we can write $g = (s, a)$ accordingly to the decomposition $W_b \overset{sets}{=} U_b \times A_b$ ($U_b \cong {\mathbb {R}}$, $A_b \cong {\mathbb {C}}^{n-1}$), and we have, for any $(w', w_n) \in S_b^{(N)}$:

$$\begin{aligned} g \cdot (z', z_n) = (z' + a, z_n + i||a||^2 + 2 i {\overline{a}} \cdot z' + s). \end{aligned}$$

(A.1.2)

We check easily that $S^{(N)}_b$ is preserved by $W_b$. Also, for any $g \in L_b \simeq U(n-1) \times {\mathbb {R}}$, we can write $g = (r, t)$, and we then have

$$\begin{aligned} g \cdot (z', z_n) = (e^{t} (r \cdot z'), e^{2t} z_n). \end{aligned}$$

(A.1.3)

We are now ready to describe the quotients of $S_b^{(N)}$ by the action of $\Gamma ' \cap N_b$. Note first that since $\Gamma '$ is neat, we have $\Gamma ' \cap N_b \subset W_b$. Then, by the discussion above, we obtain a decomposition as sets $N_b \overset{sets}{=} ({\mathbb {C}}^{n-1} \times {\mathbb {R}}) \times (U(n-1) \times {\mathbb {R}})$, in which the elements of $\Gamma ' \cap N_b$ can be written as $(a, t, \mathrm{Id}, 0)$. It also follows from [44] that $\Gamma ' \cap U_b = {\mathbb {Z}} \tau$ for some $\tau \in U_b \simeq {\mathbb {R}}$. This last fact permits to form the quotient $G_b^{(N)} = \frac{S_b^{(N)}}{U_b \cap \Gamma '}$; using (A.1.1), we can also express the latter quotient as an open subset of ${\mathbb {C}}^{n-1} \times {\mathbb {C}}^*$:

$$\begin{aligned} G_b^{(N)} = \{ (w', w_n) \in {\mathbb {C}}^{n-1} \times {\mathbb {C}}^*\;\ | \;\; |w_n| e^{\frac{2\pi }{\tau } ||w'||^2} < e^{- \frac{2\pi }{\tau }N} \}, \end{aligned}$$

and the quotient is then realized by the map $(z', z_n) \in S_b^{(N)} \rightarrow (z', e^{\frac{2i\pi }{\tau } z_n}) \in G_b^{(N)}$.

The group $\Lambda _b := \pi (\Gamma ' \cap W_b) \subset {\mathbb {C}}^{n-1}$ is an abelian lattice, acting on $G_b^{(N)} \subset {\mathbb {C}}^{n-1} \times {\mathbb {C}}^*$ as

$$\begin{aligned} a \cdot (z', z_n) = (z' + a, e^{-\frac{2\pi }{\tau } ||a||^2 - \frac{4\pi }{\tau } {\overline{a}} \cdot z'} z_n), \end{aligned}$$

Clearly, the closure $\overline{G_b^{(N)}}$ in ${\mathbb {C}}^n$ is an open neighborhood of ${\mathbb {C}}^{n-1} \times \{0\}$. We can form the quotient

$$\begin{aligned} \Omega ^{(N)}_b = \frac{\overline{G_b^{(N)}}}{\Lambda _b} \end{aligned}$$

which is then isomorphic to a tubular neighborhood of the abelian variety $\frac{{\mathbb {C}}^{n-1}}{\Lambda _b}$ in some negative line bundle. Finally, the toroidal compactification $X'$ can be obtained by glueing the open varieties $\Omega _b^{(N)}$ to $U'$ (as b runs among a system of representatives of the rational boundary components, and N is large enough).

Our claims about X can be derived from the following lemma.

Lemma A.2

Let $b \in \partial {\mathbb {B}}^n$ be a $\Gamma '$-rational boundary component, and let $g \in \Gamma$. Then the point $b' = g \cdot b$ is also $\Gamma '$-rational, and there exists $N, N' > 0$, for which g induces an isomorphism $S_b^{(N)} \overset{g}{\rightarrow } S_{b'}^{(N')}$, yielding in turn a unique compatible biholomorphism $\Omega _b^{(N)} \rightarrow \Omega _{b'}^{(N')}$.

Proof

As $\Gamma '$ is torsion free, a point $z \in \partial {\mathbb {B}}^n$ is $\Gamma '$-rational if and only if $W_b \cap \Gamma ' \ne \{ e \}$ (see [44, §1.3]). Since g normalizes $\Gamma '$, we have $g(W_b \cap \Gamma ')g^{-1} \subset W_{b'} \cap \Gamma '$ so $b'$ is $\Gamma '$-rational if b is.

As for our second claim, since the set of horoballs is preserved by the action of PU(n, 1), we may find $N, N'$ such that g induces a isomorphism $S_b^{(N)} \rightarrow S_{b'}^{(N)}$. Let $(x', x_n)$ (resp. $(y', y_n)$) be standard coordinates on $S_b^{(N)}$ (resp. $S_b^{(N')}$) as in (A.1.1), chosen so that $(y', y_n) = (x', x_n) \circ u$ for some $u \in U(n)$ satisfying $u \cdot b' = b$. Then $ug \in N_b$, and by (A.1.2) and (A.1.3), we have $(y', y_n) \circ g = f(x', x_n)$ for some affine map f.

Since g normalizes $\Gamma '$, we have $g (\Gamma ' \cap U_b) g^{-1} = \Gamma ' \cap U_{b'}$, so the map $S_b^{(N)} \overset{g}{\rightarrow } S_{b'}^{(N')}$ passes to the quotient to give a map ${\widetilde{g}} : G_b^{(N)} {\rightarrow } G_{b'}^{(N')}$. Using an explicit expression for the affine map f, we find an (a priori multivaluate) expression for ${\widetilde{g}}$ as

$$\begin{aligned} (z', z_n) \in G_b^{(N)} \overset{{\widetilde{g}}}{\mapsto } (A \cdot z' + u \log z_n + z_0', \; C\, z_n^a \, e^{b \cdot z'}) \in G_{b'}^{(N')} \end{aligned}$$

for some $A \in M_{n-1}({\mathbb {C}})$, some vectors $u, b, z_0' \in {\mathbb {C}}^{n-1}$ and $C, a \in {\mathbb {C}}$. The formula above induces a well-defined, invertible map $G_b^{(N)} \rightarrow G_{b'}^{(N')}$, so we have $u=0, a=\pm 1$. This implies that ${\widetilde{g}}$ extends holomorphically to ${\widetilde{g}} : \overline{G_b^{(N)}} \rightarrow \overline{G_{b'}^{(N')}}$. Finally, as g normalizes $\Gamma '$, ${\widetilde{g}}$ passes to the quotient by $\Lambda _{b}$ and $\Lambda _{b'}$, giving a biholomorphism $\Omega _b^{(N)} \rightarrow \Omega _{b'}^{(N')}$. $\square$

Going back to the proof of Proposition A.1, we see that Lemma A.2 permits to define a unique action of the quotient $G = \frac{\Gamma }{\Gamma '}$ on $X'$, compatible with its natural action on $U'$. We then let $X:= \frac{X'}{G}$. The following lemma ends the proof of Proposition A.1, and clarifies the link with the construction of [2].

Lemma A.3

The variety X defined above does not depend on the choice of $\Gamma '$. When $\Gamma$ is arithmetic, X coincides with the toroidal compactification of U constructed in [2].

Proof

Let $\Gamma ', \Gamma '' \subset \Gamma$ be two neat lattices of finite index, and let us show that the varieties constructed from $\Gamma '$ and $\Gamma ''$ are the same. Since $\Gamma \cap \Gamma '$ also has finite index in $\Gamma$, we may assume $\Gamma '' \subset \Gamma '$. By Lemma A.2, the action of two lattices $\Gamma '' \subset \Gamma '$ are compatible with each other over each set $\Omega _b^{(N)}$, which suffices to prove the first point.

Let us prove the second point. The construction of the toroidal compactification of [2] depends on a certain choice of $\Gamma$-admissible polyhedra for each rational boundary component (see [2, Definition 5.1]). In the case of the ball, since $\dim _{{\mathbb {R}}} U_b = 1$ for any $b \in \partial {\mathbb {B}}^n$, there is only one such possible choice (cf. [loc. cit., Theorem 4.1.(2)]). The claim now follows from the functoriality of compatible toroidal compactifications (see [28, Lemma 2.6]), since “choices” of polyhedra admissible for two lattices $\Gamma ' \subset \Gamma$ are thus automatically compatible with each other. $\square$

Note that even though this construction of X is well adapted to our purposes, it should not be used to define X as an orbifold, as it has the drawback of producing artificial ramification orders along the boundary components of X. As explained in [21], a better way of proceeding is to construct directly open neighborhoods of the components of $X - U$ as stacks, before glueing them to U.

1.2 A.2. Main results

Let us first begin with the following lemma.

Lemma A.4

Let Y be the toroidal compactification of the ball quotient $U:=\frac{{\mathbb {B}}^n}{\Gamma }$ by a torsion free lattice $\Gamma \subset PU(n,1)$ whose parabolic isometries are all unipotent. Let X be another projective compactification of U, and assume that $X$ has at most klt singularities.

Then the identity map of U extends to a birational morphism $f:X\rightarrow Y$.

Proof

The identity map of U defines a birational map $f:X\dashrightarrow Y$. Assume by contradiction that f is not regular. One can take a resolution of indeterminacies $\mu :{\tilde{X}}\rightarrow X$ for f so that $\mu |_{\mu ^{-1}(U)}:\mu ^{-1}(U)\xrightarrow {\sim }U$ is an isomorphism:

By the rigidity result (see [17, Chapter 3, Lemma 1.15]), there is a fiber $\mu ^{-1}(z)$ with $z\in D$ which is not contracted by ${\tilde{f}}$. Clearly, we have ${\tilde{f}}(\mu ^{-1}(z)) \subset Y - U$.

Since X has klt singularities, the work of Hacon–McKernan [29] implies that every fiber of $\mu$ is rationally chain connected. Thus, ${\widetilde{f}}(\mu ^{-1}(z))$ is a point since abelian varieties do not contain rational curves. This gives a contradiction. $\square$

Remark A.5

If we make the more restrictive hypothesis that $X$ has at most quotient singularities, we can replace the use of [29] by the work of Kollar [36], which implies that each fiber of $\mu$ is simply connected. As $Y-U$ is a disjoint union of abelian varieties, this also implies that the image of ${\tilde{f}} : \mu ^{-1}(z) \rightarrow Y - U$ must be a point.

Let us introduce a natural class of pairs under which our rigidity theorem will hold.

Definition A.6

Let (X, D) be a pair consisting of normal algebraic variety and a reduced divisor. We say that (X, D) has algebraic quotient singularities if it admits a finite affine cover $(X_i)_{i \in I}$, such that each $(X_i, D \cap X_i)$ is the quotient of a smooth SNC pair $(U_i, D_i)$ by a finite group $G_i$ leaving $D_i$ invariant.

Note that for any lattice $\Gamma \subset \mathrm{Aut}({\mathbb {B}}^n)$, if X is the toroidal compactification of $U = \frac{{\mathbb {B}}^n}{\Gamma }$ described in Section 1, then $(X, X - U)$ has algebraic quotient singularities.

We can now state our main result as follows.

Theorem A.7

Let $U:=\frac{{\mathbb {B}}^n}{\Gamma }$ be an n-dimensional ball quotient by a torsion free lattice $\Gamma \subset PU(n,1)$. Let X be a klt compactification of U, and let $D:=X-U$.

Let $D^{(1)} \subset D$ be the divisorial part of D. If the Kähler–Einstein metric $\omega$ on the bundle $T_{X}(-\log D^{(1)})|_{U}$ is adapted to log order near the generic point of any component of $D^{(1)}$, then (X, D) identifies with the toroidal compactification of U.

Remark A.8

• (1) Under the more restrictive assumption that (X, D) has algebraic quotient singularities, the use of Lemma A.4 in our proof below can be made without appealing to the difficult result of [29] (see Remark A.5).

• (2) As an easy consequence of Theorem A.7, we can remark that there is no klt compactification X of U such that $X - U$ has codimension $\ge 2$.

Corollary A.9

With the same assumptions as in Theorem A.7, if X is smooth and D has simple normal crossings, then D is in fact smooth, and each component is a smooth quotient of an abelian variety A by some finite group acting freely on A.

Let us prove Theorem A.7. Let $\Gamma '\subset \Gamma$ be a subgroup of finite index so that all parabolic elements of $\Gamma '$ are unipotent. Writing $U':=\frac{{\mathbb {B}}^n}{\Gamma '}$, this gives a finite étale surjective morphism $U'\rightarrow U$.

Let $X'$ be the normalization of X in the function field of $U'$: this is a normal projective variety $X'$ compactifying $U'$, with a compatible finite surjective morphism $\mu :X'\rightarrow X$ (see e.g. [1, Chapter 12, §9]). Since klt singularities are preserved under finite surjective morphisms, the variety $X'$ has at most klt singularities (see [31, Corollary 5.20]).

Remark A.10

If $(X, D)$ has algebraic quotient singularities, one sees easily that this is also the case for $X'$. To see this, form the fiber product $Z' = Z \times _{X} X'$, where $Z \rightarrow X$ is an affine covering as in Definition A.6. By [37, Theorem 2.23], the variety $Z'$, endowed with it natural boundary divisor, has algebraic quotient singularities. Finally, Lemma A.14 shows that $Z' \rightarrow X'$ is a quotient map, which gives the result.

Let $Y'$ be the toroidal compactification of $U'$, so that the boundary $A := Y' - U'$ is a smooth divisor.

Lemma A.11

The identity map on $U'$ extends as an isomorphism $f : X' \rightarrow Y'$. In particular, there is a finite surjective morphism $g:Y'\rightarrow X,$ which identifies with the étale and surjective map $U' \rightarrow U$ over $X-D$.

Proof

Since $X'$ is klt, Lemma A.4 shows that the identity map of $U'$ extends to a birational morphism $f:X'\rightarrow Y'$. Assume by contradiction that f is not an isomorphism. As $Y'$ is smooth, it follows from [31, Corollary 2.63] that the exceptional set $\mathrm{Ex}(f)$ is of pure codimension one. Thus, the birational morphism f must contract an irreducible divisorial component $E$ of the boundary $D':= X'-U'$.

Denote by $D^{\mathrm{sing}}$ the singular locus of D, and let $\omega ':=\mu ^*\omega$, be the canonical Kähler Einstein metric on $U'$. Lemma A.12 below shows that $\omega '$ is adapted to log-order for $T_{X'^\circ }(-\log E^\circ )$, where $X'^\circ :=\mu ^{-1}(X-D^{\mathrm{sing}})$, and $E^\circ :=X'^\circ \cap E$. We are going to derive a contradiction with the fact the E is contracted. Let $A_1$ be the component of A containing f(E). We can take admissible coordinates $({\mathcal {W}};z_1,\ldots ,z_n)$ and $({\mathcal {U}};w_1,\ldots ,w_n)$ centered at some well-chosen $x'\in E\cap X'^\circ$ and $y:=f(x')\in A_1$ respectively so that $f({\mathcal {W}})\subset {\mathcal {U}}$, and $f|_{E}:E\rightarrow f(E)$ is smooth at $x'$. Denote by $(f_1(z),\ldots ,f_n(z))$ the expression of f within these coordinates. Then if the admissible coordinates are chosen properly, one has

$$\begin{aligned} (f_1(z),\ldots ,f_n(z))=(z_1^{m_{1}}g_1(z),\ldots ,z_1^{m_k}g_k(z),g_{k+1},\ldots ,g_{n} ) \end{aligned}$$

where $g_1(z),\ldots ,g_k(z)$ are holomorphic functions defined on ${\mathcal {W}}$ so that $g_i(z)\ne 0$ and $m_{i}\ge 1$ for $i=1,\ldots ,k$. Since E is exceptional, one has $k\ge 2$. By the norm estimate in [44, eq. (8) on p. 338], the Kähler–Einstein metric $\omega$ for $T_{Y}(-\log A)|_{U}$ is adapted to log order. More precisely, one has

$$\begin{aligned} |d w_2|_{\omega ^{-1}}^2\sim (-\log |w_1|^2). \end{aligned}$$

Since $f^*d\log w_2=m_2d\log z_1+d\log g_2(w),$ one thus has the following norm estimate

$$\begin{aligned} |d\log z_1|_{\omega '^{-1}}^2\ge \frac{1}{m^2_2}\mu ^*|d\log w_2|_{\omega ^{-1}}^2-\frac{1}{m^2_2}\mu ^*|\frac{dg_2}{g_2}|_{\omega ^{-1}}^2\ge \frac{C(-\log |z_1|^2)}{|z_1|^{2m_2}} \end{aligned}$$

for some constants $C>0$. Since $d\log z_1$ is a local nowhere vanishing section for $\Omega _{X'}^1(\log D')$, we conclude that the metric $\omega '^{-1}$ for $\Omega _{X'^\circ }^{1}(-\log D'^\circ )$ is not adapted to log order, and so is $\omega '$ for $T_{X'^\circ }(-\log D'^\circ )$. This gives a contradiction, and ends the proof of the lemma. $\square$

Lemma A.12

With the notations of the proof of Lemma A.11, the metric $\omega '$ is adapted to log-order for $T_{X'^\circ }(-\log E^\circ )$.

Proof

Write ${\mathcal {W}}:=\mu ^{-1}({\mathcal {V}})$. Since $\mu |_{{\mathcal {W}}-D'}:{\mathcal {W}}-D'\rightarrow {\mathcal {V}}-D$ is a finite unramified cover, the image of $(\mu |_{{\mathcal {W}}-D'})_*\big (\pi _1({\mathcal {W}}-D')\big )$ is a subgroup of $\pi _1({\mathcal {W}}-D)\simeq {\mathbb {Z}}$ of index m. Letting $\nu (z_{1}, \cdots , z_{n}) = (z_{1}^{m}, z_{2}, \ldots , z_{n})$, one has thus the following commutative diagram

so that $h^\circ _{\Delta ^*\times \Delta ^{n-1}}:\Delta ^*\times \Delta ^{n-1}\rightarrow {\mathcal {W}}\cap U'$ is an isomorphism. By the Riemann removable singularities theorem, h extends to a holomorphic map $h:\Delta ^n\rightarrow {\mathcal {W}}$, which is easily seen to be surjective with finite fibers. Hence h is moreover biholomorphic. $({\mathcal {W}};z_1,\ldots ,z_n;h)$ is therefore an admissible coordinate centered at $x'$ with $(z_1=0)={\mathcal {W}}\cap D'$ so we can now identify $\mu$ with $\nu$. Hence,

$$\begin{aligned} \mu ^*d\log x_1=md\log z_1, \mu ^*dx_2=d z_2, \ldots , \mu ^*dx_n=dz_n, \end{aligned}$$

and the frame $(d\log z_1,dz_2,\ldots ,dz_n)$ for $\Omega _{X'}^1(\log D')|_{{\mathcal {W}}}$ is adapted to log order. This shows that the metric $\omega '$ is adapted to log order for $T_{X'^\circ }(-\log D'^\circ )$. $\square$

We can now conclude the case discussed in Corollary A.9, where (X, D) is assumed to be a smooth log-pair. Since the boundary of $Y' - U'$ is smooth, this implies that D must also be smooth. Moreover, for each connected component $A_i$ of A, there is a connected component $D_j$ of D so that $g|_{A_i}:A_i\rightarrow D_j$ is a finite surjective morphism, which is also étale by the local description of $\mu$ given in the proof of Lemma A.12. Hence in this case, $D_i$ is a smooth quotient of an abelian variety by the free action of some finite group $G_i$. This suffices to establish Corollary A.9.

The proof of Theorem A.7 will be complete with the following lemma.

Lemma A.13

The variety X identifies with the quotient of $Y'$ by the natural action of $G = \frac{\Gamma }{\Gamma '}$. In particular, $X \cong Y$.

Proof

This result comes right away from Lemma A.14 below, taking $M = Y'$, $N = X$, and $G = {\mathcal {G}}$. Remark that we have $R(Y')^G = R(U')^G = R(U) = R(X)$ since $U = \frac{U'}{G}$. For the second statement, remark that by Proposition A.1, the toroidal compactification Y of U also identifies with the quotient $\frac{Y'}{G}$. Thus, there is an isomorphism $Y \cong X$ compatible with the identity on U. Theorem A.7 is proved. $\square$

Lemma A.14

Let $f : M \rightarrow N$ be a finite surjective morphism between two normal reduced schemes. Assume that M is acted upon by a finite groupoid ${\mathcal {G}}$, and that f is ${\mathcal {G}}$-invariant. Suppose in addition that $R(M)^{{\mathcal {G}}} = R(N)$, where R(M), R(N) are the rings of rational functions on M, N. Then N is the quotient of M by ${\mathcal {G}}$.

Proof

It suffices to show that $f_*({\mathcal {O}}_M)^{{\mathcal {G}}} = {\mathcal {O}}_N$. This is a local statement on the base, so we may assume that $N = \mathrm{Spec}\, A$, $M = \mathrm{Spec}\, B$, with A is integrally closed. We have $B^{{\mathcal {G}}} \subset R(B)^{{\mathcal {G}}} = R(A)$ by assumption. Since $A \subset B$ is finite, and A is integrally closed, this implies $B^{{\mathcal {G}}} \subset A$, as required. $\square$