On the Kodaira dimension of orthogonal modular varieties

Abstract

We prove that up to scaling there are only finitely many integral lattices L of signature (2, n) with $n\ge 21$ or $n=17$ such that the modular variety defined by the orthogonal group of L is not of general type. In particular, when $n\ge 108$, every modular variety defined by an arithmetic group for a rational quadratic form of signature (2, n) is of general type. We also obtain similar finiteness in $n\ge 9$ for the stable orthogonal groups. As a byproduct we derive finiteness of lattices admitting reflective modular form of bounded vanishing order, which proves a conjecture of Gritsenko and Nikulin.

Appendix A. Singularities over the 0-dimensional cusps

Let L be a lattice of signature (2, n). Let $\Gamma$ be a finite index subgroup of ${\mathrm{O}^+}(L)$ and $\mathcal {F}(\Gamma )=\Gamma \backslash {\mathcal {D}_{L}}$ the associated modular variety. For simplicity we assume $-1\in \Gamma$, which does not affect $\mathcal {F}(\Gamma )$.

0-dimensional cusps of the Baily–Borel compactification of $\mathcal {F}(\Gamma )$ correspond to primitive isotropic vectors l in L up to the $\Gamma$-action. We write $M_l=l^{\perp }\cap L/{\mathbb {Z}}l$. Let $N(l)_{{\mathbb {Q}}}$ be the stabilizer of l in ${\mathrm{O}^+}(L_{{\mathbb {Q}}})$. The unipotent radical $U(l)_{{\mathbb {Q}}}$ of $N(l)_{{\mathbb {Q}}}$ consists of the Eichler transvections $E_{l,m}$, $m\in (M_l)_{{\mathbb {Q}}}$, which are defined by (see, e.g., [33] §3.7 or [15] §3.1)

$$\begin{aligned} E_{l,m}(v) = v - (\tilde{m}, v)l + (l, v)\tilde{m} - \frac{1}{2}(m, m)(l, v)l, \qquad v\in L_{{\mathbb {Q}}}, \end{aligned}$$

where $\tilde{m}\in l^{\perp }\cap L_{{\mathbb {Q}}}$ is a lift of m. Thus $U(l)_{{\mathbb {Q}}}$ is canonically identified with $(M_l)_{{\mathbb {Q}}}$. We have the fundamental exact sequence

$$\begin{aligned} 0 \rightarrow U(l)_{{\mathbb {Q}}} \rightarrow N(l)_{{\mathbb {Q}}} {\mathop {\rightarrow }\limits ^{\pi }} {\mathrm{O}^+}((M_l)_{{\mathbb {Q}}})\rightarrow 1. \end{aligned}$$

If we choose a splitting $f:L_{{\mathbb {Q}}}\simeq U_{{\mathbb {Q}}} \oplus (M_l)_{{\mathbb {Q}}}$ with $f(l)\in U_{{\mathbb {Q}}}$, we obtain a section of $\pi$ and thus a non-canonical isomorphism

$$\begin{aligned} \varphi _{f} : N(l)_{{\mathbb {Q}}} {\mathop {\rightarrow }\limits ^{\simeq }} {\mathrm{O}^+}((M_l)_{{\mathbb {Q}}})\ltimes U(l)_{{\mathbb {Q}}} = {\mathrm{O}^+}((M_l)_{{\mathbb {Q}}})\ltimes (M_l)_{{\mathbb {Q}}}. \end{aligned}$$

(A.1)

We write $N(l)_{{\mathbb {Z}}}=N(l)_{{\mathbb {Q}}}\cap \Gamma$, $U(l)_{{\mathbb {Z}}}=U(l)_{{\mathbb {Q}}}\cap \Gamma$ and $\overline{N(l)}_{{\mathbb {Z}}}= N(l)_{{\mathbb {Z}}}/U(l)_{{\mathbb {Z}}}$. For instance, when $\Gamma ={\widetilde{\mathrm{O}}^+}(L)$ with L even, we have $U(l)_{{\mathbb {Z}}}=M_l$.

Choose representatives $l_1, \ldots , l_N\in L$ of primitive isotropic vectors modulo $\Gamma$. We put a ${\mathbb {Z}}$-structure on $(M_i)_{{\mathbb {R}}}=(M_{l_i})_{{\mathbb {R}}}$ by $U(l_i)_{{\mathbb {Z}}}$. Let $\mathcal {C}_i$ be the union of the positive cone $(M_i)_{{\mathbb {R}}}^{+}$ of $(M_i)_{{\mathbb {R}}}$ and the rays ${\mathbb {R}}_{\ge 0}m$ for $m\in (M_i)_{{\mathbb {Q}}}$ in the boundary of $(M_i)_{{\mathbb {R}}}^{+}$. According to [1], toroidal compactification of $\mathcal {F}(\Gamma )$ can be constructed by choosing for each i an $\overline{N(l_{i})}_{{\mathbb {Z}}}$-admissible fan $\Sigma _i$ in $(M_i)_{{\mathbb {R}}}$ with $|\Sigma _i|=\mathcal {C}_i$. (There is no ambiguity of choice at the 1-dimensional cusps, and the choices of fan at each i are independent.) By [1], we can choose $\Sigma _i$ to be regular with respect to $U(l_i)_{{\mathbb {Z}}}$.

Our purpose in this appendix is to supplement a proof of the following

Theorem A.1

([12]) When the fans $\Sigma _i$ are regular, the toroidal compactification $\mathcal {F}(\Gamma )^{\Sigma }$ associated to $\Sigma =(\Sigma _i)$ has canonical singularities at the points lying over the 0-dimensional cusps.

This theorem was first found by Gritsenko–Hulek–Sankaran ([12] §2.2), but as we explain later (Remark A.8), their proof needs to be modified.

Since Tai [38], proof of such a statement consists of the following steps:

• (1) find finite linear quotient models V / G of the singularities;

• (2) the Reid–Shepherd-Barron–Tai criterion [32, 38] tells whether V / G has canonical singularities in terms of the eigenvalues of each element g of G;

• (3) so we are reduced to analysing V as a representation of the cyclic group $\langle g\rangle$ for each $g\in G$.

In Sect. A.1 we first present a certain class of representations V of the cyclic groups ${\mathbb {Z}}/m$ and show that $V/({\mathbb {Z}}/m)$ has canonical singularities by the RST criterion. This part is elementary linear algebra and independent of modular varieties. We then study local model V / G of the toroidal compactification and show (Sect. A.3) that for each $g\in G$, $V|_{\langle g\rangle }$ belongs to the class of representations we have studied in advance.

1.1 Some cyclic quotients

Let $G={\mathbb {Z}}/m$ be the standard cyclic group of order $m>1$. By a representation of G we always mean a finite-dimensional complex representation. For $\mu \in \frac{1}{m}{\mathbb {Z}}/{\mathbb {Z}}$ we denote by $\chi _{\mu }$ the character $G\rightarrow {\mathbb {C}}^{\times }$ that sends $\bar{1}\in G$ to $e(\mu )=\mathrm{exp}(2\pi i \mu )$. For d|m we write

$$\begin{aligned} V_d= \bigoplus _{k\in ({\mathbb {Z}}/d)^{\times }}\chi _{k/d}. \end{aligned}$$

It is classical that a representation of G defined over ${\mathbb {Q}}$ is isomorphic to $\oplus _{i}V_{d_i}$ for some $d_i|m$ (see [36] §13.1). When $m=m'm''$, we can view ${\mathbb {Z}}/m'$ as a subgroup of ${\mathbb {Z}}/m$ of index $m''$ by multiplication by $m''$:

$$\begin{aligned} {\mathbb {Z}}/m' \simeq m''{\mathbb {Z}}/m \subset {\mathbb {Z}}/m. \end{aligned}$$

If we put $d''=(d, m'')$ and $d'=d/d''$, the restriction of $V_d$ to ${\mathbb {Z}}/m'\subset {\mathbb {Z}}/m$ is isomorphic to a direct sum of copies of $V_{d'}$.

If d|m and $\mu \in \frac{1}{m}{\mathbb {Z}}/{\mathbb {Z}}$, we write $W_{d,\mu }$ for the G-representation

$$\begin{aligned} W_{d,\mu } = {\mathbb {C}}[{\mathbb {Z}}/d]\otimes \chi _{\mu } = \bigoplus _{k\in {\mathbb {Z}}/d} \chi _{k/d} \otimes \chi _{\mu }. \end{aligned}$$

The eigenvalues of $\bar{1}\in G$ on $W_{d,\mu }$ are the $e(\mu )$-shift of the d-th roots of 1. The restriction rule is as follows.

Lemma A.2

Let $m=m'm''$. We put $\mu '=m''\mu$, $d''=(d, m'')$ and $d'=d/d''$. The restriction of $W_{d,\mu }$ to ${\mathbb {Z}}/m' \subset {\mathbb {Z}}/m$ is isomorphic to $(W_{d',\mu '})^{\oplus d''}$.

Proof

We have $\chi _{\mu }|_{{\mathbb {Z}}/m'}=\chi _{\mu '}$. The image of ${\mathbb {Z}}/m'$ by the reduction map ${\mathbb {Z}}/m \rightarrow {\mathbb {Z}}/d$ is $d''{\mathbb {Z}}/d \simeq {\mathbb {Z}}/d'$, and ${\mathbb {C}}[{\mathbb {Z}}/d]|_{{\mathbb {Z}}/d'} \simeq {\mathbb {C}}[{\mathbb {Z}}/d']^{\oplus d''}$. $\square$

Example A.3

Let $g\in \mathrm{GL}_{d}({\mathbb {C}})$ be the linear transformation

$$\begin{aligned} g = \mathrm{diag}(e(\alpha _1), \ldots , e(\alpha _d)) \circ (2, 3, \ldots , d, 1) \end{aligned}$$

where $\alpha _i\in {\mathbb {C}}/{\mathbb {Z}}$. Let $m=\mathrm{ord}(g)<\infty$. The eigenpolynomial of g is $x^d-e(\sum _{i}\alpha _i)$. If $\mu \in {\mathbb {Q}}/{\mathbb {Z}}$ is an element with $d\mu =\sum _{i}\alpha _i$, it follows that ${\mathbb {C}}^d\simeq W_{d,\mu }$ as representations of $\langle g \rangle \simeq {\mathbb {Z}}/m$. When $m=m'm''$, the restriction of the cyclic permutation $(2,\ldots , d, 1)$ to $\langle g^{m''} \rangle \simeq {\mathbb {Z}}/m'$ splits into $d''$ copies of cyclic permutation of length $d'$. In Sect. A.3, $W_{d,\mu }$ and Lemma A.2 will appear in this form.

Based on Lemma A.2, we make the following definition.

Definition A.4

Let U be a representation of G defined over ${\mathbb {Q}}$. Let $\{ (d_i, \mu _i) \}_i$ be a finite set of pairs $(d_i, \mu _i)$ with $d_i|m$ and $\mu _i\in \frac{1}{m}{\mathbb {Z}}/{\mathbb {Z}}$. We say that $\theta =(U, (d_i, \mu _i)_i)$ is an admissible data for G if for every nontrivial subgroup $G'\simeq {\mathbb {Z}}/m'$ of G of index $m''$, either $U|_{G'}$ is nontrivial or $d_i':=d_i/(d_i, m'')>1$ for some i.

If $G'\simeq {\mathbb {Z}}/m'$ is a subgroup of G of index $m''$, we define the restriction of the data $\theta =(U, (d_i, \mu _i)_i)$ to $G'$ by

$$\begin{aligned} \theta |_{G'} = ( U|_{G'}, ((d_i', \mu _i')^{\times d_i''})_i ), \end{aligned}$$

(A.2)

where $\mu _i'=m''\mu _i$, $d_{i}''=(d_i, m'')$ and $d_{i}'=d_i/d_{i}''$. Admissibility condition says that $\theta |_{G'}$ is not $((V_{1})^{\oplus a}, (1, \lambda _j)_j)$ if $m'>1$. We have $(\theta |_{G'})|_{G''}=\theta |_{G''}$ for $G''\subset G' \subset G$. Hence admissibility of the data $\theta$ for G implies that of the data $\theta |_{G'}$ for $G'$.

To an admissible data $\theta =(U, (d_i, \mu _i)_i)$ we associate the G-representation

$$\begin{aligned} V_{\theta } = U \oplus \bigoplus _{i} W_{d_i,\mu _i}. \end{aligned}$$

Lemma A.2 shows that $(V_{\theta })|_{G'}\simeq V_{(\theta |_{G'})}$ as $G'$-representations.

Recall that a linear transformation of finite order is called quasi-reflection (or pseudo-reflection) if all but one of its eigenvalues are 1.

Lemma A.5

Let $\theta =(U, (d_i, \mu _i)_i)$ be an admissible data for $G={\mathbb {Z}}/m$. Suppose that G contains an element g acting by quasi-reflection on $V_{\theta }$. Let $m'=\mathrm{ord}(g)$ and $m''=m/m'$. Then g acts on $V_{\theta }$ by reflection, so $m'=2$, and $m''$ is odd. The reflective vector $\delta \in V_{\theta }$ of g is also an eigenvector of G, and contained in either U or $W_{d_i,\mu _i}$ for some i. When $\delta \in U$, we have ${\mathbb {C}}\delta \simeq V_2$ as G-representations. When $\delta \in W_{d_i, \mu _i}$, we have $d_i=2$.

Proof

We can write $g=g_0^{m''}$ for a generator $g_0$ of G. There is only one eigenvalue $\lambda$ of $g_0$ such that $\lambda ^{m''}\ne 1$, and the remaining eigenvalues of $g_0$ are $m''$-th roots of 1. In particular, $\lambda$ has multiplicity 1. Let $\delta$ be a generator of the 1-dimensional $\lambda$-eigenspace of $g_0$. Since every eigenvalue of $g_0$ occurs in U or one of $W_{d_i,\mu _i}$, the multiplicity one property implies that $\delta \in U$ or $\delta \in W_{d_i,\mu _i}$ for some i.

First consider the case $\delta \in U$. Again by the multiplicity one, $\delta$ is contained in a sub G-representation isomorphic to $V_d$ for some d|m. Since $V_d|_{\langle g \rangle } \simeq (V_{d'})^{\oplus a}$ for $d'=d/(d, m'')$ while g acts on this space by quasi-reflection, we must have $d'=2$ and $a=1$. Hence $d=2$, namely ${\mathbb {C}}\delta \simeq V_2$ as G-representations. Since $(-1)^{m''}=-1$, $m''$ is odd.

Next consider the case $\delta \in W_{d_i,\mu _i}$. Since g acts trivially on U and $W_{d_j,\mu _j}$ for $j\ne i$, the admissibility condition says that we must have $d_i'>1$ in $W_{d_i,\mu _i}|_{\langle g \rangle }\simeq (W_{d_i',\mu _i'})^{\oplus d_i''}$. On the other hand, g has only one $\ne 1$ eigenvalue on $W_{d_i,\mu _i}$, so $d_i'=2$, $d_i''=1$ and $\mu _i'=0$ or 1 / 2. Hence $d_i=2$ and g acts by reflection. Since $W_{2,\mu _i}|_{\langle g\rangle } \simeq W_{2,\mu _i'}$, $m''$ is odd. $\square$

We can now present the main result of this subsection.

Proposition A.6

Let $\theta =(U, (d_i, \mu _i)_i)$ be an admissible data for $G={\mathbb {Z}}/m$. Then $V_{\theta }/G$ has canonical singularities.

Proof

If V is a representation of G and $g\in G$ has eigenvalues $e(\alpha _1), \ldots , e(\alpha _n)$ with $0\le \alpha _i <1$, the Reid-Tai sum of g is defined by

$$\begin{aligned} \Sigma _V(g) = \sum _{i=1}^{n} \alpha _i. \end{aligned}$$

(Similar invariant appears in the dimension formula for modular forms: see [3, 37].) The Reid–Shepherd-Barron–Tai criterion [32, 38] says that when G contains no quasi-reflection, V / G has canonical singularities if and only if $\Sigma _V(g)\ge 1$ for every $g\ne \mathrm{id} \in G$. We apply this to $V=V_{\theta }$ or its variation.

We first consider the case G contains no reflection on $V_{\theta }$.

Lemma A.7

Let $\theta =(U, (d_i, \mu _i)_i)$ be an admissible data for $G=\langle g \rangle = {\mathbb {Z}}/m$. Assume that g does not act as reflection on $V_{\theta }$. Then $\Sigma _{V_{\theta }}(g)\ge 1$.

Proof

Let $W=\bigoplus _i W_{d_i,\mu _i}$. It is clear that $\Sigma _{V_{\theta }}(g)\ge 1$ in the following cases:

• U contains $V_d$ with $d\ge 3$ or $(V_2)^{\oplus 2}$;

• W contains $W_{d,\mu }$ with $d\ge 3$ or $W_{2,\mu }\oplus W_{2,\lambda }$;

• U contains $V_2$ and W contains $W_{2,\mu }$.

The remaining cases are

• (1) $U=V_2\oplus (V_1)^{\oplus a}$ and $W=\bigoplus _{i}W_{1,\mu _i}$;

• (2) U is trivial and $W=W_{2,\mu }\oplus \bigoplus _i W_{1,\mu _i}$.

In both cases m must be even, say $m=2m'$. If $m'=1$, the eigenvalue $-1$ has multiplicity at least 2 because g is not a reflection. Then $\Sigma _{V_{\theta }}(g)\ge 1$. We show that the case $m'>1$ does not occur. Consider the restriction of $\theta =(U, (d_i, \mu _i)_i)$ to the subgroup $G'=\langle g^2 \rangle \simeq {\mathbb {Z}}/m'$. Then $U|_{G'}$ is trivial. On the other hand, we have $W|_{G'}\simeq \bigoplus _i W_{1,2\mu _i}$ in case (1) and $W|_{G'}\simeq (W_{1,2\mu })^{\oplus 2}\oplus \bigoplus _{i} W_{1,2\mu _i}$ in case (2), in the sense of the restriction of $(d_i, \mu _i)_i$ as defined by the equation (A.2). By admissibility, we must have $m'=1$. $\square$

When G contains no reflection, we can apply this lemma to all subgroups $G'$ of G and their generators because $\theta |_{G'}$ is admissible for $G'$. By the RST criterion we obtain Proposition A.6 in this case.

We next consider the case G contains an element g acting as reflection on $V_{\theta }$. We may assume $G\ne \langle g\rangle$. Let $m''=m/2>1$ be the index of $\langle g \rangle$ in G, and $\delta$ a reflective vector of g. By Lemma A.5, $m''$ is odd, and $\delta$ is an eigenvector for G contained in U or some $W_{d_i,\mu _i}$. We write $\bar{G}<G$ for the subgroup of order $m''$. We have the decomposition $G=\bar{G}\oplus \langle g\rangle$ and $\bar{G}$ is canonically identified with $G/\langle g\rangle$. We set $\bar{V}=V_{\theta }/\langle g\rangle$, which is a $\bar{G}$-representation. We have $V_{\theta }/G\simeq \bar{V}/\bar{G}$, and we want to apply the previous step to $(\bar{V}, \bar{G})$. Note that $\bar{G}$ cannot contain a reflection because its order $m''$ is odd.

When $\delta \in U$, consider the G-decomposition $V_{\theta }=V'\oplus {\mathbb {C}}\delta$. By Lemma A.5, ${\mathbb {C}}\delta \simeq V_2$ as G-representations. Then as $\bar{G}$-representations

$$\begin{aligned} \bar{V} = V'\oplus ({\mathbb {C}}\delta )^{\otimes 2} \simeq V'\oplus V_1 \simeq V_{\theta }. \end{aligned}$$

Since $\theta |_{\bar{G}}$ is admissible for $\bar{G}$, $\bar{V}/\bar{G}\simeq V_{\theta }/\bar{G}$ has canonical singularities by the previous step.

When $\delta \in W_{d_i,\mu _i}$, we have $d_i=2$ by Lemma A.5. Since $W_{2,\mu _i}|_{\bar{G}}\simeq (W_{1,2\mu _i})^{\oplus 2}$, then $\eta = (U, (d_j, \mu _j)_{j\ne i})|_{\bar{G}}$ must be admissible for $\bar{G}$. Hence $\Sigma _{V_{\eta }}(h)\ge 1$ for every $h\ne \mathrm{id} \in \bar{G}$ by Lemma A.7. Since $V_{\eta }$ is a direct summand of $\bar{V}$, we have $\Sigma _{\bar{V}}(h)\ge 1$. Hence $\bar{V}/\bar{G}$ has canonical singularities. This finishes the proof of Proposition A.6. $\square$

1.2 Toroidal compactification

We go back to modular varieties and explain toroidal compactification over 0-dimensional cusps. We keep the notation from the beginning of this appendix. Let $l\in L$ be a primitive isotropic vector and $\mathcal {D}_l = (M_l)_{{\mathbb {R}}}+i(M_l)_{{\mathbb {R}}}^{+}$ the tube domain associated to l. We choose a vector $l'\in L_{{\mathbb {Q}}}$ with $(l, l')=1$ and identify $(M_l)_{{\mathbb {Q}}}$ with $\langle l, l' \rangle ^{\perp }\cap L_{{\mathbb {Q}}}$. As explained in Sect. 2, this induces the tube domain realization

$$\begin{aligned} \iota _{l'} : \mathcal {D}_l {\mathop {\rightarrow }\limits ^{\simeq }} {\mathcal {D}_{L}}, \qquad v\mapsto {\mathbb {C}}(l'+v-\frac{1}{2}((v, v)+(l', l'))l), \end{aligned}$$

which depends on $l'$. Via this, $U(l)_{\mathbb {Q}}\simeq (M_l)_{{\mathbb {Q}}}$ acts on $\mathcal {D}_l$ by parallel transformation. If we form the torus $T_l=(M_l)_{{\mathbb {C}}}/U(l)_{{\mathbb {Z}}}$, then $\iota _{l'}^{-1}$ maps $X_l = {\mathcal {D}_{L}}/U(l)_{{\mathbb {Z}}}$ isomorphically to the open set $\mathcal {D}_l/U(l)_{{\mathbb {Z}}} = \mathrm{ord}^{-1}( (M_l)_{{\mathbb {R}}}^{+})$ of $T_l$. The group $\overline{N(l)}_{{\mathbb {Z}}}$ acts on $X_l$ through the $N(l)_{{\mathbb {Z}}}$-action on ${\mathcal {D}_{L}}$.

The action of $N(l)_{{\mathbb {Z}}}$ on $U(l)_{{\mathbb {Q}}}\simeq (M_l)_{{\mathbb {Q}}}$ preserves the lattice $U(l)_{{\mathbb {Z}}}$. Hence if $\pi :N(l)_{{\mathbb {Q}}}\rightarrow {\mathrm{O}^+}((M_l)_{{\mathbb {Q}}})$ is the natural map, $N(l)_{{\mathbb {Z}}}$ is contained in $\pi ^{-1}({\mathrm{O}^+}(U(l)_{{\mathbb {Z}}}))$, of which $U(l)_{{\mathbb {Z}}}$ is a normal subgroup. Thus $\overline{N(l)}_{{\mathbb {Z}}}$ is canonically a subgroup of $\pi ^{-1}({\mathrm{O}^+}(U(l)_{{\mathbb {Z}}}))/U(l)_{{\mathbb {Z}}}$. By (A.1), the splitting $L_{{\mathbb {Q}}}=\langle l, l' \rangle _{{\mathbb {Q}}}\oplus (M_l)_{{\mathbb {Q}}}$ given by $l'$ induces an isomorphism

$$\begin{aligned} \varphi _{l'} : \pi ^{-1}({\mathrm{O}^+}(U(l)_{{\mathbb {Z}}}))/U(l)_{{\mathbb {Z}}} \rightarrow {\mathrm{O}^+}(U(l)_{{\mathbb {Z}}})\ltimes (U(l)_{{\mathbb {Q}}}/U(l)_{{\mathbb {Z}}}). \end{aligned}$$

The right side group is canonically a subgroup of

$$\begin{aligned} \mathrm{GL}(U(l)_{\mathbb {Z}})\ltimes (U(l)_{{\mathbb {Q}}}/U(l)_{{\mathbb {Z}}}) = \mathrm{Aut}(T_l)\ltimes (T_l)_{tor} \subset \mathrm{Aut}(T_l)\ltimes T_l. \end{aligned}$$

We thus obtain an embedding depending on $l'$

$$\begin{aligned} \varphi _{l'} : \overline{N(l)}_{{\mathbb {Z}}} \hookrightarrow \mathrm{Aut}(T_l)\ltimes T_l. \end{aligned}$$

By the definition of $\overline{N(l)}_{\mathbb {Z}}$, the projection $\varphi _{l'}(\overline{N(l)}_{{\mathbb {Z}}})\rightarrow \mathrm{Aut}(T_l)$ is injective. If we express $\varphi _{l'}(g)=(\gamma , a)\in \mathrm{Aut}(T_l)\ltimes T_l$ for $g\in \overline{N(l)}_{\mathbb {Z}}$, then $\gamma =\pi (\tilde{g})$ and $a=[\tilde{g}(l')-l']$ where $\tilde{g}\in N(l)_{{\mathbb {Z}}}$ is a lift of g.

The affine group $\mathrm{Aut}(T_l)\ltimes T_l$ acts on $T_l$ naturally: $\mathrm{Aut}(T_l)$ by torus automorphisms (fixing the identity), and $T_l$ by translation. The $\overline{N(l)}_{{\mathbb {Z}}}$-action on $X_l$ is the restriction of the action of $\mathrm{Aut}(T_l)\ltimes T_l$ on $T_l$ through $\varphi _{l'}$ and $\iota _{l'}$.

Remark A.8

In [12] p. 534, Gritsenko–Hulek–Sankaran implicitly assume that $\varphi _{l'}(\overline{N(l)}_{{\mathbb {Z}}})$ is contained in $\mathrm{Aut}(T_l)$ for some $l'\in L_{{\mathbb {C}}}$ so that the translation component $a=a_g$ is trivial for every g. If this holds, $N(l)_{{\mathbb {Z}}}$ will decompose into $\overline{N(l)}_{{\mathbb {Z}}}\ltimes U(l)_{{\mathbb {Z}}}$. However, this assumption seems to be too strong in general. For each g, $a_g$ varies holomorphically with $l'$ so that it is not 1 for generic $l'$, and it seems highly nontrivial or even impossible for general $\Gamma$ that one can find a specific $l'$ such that $a_g=1$ for all g. (Note that the isomorphism $\mathcal {D}_{L}(F)\simeq U(F)_{{\mathbb {C}}}$ in [12] p. 534 depends on the choice of a base point ${\mathbb {C}}\omega$ of $\mathcal {D}_{L}(F)$. This isomorphism is the extension of $\iota _{l'}$, and ${\mathbb {C}}\omega$ is another intersection point of ${{\mathbb P}}\langle l, l'\rangle _{{\mathbb {C}}}$ with the isotropic quadric.)

On the other hand, in the important example $\Gamma ={\widetilde{\mathrm{O}}^+}(L)$ with L even, $\varphi _{l'}(\overline{N(l)}_{{\mathbb {Z}}})$ is indeed contained in $\mathrm{Aut}(T_l)$ if $l'$ is taken from $L^{\vee }$. Hence in this case the proof of [12] works.

Now let $\Sigma _l$ be the $\overline{N(l)}_{{\mathbb {Z}}}$-admissible regular fan in $(M_l)_{{\mathbb {R}}}$ we have chosen for l. This defines a torus embedding $T_l\hookrightarrow T_{\Sigma _l}$. The partial compactification $X_{\Sigma _l}$ of $X_l$ in the direction of l is by definition the interior of the closure of $X_l$ in $T_{\Sigma _l}$. The group $\overline{N(l)}_{{\mathbb {Z}}}$ acts on $X_{\Sigma _l}$ properly discontinuously. We have a natural map

$$\begin{aligned} X_{\Sigma _l}/\overline{N(l)}_{{\mathbb {Z}}} \rightarrow \mathcal {F}(\Gamma )^{\Sigma }, \end{aligned}$$

which is locally isomorphic at the points lying over the 0-dimensional cusp ${\mathbb {C}}l$ (see [1] p. 175). Hence Theorem A.1 reduces to the following assertion (cf. [12] Theorem 2.17, where the case $G\subset \mathrm{Aut}(T)$ is considered).

Theorem A.9

Let N be a free abelian group of finite rank and $T=T_N$ be the associated torus. Let G be a finite subgroup of $\mathrm{Aut}(T)\ltimes T$ such that $G\rightarrow \mathrm{Aut}(T)$ is injective. Let $\Sigma$ be a regular fan in $N_{{\mathbb {R}}}$ preserved by G, and $T_{\Sigma }=T_{N,\Sigma }$ the torus embedding defined by $\Sigma$. Then $T_{\Sigma }/G$ has canonical singularities.

In the next subsection we prove this by reducing it to Proposition A.6. Note that the injectivity condition on $G\rightarrow \mathrm{Aut}(T)$ is essential: consider the extreme situation $G\subset T$, where one loses control of the Reid-Tai sum.

1.3 Proof of Theorem A.9

Let x be a point of $T_{\Sigma }$ and $G_x\subset G$ be the stabilizer of x. It suffices to prove that $T_xT_{\Sigma }/G_x$ has canonical singularities. By the well-known cyclic reduction [32, 38], this reduces to showing that $T_xT_{\Sigma }/\langle g\rangle$ has canonical singularities for every $g\in G_x$. We write m for the order of g. Let $\mathrm{orb}(\sigma )$ be the T-orbit x belongs to, where $\sigma$ is a regular cone in $\Sigma$. Write $g=(\gamma , a)\in \mathrm{Aut}(T)\ltimes T$. Since g preserves $\mathrm{orb}(\sigma )$, $\gamma$ preserves the cone $\sigma$, permuting its rays. The open embedding $T_{\sigma }\hookrightarrow T_{\Sigma }$ is g-equivariant, hence $T_xT_{\Sigma }=T_xT_{\sigma }$ as $\langle g\rangle$-representations. We are thus reduced to showing that $T_xT_{\sigma }/\langle g\rangle$ has canonical singularities.

Since g has finite order, we have the g-decomposition

$$\begin{aligned} T_xT_{\sigma } = T_x(\mathrm{orb}(\sigma )) \oplus N_x(\mathrm{orb}(\sigma )). \end{aligned}$$

Let $N_0={\mathbb {Z}}(\sigma \cap N)$ and $N_1=N/N_0$, which are free $\gamma$-modules. We have a natural isomorphism $\mathrm{orb}(\sigma )\simeq T_{N_1}$ so that $T_x(\mathrm{orb}(\sigma ))\simeq (N_1)_{{\mathbb {C}}}$. The rays of $\sigma$ define a basis of $N_0$, and $\gamma$ acts on $N_0$ by permuting these basis vectors. Let $(d_1, \ldots , d_N)$ be the cyclic type of this permutation ($\sum _i d_i=\mathrm{rk}(N_0)$).

Proposition A.10

• (1) Via the isomorphism $T_x(\mathrm{orb}(\sigma ))\simeq (N_1)_{{\mathbb {C}}}$, the g-action on $T_x(\mathrm{orb}(\sigma ))$ is identified with the $\gamma$-action on $(N_1)_{{\mathbb {C}}}$. In particular, it is defined over ${\mathbb {Q}}$.

• (2) As a representation of $\langle g\rangle \simeq {\mathbb {Z}}/m$, the normal space $N_x(\mathrm{orb}(\sigma ))$ is isomorphic to $\bigoplus _{i=1}^{N}W_{d_i,\mu _i}$ for some $\mu _1, \ldots , \mu _N\in {\mathbb {Q}}/{\mathbb {Z}}$.

• (3) The data $(T_x(\mathrm{orb}(\sigma )), (d_i, \mu _i)_i)$ for $\langle g\rangle \simeq {\mathbb {Z}}/m$ is admissible in the sense of Definition A.4.

Theorem A.9 follows from the assertion (3) and Proposition A.6.

Proof

We first show that (3) follows from (1) and (2). Suppose we have a factorization $m=m'm''$ with $m'\ne 1$ and consider the restriction of $((N_1)_{{\mathbb {C}}}, (d_i, \mu _i)_i)$ to the subgroup $\langle g^{m''} \rangle \simeq {\mathbb {Z}}/m'$ of $\langle g\rangle \simeq {\mathbb {Z}}/m$. As explained in Example A.3, the restriction of the cyclic permutation $(2, \ldots , d_i, 1)$ to ${\mathbb {Z}}/m'\subset {\mathbb {Z}}/m$ splits into copies of $(2, \ldots , d_i', 1)$ where $d_i'=d_i/(d_i, m'')$. Therefore, if $d_i'=1$ for all $1\le i \le N$, the $\gamma ^{m''}$-action on $N_0$ must be trivial. If furthermore $\gamma ^{m''}$ acts on $N_1$ trivially, then $\gamma ^{m''}=\mathrm{id}$. By the injectivity of $\langle g\rangle \rightarrow \mathrm{GL}(N)$, we have $g^{m''}=\mathrm{id}$, so $m'=1$. This shows that $((N_1)_{{\mathbb {C}}}, (d_i, \mu _i)_i)$ is admissible.

We check (1). We write $T_1=T_{N_1}$. We have a canonical isomorphism $T_yT_1\simeq (N_1)_{{\mathbb {C}}}$ for every $y\in T_{1}$. Via this, $\gamma :T_xT_1\rightarrow T_{\gamma x}T_1$ is identified with $\gamma :(N_1)_{{\mathbb {C}}}\rightarrow (N_1)_{{\mathbb {C}}}$, and the translation $t_a:T_{\gamma x}T_1\rightarrow T_xT_1$ is identified with the identity of $(N_1)_{{\mathbb {C}}}$.

We verify (2). We write $T_0=T_{N_0}$. Via the generators of the rays of $\sigma$, $T_0\subset (T_0)_{\sigma }$ is isomorphic to $({\mathbb {C}}^{\times })^r \subset {\mathbb {C}}^r$, and $\gamma$ acts on $(T_0)_{\sigma }\simeq {\mathbb {C}}^r$ by permuting the basis vectors. We have a canonical isomorphism $T_{\sigma }\simeq T\times _{T_0}(T_0)_{\sigma }$ which makes $T_{\sigma }$ a vector bundle over $T_1$ with zero section $\mathrm{orb}(\sigma )$. Let $\pi :T_{\sigma } \rightarrow T_1\simeq \mathrm{orb}(\sigma )$ be the projection. If $y\in T$, the $\pi$-fiber through y is isomorphic to $(T_0)_{\sigma }$ by

$$\begin{aligned} \varphi _y : \pi ^{-1}(\pi (y)) \rightarrow (T_0)_{\sigma }, \qquad [(y, z)]\mapsto z. \end{aligned}$$

This trivialization depends on y: if we replace y by $y'=b^{-1} y$ where $b\in T_0$, then $\varphi _{y'}\circ \varphi _{y}^{-1}$ acts on $(T_0)_{\sigma }$ by the torus action by b.

Now take a point $y\in T$ with $\pi (y)=x$, the fixed point of $g=t_a\circ \gamma$ in question. Via $\varphi _y$ and $\varphi _{\gamma y}$ the map $\gamma :\pi ^{-1}(x) \rightarrow \pi ^{-1}(\gamma x)$ is identified with the permuting action of $\gamma$ on $(T_0)_{\sigma }$, and via $\varphi _{\gamma y}$ and $\varphi _y$ the map $t_a:\pi ^{-1}(\gamma x) \rightarrow \pi ^{-1}(x)$ is identified with the torus action of an element of $T_0$ on $(T_0)_{\sigma }$. Via the trivialization $(T_0)_{\sigma }\simeq {\mathbb {C}}^r$, the last action is expressed by a diagonal matrix. Hence via $\varphi _y$ and $(T_0)_{\sigma }\simeq {\mathbb {C}}^r$, the map $g:\pi ^{-1}(x)\rightarrow \pi ^{-1}(x)$ is expressed by a direct sum of linear transformations of the form

$$\begin{aligned} \mathrm{diag}(e(\alpha _{?}), \ldots , e(\alpha _{?})) \circ (2, 3, \ldots , d_i, 1) \end{aligned}$$

over $i=1, \ldots , N$. In view of Example A.3, this proves our assertion. $\square$

1.4 No ramifying boundary divisor

We keep the notation in Sect. A.2. In [12], Gritsenko–Hulek–Sankaran also proved the following.

Proposition A.11

The natural projection $X_{\Sigma _l}\rightarrow \mathcal {F}(\Gamma )^{\Sigma }$ has no ramification divisor at the boundary.

This is equivalent to saying that no nontrivial element of $\overline{N(l)}_{{\mathbb {Z}}}$ fixes a boundary divisor of $X_{\Sigma _l}$. For the same reason the proof of this assertion also needs to be modified, but this is easier than Theorem A.1. It suffices to check the following.

Lemma A.12

Let N and T be as in Theorem A.9. Let $g=(\gamma , a)$ be a finite order element of $\mathrm{Aut}(T)\ltimes T$ such that $\gamma \ne \mathrm{id}$. Let $\sigma \subset N_{{\mathbb {R}}}$ be a ray fixed by $\gamma$. Then the g-action on $T_{\sigma }$ does not fix the boundary divisor $\mathrm{orb}(\sigma )$.

Proof

Let $N_0={\mathbb {Z}}(\sigma \cap N)$ and $N_1=N/N_0$. Via the natural isomorphism $\mathrm{orb}(\sigma )\simeq T_{N_1}$, g acts on $\mathrm{orb}(\sigma )$ by $t_{\bar{a}}\circ \bar{\gamma }$ where $\bar{a}\in T_{N_1}$ is the image of a and $\bar{\gamma }$ is the $\gamma$-action on $N_1$. If this is the identity, then $\bar{a}=1$ and $\bar{\gamma }=\mathrm{id}$. Hence $\gamma$ acts on both $N_0$ and $N_1$ trivially, so $\gamma =\mathrm{id}$. $\square$