lemmatheorem lemma conjecturetheorem conjecture propositiontheorem proposition
corollarytheorem corollary problemtheorem problem questiontheorem question
claimtheorem claim definitiontheorem definition exampletheorem example
remarktheorem remark remarkstheorem remarks

# Non-existence of Shimura curves of Mumford type generically in the non-
hyperelliptic locus

Xin Lu  School of Mathematical Sciences, Key Laboratory of Mathematics and
Engineering Applications (Ministry of Education) & Shanghai Key Laboratory of
Pure Mathematics and Mathematical Practice East China Normal University,
Shanghai 200241, China<EMAIL_ADDRESS>,  Shengli Tan  School of
Mathematical Sciences, Key Laboratory of Mathematics and Engineering
Applications (Ministry of Education) & Shanghai Key Laboratory of Pure
Mathematics and Mathematical Practice East China Normal University, Shanghai
200241, China<EMAIL_ADDRESS>and  Kang Zuo  School of Mathematics and
Statistics, Wuhan University, Wuhan, China; Institut für Mathematik,
Universität Mainz, Mainz, Germany<EMAIL_ADDRESS>In memory of Gang Xiao

###### Abstract.

We show that there does not exist any Shimura curve with strictly maximal
Higgs field generically in the Torelli locus of non-hyperelliptic curves of
genus $g\geq 4$. In particular, Shimura curves of Mumford type are not
generically in the Torelli locus of non-hyperelliptic curves of genus $g\geq
4$.

###### Key words and phrases:

Shimura curve, Coleman-Oort conjecture, fibred surface

###### 2010 Mathematics Subject Classification:

14J10,14E30

This work is supported by National Natural Science Foundation of China,
Fundamental Research Funds for central Universities, and Science and
Technology Commission of Shanghai Municipality (No. 22DZ2229014)

##  1\. Introduction

Shimura varieties are classically constructed as quotients of bounded
symmetric domains by arithmetic groups. A large class of Shimura varieties,
called ‘of Hodge type’, parametrizes families of abelian varieties with
certain properties. A characterization of Shimura varieties to be of Hodge
type can be found in [Mum69]. In the same paper, Mumford also constructs an
interesting and surprising example of Shimura curves
$C_{0}\subseteq\mathcal{A}_{4}$ of Hodge type which are not of PEL type (i.e.
not characterized by the existence of algebraic endomorphisms on the abelian
varieties), where $\mathcal{A}_{4}$ is the moduli space of principally
polarized abelian 4-folds. Such Shimura curves $C_{0}\subseteq\mathcal{A}_{4}$
will be referred as Shimura curves of Mumford type in $\mathcal{A}_{4}$. A
general fiber over such a Shimura curve $C_{0}\subseteq\mathcal{A}_{4}$ is an
abelian $4$-fold admitting an endomorphism ring equal to $\mathbb{Z}$. The
properties of such Shimura curves of Mumford type have been extensively
studied by many mathematicians, cf. [Noo00, Noo01, VZ04]. In this paper, we
are mainly interested in the intersection of the Shimura curves of Mumford
type with the Torelli locus of algebraic curves.

Fixing $n\geq 3$ an integer, we have $\mathcal{M}_{g,[n]}$ the moduli space of
smooth projective curves over complex number $\mathbb{C}$ of genus $g\geq 2$
with a full level $n$-structure, and $\mathcal{A}_{g,[n]}$ the moduli space of
$g$-dimensional principally polarized abelian varieties over $\mathbb{C}$ with
a full level-$n$ structure. No specific choice of the level $n(\geq 3)$ is
made because it is only imposed to assure the representability, which plays no
essential role in our study. For simplicity, the subscript ‘$[n]$’ is omitted
without causing much confusion. The Torelli morphism

$j^{\circ}:~{}\mathcal{M}_{g}\longrightarrow\mathcal{A}_{g}$

sends a curve to its Jacobian with its canonical principal polarization and
level structure. The image of $j^{\circ}$, denoted as
$\mathcal{T}_{g}^{\circ}$, is a locally closed subvariety in $\mathcal{A}_{g}$
and called the open Torelli locus. Its closure will be denoted as
$\mathcal{T}_{g}$, and called the Torelli locus. Denote by
$\mathcal{TH}_{g}^{\circ}=j^{\circ}(\mathcal{H}_{g})$ the image of the
hyperelliptic locus, and $\mathcal{TH}_{g}\subseteq\mathcal{A}_{g}$ its
closure which we call the Torelli locus of hyperelliptic curves, where
$\mathcal{H}_{g}\subseteq\mathcal{M}_{g}$ is the locus corresponding to the
hyperelliptic curves.

A closed subvariety $Z\subseteq\mathcal{A}_{g}$ (of positive dimension) is
said to be contained generically in the Torelli locus (resp. the Torelli locus
of hyperelliptic curves), written as $Z\Subset\mathcal{T}_{g}$ (resp.
$Z\Subset\mathcal{TH}_{g}$), if $Z\subseteq\mathcal{T}_{g}$ and
$Z\cap\mathcal{T}_{g}^{\circ}\neq\emptyset$ (resp.
$Z\subseteq\mathcal{TH}_{g}$ and $Z\cap\mathcal{T}_{g}^{\circ}\neq\emptyset$).
Meanwhile, we will say a subvariety $Z\subseteq\mathcal{A}_{g}$ is contained
generically in the Torelli locus of non-hyperelliptic curves, if
$Z\Subset\mathcal{T}_{g}$ but $Z\not\Subset\mathcal{TH}_{g}$. It is easy to
see that $Z\cap j^{\circ}(\mathcal{NH}_{g})$ is Zariski dense in $Z$ for any
closed subvariety $Z$ contained generically in the Torelli locus of non-
hyperelliptic curves, where $\mathcal{NH}_{g}\subseteq\mathcal{M}_{g}$ is the
locus corresponding to the non-hyperelliptic curves.

Note that the Shimura curve $C_{0}\subseteq\mathcal{A}_{4}$ of Mumford type
constructed in [Mum69] is complete. Recall also that the Torelli locus
$\mathcal{T}_{4}\subseteq\mathcal{A}_{4}$ is a subvariety of codimension one,
which is in fact ample as a divisor by [Igu81]. It follows immediately that
$C_{0}\cap\mathcal{T}_{4}\neq\emptyset$ for any Shimura curve
$C_{0}\subseteq\mathcal{A}_{4}$ of Mumford type. Oort [Oor19, § 7] (see also
[MO13, Question 6.5]) has asked the following question.

######  Question .

Is there any Shimura curve of Mumford type contained generically in
$\mathcal{T}_{4}$?

Recently, Baldi-Klingler-Ullmo [BKU24, Theorem 3.17] made a progress on the
above question by showing that there exists a Shimura curve
$C_{0}\subseteq\mathcal{A}_{4}$ of Mumford type such that
$C_{0}\cap\mathcal{T}_{4}^{\circ}$ is a non-CM point. Here we recall that
$p\in\mathcal{A}_{g}$ is a CM point if the corresponding abelian variety
$A_{p}$ admits a ’large’ endomorphism ring, i.e.,
$\mathrm{End}(A_{p})\otimes\mathbb{Q}$ contains a commutative semi-simple
algebra of $\text{{\rm rank\,}}=2g$. Our main result is the following.

######  Theorem 1.1.

There does not exist any Shimura curve of Mumford type contained generically
in the Torelli locus of non-hyperelliptic curves of genus $g=4$.

The proof relies on the characterization of Shimura curves by Viehweg-Zuo
[VZ04]. Recall that the special subvarieties in $\mathcal{A}_{g}$ are just
totally geodesic subvarieties containing CM points, cf. [Moo98]. They are of
particular interest because on one hand they are locally symmetric in the
sense of differential geometry, and on the other hand they parameterize
abelian varieties with prescribed Hodge classes. Special subvarieties of
dimension zero are exactly CM points, and special curves will be also called
as Shimura curves in this paper. The Coleman-Oort conjecture [Col87, Oor97]
predicts the non-existence of special subvarieties of positive dimension
contained generically in the Torelli locus.

######  Conjecture  (Coleman-Oort conjecture).

For any sufficiently large $g$, there does not exist any positive-dimensional
special subvariety $Z\Subset\mathcal{T}_{g}$; or equivalently, there exist at
most finitely many CM points contained in $\mathcal{T}_{g}^{\circ}$ if $g$ is
sufficiently large.

We refer to [MO13] and the references therein for an introduction to the
Coleman-Oort conjecture. We remark also that for small genera ($g\leq 7$),
there indeed exist positive-dimensional special subvarieties
$Z\Subset\mathcal{T}_{g}$.

In [VZ04], it is proved that an algebraic curve
$C_{0}\subseteq\mathcal{A}_{g}$ is Shimura if and only if $C_{0}$ contains at
least one CM point and the Higgs bundle associated to $C_{0}$ admits maximal
Higgs field; see also Section 2.2 for more details. Based on such a
characterization, we proved the non-exsistence of Shimura curves generically
in the Torelli locus of hyperelliptic curves if $g\geq 8$ in [LZ17].
Furthermore, we [CLZ21] showed that there are at most finitely many CM points
contained in $j^{\circ}(\mathcal{S}_{g})$ if $g\geq 8$, where
$\mathcal{S}_{g}\subseteq\mathcal{M}_{g}$ is the locus corresponding to the
superelliptic curves; in particular, $\mathcal{TH}_{g}^{\circ}$ contains at
most finitely many CM points if $g\geq 8$.

Our main interest here is the Shimura curves of Mumford type in
$\mathcal{A}_{4}$, as well as their generalizations in [VZ04], which will be
also referred as Shimura curves of Mumford type (cf. [LZ19, § 2.2]). The Higgs
bundle associated to a Shimura curve $C_{0}\subseteq\mathcal{A}_{g}$ of
Mumford type admits strictly maximal Higgs field [VZ04]. Thus Theorem 1.1
follows immediately from the next theorem.

######  Theorem 1.2.

There does not exist any Shimura curve with strictly maximal Higgs field
generically in the Torelli locus of non-hyperelliptic curves of genus $g\geq
4$.

When $C_{0}\Subset\mathcal{TH}_{g}$ is a Shimura curve with strictly maximal
Higgs field, then the family of semi-stable curves representing $C_{0}$ admits
also strictly strictly maximal Higgs field, cf. Section 2.3. Combining the
above theorem together with [LZ19, Theorem D], we obtain

######  Corollary .

Let $C_{0}\subseteq\mathcal{A}_{g}$ be a Shimura curve with strictly maximal
Higgs field. If $g>4$, then $C_{0}\not\Subset\mathcal{T}_{g}$.

The paper is organized as follows. In Section 2, we recall some known facts
about the Shimura curves of Mumford type, Higgs bundles and semi-stable
families representing Shimura curves generically in the TOrelli locus. In
Section 3 we prove a slope inequality related to the family of semi-stable
curves representing a Shimura curve of Mumford type, and the main result will
be proved in Section 4.

##  2\. Preliminaries

###  2.1. The Shimura curves of Mumford type

In this subsection, we briefly recall the construction of the Shimura curves
of Mumford type in $\mathcal{A}_{4}$ by Mumford [Mum69], and refer to [VZ04]
(see also [LZ19, § 2.2]) for the generalizations in $\mathcal{A}_{g}$ with $g$
being large.

Let $F$ be a totally real cubic field over $\mathbb{Q}$ with $3$ distinct real
embeddings $\sigma_{1},\sigma_{2},\sigma_{3}$. Let $R$ be a quaternion algebra
over $F$ such that

$\rho_{1}:R\otimes_{\sigma_{1}}\mathbb{R}\simeq\mathrm{Mat}_{2}(\mathbb{R}),\qquad\rho_{i}:R\otimes_{\sigma_{i}}\mathbb{R}\simeq\mathbb{H},\
(i=2\text{~{}or~{}}3),$

where $\mathbb{H}$ is Hamilton’s quaternion algebra over $\mathbb{R}$. The
corestriction $D:=\mathrm{Cor}_{F/\mathbb{Q}}R$ is a central simple algebra
over $\mathbb{Q}$. Suppose $R$ is chosen such that the corestriction $D$
splits, i.e.,

$D\cong\mathrm{Mat}_{8}(\mathbb{Q}).$

Denote by $K$ the kernel of the reduced norm

${\rm Nm}:~{}R^{\times}\longrightarrow F^{\times}.$

The $\mathbb{Q}$-group $\mathbf{G}^{\prime}$ associated to $K$ is connected
and semi-simple with

$\mathbf{G}^{\prime}_{\mathbb{R}}\simeq\mathrm{SL}_{2}(\mathbb{R})\times\mathrm{SU}_{2}(\mathbb{R})\times\mathrm{SU}_{2}(\mathbb{R}).$

The homomorphism

$R^{\times}\longrightarrow D\cong\mathrm{Mat}_{8}(\mathbb{Q})$

defines a representation of $K$ which preserves a symplectic form on
$V=\mathbb{Q}^{8}$. Enlarge $\mathbf{G}^{\prime}$ to a connected reductive
$\mathbb{Q}$-group $\mathbf{G}$ which only differs from $\mathbf{G}^{\prime}$
by the split center $\mathbb{G}_{\mathrm{m},\mathbb{Q}}$. It is the
$\mathbb{Q}$-group associated to

$R^{*}:=\\{a\in R:{\rm Nm}(a)\in\mathbb{Q}^{\times}\subset F^{\times}\\}.$

We thus obtain the Shimura datum $(\mathbf{G},X)$, where $X$ is the
$\mathbf{G}(\mathbb{R})$-conjugacy class of the following homomorphism
$h:\mathbb{S}\rightarrow\mathbf{G}_{\mathbb{R}}$ given by

$z=\rho\exp(\sqrt{-1}\theta)\mapsto\bigg{(}\left[\begin{array}[]{cc}\rho\cos\theta&\rho\sin\theta\\\
-\rho\sin\theta&\rho\cos\theta\end{array}\right],\,I_{2},I_{2}\bigg{)}.$

It is clear that $X$ is isomorphic to $\mathcal{H}_{1}^{\pm}$, and
$(\mathbf{G},X)$ is a subdatum of $(\mathrm{GSp}_{8},\mathcal{H}^{\pm}_{4})$
by the representation of $\mathbf{G}$ induced by
$R^{\times}\rightarrow\mathrm{Mat}_{8}(\mathbb{Q})$, where
$\mathcal{H}^{\pm}_{g}$ is the Siegel double half spaces of complex symmetric
$g\times g$ matrices with definite imaginary part. It follows that the Shimura
datum $(\mathbf{G},X)$ defines a Shimura curve
$C_{0}\subseteq\mathcal{A}_{4}$.

By construction, the representation of $G^{\prime}$ on the space
$V=\mathbb{Q}^{8}$ is absolutely irreducible. Hence $A_{p}$ admits no non-
trivial endomorphism for any general point $p\in C_{0}$, where $A_{p}$ is the
abelian $4$-fold corresponding to $p\in\mathcal{A}_{4}$.

###  2.2. The Higgs bundle associated to a Shimura curve

Let $C_{0}\subseteq\mathcal{A}_{g}$ be any closed curve. We assume that
$C_{0}$ is smooth by normalization if necessary. Denote by
$\mathfrak{h}_{0}:\,\mathfrak{X}_{0}\to C_{0}$ the universe family of
$g$-dimensional abelian varieties over $C_{0}$, which is assumed to be
extended to a semi-stable family $\mathfrak{h}:\,\mathfrak{X}\to C$ over the
compactification $C$ by arising the level structure if necessary. The local
system
$\mathbb{V}_{C_{0}}:=R^{1}(\mathfrak{h}_{0})_{*}\mathbb{Z}_{\mathfrak{X}_{0}}$
admits unipotent local monodromy around the boundary $\Delta_{C}:=\,C\setminus
C_{0}$. By taking the grading of the extended Hodge filtration, one obtains a
logarithmic system of Hodge bundles

$\left(E_{C},\,\theta_{C}\right)=\big{(}E^{1,0}_{C}\oplus
E^{0,1}_{C},\,\theta_{C}\big{)}=\Big{(}\mathfrak{h}_{*}\big{(}\Omega^{1}_{\mathcal{X}/C}(\log\Upsilon_{C})\big{)}\oplus
R^{1}\mathfrak{h}_{*}\mathcal{O}_{\mathcal{X}},\,\theta_{C}\Big{)},$

where $\Upsilon_{C}=\mathfrak{h}^{-1}(\Delta_{C})$, and the Higgs field

$\theta_{C}:~{}E^{1,0}_{C}\longrightarrow
E^{0,1}_{C}\otimes\Omega^{1}_{C}(\log\Delta_{C})$ (2.1)

is just the extension of the following Kodaira-Spencer map on the Hodge
bundles

$\theta_{0}:~{}(\mathfrak{h}_{0})_{*}\big{(}\Omega^{1}_{\mathcal{X}_{0}/C_{0}}\big{)}\longrightarrow
R^{1}(\mathfrak{h}_{0})_{*}\mathcal{O}_{\mathcal{X}_{0}}\otimes\Omega^{1}_{C_{0}}.$

We will call $\left(E_{C},\,\theta_{C}\right)$ the logarithmic Higgs bundle
associated to $C_{0}$ (or $C$). The logarithmic Higgs bundle admits the
following Fujita decomposition (cf. [Fuj78]):

$\left(E_{C}^{1,0}\oplus
E_{C}^{0,1},~{}\theta_{C}\right)=\left(A_{C}^{1,0}\oplus
A_{C}^{0,1},~{}\theta_{C}\big{|}_{A_{C}^{1,0}}\right)\oplus\left(F_{C}^{1,0}\oplus
F_{C}^{0,1},~{}0\right),$ (2.2)

where $A_{C}^{1,0}$ is an ample vector bundle, while $F_{C}^{1,0}$ and
$F_{C}^{0,1}$ are flat vector bundles associated to unitary local subsystems
$\mathbb{F}_{C}^{1,0}\oplus\mathbb{F}_{C}^{0,1}\subseteq\mathbb{V}_{C_{0}}\otimes\mathbb{C}.$

######  Definition  ([VZ03]).

The Higgs bundle $(E_{C}^{1,0}\oplus E_{C}^{0,1},~{}\theta_{C})$ is said to
admit maximal Higgs field if the map

$\theta_{C}\big{|}_{A_{C}^{1,0}}:\,A_{C}^{1,0}\longrightarrow
A_{C}^{0,1}\otimes\Omega^{1}_{C}(\log\Delta_{C})$

is an isomorphism; and to admit strictly maximal Higgs field if furthermore
$F_{C}^{1,0}\oplus F_{C}^{0,1}=0$, namely the map $\theta_{C}$ in (2.1) is an
isomorphism.

By [VZ03] the Higgs field $\theta_{C}$ is strictly maximal (resp. maximal) if
and only if the following Arakelov equality holds

$\displaystyle\deg E^{1,0}_{C}$ $\displaystyle={g\over
2}\cdot\deg\Omega^{1}_{C}(\log\Delta_{C}),$ (2.3)
$\displaystyle\Big{(}\text{resp.}\quad\deg E^{1,0}_{C}$
$\displaystyle={\text{{\rm rank\,}}A_{C}^{1,0}\over
2}\cdot\deg\Omega^{1}_{C}(\log\Delta_{C})~{}\Big{)}.$

Recall also that if the Higgs bundle $\theta_{C}$ is maximal, then the ample
part $A_{C}^{1,0}$ must be semi-stable. In particular, if $\theta_{C}$ is
strictly maximal, then $E^{1,0}_{C}$ is semi-stable.

The following theorem gives numerical characterizations of Shimura curves, and
more generally of totally geodesic curves.

######  Theorem 2.1 ([Möl11, VZ04]).

Let $C_{0}\subseteq\mathcal{A}_{g}$ be a closed curve, and $(E_{C}^{1,0}\oplus
E_{C}^{0,1},~{}\theta_{C})$ the associated logarithmic Higgs bundle as above.
Then

  1. (1)

The Higgs field $\theta_{C}$ is maximal if and only if $C_{0}$ is a totally
geodesic curve.

  2. (2)

The Higgs field $\theta_{C}$ is maximal if and only if either $C_{0}$ is a
Shimura curve of Mumford type or $C_{0}$ parametrizes abelian varieties
isogeneous to the self-product of a modular family of elliptic curves.

In particular, if $C_{0}\subseteq\mathcal{A}_{g}$ is a Shimura curve of
Mumford type with $(E_{C}^{1,0}\oplus E_{C}^{0,1},~{}\theta_{C})$ the
associated logarithmic Higgs bundle as above, then the Higgs field
$\theta_{C}$ is strictly maximal. Note also that Shimura curves of Mumford
type is itself compact, from which it follows that $C=C_{0}$ is this case.
Therefore, the Arakelov equality (2.3) on a Shimura curve
$C_{0}\subseteq\mathcal{A}_{g}$ of Mumford type becomes

$\deg E^{1,0}_{C}={g\over 2}\cdot\deg\Omega^{1}_{C}.$ (2.4)

###  2.3. The family of semi-stable curves representing a Shimura curve
generically in the Torelli locus

Given a smooth closed curve $C_{0}\Subset\mathcal{T}_{g}$ with a suitable
smooth compactification $C$ by joining a finite set of cusps
$\Delta_{C}=C\setminus C_{0}$, we have constructed a family $f:\,S\to B$ of
semi-stable curves representing $C_{0}$ in [LZ19, § 3]. For convenience, we
briefly recall here the construction.

Fix an integer $n$, let
$\mathcal{M}^{ct}_{g}=\mathcal{M}^{ct}_{g,[n]}\supseteq\mathcal{M}_{g}=\mathcal{M}_{g,[n]}$
be the partial compactification of the moduli space of smooth projective
genus-$g$ curves with level-$n$ structure by adding stable curves with compact
Jacobians. When $n\geq 3$, it carries a universal family of stable curves with
compact Jacobians

$\mathfrak{f}:~{}\mathcal{S}_{g}^{ct}\longrightarrow{\mathcal{M}}^{ct}_{g}.$
(2.5)

The Torelli morphism $j^{\rm o}$ can be naturally extended to
$\mathcal{M}^{ct}_{g}$:

$j:~{}\mathcal{M}^{ct}_{g}\longrightarrow\mathcal{A}_{g},\qquad\text{with~{}}\mathcal{T}_{g}=j\big{(}\mathcal{M}^{ct}_{g}\big{)}.$

The morphism $j^{\rm o}$ is 2:1 and ramified exactly over the locus of
hyperelliptic curves. However, the relative dimension of $j$ is positive along
the boundary $\mathcal{T}_{g}\setminus\mathcal{T}^{\rm o}_{g}$.

Given a smooth closed curve $C_{0}\Subset\mathcal{T}_{g}$, let $B_{0}$ be the
normalization of the strict inverse image $j^{-1}(C)$ of $C$, and denote by
$\pi_{0}:\,B_{0}\to C_{0}$ the induced morphism. If $B_{0}$ is reducible, then
we replace $B_{0}$ by any one of the two irreducible components. By pulling
back the universal family over $\mathcal{M}^{ct}_{g}$ to $B_{0}$ and resolving
the singularities, one gets a family $f_{0}:S_{0}\to B_{0}$ of semi-stable
curves that extends uniquely to a family $f:S\to B$ of semi-stable curves over
the smooth completion $B\supseteq B_{0}$.

######  Definition .

The family $f:\,S\to B$ is called the family of semi-stable curves
representing $C_{0}\subseteq\mathcal{T}_{g}$ via the Torelli morphism.

By construction, any singular fiber over the boundary
$\Delta_{nc}:=\,B\setminus B_{0}$ admits non-compact Jacobian. Moreover, the
local system
$\mathbb{V}_{B_{0}}:=R^{1}(f_{0})_{*}\mathbb{Z}_{S_{0}}=\pi_{0}^{*}\mathbb{V}_{C_{0}}$.
It is well-known that the logarithmic Higgs bundle associated to
$\mathbb{V}_{B_{0}}$ has the form

$\left(E_{B}^{1,0}\oplus
E_{B}^{0,1},~{}\theta_{B}\right)=\left(f_{*}\omega_{S/B}\oplus
R^{1}f_{*}\mathcal{O}_{S},~{}\theta_{B}\right),$ (2.6)

and it admits a decomposition of Higgs bundles similar to (2.2):

$\left(E_{B}^{1,0}\oplus E_{B}^{0,1},\theta_{B}\right)=\left(A_{B}^{1,0}\oplus
A_{B}^{0,1},~{}\theta_{B}\big{|}_{A_{B}^{1,0}}\right)\oplus\left(F_{B}^{1,0}\oplus
F_{B}^{0,1},~{}0\right).$ (2.7)

Let $\pi:\,B\to C$ be the induced map. Then
$\Delta_{nc}=\pi^{-1}(\Delta_{C})$, and

$\pi^{*}\left(E^{1,0}_{C}\oplus
E^{0,1}_{C},~{}\theta_{C}\right)=\left(E^{1,0}_{B}\oplus
E^{0,1}_{B},~{}\theta_{B}\right),$

which is compatible with the Fujita decompositions (2.2) on $C$ and (2.7) on
$B$.

######  Definition .

Let $f:\,S\to B$ be any family of semi-stable curves of genus $g\geq 2$ and
$\Upsilon_{nc}\to\Delta_{nc}$ the singular fibres with non-compact Jacobians.
Then $f$ is said to admit maximal Higgs field if

$\theta_{B}\big{|}_{A_{B}^{1,0}}:\,A_{B}^{1,0}\longrightarrow
A_{B}^{0,1}\otimes\Omega^{1}_{B}(\log\Delta_{nc})$

is an isomorphism; and to be with strictly maximal Higgs field if furthermore
$F_{B}^{1,0}\oplus F_{B}^{0,1}=0$. Similar to (2.3), $f$ has strictly maximal
(resp. maximal) Higgs field if and only if

$\displaystyle\deg f_{*}\omega_{S/B}$ $\displaystyle={g\over
2}\cdot\deg\Omega^{1}_{B}(\log\Delta_{nc}),$ (2.8)
$\displaystyle\Big{(}\text{resp.}\quad\deg f_{*}\omega_{S/B}$
$\displaystyle={\text{{\rm rank\,}}A^{1,0}_{B}\over
2}\cdot\deg\Omega^{1}_{B}(\log\Delta_{nc})~{}\Big{)}.$

######  Proposition  ([LZ19, Proposition 3.4]).

Let $f:\,S\to B$ be the family of semi-stable curves representing a closed
curve $C_{0}\Subset\mathcal{T}_{g}$ as above. Then

(i). if $C\Subset\mathcal{TH}_{g}$, then

$\deg f_{*}\omega_{S/B}=\deg
E^{1,0}_{C},\quad\deg\Omega_{B}^{1}(\log\Delta_{nc})=\deg\Omega_{C}^{1}(\log\Delta_{C});$
(2.9)

(ii). if $C\not\Subset\mathcal{TH}_{g}$, then either $\pi:\,B\to C$ is an
isomorphism, in which case (2.9) holds, or $\pi:\,B\to C$ is double cover, in
which case we have

$\deg f_{*}\omega_{S/B}=2\deg
E^{1,0}_{C},\quad\deg\Omega_{B}^{1}(\log\Delta_{nc})=2\deg\Omega_{C}^{1}(\log\Delta_{C})+|\Lambda|,$

where $\Lambda$ is the ramification locus of the morphism $\pi_{0}:\,B_{0}\to
C_{0}$.

######  Corollary .

Let $f:\,S\to B$ be the family of semi-stable curves representing a Shimura
curve $C_{0}\Subset\mathcal{T}_{g}$ with strictly maximal Higgs field. Suppose
that $C_{0}\not\Subset\mathcal{TH}_{g}$. Then either $\pi:\,B\to C$ is an
isomorphism, in which case $f$ has strictly maximal Higgs field, i.e.,

$\deg f_{*}\omega_{S/B}=\frac{g}{2}\deg\Omega_{B}^{1}(\log\Delta_{nc});$
(2.10)

or $\pi:\,B\to C$ is a double cover, in which case $f_{*}\omega_{S/B}$ is
semi-stable with

$\deg f_{*}\omega_{S/B}=\frac{g}{2}\deg\Omega_{B}^{1}(\log\Delta_{nc})-{g\over
2}\cdot|\Lambda|,$ (2.11)

where $\Lambda$ is the ramification locus of the morphism $\pi_{0}:\,B_{0}\to
C_{0}$.

##  3\. Slope inequality

In this section, we would like to consider the slope inequality for a family
$f:\,S\to B$ of semi-stable curves representing a closed curve
$C_{0}\Subset\mathcal{T}_{g}$ as in Section 2.3. We will suppose in this
section that $C_{0}$ is not contained generically in the Torelli locus
$\mathcal{TH}_{g}$ of hyperelliptic curves, and that the induced map
$\pi_{0}:\,B_{0}\to C_{0}$ is a double cover with ramification locus
$\Lambda\subseteq B_{0}$.

Let $\tau_{0}:\,B_{0}\to B_{0}$ be the involution on $B_{0}$ induced by the
double cover $\pi_{0}:\,B_{0}\to C_{0}$. Then there is an involution
$\sigma_{0}$ acting on $S_{0}$ with the following diagram (cf. [LZ19, Lemma
A.1])

$\textstyle{S_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{0}}$$\scriptstyle{f_{0}}$$\textstyle{S_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\textstyle{B_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{0}}$$\textstyle{B_{0}}$

Let $\Upsilon_{ct}\to\Delta_{ct}$ be the singular locus of the family
$f_{0}:\,S_{0}\to B_{0}$. By construction, any singular fiber over
$\Delta_{ct}$ admits a compact Jacobian. According to [LZ19, Lemma A.1], the
fiber over any point $p\in\Lambda$ is hyperelliptic(may be singular), and
$\sigma_{0}|_{F_{p}}$ is just the hyperelliptic involution of the fiber
$F_{p}$ (i.e., the quotient $F_{p}/\langle\sigma_{0}|_{F_{p}}\rangle$ is a
tree of rational curves). For convenience, let

$\Lambda_{1}=\Lambda\cap\Delta_{ct},\quad\text{~{}and~{}}\Lambda_{2}=\Lambda\setminus\Lambda_{1}.$

By definition, the fiber $F_{p}$ over any point $p\in\Lambda_{1}$ (resp.
$p\in\Lambda_{2}$) is a singular hyperelliptic curve of compact type (resp.
smooth hyperelliptic curve).

It is clear that the involution $\tau_{0}$ (resp. $\sigma_{0}$) can be
extended to an involution $\tau$ on $B$ (resp. $\sigma$ on $S$) with the
following diagram:

$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{f}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{B}$

Our main result in the section is the following.

######  Theorem 3.1.

Let $f:\,S\to B$ be the family of semi-stable curves representing a closed
curve $C_{0}\Subset\mathcal{T}_{g}$ as in Section 2.3. Suppose that
$C_{0}\not\Subset\mathcal{TH}_{g}$ with $\pi_{0}:\,B_{0}\to C_{0}$ being a
double cover with ramification locus $\Lambda\subseteq B_{0}$, and that
$f_{*}\omega_{S/B}$ is semi-stable. Then

$\displaystyle\omega_{S/B}^{2}\geq$ $\displaystyle\frac{5g-6}{g}\deg
f_{*}\omega_{S/B}+3(g-2)|\Lambda|$ (3.1)
$\displaystyle+\sum_{p\in\Lambda_{1}}4\big{(}l_{h}(F_{p})+l_{1}(F_{p})-1\big{)}+\sum_{p\in\Delta_{ct}\setminus\Lambda_{1}}\big{(}3l_{h}(F_{p})+2l_{1}(F_{p})-3\big{)},$

where $l_{i}(F_{p})$ is the number of irreducible components of genus $i$ in
$F_{p}$ for any $i\geq 1$, and $l_{h}(F_{p})=\sum\limits_{i\geq
2}l_{i}(F_{p})$.

###### Proof.

Firstly, we may assume that $\Lambda\neq\emptyset$; otherwise, the theorem
follows from [LZ19, Theorem 5.2]. We claim that

######  Claim .

There exists a unramified base change $\psi:\,\widetilde{B}\to B$ with the
following properties.

  1. (1)

The involution $\tau$ lifts to an involution $\tilde{\tau}$ on $\widetilde{B}$
such that the fixed locus of $\tilde{\tau}$ is exactly
$\widetilde{\Lambda}=\psi^{-1}(\Lambda)$.

  2. (2)

The involution $\sigma$ lifts to an involution $\tilde{\sigma}$ on
$\widetilde{S}$ with following diagram, where $\widetilde{S}$ is the
relatively minimal model of $S\times_{B}\widetilde{B}$ over $\widetilde{B}$.

$\textstyle{\widetilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\sigma}}$$\scriptstyle{\tilde{f}}$$\textstyle{\widetilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}}$$\textstyle{\widetilde{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\tau}}$$\textstyle{\widetilde{B}}$
(3.2)

  3. (3)

For any $p\in\widetilde{\Lambda}=\psi^{-1}(\Lambda)$, the fiber
$\widetilde{F}_{p}=\tilde{f}^{-1}(p)$ is hyperelliptic and the restriction
$\tilde{\sigma}|_{\widetilde{F}_{p}}$ is the hyperelliptic involution of
$\widetilde{F}_{p}$.

###### Proof of Section 3.

Let $R\subseteq B$ be the ramification divisor of the double cover $\pi:\,B\to
C$. By assumption, $R\cap B_{0}=\Lambda=\Lambda_{1}+\Lambda_{2}$. Let
$R_{nc}=R\cap\Delta_{nc}$. Suppose first that

$\big{|}R_{nc}\big{|}\qquad\text{is even}.$ (3.3)

Let

$\varphi:\,\widetilde{C}\to C$

be a double cover ramified over $\pi(R_{nc})$. Such a double cover $\varphi$
exists, since we have assumed that $\big{|}R_{nc}\big{|}$ is even and $R_{nc}$
is contained in the ramification divisor $R$, which implies that
$\big{|}\pi(R_{nc})\big{|}$ is also even. Let $\tau^{\prime}$ be the
involution on $\widetilde{C}$ induced by the double cover $\varphi$ above, and
$\widetilde{B}$ be the normalization of $B\times_{C}\widetilde{C}$ with the
induced double cover $\psi:\,\widetilde{B}\to B$. Then the involution
$(\tau,id)$ (resp. $(id,\tau^{\prime})$) on the product
$B\times_{C}\widetilde{C}$ induces an involution $\tau_{1}$ (resp. $\tau_{2}$)
on $\widetilde{B}$ such that
$\widetilde{B}/\langle\tau_{1}\rangle=\widetilde{C}$ (resp.
$\widetilde{B}/\langle\tau_{2}\rangle=B$). By construction, we have the
following diagram

$\textstyle{\widetilde{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{\tilde{\pi}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\widetilde{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{C}$

Note that $\pi:\,B\to C$ (resp. $\varphi:\,\widetilde{C}\to C$) is branched
over $\pi(R)$ (resp. $\pi(R_{nc})$). Since $\pi(R_{nc})\subseteq\pi(R)$, the
double cover $\psi$ is unramified and the double cover $\tilde{\pi}$ is
ramified over
$\tilde{\pi}^{-1}\big{(}\varphi^{-1}(\pi(\Lambda))\big{)}=\psi^{-1}(\Lambda)$.
Let $\tilde{\tau}=\tau_{1}$. Then the fixed locus $\widetilde{\Lambda}$ of
$\tilde{\tau}$ is nothing but the ramified divisor of $\tilde{\pi}$; namely
$\widetilde{\Lambda}=\psi^{-1}(\Lambda)$. This proves the first statement.

Let $\tilde{\sigma}$ be the involution on $\widetilde{S}$ induced by the
involution $(\sigma,\tilde{\tau})$ on the fiber-product
$S\times_{B}\widetilde{B}$. Then the involution $\tilde{\sigma}$ satisfies the
required commutative diagram (3.2). Let $\Psi:\,\widetilde{S}\to S$ be the
induced map. Then the following diagram commutes

$\textstyle{\widetilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\scriptstyle{\tilde{f}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\widetilde{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{B}$

Thus for any $p\in\widetilde{\Lambda}$, the restricted map

$\Psi|_{\widetilde{F}_{p}}:\,\widetilde{F}_{p}\longrightarrow F_{\psi(p)}$

gives an isomorphism between $\widetilde{F}_{p}$ and $F_{\psi(p)}$, and under
this isomorphism, $\tilde{\sigma}|_{\widetilde{F}_{p}}$ is the same as
$\sigma|_{F_{\psi(p)}}$. In other word, the restriction
$\tilde{\sigma}|_{\widetilde{F}_{p}}$ is the hyperelliptic involution of
$\widetilde{F}_{p}$ as required.

The situation would be a little more subtle if (3.3) does not hold, i.e., if
$\big{|}R_{nc}\big{|}$ is odd. Choose a subset $D_{1}\subseteq R_{nc}$ such
that $|D_{1}|$ is even, and let $D_{2}=R_{nc}\setminus D_{1}$. Let
$\varphi_{1}:\,\widetilde{C}_{1}\to C$ be a double cover ramified over
$\pi(D_{1})$, $\widetilde{B}$ be the normalization of
$B\times_{C}\widetilde{C}_{1}$, and $\psi_{1}:\,\widetilde{B}_{1}\to B$. Then
similar to the above arguments, one shows that
$\psi_{1}:\,\widetilde{B}_{1}\to B$ is unramified with the following
properties.

  1. (1)

The involution $\tau$ lifts to an involution $\tilde{\tau}_{1}$ on
$\widetilde{B}_{1}$ such that the fixed locus of $\tilde{\tau}_{1}$ is exactly
$\widetilde{\Lambda}_{1}=\psi_{1}^{-1}(R\setminus D_{1})$.

  2. (2)

The involution $\sigma$ lifts to an involution $\tilde{\sigma}_{1}$ on
$\widetilde{S}_{1}$ with following diagram, where $\widetilde{S}_{1}$ is the
relatively minimal model of $S\times_{B}\widetilde{B}_{1}$ over
$\widetilde{B}_{1}$.

$\textstyle{\widetilde{S}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\sigma}_{1}}$$\scriptstyle{\tilde{f}_{1}}$$\textstyle{\widetilde{S}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}_{1}}$$\textstyle{\widetilde{B}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\tau}_{1}}$$\textstyle{\widetilde{B}_{1}}$

  3. (3)

For any $p\in\psi_{1}^{-1}(\Lambda)$, the fiber
$\widetilde{F}_{p}=\tilde{f}^{-1}(p)$ is hyperelliptic and the restriction
$\tilde{\sigma}|_{\widetilde{F}_{p}}$ is the hyperelliptic involution of
$\widetilde{F}_{p}$.

  4. (4)

The number $|\psi_{1}^{-1}(D_{2})|$ is even.

Let
$\pi_{1}:\,\widetilde{B}_{1}\to\widetilde{C}_{1}\triangleq\widetilde{B}_{1}/\langle\tilde{\tau}_{1}\rangle$,
$\varphi_{2}:\,\widetilde{C}_{2}\to\widetilde{C}_{1}$ be a double cover
ramified over $\pi_{1}\big{(}\psi^{-1}(D_{2})\big{)}$, and $\widetilde{B}$ be
the normalization of
$\widetilde{B}_{1}\times_{\widetilde{C}_{1}}\widetilde{C}_{2}$ with the
induced unramified cover $\psi_{2}:\,\widetilde{B}\to\widetilde{B}_{1}$.
Repeating the above construction, one shows that the composition
$\psi=\psi_{1}\circ\psi_{2}:\,\widetilde{B}\to B$ would satisfy the
requirements of the claim. ∎

Let $\widetilde{X}$ be the desingularization of
$\widetilde{S}/\langle\tilde{\sigma}\rangle$, and
$\tilde{h}:\,\widetilde{X}\to\widetilde{C}$ the induced fibration. Then we
have the following diagram:

|
---|---
$\textstyle{\widetilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}}$$\textstyle{\widetilde{S}/\langle\tilde{\sigma}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{h}}$$\textstyle{\widetilde{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\pi}}$$\textstyle{\widetilde{C}}$

We will prove

######  Claim .

Let $\tilde{h}:\,\widetilde{X}\to\widetilde{C}$ be as above.

  1. (1)

The direct image sheaf $\tilde{h}_{*}\omega_{\widetilde{X}/\widetilde{C}}$ is
semi-stable, and

$\left\\{\begin{aligned}
\omega_{\widetilde{X}/\widetilde{C}}^{2}&\,\leq\frac{\omega_{\widetilde{S}/\widetilde{B}}^{2}}{2}+\sum_{q\in\psi^{-1}(\Lambda)}\big{(}2g-2-2(l_{h}(\widetilde{F}_{q})+l_{1}(\widetilde{F}_{q})-1)\big{)},\\\
\deg\tilde{h}_{*}\omega_{\widetilde{X}/\widetilde{C}}&\,=\frac{\deg
f_{*}\omega_{\widetilde{S}/\widetilde{B}}}{2}+\frac{g}{2}\,\big{|}\psi^{-1}(\Lambda)\big{|},\end{aligned}\right.$
(3.4)

where $\widetilde{F}_{q}$ is the fiber $\tilde{f}^{-1}(q)$ for any
$q\in\psi^{-1}(\Lambda)$.

  2. (2)

The following slope inequality holds

$\omega_{\widetilde{X}/\widetilde{C}}^{2}\geq\frac{5g-6}{g}\deg\tilde{h}_{*}\omega_{\widetilde{X}/\widetilde{C}}+d(g-2)\,|\Lambda|+\sum_{p\in\Delta_{ct}\setminus\Lambda_{1}}\frac{d}{2}\,\big{(}3l_{h}(F_{p})+2l_{1}(F_{p})-3\big{)}.$
(3.5)

where $d=\deg\psi$ is the degree of the map $\psi:\,\widetilde{B}\to B$
constructed in Section 3.

###### Proof of Section 3.

(1) We first prove that the direct image sheaf
$\tilde{h}_{*}\omega_{\widetilde{X}/\widetilde{C}}$ is semi-stable based on
(3.4). By the construction, the family
$\tilde{f}:\,\widetilde{S}\to\widetilde{B}$ is just the relatively minimal
model of $\widetilde{X}\times_{\widetilde{C}}\widetilde{B}\to\widetilde{B}$.
That is, we have the following diagram:

$\textstyle{\widetilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}}$$\textstyle{\widetilde{S}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{\prime}}$$\scriptstyle{\tilde{f}^{\prime}}$$\scriptstyle{\beta}$$\textstyle{\widetilde{X}\times_{\widetilde{C}}\widetilde{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{h}}$$\textstyle{\widetilde{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\pi}}$$\textstyle{\widetilde{C}}$

where
$\beta:\,\widetilde{S}^{\prime}\to\widetilde{X}\times_{\widetilde{C}}\widetilde{B}$
is the desingularization. By [Har77, Remark 9.3.1], we have a natural map

$\alpha:\,\tilde{\pi}^{*}\tilde{h}_{*}\omega_{\widetilde{X}/\widetilde{C}}\longrightarrow\tilde{f}^{\prime}_{*}\tilde{\beta}^{*}\omega_{\widetilde{X}/\widetilde{C}},$
(3.6)

which is clear injective, where
$\tilde{\beta}:\,\widetilde{S}^{\prime}\to\widetilde{X}$ is the composition
map. Let $\mathcal{E}\subseteq\widetilde{S}^{\prime}$ be the reduced divisor
consisting of the exceptional curves of
$\beta^{\prime}:\,\widetilde{S}^{\prime}\to\widetilde{S}$, and let
$R_{\tilde{\beta}}\subseteq\widetilde{S}^{\prime}$ be the ramification divisor
of $\tilde{\beta}:\,\widetilde{S}^{\prime}\to\widetilde{X}$. Then
$\mathcal{E}\subseteq R_{\tilde{\beta}}$. Hence

$(\beta^{\prime})^{*}\omega_{\widetilde{S}}\otimes\mathcal{O}_{\widetilde{S}^{\prime}}(\mathcal{E})=\omega_{\widetilde{S}^{\prime}}=\tilde{\beta}^{*}\omega_{\widetilde{X}}\otimes\mathcal{O}_{\widetilde{S}^{\prime}}(R_{\tilde{\beta}}+D),$

where $D\geq 0$ is supported on the curves contracted by $\tilde{\beta}$. In
particular,
$\tilde{\beta}^{*}\omega_{\widetilde{X}}\hookrightarrow(\beta^{\prime})^{*}\omega_{\widetilde{S}}$.
Note also that
$\omega_{\widetilde{B}}=\tilde{\pi}^{*}\omega_{\widetilde{C}}\otimes\mathcal{O}_{\widetilde{B}}\big{(}\psi^{-1}(\Lambda)\big{)}$.
It follows that

$\alpha_{1}:\,\tilde{f}^{\prime}_{*}\tilde{\beta}^{*}\omega_{\widetilde{X}/\widetilde{C}}\hookrightarrow\tilde{f}^{\prime}_{*}\left((\beta^{\prime})^{*}\omega_{\widetilde{S}/\widetilde{B}}\otimes(\tilde{f}^{\prime})^{*}\mathcal{O}_{\widetilde{B}}\big{(}\psi^{-1}(\Lambda)\big{)}\right)=\tilde{f}_{*}\omega_{\widetilde{S}/\widetilde{B}}\otimes\mathcal{O}_{\widetilde{B}}\big{(}\psi^{-1}(\Lambda)\big{)}.$
(3.7)

In particular,

$\deg
f^{\prime}_{*}\tilde{\beta}^{*}\omega_{\widetilde{X}/\widetilde{C}}\leq\deg\tilde{f}_{*}\omega_{\widetilde{S}/\widetilde{B}}+g\cdot\big{|}\psi^{-1}(\Lambda)\big{|}.$

On the other hand,

$\deg\tilde{\pi}^{*}\tilde{h}_{*}\omega_{\widetilde{X}/\widetilde{C}}=2\deg\tilde{h}_{*}\omega_{\widetilde{X}/\widetilde{C}}=\deg\tilde{f}_{*}\omega_{\widetilde{S}/\widetilde{B}}+g\cdot\big{|}\psi^{-1}(\Lambda)\big{|}.$

The last equality follows from (3.4). Hence both the map $\alpha$ in (3.6) and
$\alpha_{1}$ in (3.7) are isomorphisms. Since
$\tilde{f}_{*}\omega_{\widetilde{S}/\widetilde{B}}=\psi^{*}f_{*}\omega_{S/B}$
is semi-stable, so is the sheaf

$\tilde{f}_{*}\omega_{\widetilde{S}/\widetilde{B}}\otimes\mathcal{O}_{\widetilde{B}}\big{(}\psi^{-1}(\Lambda)\big{)}=\tilde{f}^{\prime}_{*}\tilde{\beta}^{*}\omega_{\widetilde{X}/\widetilde{C}}\cong\tilde{\pi}^{*}\tilde{h}_{*}\omega_{\widetilde{X}/\widetilde{C}}.$

This implies that $\tilde{h}_{*}\omega_{\widetilde{X}/\widetilde{C}}$ is also
semi-stable.

It remains to prove (3.4). According to Section 3, the quotient
$\widetilde{S}/\langle\tilde{\sigma}\rangle$ is smooth over
$\widetilde{C}\setminus\tilde{\pi}(\widetilde{\Lambda})=\widetilde{C}\setminus\tilde{\pi}\big{(}\psi^{-1}(\Lambda)\big{)}$.
Moreover, for any $q\in\psi^{-1}(\Lambda)$, the restriction
$\tilde{\sigma}|_{\widetilde{F}_{q}}$ is the hyperelliptic involution of
$\widetilde{F}_{q}$. According to [LZ19, Lemma 4.7], each irreducible
component contained in the fiber $\widetilde{F}_{q}$ is either of positive
genus or rational curve with self-intersection equal to $-2$. In particular,
each component of $\widetilde{F}_{q}$ is mapped to itself under
$\tilde{\sigma}|_{\widetilde{F}_{q}}$, and $\tilde{\sigma}|_{C}$ is the
hyperelliptic involution for each component $C$ with $g(C)\geq 1$ (if
$g(C)=1$, we understand $\tilde{\sigma}|_{C}$ also as hyperelliptic involution
for convenience since
$C/\langle\tilde{\sigma}|_{C}\rangle\cong\mathbb{P}^{1}$). Let $D$ be the
fixed locus of $\tilde{\sigma}|_{\widetilde{F}_{q}}$. Then $D$ consists of
rational curves and the fixed locus
$\mathrm{Fix}\big{(}\tilde{\sigma}|_{C}\big{)}$ with $g(C)\geq 1$. Therefore,
the corresponding fiber $\widetilde{\Gamma}_{p}$ is normal crossing, and the
multiplicity of each component $C\subseteq\widetilde{\Gamma}_{p}$ is with one
or two, where $p=\tilde{\pi}(q)$ and
$\widetilde{\Gamma}_{p}=\tilde{h}^{-1}(p)$. Moreover, the components with
multiplicity two are exactly the image of the components in
$\widetilde{F}_{q}$ with positive genus. Let $C\subseteq\widetilde{F}_{q}$ be
a component with $g(C)\geq 1$, and $C^{\prime}$ the corresponding image in
$\widetilde{\Gamma}_{p}$. Then all the components in $\widetilde{\Gamma}_{p}$
is of multiplicity one and there are exactly $2g(C)+2$ such components. Hence
for any such component $C\subseteq\widetilde{F}_{q}$ with the corresponding
component $C^{\prime}\subseteq\widetilde{\Gamma}_{p}$,

$(C^{\prime})^{2}=-\frac{2g(C)+2}{2}=-\big{(}g(C)+1\big{)},\qquad\omega_{\widetilde{X}/\widetilde{C}}\cdot
C^{\prime}=-2-(C^{\prime})^{2}=g(C)-1.$

Let $\widetilde{\Gamma}_{p,red}$ be the reduced part of
$\widetilde{\Gamma}_{p}$. Then

$\displaystyle\widetilde{\Gamma}_{p,red}^{2}$
$\displaystyle\,=\big{(}\widetilde{\Gamma}_{p}-\sum_{C^{\prime}}C^{\prime}\big{)}^{2}=\sum_{C^{\prime}}(C^{\prime})^{2}=\sum_{C^{\prime}}\big{(}g(C)+1\big{)}=g+l_{h}(\widetilde{F}_{q})+l_{1}(\widetilde{F}_{q}),$
$\displaystyle\omega_{\widetilde{X}/\widetilde{C}}\cdot\widetilde{\Gamma}_{p,red}$
$\displaystyle\,=\omega_{\widetilde{X}/\widetilde{C}}\cdot\big{(}\widetilde{\Gamma}_{p}-\sum_{C^{\prime}}C^{\prime}\big{)}=g-1+\big{(}l_{h}(\widetilde{F}_{q})+l_{1}(\widetilde{F}_{q})-1\big{)},$

where the sum is taken over all components
$C^{\prime}\subseteq\widetilde{\Gamma}_{p}$ of multiplicity two. By [Tan96,
Theorem 3.1], one computes easily that

$\displaystyle c_{1}^{2}(\widetilde{\Gamma}_{p})$
$\displaystyle=4g-4-2\,\omega_{\widetilde{X}/\widetilde{C}}\cdot\widetilde{\Gamma}_{p,red}-\widetilde{\Gamma}_{p,red}^{2}-\eta_{\widetilde{\Gamma}_{p}}$
(3.8)
$\displaystyle=4g-4-2\left(g-1+\big{(}l_{h}(\widetilde{F}_{q})+l_{1}(\widetilde{F}_{q})-1\big{)}\right)$
$\displaystyle\quad+\big{(}g+l_{h}(\widetilde{F}_{q})+l_{1}(\widetilde{F}_{q})\big{)}-\big{(}g+l_{h}(\widetilde{F}_{q})+l_{1}(\widetilde{F}_{q})+\epsilon\big{)}$
$\displaystyle=2g-2-2\big{(}l_{h}(\widetilde{F}_{q})+l_{1}(\widetilde{F}_{q})-1\big{)}-\epsilon,$

where

$\eta_{\widetilde{\Gamma}_{p}}=\sum_{x\in
N(\widetilde{\Gamma}_{p,red})}\frac{\big{(}\gcd(a_{x},b_{x})\big{)}^{2}}{a_{x}b_{x}}=\sum_{C^{\prime}}\big{(}g(C)+1\big{)}+\epsilon=g+l_{h}(\widetilde{F}_{q})+l_{1}(\widetilde{F}_{q})+\epsilon,$

where $N(\widetilde{\Gamma}_{p,red})$ is the set of nodes in
$\widetilde{\Gamma}_{p,red}$, $a_{x},b_{x}$ are the multiplicities of the two
local components of $\widetilde{\Gamma}_{p}$ through $x\in
N(\widetilde{\Gamma}_{p,red})$, the sum is taken over all components
$C^{\prime}\subseteq\widetilde{\Gamma}_{p}$ of multiplicity two, and
$\epsilon$ is the number of nodes in $N(\widetilde{\Gamma}_{p,red})$ where the
multiplicities of the two local components are both equal to two. Similarly,
one obtains that

$\chi_{\widetilde{\Gamma}_{p}}=\frac{1}{12}\left(6g-6-3\,\omega_{\widetilde{X}/\widetilde{C}}\cdot\widetilde{\Gamma}_{p,red}-2\widetilde{\Gamma}_{p,red}^{2}+\big{|}N(\widetilde{\Gamma}_{p,red})\big{|}-\eta_{\widetilde{\Gamma}_{p}}\right)=\frac{g}{2}.$
(3.9)

Note that the family $\tilde{f}:\,\widetilde{S}\to\widetilde{B}$ is semi-
stable, and that $\tilde{\pi}$ is ramified over
$\tilde{\pi}\big{(}\psi^{-1}(\Lambda)\big{)}$. It follows from (3.8) that

$\omega_{\widetilde{X}/\widetilde{C}}^{2}=\frac{\omega_{\widetilde{S}/\widetilde{B}}^{2}}{2}+\sum_{q\in\tilde{\pi}\big{(}\psi^{-1}(\Lambda)\big{)}}c_{1}^{2}(\widetilde{F}_{q})\leq\frac{\omega_{\widetilde{S}/\widetilde{B}}^{2}}{2}+\sum_{q\in\psi^{-1}(\Lambda)}\big{(}2g-2-2(l_{h}(\widetilde{F}_{q})+l_{1}(\widetilde{F}_{q})-1)\big{)}.$

By (3.9), one can similarly prove the second equality in (3.4).

(2) Consider the natural map

$\rho:\,S^{2}\tilde{h}_{*}\omega_{\widetilde{X}/\widetilde{C}}\longrightarrow\tilde{h}_{*}\big{(}\omega_{\widetilde{X}/\widetilde{C}}^{\otimes
2}\big{)}.$

Since $\tilde{h}$ is non-hyperelliptic by construction, the map $\rho$ is
generically surjective. Let $\mathcal{S}$ be the cokernel of $\rho$, which is
torsion as $\rho$ is generically surjective. Since $\psi:\,\widetilde{B}\to B$
is unramified over $B_{0}$ and $\tilde{\pi}:\,\widetilde{B}\to\widetilde{C}$
is ramified exactly over $\psi^{-1}(\Lambda)$, it follows that

$\big{|}\tilde{\pi}(\psi^{-1}(\Lambda))\big{|}=d\,|\Lambda|,\qquad\big{|}\tilde{\pi}(\psi^{-1}(\Delta_{ct}\setminus\Lambda_{1}))\big{|}=\frac{d}{2}\,|\Delta_{ct}\setminus\Lambda_{1}|.\quad$
(3.10)

Moreover, locally around any point
$q\in\tilde{\pi}(\psi^{-1}(\Delta_{ct}\setminus\Lambda_{1}))$,

the local family $\tilde{h}:\,\widetilde{X}|_{U_{q}}\to U_{q}$ is isormorphic
to the local family $f:\,S|_{U_{p}}\to U_{p}$, (3.11)

where $p=\psi(\tilde{\pi}^{-1}(q))\in B$, and $U_{p}\subseteq B$ and
$U_{q}\subseteq\widetilde{C}$ are local neighborhoods of $p$ and $q$
respectively (Here we remark that $\psi(\tilde{\pi}^{-1}(q))$ consists of only
one point). It follows that

$3l_{h}(\widetilde{F}_{q})+2l_{1}(\widetilde{F}_{q})-3=3l_{h}(F_{p})+2l_{1}(F_{p})-3,\quad\forall\,q\in\tilde{\pi}(\psi^{-1}(\Delta_{ct}\setminus\Lambda_{1}))\text{~{}and~{}}p=\psi(\tilde{\pi}^{-1}(q)).$

Therefore, similar to the proof of [LZ19, Theorem 5.2], it suffices to prove
that

$\mathrm{length\,}\mathcal{S}_{q}\geq\left\\{\begin{aligned}
&3l_{h}(\widetilde{F}_{q})+2l_{1}(\widetilde{F}_{q})-3,&&\forall\,q\in\tilde{\pi}(\psi^{-1}(\Delta_{ct}\setminus\Lambda_{1}));\\\
&g-2,&&\forall\,q\in\tilde{\pi}(\psi^{-1}(\Lambda)).\end{aligned}\right.$
(3.12)

Applying (3.11), we see that the first inequality in (3.12) is in fact already
proved in [LZ19, Lemmas 5.3]. It remains to prove the last inequality of
(3.12). But this is clear, since the fiber $\widetilde{\Gamma}_{q}$ over
$q\in\tilde{\pi}(\psi^{-1}(\Lambda))$ is hyperelliptic, and hence

$\mathrm{length\,}\mathcal{S}_{q}\geq\dim\mathrm{coker}(\nu_{q})\geq g-2,$

where

$\nu_{q}:\,S^{2}H^{0}(\widetilde{\Gamma}_{q},\omega_{\widetilde{\Gamma}_{q}})\longrightarrow
H^{0}\big{(}\widetilde{\Gamma}_{q},\omega_{\widetilde{\Gamma}_{q}}^{\otimes
2}\big{)},$

is the natural multiplication map. This proves (3.12). ∎

We now come back to the proof of Theorem 3.1. Since $\psi$ is unramified,

$\omega_{\widetilde{S}/\widetilde{B}}^{2}=d\,\omega_{S/B}^{2},\qquad\deg\tilde{f}_{*}\omega_{\widetilde{S}/\widetilde{B}}=d\deg
f_{*}\omega_{S/B},\qquad\text{where $d=\deg\psi$}.$

Combining this with (3.4) and (3.5), one proves (3.1). ∎

##  4\. The proof of Theorem 1.2

In this section, we prove out main result.

###### Proof of Theorem 1.2.

We prove by contradiction. Suppose that there exists a Shimura curve
$C_{0}\Subset\mathcal{T}_{g}$ with strictly maximal Higgs field generically in
the Torelli locus of non-hyperelliptic curves of genus $g\geq 4$.

Let $B_{0}\subseteq\mathcal{M}_{g}^{ct}$ be the inverse image curve, and let
$f:\,S\to B$ be the family of semi-stable curves representing $C_{0}$ as in
Section 2.3. Since $C_{0}$ is contained generically in the Torelli locus of
non-hyperelliptic curves, the family $f$ is non-hyperelliptic. If $B_{0}$ is
reducible, then the induced cover $\pi:\,B\to C$ is an isomorphism and the
family $f$ has strictly maximal Higgs field by Section 2.3. This contradicts
[LZ19, Theorem 7.3]. Thus we may assume that $C_{0}$ is irreducible, in which
case, $\pi:\,B\to C$ is a double cover and $f_{*}\omega_{S/B}$ is semi-stable
satisfying (2.11) by Section 2.3. Thus by Theorem 3.1, one has

$\displaystyle\omega_{S/B}^{2}\geq$ $\displaystyle\frac{5g-6}{g}\deg
f_{*}\omega_{S/B}+3(g-2)|\Lambda|$ (4.1)
$\displaystyle+\sum_{p\in\Lambda_{1}}4\big{(}l_{h}(F_{p})+l_{1}(F_{p})-1\big{)}+\sum_{p\in\Delta_{ct}\setminus\Lambda_{1}}\big{(}3l_{h}(F_{p})+2l_{1}(F_{p})-3\big{)},$

Here we recall that $\Lambda\subseteq B_{0}$ is the ramification divisor of
the double cover $\pi_{0}:\,B_{0}\to C_{0}$, $\Delta_{ct}\subseteq B_{0}$ is
the subset over which the fiber of $f_{0}:\,S_{0}\to B_{0}$ is singular,
$\Lambda_{1}=\Lambda\cap\Delta_{ct}$. On the other hand, according to [LZ19,
Theorem 4.3], one has

$\displaystyle\omega_{S/B}^{2}~{}\leq{}$
$\displaystyle(2g-2)\cdot\deg\left(\Omega^{1}_{B}(\log\Delta_{nc})\right)+\sum_{p\in\Delta_{ct}\cap\Lambda^{\prime}}\frac{3}{2}\cdot\big{(}l_{h}(F_{p})+l_{1}(F_{p})-1\big{)}$
(4.2)
$\displaystyle+\left(2g-2-\frac{g-1}{6}\right)\cdot|\Delta_{ct,b}|+\sum_{p\in\Delta_{ct,ub}}\big{(}3l_{h}(F_{p})+2l_{1}(F_{p})-3\big{)},$

where

$\displaystyle\Lambda^{\prime}\,=$ $\displaystyle\left\\{p\in
B~{}\Bigg{|}\begin{aligned} &F_{p}\text{~{}is a hyperelliptic curve with
compact Jacobian, and any}\\\ &\text{two irreducible irrational components of
$F_{p}$ do not intersect}\end{aligned}~{}\right\\};$
$\displaystyle\Delta_{ct,b}\,=$
$\displaystyle\left\\{p\in\Delta_{ct}\setminus\Lambda^{\prime}~{}\big{|}~{}\text{each
irreducible component of $F_{p}$ is of genus $0$ or $1$}\right\\};$
$\displaystyle\Delta_{ct,ub}\,=$
$\displaystyle\Delta_{ct}\setminus\left\\{\Lambda^{\prime}\cup\Delta_{ct,b}\right\\}.$

According to the proof of [LZ19, Theorem 4.2] in [LZ19, § 4.3], it follows
that $\Lambda\subseteq\Lambda^{\prime}$, and hence
$\Lambda_{1}\subseteq\Delta_{ct}\cap\Lambda^{\prime}$. Combining this with
(2.11) and (4.2), one obtains that

$\displaystyle\omega_{S/B}^{2}~{}\leq{}$ $\displaystyle\frac{4g-4}{g}\deg
f_{*}\omega_{S/B}+(2g-2)|\Lambda|+\sum_{p\in\Lambda_{1}}\frac{3}{2}\cdot\big{(}l_{h}(F_{p})+l_{1}(F_{p})-1\big{)}$
$\displaystyle+\left(2g-2-\frac{g-1}{6}\right)\cdot|\Delta_{ct,b}|+\sum_{p\in\Delta_{ct}\setminus(\Lambda_{1}\cup\Delta_{ct,b})}\big{(}3l_{h}(F_{p})+2l_{1}(F_{p})-3\big{)},$

Together with (4.1), we get

$\displaystyle\frac{g-2}{g}\deg f_{*}\omega_{S/B}\leq$
$\displaystyle(4-g)|\Lambda|-\sum_{p\in\Lambda_{1}}\frac{5}{2}\cdot\big{(}l_{h}(F_{p})+l_{1}(F_{p})-1\big{)}$
(4.3)
$\displaystyle+\sum_{p\in\Delta_{ct,b}}\left(2g-2-\frac{g-1}{6}-\big{(}3l_{h}(F_{p})+2l_{1}(F_{p})-3\big{)}\right)$
$\displaystyle<$
$\displaystyle\sum_{p\in\Delta_{ct,b}}\left(2g-2-\frac{g-1}{6}-\big{(}3l_{h}(F_{p})+2l_{1}(F_{p})-3\big{)}\right)$
$\displaystyle=$ $\displaystyle\frac{7-g}{6}\,\big{|}\Delta_{ct,b}\big{|}.$

We remark here that the last equality above follows from the definition of
$\Delta_{ct,b}$, which implies that $3l_{h}(F_{p})+2l_{1}(F_{p})-3=2g-3$; and
that the strictness of the above inequality follows from the proof of [LZ19,
Theorem 4.3] (see [LZ19, Claim 4.6(ii)]). In particular, $|\Delta_{ct,b}|>0$.

On the other hand, since each irreducible component of $F_{p}$ is of genus $0$
or $1$ for any $p\in\Delta_{ct,b}$, it follows that the $F_{p}$ admits at
least $g-1$ nodes, i.e.,

$\delta(F_{p})=\sum_{i=1}^{g/2}\delta_{i}(F_{p})\geq
g-1,\qquad\forall\,p\in\Delta_{ct,b},$

where $\delta_{i}(F_{p})$ is the number of nodes of type $i$ (we refer to
[LZ19, § 2.4] for details). Hence by Moriwaki’s slope inequality ([Mor98,
Theorem D]),

$\displaystyle(8g+4)\deg f_{*}\omega_{S/B}\geq$
$\displaystyle\sum_{F~{}\mathrm{singular}}\bigg{(}g\delta_{0}(F)+\sum_{i=1}^{g/2}4i(g-i)\delta_{i}(F)\bigg{)}$
$\displaystyle\geq$
$\displaystyle\sum_{p\in\Delta_{ct,b}}\sum_{i=1}^{g/2}4i(g-i)\delta_{i}(F_{p})$
$\displaystyle\geq$
$\displaystyle\sum_{p\in\Delta_{ct,b}}4(g-1)\delta(F_{p})\geq
4(g-1)^{2}\big{|}\Delta_{ct,b}\big{|}.$

Combining this with (4.3), one obtains

$\frac{(7-g)g}{6(g-2)}\big{|}\Delta_{ct,b}\big{|}>\deg
f_{*}\omega_{S/B}\geq\frac{(g-1)^{2}}{2g+1}\big{|}\Delta_{ct,b}\big{|},$

which gives a contradiction since $g\geq 4$. ∎

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