# Open Torelli locus and complex ball quotients

Sai-Kee Yeung  Mathematics Department, Purdue University, West Lafayette, IN
47907, USA<EMAIL_ADDRESS>
###### Abstract.

We study the problem of non-existence of totally geodesic complex ball
quotients in the open Torelli locus in a moduli space of principally polarized
Abelian varieties using analytic techniques.

The author was partially supported by a grant from the National Science
Foundation

§1. Introduction

1.1 Let ${\mathcal{M}}_{g}$ be the moduli space or stack of Riemann surfaces
of genus $g\geqslant 2$. Let $\overline{\mathcal{M}}_{g}$ be the Deligne-
Mumford compactification of ${\mathcal{M}}_{g}$. Let ${\mathcal{A}}_{g}$ be
the moduli space of principally polarized Abelian varieties of complex
dimension $g$. We know that
${\mathcal{A}}_{g}={\mathcal{S}}_{g}/Sp(2g,{\mathbb{Z}})$ is the quotient of
the Siegel Upper Half Space ${\mathcal{S}}_{g}$ of genus $g$. Let
$\overline{{\mathcal{A}}}_{g}$ be the Bailey-Borel compactification of
${\mathcal{A}}_{g}$. Associating a smooth Riemann surface represented by a
point in ${\mathcal{M}}_{g}$ to its Jacobian, we obtain the Torelli map
$j_{g}:{\mathcal{M}}_{g}\rightarrow{\mathcal{A}}_{g}$. The Torelli map extends
to $j_{g}:\overline{\mathcal{M}}_{g}\rightarrow\overline{{\mathcal{A}}}_{g}$.
The image $T_{g}^{o}:=j_{g}({\mathcal{M}}_{g})$ is called the open Torelli
locus of ${\mathcal{A}}_{g}$. It is well-known that the Torelli map $j_{g}$ is
injective on ${\mathcal{M}}_{g}$. As a mapping between stacks, the mapping
$t_{g}|_{{\mathcal{M}}_{g}}$ is known to be an immersion apart from the
hyperelliptic locus, which is denoted by $H_{g}$. $H_{g}$ is the set of points
in ${\mathcal{M}}_{g}$ parametrizing hyperelliptic curves of genus $g$.

It is a natural problem to study ${\mathcal{M}}_{g}$, $j_{g}$ and to
characterize the Torelli locus in ${\mathcal{A}}_{g}$. There are many
interesting directions and approaches to the problems. Our motivation comes
from the following conjecture in the literature.

######  Conjecture 1.

(Oort [O]) Let $T_{g}^{o}$ be the open Torelli locus in the Siegel modular
variety ${\mathcal{A}}_{g}.$ Then for $g$ sufficiently large, the intersection
of $T_{g}^{o}$ with any Shimura variety $M\subset{\mathcal{A}}_{g}$ of
strictly positive dimension is not Zariski dense in $M$.

The problem is related to a conjecture of Coleman [C] that the cardinality of
CM points on ${\mathcal{M}}_{g}$ cannot be infinite if $g$ is sufficiently
large. Shimura varieties are arithmetic locally Hermitian symmetric spaces.
Hence we may consider a geometrically slightly more general question of
whether there exists a locally Hermitian symmetric space in
${\mathcal{A}}_{g}$ with a Zariski open set in $T_{g}^{o}$. In such case, the
lattice $\Gamma$ involved in the complex rank one case, namely complex balls
$B^{n}_{{\mathbb{C}}}=PU(n,1)/P(U(n)\times U(1))$, may not be arithmetic.

Conjecture 1 is open, but there are quite a few interesting partial results.
First of all there is the result of Hain [Ha] that for $M$ locally Hermitian
symmetric of rank at least $2$, it cannot happen that $M\subset
T_{g}(\overline{\mathcal{M}}_{g})-(t_{g}(H_{g})\cup
t_{g}(\overline{\mathcal{M}}_{g}\backslash{\mathcal{M}}_{g}))$, cf. also de
Jong-Zhang [dJZ] for precise formulation and some results. On the other hand,
there is the result of de Jong-Noot [dJN] that there are examples of Shimura
curves in ${\mathcal{M}}_{g}$ for $g=4$ and $6$. A more systematic and
complete treatment in this direction can be found in Moonen [Moo]. To the
knowledge of the author, not much is known for complex ball quotients of
dimension $n\geqslant 2$, apart from some restrictions in terms of Higgs
bundles given in Chen-Lu-Tan-Zuo [CLTZ]. A possible reason is that on the one
hand rigidity results in general are not strong enough for super-rigidity
properties in the rank one complex cases, and on the other hand the problem is
not concrete enough to be handled by geometric techniques developed for
specific Riemann surfaces.

1.2 The goal of this paper is to provide a method which is applicable to
locally Hermitian symmetric spaces and in particular to all complex ball
quotients of dimension $n\geqslant 2$. It is a weaker statement but is in
support of Conjecture 1.

######  Theorem 1.

The set $T_{g}^{o}-j_{g}(H_{g})\subset{\mathcal{A}}_{g}$ for $g\geqslant 2$
does not contain any complex hyperbolic complex ball quotient, compact or non-
compact with finite volume, of complex dimension at least $2$ as a totally
geodesic complex suborbifold of ${\mathcal{A}}_{g}$.

Remarks
The method of proof applies immediately to other locally Hermitian symmetric
spaces of complex dimension at least $2$. We refer the readers to Section 3.5
and 4.4 for more details. However, the end result for
$\mbox{rank}_{{\mathbb{R}}}M\geqslant 2$ followed already from the results of
[Ha], where the proof is completely different.

Combining Theorem 1, the results of [Ha], the results of [Mö] and the very
recent result of [AN], we have now a rather complete picture for Shimura
varieties on the complement of the hyperelliptic locus in the open Torelli
locus.

######  Theorem 2.

Let $g\geqslant 2$. The space $T_{g}^{o}-j_{g}(H_{g})\subset{\mathcal{A}}_{g}$
in the Siegel modular variety ${\mathcal{A}}_{g}$ does not contain any Shimura
subvariety of ${\mathcal{A}}_{g}$, except when $M$ is the Torelli image of a
Riemann surface with genus $g=3,4$. There is only one curve for each of
$g=3,4$, with universal families given by $y^{4}=x(x-1)(x-t)$ and
$y^{6}=x(x-1)(x-t)$ respectively.

It is a pleasure for the author to thank Ngaiming Mok for very helpful
comments and suggestions. The author is also indebted to the referee for
pointing out mistakes in earlier drafts of the paper and for other useful
comments.

§2. Preliminaries and rigidity

2.1 The approach we take is complex analytic, trying to compare various
Kobayashi metrics making use of Schwarz Lemma and results in rigidity. The
reader may refer to 3.1 for a brief summary of facts needed about Kobayashi
metric.

As mentioned in the introduction, we may consider either $M$ as a smooth
submanifold of ${\mathcal{A}}_{g}$, or a suborbifold. In the latter case the
Kobayashi metrics are considered to be in orbifold sense as follows. Recall
that all the singularities of ${\mathcal{A}}_{g}$ are orbifold singularities
since ${\mathcal{A}}_{g}$ is a quotient of the Siegel Upper Half Space
${\mathcal{S}}_{g}$ by a discrete group and ${\mathcal{S}}_{g}$ is smooth. By
an orbifold embedding (resp. mapping) $f:M\rightarrow{\mathcal{A}}_{g}$, we
mean that there is a finite covering
$\pi:{\mathcal{A}}_{g}^{\prime}\rightarrow{\mathcal{A}}_{g}$ so that
${\mathcal{A}}_{g}^{\prime}$ is smooth and there is an embedding (resp.
mapping) $f^{\prime}:M^{\prime}\rightarrow{\mathcal{A}}^{\prime}$ for which
$M^{\prime}$ is smooth and $f\circ\pi=\pi\circ f^{\prime}$. Since Kobayashi
metric is invariant under a biholomorphism and in particular invariant under a
local holomorphic covering map, the argument throughout the article would be
independent of the local uniformization taken at each orbifold singularity.

Hence the Kobayashi metric studied throughout the article is in orbifold sense
as explained.

2.2 The proof of Theorem 1 makes use of the following result in [A].

######  Proposition 1.

([A] Theorem 1.1) Let $\widetilde{M}=B_{{\mathbb{C}}}^{N}$ be the complex unit
ball of dimension $N\geqslant 2$. There is no holomorphic embedding of
$\widetilde{M}$ into ${\mathcal{T}}_{g}$ which is isometric with respect to
the Kobayashi metrics on $\widetilde{M}$ and ${\mathcal{T}}_{g}$.

§3. Totally geodesic embeddings from complex balls to Siegel Upper spaces

3.1 Our approach relies on basic properties concerning totally geodesic
embeddings of complex balls in the Siegel Upper Half Space. The purpose of
this section is to explain results in this direction relevant to our purpose.
For this purpose, it is more convenient to consider the bounded model of
${\mathcal{S}}_{g}$, namely, the classical bound domain $III_{g}$. The main
technical result of this paper is Proposition 3 stated in 3.4.

Basic properties of classical domains can be found in [He], [Mok1] and [Sat].
Since we are going to use the results of Satake, we follow closely the
exposition in [Sat]. First let us explain briefly classical domains of type
$I_{p,q}$ and $III_{k}$ according to 1.2, 1.3 of [Sat], of which the
terminology is to be used in later parts of this section.

(i). $I_{p,q}$: Consider $V$ a vector space over ${\mathbb{C}}$ equipped with
a non-degenerate Hermitian form $F$ of signature $(p,q)$ with $p>q>0$.
$I_{p,q}={\mathcal{D}}(V,F)=:{\mathcal{D}}$ is the space of $q$ dimensional
complex subspace $V_{-}$ of $V$ so that $F|_{V_{-}}$ is negative definite. Let
$V_{+}$ be the orthogonal complement of $V_{-}$ in $V$, so that $F|_{V_{+}}$
is positive definite. A point in ${\mathcal{D}}$ is determined by $V_{-}$, or
the pair $(V_{+},V_{-})$. Let $z_{0}$ be a fixed point ${\mathcal{D}}$,
determined by $(V_{+}^{(0)},V_{-}^{(0)})$ and for convenience can be chosen to
be the origin. Let $(e_{1},\cdots,e_{p})$ and $(e_{p+1},\cdots,e_{p+q})$ be
orthonormal basis of $V_{+}^{(0)}$ and $V_{-}^{(0)}$ respectively, so that
together they form a basis of $V$. A point $z\in{\mathcal{D}}$ is now
determined by $(V_{+},V_{-})$ with $V_{-}$ spanned by the basis

(1) $\sum_{i=1}^{p}e_{i}z_{ij}+e_{p+j},\ 1\leqslant j\leqslant q,$

where the $(p,q)$ matrix $Z=(z_{ij})$ satisfies $I_{q}-^{t}\overline{Z}Z>0$.
Denote by $M_{p,q}$ the space of all $p\times q$ matrices with entries in
${\mathbb{C}}$. Identifying ${\mathcal{D}}$ with $\\{Z\in
M_{p,q}:I_{q}-\overline{Z}^{t}Z>0\\}$, we realize $I_{p,q}$ as a bounded
domain in ${\mathbb{C}}^{pq}$.

The complex ball $B_{{\mathbb{C}}}^{n}$ in ${\mathbb{C}}^{n}$ is just
$I_{n,1}$.

(ii). $III_{k}$: Consider $V_{{\mathbb{R}}}$ a vector space over
${\mathbb{R}}$ of dimension $2k$ equipped with a non-degenerate alternating
bilinear form $A$. Let $V=V_{{\mathbb{C}}}$ be the complexification of
$V_{{\mathbb{R}}}$. $A$ extends naturally to $V$. The Hermitian form defined
by

(2) $F(x,y):=iA(\bar{x},y)$

has signature $(k,k)$ on $V$. $III_{k}={\mathcal{D}}(V_{{\mathbb{R}}},A)$, or
simply, ${\mathcal{D}}$, is the space of all complex structures $I$ on
$V_{{\mathbb{R}}}$ so that the bilinear form $A(x,Iy)$ is symmetric and
positive definite. Let $W=\\{x\in V|Ix=ix\\}$ so that $V=W+\overline{W}$. It
follows that

(3) $A|_{W}=0,\ \ F|_{W}>0$

and $\overline{W}$ is the orthogonal complement of $W$ in $V$ with respect to
$F$. Hence $I\in{\mathcal{D}}$ is determined by $W$ or the pair
$(W,\overline{W})$ satisfying (3). Fix a point $z_{o}\in{\mathcal{D}}$
corresponding to $W^{o}$. Let $(e_{1},\cdots,e_{p})$ be an orthonormal basis
of $W^{o}$ with respect to $F$ and let

(4) $e_{k+i}=\overline{e}_{i},\ 1\leqslant i\leqslant k.$

As described in $I_{p,q}$ with $p=q=k$ in the description of
$W\in{\mathcal{D}}$, it follows that $W$ is described by a $k\times k$
symmetric complex matrices $Z$ with $I_{k}-\overline{Z}^{t}Z>0$. Hence
${\mathcal{D}}$ is identified with the bounded domain $\\{Z\in
M_{k,k}:Z=Z^{t},I_{k}-\overline{Z}^{t}Z>0\\}\subset{\mathbb{C}}^{n(n+1)/2}$.

The Siegel Upper Half Space ${\mathcal{S}}_{g}$ is biholomorphic to $III_{g}$.

3.2 Here we recall briefly the classification of holomorphic totally geodesic
embeddings of a Hermitian symmetric domain into another. In general, the
classification of holomorphic totally geodesic embedding of a Hermitian
symmetric space $N_{1}=G_{1}/K_{1}$ into another Hermitian symmetric space
$N_{2}=G_{2}/K_{2}$ with respect to the Bergman metrics has been given by
Satake [Sat] and Ihara [I], where $G_{i}$ is a semi-simple Lie group and
$K_{i}$ a maximal compact subgroup for $i=1,2$. Since the manifolds involved
are symmetric, the classification of totally geodesic embeddings is reduced to
the classification of injective Lie algebra homomorphisms
$\rho:\mathfrak{g_{1}}\rightarrow\mathfrak{g_{2}}$ for the corresponding Lie
groups. The invariant complex structure on $N_{i}$ is given by an element
$H_{oi}\in K_{i}$. The totally geodesic embedding is holomorphic if the
condition $(H_{1})$, namely,
$\rho\circ\mbox{ad}(H_{o1})=\mbox{ad}(H_{o2})\circ\rho$ is satisfied. The
condition $(H_{1})$ is a consequence of the condition $(H_{2})$, namely,
$\rho(H_{o1})=H_{o2}$. This is explained on page 427 of [Sat]. Some
explanation in terms of root system and Dynkin diagrams by Ihara in [I]. We
refer the reader to [Sat] for any unexplained notation and terminology.

3.3 We consider now the specific situation of classification of holomorphic
totally geodesic embeddings of $I_{N,1}$ into $III_{g}$. Let $0$ be the origin
in $B^{N}_{{\mathbb{C}}}$, which may be assumed to be mapped to the origin $0$
of ${\mathcal{S}}_{g}$ realized as the bounded domain $III_{g}$ as above,
since the spaces involved are homogeneous. The classification is described in
Theorem 1, 3.2 and the Table on page 460 of [Sat]. From the description there,
any holomorphic embedding of $I_{N,1}$ to $III_{g}$ is a direct sum of a
number of compositions of the following types of totally geodesic mappings.

Type 1, standard embeddings: This includes embedding
$i_{1a}:I_{N,1}\rightarrow I_{p,q}$, for $N\leqslant p$ and
$i_{1b}:III_{k}\rightarrow III_{l}$, for $k<l$, given by the standard
representation or standard embedding.

Type 2, connecting embeddings: The embedding
$i_{2}=\iota_{p,q}:I_{p,q}\rightarrow III_{p+q}$ is given in page 432-433 of
[Sat],

$I_{p,q}\ni Z\longmapsto\left(\begin{array}[]{cc}0&{}^{t}Z\\\
Z&0\end{array}\right)\in III_{g}\cong{\mathcal{S}}_{p+q}.$

Type 3, absolutely irreducible embeddings: The embedding
$i_{3}:I_{p,1}\rightarrow I_{r,s},r={p\choose m},s={p\choose m-1}$, and
$i_{3}:I_{p,1}\rightarrow I_{r}$ in the case of $r=s$, which happens when
$p\equiv 1\pmod{4}$ and $m=\frac{p+1}{2}$. This corresponds to the skew-
symmetric tensor representations of degree $m$ as explained in 3.2 and the
Table on page 460 of [Sat].

The construction is described in page 448 of [Sat]. The representation is
given by $\rho=\Lambda_{m}$ of $G\cong SU(p,1)$, corresponding to skew-
symmetric tensors of degree $m$. In terms of a fundamental system of roots of
the Lie algebra involved, the highest weight $\lambda_{\rho}$ of $\rho$ is
given by

$\lambda_{\rho}=(\overbrace{1,1,\cdots,1}^{\text{m}},0,\cdots 0)$

with $1\leqslant m\leqslant p$. Recall the setting given in (3.1)(i) in the
case of $I_{p,1}$. Let $(e_{i})_{i=1,\cdots,p+1}$ be an orthonormal basis of
$V$, a basis of the exterior algebra $\Lambda^{m}$ is given by $e_{i_{1}\cdots
i_{m}}=e_{i_{1}}\wedge\cdots\wedge e_{i_{m}}$, $i_{1}<i_{2}<\cdots<i_{m}$. As
in (3.1)(i), there is a Hermitian form $F$ on $V$, which induces $F^{(m)}$ on
$\Lambda^{m}$ given by

$F^{(m)}(x_{1}\wedge\cdots\wedge x_{m},y_{1}\wedge\cdots\wedge
y_{m})=\det(F(x_{i},y_{j}))$

for $x_{i},y_{j}\in V$ and is invariant under $\rho(\mathfrak{g}_{su_{p,1}})$.
Since

$F^{(m)}(e_{i_{1}\cdots i_{m}},e_{i_{1}\cdots
i_{m}})=\left\\{\begin{array}[]{ll}1&\ i_{m}<p+1,\\\ -1&\
i_{m}=p+1,\end{array}\right.$

the Hermitian for $F^{(m)}$ has signature $(r,s)$ with $r={p\choose
m},s={p\choose m-1}$.

The totally geodesic isometry of the symmetric domains $f:I_{p,1}\rightarrow
I_{r,s}$ described here is given in page 448 of [Sat] by

(5)
$\begin{array}[]{ccccccc}{\mathcal{D}}(V,F)&\ni&(V_{+},V_{-})&\longmapsto&(\Lambda_{m}(V_{+})\otimes
1,\Lambda_{m-1}(V_{+})\otimes
V_{-})&\in&{\mathcal{D}}(\Lambda_{m}(V),F^{(m)}).\\\
\parallel&&\parallel&&\parallel&&\parallel\\\
I_{p,1}&\ni&z&\longmapsto&z^{\prime}&\in&I_{r,s}\end{array}$

In the case of representation in $III_{r}\subset I_{r,r}$, this corresponds to
the above discussion with $r=s$ and hence $m=\frac{p+1}{2}$ and $p\equiv
1\pmod{2}$. In such case, we can define a Bilinear form $B$ on
$\Lambda_{m}(V)\times\Lambda_{m}(V)$ by

$x\wedge y=B(x,y)e_{1\cdots p+1}$

which satisfies

$B(y,x)=(-1)^{m^{2}}B(x,y).$

Hence if $m\equiv 1\pmod{2}$ or $p\equiv 1\pmod{4}$, the bilinear form
$B(x,y)$ is an alternating bilinear form. Furthermore, there is a semi-linear
transformation $\sigma$ on $\Lambda_{m}(V)$ so that

$F^{(m)}(x,y)=iB(x^{\sigma},y),$

where $\sigma$ satisfying $\sigma^{2}=1$ is explicitly written in [Sat], page
449, as follows. Let $M=(i_{1},\cdots,i_{m})$ be an oriented subset of
$(1,2,\cdots,p+1)$ and $M^{c}$ the complement. Then

(6) $e_{M}^{\sigma}=a(M)e_{M^{c}},\ a(M)=-i\epsilon(M^{c},M)\eta(M),$

where $\epsilon(M^{c},M)=\pm 1$ is the signature of the permutation of
$(M^{c},M)$ with respect to $(1,2,\cdots,p+1)$ and $\eta(M)=-1$ (resp. $1$) if
$p+1\in M$ (resp. $p+1\not\in M$). Hence $a(M)=\pm i$. $\sigma$ serves and
complex conjugate as in 3.1(ii). In this case, the bilinear form $B$ serves as
the bilinear form $A$ as needed in equation (2) in 3.1(ii) for the definition
of $III_{r}$. The totally geodesic isometry of the symmetric domains
$f:I_{p,1}\rightarrow III_{r}$ is given by (5) with $r=s$.

We summarize the result of [Sat], which is relevant to us, from Theorem 1 and
the Table on page 460 in [Sat]. As explained in 3.2, totally geodesic
holomorphic embedding corresponds to condition $(H_{1})$, which by (a) below
reduces the problem to representations satisfying condition $(H_{2})$. The
Table on page 460 in [Sat] summarizes the representations satisfying condition
$(H_{2})$ determined in Section 3 of [Sat].

######  Proposition 2.

(Satake)
(a). Let $\rho$ be a representation of $\mathfrak{g}=\mathfrak{su(m,1)}$ into
$III_{g}$ satisfying condition $(H_{1})$. Then there exists absolutely
irreducible representations $\rho_{i}$ $(1\leqslant i\leqslant r_{1})$ of
$\mathfrak{g}$ into $(III)_{p_{i}}$ $(p_{i}>0)$ satisfying $(H_{2})$ and
absolutely irreducible representations $\rho_{i}$ $(r_{1}\leqslant i\leqslant
r_{1}+r_{2})$ of $\mathfrak{g}$ into $(I)_{p_{i},q_{i}}$
$(p_{i},q_{i}\geqslant 0,p_{i}+q_{i}>0)$ satisfying $(H_{2})$ such that $\rho$
is $k$ equivalent to the direct sum of representations
$\sum_{i=1}^{r_{1}}\rho_{i}+\sum_{i=r_{1}+1}^{r_{1}+r_{2}}\iota_{p_{i},q_{i}}\circ\rho_{i}$
up to a trivial representation, where
$\sum_{i=1}^{r_{1}}p_{i}+\sum_{i=r_{1}+1}^{r_{2}}(p_{i}+q_{i})\leqslant g$.
(b). An absolutely irreducible representation $\rho$ of $\mathfrak{g}$ into
$I_{r,s}$ or $III_{r}$ satisfying $(H_{2})$ corresponds to embeddings of type
$i_{3}$ described earlier.

We refer the reader to the original source [Sat] for any unexplained
terminology. In particular, the notion of $k$-equivariant and direct sum are
described in §1 and §2 of [Sat]. The result of Satake applies to any semi-
simple Lie algebra $\mathfrak{g}$ of Hermitian type.

Let us now describe the mapping given in (5) more carefully, which is to be
used later.

######  Lemma 1.

In terms of standard coordinates, the totally geodesic mapping
$f:B_{n}\rightarrow I_{r,s}\subset{\mathbb{C}}^{N}$ or
$III_{r}\subset{\mathbb{C}}^{N}$ for $r=s$ with $f(0)=0$ with
$z^{\prime}=f(z)$ as described above is linear in $z$, with image given by the
intersection $f(B_{n})$ with a subspace of ${\mathbb{C}}^{N}$ of appropriate
dimension.

Proof We remark that the standard coordinates as used in [Sat] in the
description above are also the Harish-Chandra coordinates.

Consider first $f:I_{p,1}\rightarrow I_{r,s}$ as given by (5). In terms of (1)
for $I_{p,1}$, the point $z$ in (5) corresponds to $V_{+}$ being spanned by
$\sum_{i=1}^{p}e_{i}z_{i}+e_{p+1}$. Since it is a holomorphic totally geodesic
embedding, the mapping is equivariant with respect to the action of $G$ and in
particular invariant under the action of the isotropy group $K=S(U(p)\times
U(1))$. In particular, it suffices for us to investigate the image $f(z)$ for
$z=(z_{1},0,\cdots,0)$ with $|z_{1}|^{2}<1$. Again, we use
$(e_{1},\cdots,e_{p})$ and $(e_{p+1})$ to denote some orthonormal basis of
$V_{+}^{(0)}$ and $V_{-}^{(0)}$ respectively. With $z$ as described, an
orthonormal basis of $V_{-}$ and $V_{+}$ at $z$ are $e_{p+1}^{\prime}$ and
$(e_{1}^{\prime},e_{2},\cdots,e_{p})$ respectively, where

(7) $e_{p+1}^{\prime}=\frac{1}{\sqrt{1+|z|^{2}}}(z_{1}e_{1}+e_{p+1}),\
e_{1}^{\prime}=\frac{1}{\sqrt{1+|z|^{2}}}(e_{1}-\overline{z}_{1}e_{p+1}).$

To describe $z^{\prime}$ in the image of $f$ in (5), we need to investigate
$\Lambda_{m-1}(V_{+})\otimes V_{-}$ where $F^{(m)}$ is negative definite, and
express them in terms of base vectors of
$\Lambda_{m}^{(0)}=\Lambda_{m}(V_{+}^{(0)})\oplus\Lambda_{m-1}(V_{+}^{(0)})\otimes
V_{-}^{(0)}$. From definition, $\Lambda_{m-1}(V_{+}^{(0)})\otimes V_{-}^{(0)}$
is generated by a basis consists of the following two types of elements,
(i) $e_{i_{1}\cdots i_{m-1}}\wedge e_{p+1}^{\prime}$, where
$1<i_{1}<\cdots<i_{m-1}\leqslant p$,
(ii) $e_{1}^{\prime}\wedge e_{i_{1}\cdots i_{m-2}}\wedge e_{p+1}^{\prime}$,
where $1<i_{1}<\cdots<i_{m-2}\leqslant p.$

From the formula of $e_{p+1}^{\prime}$ in (7), we compute in case (i) that

(8) $e_{i_{1}\cdots i_{m-1}}\wedge
e_{p+1}^{\prime}=\frac{1}{\sqrt{1+|z|^{2}}}((-1)^{m-1}z_{1}e_{1i_{1}\cdots
i_{m-1}}+e_{i_{1}\cdots i_{m-1}(p+1)})$

which is proportional to

(9) $(-1)^{m-1}z_{1}e_{1i_{1}\cdots i_{m-1}}+e_{i_{1}\cdots i_{m-1}(p+1)}.$

Notice that we need the coefficient of $e_{i_{1}\cdots i_{m-1}(p+1)}$ to be
$1$ in the format of (1).

In case (ii), (7) gives

$e_{1}^{\prime}\wedge e_{p+1}^{\prime}=e_{1}\wedge e_{p+1}$

and hence

(10) $e_{1}^{\prime}\wedge e_{i_{1}\cdots i_{m-2}}\wedge
e_{p+1}^{\prime}=e_{1i_{1}\cdots i_{m-2}(p+1)}.$

Let $u={p-1\choose m-1}$. It follows from the above explicit computation that
for $f:I_{p,1}\rightarrow I_{r,s}$ as given in (5), the coordinates of
$z^{\prime}=f(z)$ are given by

(11) $z=(z_{1},0,\cdots,0)\mapsto f(z)=z^{\prime}\ \mbox{with}\
z^{\prime}_{ij}=\left\\{\begin{array}[]{cc}(-1)^{m-1}z_{1},&1\leqslant
i\leqslant t,1\leqslant j\leqslant u\\\ 0&\mbox{otherwise}\end{array}\right.$

Similar constructions apply to $f:I_{p,1}\rightarrow III_{r}\subset I_{r,r}$
corresponding (5) with $r=s$. We recall that the coordinates $z^{\prime}$ in
$III_{r}$ in the image of $f$ is determined according to (1) in (3.1) with
respect to a corresponding basis of vectors in $(\Lambda_{m}(V_{+})\otimes
1,\Lambda_{m-1}(V_{+})\otimes V_{-})$. Moreover, the choice of the base
vectors of is given by (4), choosing $e_{r+i}$ to be the complex conjugate of
$e_{i}$. In our case of $\Lambda_{m}^{(0)}$, the complex conjugate is given by
$\sigma$ in the setting of (5) and $e_{I_{m}}^{\sigma}=a(I_{m})e_{I_{m}^{c}}$
in terms of earlier notations, here $I_{m}=i_{1}\cdots i_{m}$ is an index set.
Hence $z^{\prime}=(z^{\prime}_{ij})_{1\leqslant i,j\leqslant r}$ are
determined by having a basis of $\Lambda_{m-1}(V_{+})\otimes V_{-}$ of the
form

(12)
$\sum_{i=1}^{r}e^{\sigma}_{I_{i}(p+1)}z^{\prime}_{ij}+e_{I_{j}(p+1)}=\sum_{i=1}^{r}a(I_{i}(p+1))e_{(I_{i}(p+1))^{c}}z^{\prime}_{ij}+e_{I_{j}(p+1)},\
1\leqslant j\leqslant r$

where $a(I_{i}(p+1))=\pm 1$. From the expressions in (9) and (10), we see that
(11) still applies in the sense that $z^{\prime}_{ij}$ are either $0$ or
$(-1)^{m}z_{1}$. This computation shows that in terms of the standard
coordinates of the bounded domains in ${\mathbb{C}}^{N}$ as described in 1.3,
1.4 of [Sat] or Chapter 4 §2 of [Mok 1] for the classical domains that the
image is the intersection of a line in ${\mathbb{C}}^{n}$ with $I_{r,s}$ or
$I_{r}$. As mentioned earlier, this works for any complex direction obtained
under the action of the isotropy group at $0$ on the domain. Since the mapping
$f$ is a totally geodesic mapping and is hence equivariant under the action of
the isotropy group, the lemma follows.

∎

3.4 We now state the main result of this section.

######  Proposition 3.

Let $i:B^{N}_{{\mathbb{C}}}\rightarrow III_{g}$ be a totally geodesic
embedding. Then there exists a holomorphic map $p:III_{g}\rightarrow
B^{N}_{{\mathbb{C}}}$ so that $p\circ i$ is the identity mapping on
$B^{N}_{{\mathbb{C}}}$.

Proof Again, we let $0$ be the origin in $B^{N}_{{\mathbb{C}}}$, which may be
assumed to be mapped to the origin $0$ of $III_{g}$, a bounded domain
realization of ${\mathcal{S}}_{g}$, as discussed earlier. To streamline the
presentation, let us consider each simple type of presentations
$i_{1},i_{2},i_{3}$ in details before the general case described in
Proposition 2.

Type 1: This includes embedding is given by $i_{1}:I_{N,1}\rightarrow
I_{p,q}$, for $N\leqslant p$ or $i_{1b}:III_{k}\rightarrow III_{l}$, for
$k<l$, given by the standard embedding into the corresponding upper left hand
corner of the image.

The classical domain $I_{p,q}$ is given as a symmetric space $G/K$ with
$G=SU(p,q)$ and $K=S(U(p)\times U(q))$, where $K$ is the isotopy group at $0$.
For a holomorphic totally geodesic embedding $i_{1a}:B^{N}_{{\mathbb{C}}}\cong
I_{N,1}\rightarrow I_{p,q}$, conjugating by $K$ if necessary, we may assume
that $\frac{\partial}{\partial
z_{11}}\in(i_{1})_{*}(T_{B^{N}_{{\mathbb{C}}}})$. Now we observe that
$\frac{\partial}{\partial z_{ij}}$ with $i>1$ and $j>1$ cannot lie in
$(i_{1a})_{*}(T_{B^{N}_{{\mathbb{C}}}})$, for otherwise the image
$i_{1a}(B^{N}_{{\mathbb{C}}})$ as a symmetric space would have real rank at
least $2$ considering the tangent vectors $\frac{\partial}{\partial z_{11}}$
and $\frac{\partial}{\partial z_{ij}}$, contradicting the fact that $I_{1,N}$
has real rank $1$. It follows that $i_{1a}(B^{N}_{{\mathbb{C}}})$ has to lie
in one of the following two subspaces of $I_{p,q}$

$I_{1,q}=\\{z=[z_{ij}]\in I_{p,q}|z_{ij}=0\ \mbox{for}\ i\geqslant 2\\}\
\mbox{or}\ I_{p,1}=\\{z=[z_{ij}]\in I_{p,q}|z_{ij}=0\ \mbox{for}\ j\geqslant
2\\}.$

In either case, $N\leqslant\max(p,q)=q.$

Suppose $i_{1a}(B^{N}_{{\mathbb{C}}})\subset I_{1,q}$. Conjugating by some
elements in $K$ if necessary, we may assume that mapping $i_{1a}$ is given by

$i_{1a}(z_{1},\cdots,z_{N})=\left[\begin{array}[]{cccccc}z_{1}&\cdots&z_{N}&0&\cdots&0\\\
0&\cdots&0&0&\cdots&0\\\ \vdots&&&&&\vdots\\\
0&\cdots&0&0&\cdots&0\end{array}\right].$

Consider the holomorphic projection map $p_{1}:I_{p,q}\rightarrow
B_{{\mathbb{C}}}^{N}$ given by

$p_{1}(\left[\begin{array}[]{cccccc}z_{11}&\cdots&z_{1N}&z_{N+1}&\cdots&z_{1q}\\\
\vdots&&&&&\vdots\\\
z_{p1}&\cdots&z_{pN}&z_{N+1}&\cdots&z_{pq}\end{array}\right])=\left[\begin{array}[]{cccccc}z_{11}&\cdots&z_{1N}&0&\cdots&0\\\
0&\cdots&0&0&\cdots&0\\\ \vdots&&&&&\vdots\\\
0&\cdots&0&0&\cdots&0\end{array}\right].$

Denote by $Y$ the matrix in the domain and $Z$ the matrix in the image. As the
$p\times q$ matrix $Y$ satisfies $I-^{t}\overline{Y}Y>0$ from definition, it
follows that $\sum_{i=1}^{N}|z_{i}|^{2}\leqslant\sum_{i=1}^{p}|z_{i}|^{2}<1$
and hence the image of $p_{1}$ lies in $B_{{\mathbb{C}}}^{N}$. Furthermore, it
follows from definition that $p_{1}\circ i_{1a}=1_{B_{{\mathbb{C}}}^{N}}.$
Hence $p_{1}$ gives us the retraction that we need.

For $i_{1b}:III_{k}\rightarrow III_{l}$, for $k<l$, given by the standard
embedding into the corresponding upper left hand corner of the image. Denote
by $p_{1b}:III_{l}\rightarrow III_{k}$ the projection onto the upper left hand
corner

$\displaystyle p_{1b}([z_{ij}]_{i,j=1,\cdots,l})$ $\displaystyle=$
$\displaystyle[z_{ij}]_{i,j=1,\cdots,k}.$

Let $Y=[z_{ij}]_{i,j=1,\cdots,l}$ and $U=[z_{ij}]_{i,j=1,\cdots,k}$. The fact
that $Y$ is symmetric implies that $U$ is symmetric. Now
$I_{g}-\overline{Y}Y>0$ implies that $I_{g^{\prime}}-\overline{U}{U}>0$ as
each column vector of $U$ is part of a column vector of $Y$. Hence the image
lies in $III_{k}$.

It is clear that $p_{1b}\circ i_{1b}|_{III_{g^{\prime}}}$ is the identity map
and hence $p_{3}$ is a projection.

Type 2: $i_{2}=\iota_{p,q}:I_{p,q}\rightarrow III_{p+q}$ with
$\iota_{p,q}(Z)=\left(\begin{array}[]{cc}0&{}^{t}Z\\\ Z&0\end{array}\right).$
Define $i_{2}:III_{p+q}\rightarrow I_{p,q}$ the projection

${III_{p+q}}\ni Y=\left(\begin{array}[]{cc}W_{1}&{}^{t}Z\\\
Z&W_{2}\end{array}\right)\longmapsto\left(\begin{array}[]{cc}0&{}^{t}Z\\\
Z&0\end{array}\right)\longmapsto Z\in{I_{p,q}}.$

Here $Y$ is symmetric. From the fact that $I_{p+q}-\overline{Y}Y>0$, it
follows that $I_{q}-^{t}\overline{W}W-^{t}\overline{Z}Z>0$ and hence
$I_{q}-^{t}\overline{Z}Z>0$. Hence the image of $i_{2}$ is really in
$I_{p,q}$. It follows from definition that $p_{2}\circ i_{2}=1_{I_{p,q}}$ is
the identity map.

Type 3: $i_{3}:I_{p,1}\rightarrow I_{r,s}$ or $III_{r}$ corresponding to the
skew-symmetric tensor representations of degree $m$. Since the second case can
be considered as a special case of the first case, it suffices for us to
consider the case of $I_{r,s}$ being the image. From Lemma 1 in 3.3, the image
of $i_{3}$ is a linear subspace $R:=\mbox{Im}(i_{3})$ passing through the
origin in $III_{r}$. We claim that the argument of Lemma 1 shows that there is
a projection $p_{3}:I_{r,s}\rightarrow f(I_{p,1})$ so that $p_{3}\circ
i_{3}|_{I_{p,1}}$ is identity on $I_{p,1}$. We actually take $p_{3}$ be the
orthogonal projection of $I_{r,s}$ to the complex linear subspace $P$ of
${\mathbb{C}}^{N}$ containing $f(I_{p,1})$ as a subdomain, after Lemma 1 To
prove the claim, since the domain involved is a convex domain in
${\mathbb{C}}^{N}$, it suffices for us to show that
$p_{3}(I_{r,s})=f(I_{p,1})$. Clearly $p_{3}(I_{r,s})$ contains $f(I_{p,1})$.
This in turn follows if we can prove the corresponding statement for the
projection of $I_{r,s}$ to $f(B^{1}_{{\mathbb{C}}})$ for a geodesic
$B^{1}_{{\mathbb{C}}}\subset B^{p}_{{\mathbb{C}}}\cong I_{p,1}$ through the
origin, since $f$ is equivariant with respect to the action of the isotropy
groups at $0$. Hence it suffices for us to show that the projection $p_{4}$ of
$I_{r,s}$ to the line in ${\mathbb{C}}^{N}$ containing
$f(B_{{\mathbb{C}}}^{1})$ is actually $f(B_{{\mathbb{C}}}^{1})$, where
$B^{1}_{{\mathbb{C}}}=\\{(z_{1},0,\dots,0):|z_{1}|<1\\}$ as studied in the
proof of Lemma 1.

Recall that $r={p\choose m},s={p\choose m-1}$ and $u={p-1\choose m-1}$. From
(11) in the proof of Lemma 1, the image of $f$ is

(13)
$f(B^{1}_{{\mathbb{C}}})=\Big{\\{}\left(\begin{array}[]{ll}z_{1}I_{u}&0_{u,a}\\\
0_{b,u}&0_{b,a}\end{array}\right),\ (z_{1},0,\dots,0)\in
B^{1}_{{\mathbb{C}}}\Big{\\}}$

where $b={p-1\choose m},a={p-1\choose m-2}$, $I_{t}$ is the identity matrix of
size $t$ and $0_{c,d}$ is the zero matrix of size $c\times d$. Clearly

$f(B^{1}_{{\mathbb{C}}})\subset
I_{r,r}\cong\left(\begin{array}[]{ll}I_{r,r}&0_{u,a}\\\
0_{b,u}&0_{b,a}\end{array}\right),$

where $I_{r.r}$ is a bounded symmetric domain of type I.

Similar to the construction of $I_{1a}$ in the standard embedding earlier, it
is clear that there is a projection $q_{1}:I_{r,s}\rightarrow I_{u,u}$ as
explained above by taking zeros in non-relevant entries. Hence it suffices for
us to show that there is a retraction
$q_{2}:I_{u,u}\rightarrow\\{(z_{1}I_{u}):|z_{1}|<1\\}$. Let $w=(w_{ij})\in
I_{u,u}$. By definition, it satisfies $I_{q}-\bar{w}^{t}w>0$ and hence
$|w_{ij}|<1$ for each $i,j$. It suffices for us to define

$q_{2}(w)=(\frac{1}{u^{2}}\sum_{i,j=1}^{u}w_{ij})I_{u}.$

Clearly $|\frac{1}{u^{2}}\sum_{i,j=1}^{u}w_{ij}|<1$ as $|w_{ij}|<1$.
Furthermore $q_{2}(\zeta I_{u})=\zeta I_{u}$ for $\zeta\in{\mathbb{C}}$,
$|\zeta|<1$, and hence $q_{2}|_{\mbox{Im}(f)}$ is the identity map. Now it
suffices for us to let $p_{4}=q_{2}\circ q_{1}.$

This concludes the discussions on embeddings of the simple types. We now
combine the results from the above discussions with those from Proposition 2.
According to Proposition 2a, the representation $\rho$ involved is of form

$\sum_{i=1}^{r_{1}}\rho_{i}+\sum_{i=r_{1}+1}^{r_{1}+r_{2}}\iota_{p_{i},q_{i}}\circ\rho_{i}.$

Suppose $(r_{1},r_{2})=(1,0)$ or $(0,1)$, that is the representation is
irreducible. Then the projection occurs from composition of projections
corresponding to Type 1, 2, 3 embeddings respectively, making use of
Proposition 2a.

Consider now the general case. Note that each of the factors above corresponds
to image lying in some Type III classical domain $III_{s_{i}}$ for $1\leqslant
i\leqslant r_{1}$ or $III_{p_{j}+q_{j}}$ for $r_{1}+1\leqslant j\leqslant
r_{1}+r_{2}$. As explained in §2 of [Sat], this corresponds to diagonal blocks
of square matrices in $III_{g}$, where
$\sum_{i=1}^{r_{1}}s_{i}+\sum_{j=r_{1}+1}^{r_{1}+r_{2}}(p_{i}+q_{i})\leqslant
g$. For simplicity of notation, let us just define $s_{j}=p_{j}+q_{j}$ for
$r_{1}+1\leqslant j\leqslant r_{1}+r_{2}$. Hence we have

$i:I_{p,1}\stackrel{{\scriptstyle
i_{a}}}{{\longrightarrow}}\left(\begin{array}[]{cccc}III_{s_{1}}&0&&\cdots\\\
0&III_{s_{2}}&&\cdots\\\
\vdots&\vdots&\dots&\end{array}\right)\stackrel{{\scriptstyle
i_{b}}}{{\lhook\joinrel\longrightarrow}}I_{g}.$

The projection

$p_{b}:I_{g}\longrightarrow\left(\begin{array}[]{cccc}III_{s_{1}}&0&&\cdots\\\
0&III_{s_{2}}&&\cdots\\\ \vdots&\vdots&\dots&\end{array}\right)$

so that $p_{b}\circ i_{b}$ is identity is constructed exactly as in those Type
$i_{1}$ embeddings discussed earlier. For each $III_{s_{i}},1\leqslant
i\leqslant r_{1}+r_{2}$, there is a projection
$p_{s_{i}}:III_{s_{i}}\rightarrow I_{p,1}$ so that $p_{s_{i}}\circ i_{1}$ is
the identity map according to earlier discussions. The projection

$p_{a}:\left(\begin{array}[]{cccc}III_{s_{1}}&0&&\cdots\\\
0&III_{s_{2}}&&\cdots\\\ \vdots&\vdots&\dots&\end{array}\right)\longrightarrow
I_{p,1}$

is then defined by identifying $I_{p,1}$ with $i_{a}(I_{p,1})$ and letting

$p_{a}=\frac{1}{r_{1}+r_{2}}p_{s_{1}}+\cdots+\frac{1}{r_{1}+r_{2}}p_{s_{r_{1}+r_{2}}},$

where addition is given in terms of the coordinate functions of the standard
realization, which is the Harish-Chandra coordinates for $I_{g}$. Clearly from
construction, $p_{a}\circ i_{a}$ is the identity. We may then define
$p=p_{a}\circ p_{b}$. It follows from construction that $p\circ i$ is the
identity. ∎

3.5 Though not really needed for this article, we mention that the argument of
Proposition 3 can be applied to other pairs of Hermitian symmetric spaces of
non-compact type, following case-by-case checking as done above for classical
domains using the results of [Sat], see also [I]. In a more inspiring way,
Ngaiming Mok [Mok3] has shown us a conceptual proof of such a result for all
Hermitian symmetric spaces of non-compact type, including those containing
factors of exceptional types without using classification results. This was
done in terms of the Lie triple system and Harish-Chandra embedding for all
pairs of Hermitian symmetric spaces of non-compact type.

§4. Proofs of the main results

4.1 Denote by $g_{V,K}$ the Kobayashi (pseudo-)metric of a variety $V$, which
is the positive semi-definite Finsler metric defined as

$\sqrt{g_{V,K}}(x,v)=\inf\\{\frac{1}{r}|\exists f:\Delta_{r}\rightarrow V\
{\mbox{holomorphic}},f(0)=x,f^{\prime}(0)=v\\},$

where $\Delta_{r}$ is the disk of radius $r$ in ${\mathbb{C}}$ centered at the
origin. It follows from definition that the metric on a manifold is the same
as on its universal covering from the lifting properties of a map from the
unit disk. Note for orbifolds, we are considering orbifold maps and orbifold
uniformization as explained in 2.1.

From Ahlfors Schwarz Lemma, it follows easily that the Kobayashi metric on a
complex ball is precisely the same as the Poincaré metric which is the same as
the Bergman metric. The reader may consult Proposition 3 of [Y2] and the
references quoted there for various forms of Schwarz Lemma. Since we are
considering quotients of bounded domains, it follows from Schwarz Lemma that
the Kobayashi metric is positive definite in this paper. Note that the
Teichmüller space ${\mathcal{T}}_{g}$ can be realized as a bounded domain in
${\mathbb{C}}^{3g-3}$ from Bers Embedding.

Furthermore, it follows immediately from the definition that $g_{V,K}$ has
decreasing properties in the following sense. Let $F:M\rightarrow N$ be a
holomorphic mapping. Then $g_{N,K}(F_{*}v)\leqslant g_{M,K}(v)$ for all $v\in
T_{x}M$. Again, the Kobayashi metric may be degenerate in general, but in our
case it is always non-degenerate. This follows from the fact that the
manifolds involved by uniformized by bounded domains in ${\mathbb{C}}^{n}$ for
some $n>0$ and the earlier discussions.

We have the following consequence of the decreasing property of the Kobayashi
metric.

######  Lemma 2.

Suppose $M=B^{N}_{{\mathbb{C}}}/\Gamma$ is a totally geodesic subvariety of
${\mathcal{A}}_{g}$. Then
$g_{M,K}=(j_{g}^{-1})^{*}g_{T_{g}^{o},K}|_{M}=g_{{\mathcal{A}}_{g},K}|_{M}$.

Proof The inclusion map $i:M\rightarrow{\mathcal{A}}_{g}$ is a holomorphic
embedding. Now we have the holomorphic mappings

(14) $M\rightarrow{\mathcal{M}}_{g}^{o}\rightarrow{\mathcal{A}}_{g}.$

The first holomorphic map in (12) comes from our assumption that $M\subset
T_{g}^{o}$ and the fact that $j_{g}^{-1}|_{T_{g}^{o}-j_{g}(H_{g})}$ is a
holomorphic map. Here we used the fact that $j_{g}$ is an injective
holomorphic map and is an immersion on ${\mathcal{M}}_{g}-H_{g}$. The second
holomorphic mapping in (12) follows from Torelli mapping. The Torelli mapping
is holomorphic by definition. Since $M$ is a complex submanifold of
$T_{g}^{o}\subset{\mathcal{A}}_{g}$, it follows from definition of the
Kobayashi metric in terms of extremal functions that

(15) $g_{M,K}\geqslant(j_{g}^{-1})^{*}g_{T_{g}^{o},K}|_{M}\geqslant
g_{{\mathcal{A}}_{g},K}|_{M}.$

On the other hand, the Kobayashi metric on a manifold $g_{V,K}$ is the same as
its lift $g_{\widetilde{V},K}$ to the universal covering $\widetilde{V}$ of
$V$. Hence in terms of a totally geodesic $B^{N}$ in Siegel
${\mathcal{S}}_{g}$, we need to compare $g_{B^{N},K}$ and
$g_{{\mathcal{S}}_{g},K}|_{B^{N}}$. It follows from Proposition 3 that there
is a holomorphic map $p:{\mathcal{S}}_{g}\rightarrow\widetilde{M}$ so that
$p\circ i$ is identity. Hence for $x\in\widetilde{M}\subset{\mathcal{S}}_{g}$
and $w\in T_{x}\widetilde{M}\subset T_{x}{\mathcal{S}}_{g}$, a holomorphic
curve $f:\Delta_{r}\rightarrow{\mathcal{S}}_{g}\ {\mbox{holomorphic}}$ with
$f(0)=x,f^{\prime}(0)=w$ gives rise to a holomorphic map $p\circ
f:\Delta_{r}\rightarrow\widetilde{M}$ holomorphic with $p\circ f(0)=x$ and
$(p\circ f)^{\prime}(0)=w$. It follows from the decreasing property of the
Kobayashi metric that $g_{{\mathcal{S}}_{g},K}|_{B^{N}}\geqslant g_{B^{N},K}$,
which is equivalent to $g_{{\mathcal{A}}_{g},K}|_{M}\geqslant g_{M,K}$ after
descending to $M$ from the universal covering as discussed earlier.

Combining the above two paragraphs, we conclude that
$g_{{\mathcal{S}}_{g},K}|_{B^{N}}=g_{B^{N},K}$ and that the two inequalities
in (13) can be replaced by equalities.

∎

4.2 Theorem 1 now follows by putting the earlier arguments together.

Proof of Theorem 1 Assume for the sake of proof by contradiction that there
exists $M=B^{N}_{{\mathbb{C}}}/\Gamma$ so that $M$ is a totally geodesic
subvariety of ${\mathcal{A}}_{g}$ with $N>1$, and $M\subset
T_{g}^{o}-j_{g}(H_{g})=j_{g}({\mathcal{M}}_{g}^{o}),$ where
${\mathcal{M}}_{g}^{o}={\mathcal{M}}_{g}-H_{g}$.

From Lemma 2,
$g_{M,K}=(j_{g}^{-1})^{*}g_{T_{g}^{o},K}|_{M}=g_{{\mathcal{A}}_{g},K}|_{M}$.
In particular, $(j_{g}^{-1})^{*}g_{{\mathcal{M}}_{g}^{o},K}|_{M}=g_{M,K}$.
This however contradicts Proposition 1.

∎

4.3 We remark that the argument in the proof of Theorem 1 can be applied to
study the non-existence of locally Hermitian symmetric space in
$T_{g}^{o}-j_{g}(H_{g})$ as a totally geodesic complex suborbifold for
$g\geqslant 2$, a result proved earlier in [Ha]. For this purpose, we observe
that an analogue of Proposition 1 is true for $\widetilde{M}$ being any
Hermitian symmetric space as given in [A]. Together with the remarks given in
3.5 and the results of [Mok3], the other parts of the proof can be applied.

4.4 Proof of Theorem 2

From the results of [Ha], we know that any symmetric variety $M$ in
$T_{g}^{o}-j_{g}(H_{g})\subset{\mathcal{A}}_{g}$ has to be of real rank $1$ as
a locally symmetric space. Alternatively, to make the article more self-
contained, this follows from 3.5, 4.3 and the proof of Theorem 1. Since $M$ is
Hermitian symmetric, we know that $M$ has to be a complex ball quotient.
Theorem 1 implies that the complex dimension of $M$ is $1$ and hence $M$ is a
hyperbolic Riemann surface. From the discussions above, such a Riemann surface
$M$ has to be a Shimura-Teichmüller in the terminology of [Mö], since the
Kobayashi metric, which is well-known to be the same as the Teichmüller metric
on Teichmüller spaces, is the same as the natural hyperbolic metric on $M$ as
it is a totally geodesic curve in $T_{g}^{o}-j_{g}(H_{g})$. In such a case,
Möller proved in [Mö] that such a Riemann surface does not exist for genus
$g\neq 3,4,5$, and the only examples for $g=3,4$ are given in the statement of
Theorem 2. Very recently, it was proved by Aulicino and Norton in [AN] that
there is no example in $g=5$. Theorem 2 follows. ∎

Theorem 2 gives a necessary and sufficient condition for the existence of a
locally Hermitian symmetric space in $T_{g}^{o}-j_{g}(H_{g})$.

References

[A] Antonakoudis, S. M., Teichmüller spaces and bounded symmetric domains do
not mix isometrically. Geom. Funct. Anal. 27 (2017), no. 3, 453-465.

[AN] Aulicino, D., Norton, C., Shimura-Teichmüller curves in genus $5$,
arXiv:1903.01625v2.

[CLTZ] Chen, K., Lu, X., Tan, S-L., Zuo, K., On Higgs bundles over Shimura
varieties of ball quotient type. Asian J. Math. 22 (2018), no. 2, 269-284.

[C] Coleman, R. F., Torsion points on curves, in Galois representations and
arithmetic algebraic geometry (Kyoto, 1985/Tokyo, 1986), Advanced Studies in
Pure Mathematics, vol. 129 (North-Holland, Amsterdam, 1987), 235-247.

[DM] Daskalopoulos, G., Mese, C., Rigidity of Teichmüller space,
arXiv:1502.03367.

[FM] Farb B., and Masur, H., Superrigidity and mapping class groups, Topology
37 (1998), 1169-1176.

[dJN] de Jong, A., Noot, R., Jacobians with complex multiplication, in:
Arithmetic Algebraic Geometry, Texel, 1989, in: Progr. Math., vol. 89,
Birkhäuser, 1991, pp. 177-192.

[dJZ] de Jong, A., Zhang, S., Generic abelian varieties with real
multiplication are not Jacobians. Diophantine geometry, 165-172, CRM Series,
4, Ed. Norm., Pisa, 2007.

[Ha] Hain, R., Locally symmetric families of curves and Jacobians. In: Moduli
of curves and abelian varieties; pp. 91-108. (C. Faber, E. Looijenga, eds.)
Aspects of Math. E33, Vieweg, Braunschweig, 1999.

[HC] Harish-Chandra, Representations of semisimple Lie groups, CI, Amer. J.
Math. 78 (1956), 564-628.

[He] Helgason, S., Differential geometry, Lie groups, and symmetric spaces,
Pure Appl. Math., 80, Academic Press, 1978.

[I] Ihara, S., Holomorphic imbeddings of symmetric domains. J. Math. Soc.
Japan 19(1967), 261-302.

[KS] Korevaar, N., Schoen, R., Global existence theorem for harmonic maps to
non-locally compact spaces. Comm. Anal. Geom. 5 (1997) 333-387.

[Mok1] Mok, N., Metric Rigidity Theorems On Hermitian Locally Symmetric
Manifolds, World Scientific, 1989.

[Mok2] Mok, N., Projective algebraicity of minimal compactifications of
complex-hyperbolic space forms of finite volume. Perspectives in analysis,
geometry, and topology, 331?354, Progr. Math., 296, Birkhäuser/Springer, New
York, 2012.

[Mok3] Mok, N., private communication.

[MSY] Mok, N., Siu, Y.-T., Yeung, S.-K., Geometric superrigidity, Invent.
Math., 113(1993), 57-84.

[Mö] Möller, M., Shimura and Teichmüller curves. J. Mod. Dyn. 5 (2011), 1-32.

[Moo] Moonen, B., Special subvarieties arising from families of cyclic covers
of the projective line, Doc. Math. 15 (2010) 793-819.

[O] Oort, F., Canonical liftings and dense sets of CM-points, in Arithmetic
geometry (Cortona, 1994), Symposia Mathematica, vol. XXXVII (Cambridge
University Press, Cambridge, 1997), 228-234.

[Sag] Sagle, A. A., A note on triple systems and totally geodesic submanifolds
in a homogeneous space. Nagoya Math. J. 32 (1968), 5-20.

[Sat] Satake, I., Holomorphic imbeddings of symmetric domains into a Siegel
space. Amer. J. Math. 87(1965), 425-461.

[Si1] Siu, Y.-T., The complex-analyticity of harmonic maps and the strong
rigidity of compact Kähler manifolds, Ann. of Math. 112 (1980), 73-111.

[Si2] Siu, Y.-T., Strong rigidity of compact quotients of exceptional bounded
symmetric domains. Duke Math. J. 48 (1981), 857-871.

[SY] Siu, Y.-T., Yau, S.-T., Compactification of negatively curved complete
Kähler manifolds of finite volume. Seminar on Differential Geometry, pp.
363-380, Ann. of Math. Stud., 102, Princeton Univ. Press, Princeton, N.J.,
1982.

[Y1] Yeung, S.-K., Uniformization of 1/4-pinched negatively curved manifolds
with special holonomy. Int. Math. Res. Not. 1995, 365-375.

[Y2] Yeung, S.-K., Quasi-isometry of metrics on Teichmüller spaces. Int. Math.
Res. Not. 2005, 239-255.