#  Volumes of Subvarieties of Complex Ball Quotients and Effective Very
Ampleness

Soheil Memarian<EMAIL_ADDRESS>Dept. of Mathematics,
University of Toronto, Toronto, Canada.

###### Abstract

Let $X=\Gamma\backslash\mathbb{B}^{n}$ be an $n$-dimensional complex ball
quotient by a torsion-free non-uniform lattice $\Gamma$ whose parabolic
subgroups are unipotent. We prove that the volumes of subvarieties of $X$ are
controlled by the systole of $X,$ which is the length of the shortest closed
geodesic of $X$. As applications, we obtain effective global generation and
very ampleness results for multiples of the canonical bundle
$K_{\overline{X}},$ where $\overline{X}$ is the toroidal compactification of
$X.$ These results follow from the bound we find for the Seshadri constant of
$K_{\overline{X}}$ in terms of the systole.

###### :

32Q45 (primary), 14E25, 32M15 (secondary).

###### keywords:

hyperbolicity, ball quotient, toroidal compactification, very ample.

##  1 Introduction

Let $X=\Gamma\backslash\mathbb{B}^{n}$ be an $n$-dimensional complex ball
quotient by a torsion-free lattice $\Gamma.$ The complex ball has an intrinsic
Hermitian metric (Bergman metric) which induces a Kähler form on $X.$ This
Kähler form also induces a Kähler form on a subvariety $V$ of $X.$ The volume
of $V$ with respect to the induced Kähler form on $X$ will be called the
induced Kähler volume and will be denoted by $\operatorname{vol}_{X}(V).$ In
the case that $X$ is compact, Hwang and To found a lower bound for
$\operatorname{vol}_{X}(V)$ in terms of a geometric invariant of $X$:

######  Theorem.

([HT99]) Let $X$ be a compact complex ball with injectivity radius $r$ and $V$
be an $m$-dimensional subvariety of $X.$ Then, the following inequality holds:

$\displaystyle\operatorname{vol_{X}(V)}$
$\displaystyle\geq\frac{(4\pi)^{m}}{m!}\sinh^{2m}(r).$

The main goal of this paper is to find an analogous lower bound for the
induced Kähler and canonical volumes of subvarieties of a non-compact ball
quotient $X$ in terms of a geometric invariant of $X.$ Note that in the case
that $X$ is not compact, the injectivity radius of $X$ is zero. So we will use
the systole, the length of the shortest closed geodesic in $X,$ as a geometric
invariant of $X$ to bound the volume of a subvariety of $X.$ For a compact
ball quotient, the systole is twice the injectivity radius. However, for a
non-compact $X$ the systole is still not zero and can be estimated by the
trace of the hyperbolic elements in a representation of $\Gamma.$

In terms of the induced Kähler volume of a subvariety we prove the following
theorem:

######  Theorem A.

(Theorem 5.2) Let $X=\Gamma\backslash\mathbb{B}^{n}$ be a complex ball
quotient by a torsion-free non-uniform lattice $\Gamma$ whose parabolic
stabilizers are unipotent. Let $V\subset X$ be a subvariety of dimension
$m>0.$ Then,

$\displaystyle\operatorname{vol}_{X}(V)$
$\displaystyle\geq\frac{(4\pi)^{m}}{m!}\sinh^{2m}\big{(}\operatorname{sys}(X)/2\big{)},$

where $\operatorname{vol}_{X}(V)$ is the volume of $V$ induced by Bergman
metric on $V$ and $\operatorname{sys}(X)$ is the length of the shortest closed
geodesic in $X.$

The assumption on the parabolic stabilizer of $\Gamma$ is mild as every neat
lattice $\Gamma$ satisfies this assumption, and moreover, every $\Gamma$ has a
finite index subgroup which satisfies this assumption (see [Hum98]). With this
assumption, $X$ admits a smooth projective toroidal compactification
$\overline{X}$ whose boundary divisor $D=\overline{X}\setminus X$ is a
disjoint union of abelian varieties with ample conormal bundle ([Mok12]).
Bakker and Tsimerman [BT18] proved that if the uniform depth of cusps of $X$
is sufficiently large, then the canonical bundle of the toroidal
compactification $K_{\overline{X}}$ is ample. We prove in Theorem 3.9 that the
systole of $X$ bounds the uniform depth of cusps from below. Therefore, if
$\operatorname{sys}(X)$ is sufficiently large, then $K_{\overline{X}}$ is
ample.

For a subvariety $V\subset\overline{X}$ of dimension $m>0,$ we denote the
degree of $V$ with respect to the line bundle $K_{\overline{X}}$ by
$\operatorname{deg}_{\overline{X}}(V):$

$\operatorname{deg}_{\overline{X}}(V):=K_{\overline{X}}^{m}\cdot V.$

Also, we study the canonical volume of a subvariety $V$ which is an intrinsic
quantity of $V$ and a priory does not depend on the ambient space
$\overline{X}.$ Let $V^{\prime}$ be a smooth variety birational to $V$ with a
canonical bundle $K_{V^{\prime}}.$ The canonical volume of the subvariety $V$
is

$\widetilde{\operatorname{vol}}_{V}:=\limsup\limits_{b\rightarrow\infty}\dfrac{h^{0}(V^{\prime},bK_{V^{\prime}})}{b^{m}/m!},$

which does not depend on the choice of $V^{\prime}.$ In particular, if $V$ is
an integral curve, then the canonical volume of $V$ is $2g-2.$ The canonical
volume of $V$ measures the asymptotic growth rate of the pluri-canonical
linear series $|bK_{V^{\prime}}|.$ The canonical volume is a non-negative real
number and it is positive if and only if the linear system $|bK_{V^{\prime}}|$
embedds $V^{\prime}$ birationally in a projective space for a large enough
$b$, i.e., $V$ is of general type.

We prove that the systole controls both the canonical volume of $V$ and its
degree with respect to $K_{\overline{X}}$ in the following sense:

######  Theorem B.

(Theorem 5.5$+$Theorem 5.7) With the same assumption on $X$ as Theorem A, let
$\overline{X}$ be the toroidal compactification of $X$ and
$V\subset\overline{X}$ be a subvariety of dimensional $m>0$ with $X\cap
V\neq\varnothing.$ Suppose that $\operatorname{sys}(X)\geq
4\ln\big{(}5n+(8\pi)^{4}\big{)}.$ Then the following inequalities hold:

$\displaystyle\widetilde{\operatorname{vol}}_{V}$
$\displaystyle>(\frac{m}{4\pi})^{m}e^{m\operatorname{sys}(X)/16},$
$\displaystyle\operatorname{deg}_{\overline{X}}(V)$
$\displaystyle>(\frac{n}{4\pi})^{m}e^{m\operatorname{sys}(X)/16}.$

Note that systole cannot decrease in a cover and for every $X$ there exists a
finite cover $X^{\prime}$ such that $\operatorname{sys}(X^{\prime})$ is
sufficiently large. As a byproduct of Theorem B, we can observe that in a
tower of covering of $X$ where the systole increases, the canonical volume of
subvarieties increases.

Combining Theorem B with the results in the adjunction theory proved by
Angehrn-Siu [AS95], Kollar [Kol97] and Ein-Lazersfeld-Nakamaye [ELN96] gives
effective results in global generation, very ampleness and separation of jets:

######  Corollary C.

(Corollary 6.13) With the same $X$ and $\overline{X}$ as Theorem B, suppose
that

$\operatorname{sys}(X)\geq
20\operatorname{max}\\{n\ln\big{(}(1+2n+n!)(n+1)\big{)},\ln\big{(}5n+(8\pi)^{4}\big{)}\\}.$

Then, the following hold

  1. 1.

$2K_{\overline{X}}$ is globally generated and very ample modulo $D.$

  2. 2.

$3K_{\overline{X}}$ is very ample.

Another implication of Theorem B is the following bound on the Seshadri
constant of $K_{\overline{X}}$:

######  Corollary D.

(Theorem 6.8) Suppose that

$\operatorname{sys}(X)\geq
20\operatorname{max}\\{n\ln\big{(}(1+2n+n!)(n+s)\big{)},\ln\big{(}5n+(8\pi)^{4}\big{)}\\}.$

Then $2K_{\overline{X}}$ separates any $s$-jets and in particular for every
$x\in X,$ we have that

$\epsilon(K_{\overline{X}},x)\geq s/2.$

As the boundary divisor $D$ is a disjoint union of abelian varieties, the
adjunction formula gives that $K_{\overline{X}|D}$ is isomorphic to the
conormal bundle $O_{D}(-D),$ which is always an ample bundle due to [Mok12].
It is classical that every ample line bundle on an abelian variety determines
a positive definite Hermitian form on that abelian variety. Let
$\operatorname{sys}(D)$ be the length of the shortest closed geodesic on $D$
with respect to the metric induced by the ample line bundle
$K_{\overline{X}|D}.$ Assuming that both $\operatorname{sys}(X)$ and
$\operatorname{sys}(D)$ are sufficiently large relative to $n,$ we get that
the bicanonical bundle $2K_{\overline{X}}$ is very ample:

######  Corollary E.

(Corollary 7.3) With the same $X$ and $\overline{X}$ as Theorem A, suppose
that $\operatorname{sys}(D)>2\sqrt{2n/\pi}$ and that

$\operatorname{sys}(X)\geq
20\operatorname{max}\\{n\ln\big{(}5n(1+2n+n!)\big{)},\ln\big{(}5n+(8\pi)^{4}\big{)}\\}.$

Then, for every $x\in\overline{X}$ we have

$\epsilon(K_{\overline{X}},x)\geq 2n,$

and in particular $2K_{\overline{X}}$ is very ample.

### Previous Results and Comparison

Besides the results mentioned above, the central purposes of our paper are the
following technical advancements on the subject:

  * $\bullet$

Our results show that the hyperbolicity properties of a non-compact ball
quotient can be controlled by its systole, and hence the injectivity radius of
the interior and depth of cusps need not to be dealt with separately (see for
example [Won18] for the other approach). In our paper, this is achieved by
proving that the systole bounds both the uniform depth of cusps (Theorem 3.9)
and injectivity radius of the thick part (§4) from below.

  * $\bullet$

Our results depend intrinsically on $X,$ and it is not required to pass to a
cover of $X$ to apply them. In particular, our results apply even in the case
that $X$ is not a normal cover of another variety (see for example [Yeu12,
Theorem 5] and [DCL21, Corollary 1.6.] in which the passage to a cover of $X$
is required. Indeed, these results are about a cover of $X$ rather than $X$
itself).

  * $\bullet$

In a tower of covering, it is easy to estimate the behavior of systoles (one
only needs to compute the trace of hyperbolic elements) and conclude the
increasing hyperbolicity behavior via our results. This kind of increasing
hyperbolicity behavior has been extensively studied for locally symmetric
spaces in towers with high ramifications at cusps (see for example [Nad89,
Bru20a, Bru20b, Cad22, AVA18, AVA17, Rou16]). However, our approach does not
require high ramifications of cusps in towers and our results can be applied
even in towers where some of the cusps do not ramify.

The following are some of the previous effective results for pluricanonical
bundles:

  1. 1.

Yeung proved in [Yeu12] that for a quasi-projective variety $M,$ there exists
a finite normal cover $M^{\prime}$ such that $L^{2}$-holomorphic sections of
$K_{M^{\prime}}$ give rise to a holomorphic immersion of $M^{\prime}$ into
some projective space.

  2. 2.

Di Cerbo and Di Cerbo proved in [DCL21, Corollary 1.6.] that for a smooth
projective $X$ with ample $K_{X}$ and large fundamental group, there exists a
normal cover $X^{\prime}$ such that $2K_{X^{\prime}}$ is very ample. In
[DCDC15, Theorem 1.3], they also proved an effective result for the multiple
of the log canonical bundle $K_{\overline{X}}+D$ of the toroidal
compactification of ball quotient: If $m\geq(\operatorname{n}+1)^{3},$ then
$m(K_{\overline{X}}+D)$ is ample modulo $D.$

  3. 3.

Hwang proved in [Hwa05, Proposition 2.1. ] that for a non-compact ball
quotient $X,$ the sections of the line bundle $\frac{n^{2}+3n+4}{2}K_{X^{*}}$
separates any two points of Siu–Yau compactification $X^{*}.$

  4. 4.

For various compact locally symmetric spaces, effective very ampleness has
been studied in [HT99, Yeu18, Yeu01, Yeu17, Yeu00, Wan15].

### Strategy of Proof

To prove Theorem A, we decompose $X=\Gamma\backslash\mathbb{B}^{n}$ into two
disjoint parts. The first part is the thin neighborhoods around the cusps.
This part consists of all points on $X$ which have a displacement less than
$\operatorname{sys}(X)/2$ with respect to a parabolic element in $\Gamma.$ The
second part is the complement of the first part, which we call the thick part.
In Proposition 4.2 we prove that the thick part is not empty. Moreover, we
prove in Theorem 4.4 that every subvariety of $\overline{X}$ which does not
intersect the boundary $D$ contains a point from the thick part. Therefore,
using the inequality proved by Hwang-To we conclude that the volume of a
subvariety $V$ of $X$ is controlled by the systole. On the other hand, we
prove in Theorem 3.9 that the uniform depth of cusps is controlled by the
systole. Combining this with an inequality proved by the author in [Mem22] it
follows that the canonical volume of a subvariety of $\overline{X}$ which
intersects with $X$ is also controlled by the systole. The bounds on the
degree of a subvariety with respect to $K_{\overline{X}}$ follow from the
ampleness of $K_{\overline{X}}$ and Theorem A.

## Acknowledgements

I would like to thank my advisor Jacob Tsimerman for many enlightening
discussions and for his comments on the earlier draft of this paper. I would
also like to thank Sai-kee Yeung for useful conversations around the effective
very ampleness problem.

##  2 Background and Notations

In this section, we collect the necessary background and notations which will
be used frequently in the sequel. We refer to [Gol99, Par98, Kap22, BT18] for
a much fuller account.

###  2.1 Geometry of complex call quotients

The complex $n$-ball $\mathbb{B}^{n}$ is defined as

$\mathbb{B}^{n}=\\{z\in\mathbb{C}^{n}\ |\ |z|^{2}<1\\}.$

$\mathbb{B}^{n}$ has an intrinsic Hermitian metric called Bergman metric. The
holomorphic isometry group of $\mathbb{B}^{n}$ with respect to this metric is
the projective unitary group

$G:=\operatorname{PU}(n,1)=\dfrac{\operatorname{U}(n,1)}{Z(\operatorname{U}(n,1))},$

where the centre $Z(\operatorname{U}(n,1))$ can be identified with the circle
group $\\{\mu I:|\mu|=1\\}.$ The group $G$ acts transitively on
$\mathbb{B}^{n}$ and acts doubly transitively on the boundary sphere
$\partial\mathbb{B}^{n}.$ The stabilizer of the center of $\mathbb{B}^{n}$ is
$\operatorname{U}(n).$ Every isometry $g\in G$ is continuous on the closed
ball $\overline{\mathbb{B}^{n}}$ and it follows from Brouwer’s fixed point
theorem that $g$ has a fixed point on the closed ball
$\overline{\mathbb{B}^{n}}$. Moreover, if there is no fixed point on
$\mathbb{B}^{n},$ there can be at most two fixed points on the boundary sphere
$\partial\mathbb{B}^{n}.$ Accordingly, an isometry $g\in G$ is classified as:

  1. 1.

Elliptic: $g$ has a fixed point $z$ in $\mathbb{B}^{n}$. After conjugating $g$
via $h\in G$ which sends $z$ to $0$,

$hgh^{-1}\in\operatorname{U}(n),$

and therefore all eigenvalues of $g$ are roots of unity.

  2. 2.

Parabolic: $g$ has a unique fixed point in $\overline{\mathbb{B}^{n}}$ and
this fixed point is on the boundary $\partial\mathbb{B}^{n}.$ Equivalently,

$\displaystyle\inf_{z\in\mathbb{B}^{n}}\ d(z,gz)=0,$

where $d(.,.)$ denotes the Bergman metric. This infimum is not realized for a
parabolic $g.$

  3. 3.

Hyperbolic: $g$ has exactly two fixed points in $\overline{\mathbb{B}^{n}}$
and both are in $\partial\mathbb{B}^{n}.$ In particular, $g$ preserves the
unique geodesic connecting these two fixed points in $\mathbb{B}^{n}$ and acts
as a translation along this geodesic. This geodesic is called the axis of $g.$
The length of a hyperbolic isometry $g\in G$ is

$\ell(g):=\displaystyle\inf_{z\in\mathbb{B}^{n}}\ d(z,gz).$

This infimum is not zero and is realized by any point on the axis of $g.$ The
work of Chen-Greenberg on the conjugacy classification of element of
$\operatorname{U}(n,1)$ [CG74, Theorem 3.4.1] implies that a hyperbolic
isometry $g$ has two eigenvalues $re^{i\theta}$ and $r^{-1}e^{i\theta}$ with
$r>1$ and $n-1$ eigenvalues with norm $1.$

Let $\Gamma\subset\operatorname{PU}(n,1)$ be a torsion-free lattice whose
parabolic elements are unipotent. [Hum98] tells us that every lattice in
$\operatorname{PU}(n,1)$ has a finite index subgroup with this property. With
this property, an element $g\in\Gamma$ is hyperbolic if and only if $g$ is
semi-simple. Therefore, we will denote the set of the hyperbolic elements in
$\Gamma$ by $\Gamma_{s}.$

Let $X=\Gamma\backslash\mathbb{B}^{n}.$ The systole of $X$ is the length of
the shortest closed geodesic with respect to the Bergman metric:

$\operatorname{sys}(X):=\displaystyle\inf_{g\in\Gamma_{s}}\ell(g)=\displaystyle\inf_{g\in\Gamma_{s}}\\{d(z,gz)|z\in\mathbb{B}^{n}\\}.$

Equivalently, the systole of $X$ is the length of the shortest hyperbolic
element in $\Gamma.$ Consider $x\in X$ with stabilizer $\Gamma_{x}$ in
$\Gamma.$ Choose a fiber
$\stackrel{{\scriptstyle\sim}}{{\smash{x}\rule{0.0pt}{4.73611pt}}}\in\mathbb{B}^{n}.$
The injectivity radius of $x$ in $X$ is defined to be

$\operatorname{inj}_{x}(X):=\frac{1}{2}\operatorname{inf}_{\gamma\in\Gamma\setminus\Gamma_{x}}d(\stackrel{{\scriptstyle\sim}}{{\smash{x}\rule{0.0pt}{4.73611pt}}},\gamma\cdot\stackrel{{\scriptstyle\sim}}{{\smash{x}\rule{0.0pt}{4.73611pt}}}),$

which is independent of choice of
$\stackrel{{\scriptstyle\sim}}{{\smash{x}\rule{0.0pt}{4.73611pt}}}.$ The
injectivity radius of $X$ is $\operatorname{inj}(X):=\operatorname{inf}_{x\in
X}\operatorname{inj}_{x}(X).$ In the case that $X$ is compact, $\Gamma$ only
has semi-simple elements and hence $\operatorname{sys}(X)=inj(X)/2.$ However,
this relation does not hold for a non-compact $X$ because of the parabolic
elements in $\Gamma.$

###  2.2 Siegel domain model

The half-plane model of $1$-dimensional complex ball quotient is generalized
by the Siegel domain model in higher dimensions. In horospherical coordinates,
the Siegel domain of (complex) dimension $n$ is
$\mathbb{S}=\mathbb{C}^{n-1}\times\mathbb{R}\times\mathbb{R}^{+}$. The points
of $\mathbb{S}$ will be written as
$(\zeta,v,u)\in\mathbb{C}^{n-1}\times\mathbb{R}\times\mathbb{R}^{+}.$ The
boundary of $\mathbb{S}$ is $H_{0}\cup\\{\infty\\},$ where $\infty$ is a
distinguished point at infinity and
$H_{0}=\mathbb{C}^{n-1}\times\mathbb{R}\times\\{0\\}.$ We will follow [Par98]
to describe $\operatorname{PU}(n,1)$ as a matrix group by embedding the Siegel
domain as a paraboloid in $\mathbb{P}(\mathbb{C}^{n,1}).$ To do so, we should
choose a Hermitian form of signature $(n,1)$ on $\operatorname{PU}(n,1).$ Let

$J_{0}:=\begin{bmatrix}0&0&1\\\ 0&I_{n-1}&0\\\ 1&0&0\\\ \end{bmatrix},$

and $Q:=z^{*}J_{0}z,$ where $z$ is a column vector in
$\mathbb{P}(\mathbb{C}^{n,1})$ and $z^{*}$ is the Hermitian transpose of $z.$
Consider the map $\psi:\overline{\mathbb{S}}\to\mathbb{P}(\mathbb{C}^{n,1})$
given by

$\displaystyle\psi:(\zeta,v,u)\longrightarrow\begin{bmatrix}\frac{1}{2}(-||\zeta||^{2}-u+iv)\\\
\zeta\\\ 1\end{bmatrix},\ \textit{for}\
(\zeta,v,u)\in\overline{\mathbb{S}}\backslash\\{\infty\\};\
\psi:\infty\longrightarrow\begin{bmatrix}1\\\ 0\\\ 0\end{bmatrix}.$ (1)

The image of this map is the set of points in $\mathbb{P}(\mathbb{C}^{n,1})$,
where the Hermitian form $Q$ is negative. Also $\psi$ is a homeomorphism of
$\partial\mathbb{S}$ onto the set of points where $Q$ is zero. We will refer
to $q_{\infty}:=\psi(\infty)$ as the cusp at $\infty$ and $q_{0}:=\psi(0)$ as
the cusp at $0.$

The holomorphic isometry group of $\mathbb{S}$ with respect to the Bergman
metric is the projective unitary group $\operatorname{PU}(Q)$ and it acts on
$\mathbb{P}(\mathbb{C}^{n,1})$ by matrix multiplication. A matrix
$h\in\operatorname{GL}(n+1,\mathbb{C})$ is in $\operatorname{PU}(Q)$ if and
only if its inverse $h^{-1}$ is given by $J_{0}h^{*}J_{0}$, where ∗ denotes
the Hermitian transpose; that is, transpose the matrix and complex conjugate
each of its entries. The general form for $h\in PU(Q)$ and its inverse is
given in [Par98, pg 438] :

$\displaystyle h=\begin{bmatrix}a&\tau^{*}&b\\\ \alpha&A&\beta\\\
c&\delta^{*}&e\end{bmatrix},\ \ \
h^{-1}=\begin{bmatrix}\bar{e}&\beta^{*}&\bar{b}\\\ \delta&A^{*}&\tau\\\
\bar{c}&\alpha^{*}&\bar{a}\end{bmatrix},$ (2)

where $A$ is an $(n-1)\times(n-1)$ matrix, $a,b,c,e\in\mathbb{C},$ and
$\tau,\delta,\alpha,\beta$ are column vectors in $\mathbb{C}^{n-1}.$

The following lemma easily follows:

######  Lemma 2.1.

Let $h$ be an element of $PU(Q)$ written in form 2.

  1. 1.

([Par97, page 7]) If $h$ swaps $q_{\infty}$ and $q_{0},$ then it must have the
following form:

$h=\begin{bmatrix}0&0&1/\overline{c}\\\ 0&A&0\\\ c&0&0\end{bmatrix},$

where $A\in U(n-1)$ and $c\in\mathbb{C}.$ Therefore, $h$ must be the following
transformation:

$h:(\zeta,u,v)\longrightarrow\Big{(}\dfrac{-2A\zeta}{c(||\zeta||^{2}+u-iv)},\dfrac{-4v}{\big{|}c\big{|}^{2}\big{|}||\zeta||^{2}+u-iv\big{|}^{2}},\dfrac{4u}{\big{|}c\big{|}^{2}\big{|}||\zeta||^{2}+u-iv\big{|}^{2}}\Big{)}.$

  2. 2.

If $h$ fixes both the cusps $q_{\infty}$ and $q_{0}$, then it must have the
following form

$h=\begin{bmatrix}a&0&0\\\ 0&A&0\\\ 0&0&1/\bar{a}\end{bmatrix},$

where $A\in U(n-1)$ and $a\in\mathbb{C}.$

For any pair of points $z_{1}=(\zeta_{1},v_{1},u_{1})$ and
$z_{2}=(\zeta_{2},v_{2},u_{2})$ in $\mathbb{S}$ the Bergman metric is given
by:

$\displaystyle
d(z_{1},z_{2})=2\cosh^{-1}\Big{(}\dfrac{1}{4u_{1}u_{2}}\big{|}||\zeta_{1}-\zeta_{2}||^{2}+u_{1}+u_{2}+iv_{1}-iv_{2}+2i\operatorname{im}\langle\zeta_{1},\zeta_{2}\rangle\big{|}^{2}\Big{)},$
(3)

where $\langle.,.\rangle$ denotes the standard positive definite Hermitian
form on $\mathbb{C}^{n-1}$. Since $\cosh^{-1}(x)$ is increasing, the following
lower bound can be obtained for the metric :

$\displaystyle
d\big{(}(\zeta_{1},v_{1},u_{1}),(\zeta_{2},v_{2},u_{2})\big{)}\geq
2\cosh^{-1}\Big{(}\dfrac{|u_{1}+u_{2}|^{2}}{4u_{1}u_{2}}\Big{)}$ (4)

The holomorphic sectional curvature of this metric is $-1$ and the sectional
curvature of this metric varies on $[-1,-\frac{1}{4}]$ (see [Gol99]). It
follows that the holomorphic bisectional curvature of this metric is bounded
above by a negative constant since the holomorphic bisectional curvature
always can be written as the sum of two sectional curvatures.

###  2.3 Toroidal Compactification

The complex ball quotient $X$ has a unique toroidal compactification
$\overline{X},$ which is a smooth projective variety (see [Mok12]). The
boundary divisor of this compactification $D:=\overline{X}\setminus X$ is a
disjoint union of abelian varieties with ample conormal bundle. The Kähler
form of the Bergman metric on $\mathbb{S}$ is given by

$\displaystyle\omega_{\mathbb{S}}:=-2i\partial\bar{\partial}\log(u)$ (5)

(see [BT18, Lemma 2.1]). Let $\omega_{X}$ be the Kähler form induced by the
Kähler form $\omega_{\mathbb{S}}.$ It follows from Mumford’s work on the
singular Hermitian metric [Mum77] that the Bergman metric on $X$ extends as a
good Hermitian metric to $\overline{X}.$ Integration against $\omega_{X}$ on
the open part represents (as a current) a multiple of the first Chern class

$\displaystyle\operatorname{c}_{1}(K_{\overline{X}}+D)=\frac{1}{2\pi}\frac{n+1}{2}[\omega_{X}]\in
H^{1,1}(\overline{X},\mathbb{R}),$ (6)

where $K_{\overline{X}}$ is the canonical bundle of $\overline{X}$ (see
[BT18]).

###  2.4 Stabilizer of cusps

We denote the parabolic stabilizer of $q_{\infty}$ in $G$ by $G_{\infty}.$
With our choice of Hermitian form, the matrices corresponding to elements of
$G_{\infty}$ are upper triangular. There is an equivalent way to identify
these matrices:

######  Lemma 2.2.

([Par98]) Let $h$ be an element of $\operatorname{PU}(Q)$ written in the form
2. Then, $h$ fixes the cusp $q_{\infty}$ if and only if the $c$ entry of $h$
is $0.$

###### Proof.

Note that

$\displaystyle h\cdot q_{\infty}=\begin{bmatrix}a&\tau^{*}&b\\\
\alpha&A&\beta\\\ c&\delta^{*}&e\end{bmatrix}\begin{bmatrix}1\\\ 0\\\
0\end{bmatrix}=\begin{bmatrix}a\\\ \alpha\\\ c\end{bmatrix},$

and therefore $h$ fixes $q_{\infty}$ projectively if and only if $c=0$ and
$\alpha=0.$ Note that if the $c$ entry of $h$ is $0,$ the multiplication of
the matrix of $h$ and $h^{-1}$ in the form 2 yields that $\alpha$ (and also
$\delta$) must be $0.$ ∎

The group $G_{\infty}$ is generated by Heisenberg isometries $I_{\infty}$ and
a one-dimensional torus $T$. Heisenberg isometries consist of Heisenberg
Rotations $U(n-1)$ and Heisenberg translations $\mathfrak{N}$. Heisenberg
Rotations $U(n-1)$ acts on $\zeta$-coordinates of $\mathbb{S}$ in the usual
way and Heisenberg translations
$\mathfrak{N}\cong\mathbb{C}^{n-1}\times\mathbb{R}$ acts on $\zeta$ and $v$
coordinates of $\mathbb{S}$ via

$(\tau,t):(\zeta,v,u)\longrightarrow\big{(}\zeta+\tau,v+t+2i\langle\tau,\zeta\rangle,u\big{)}.$

The element $(0,t)\in\mathfrak{N}$ will be called the vertical translation by
$t$, and the subgroup generated by $(0,t)$ in $G_{\infty}$ will be denoted by
$V_{\infty}$. The vertical translation $V_{\infty}$ is the center of
$G_{\infty}$ and the quotient $V_{\infty}\backslash I_{\infty}$ is isomorphic
to the unitary transformation of $\mathbb{C}^{n-1}.$

A Heisenberg translation $(\tau,t)\in\mathfrak{N}$ fixing $q_{\infty}$
corresponds to the matrix $g_{\infty}$ and a Heisenberg translation
$(\sigma,s)\in\mathfrak{N}$ fixing $q_{0}$ corresponds to the matrix $g_{0},$
where

$\displaystyle g_{\infty}=\begin{bmatrix}1&-\tau^{*}&-(|\tau|+it)/2\\\
0&I&\tau\\\ 0&0&1\\\ \end{bmatrix},\ \ \ g_{0}=\begin{bmatrix}1&0&0\\\
\sigma&I&0\\\ -(|\sigma|+is)/2&-\sigma^{*}&1\\\ \end{bmatrix}.$ (7)

With our assumption on $\Gamma,$ all parabolic stabilizers of $q_{\infty}$ in
$\Gamma,$ i.e., $\Gamma_{\infty}:=\Gamma\cap G_{\infty}$ are Heisenberg
translations.

###  2.5 Neighborhood of cusps

A horoball centered at the cusp $q_{\infty}$ with height $\tilde{u}$ is the
open set

$B_{\infty}(\tilde{u}):=\\{(\zeta,v,u)\in\mathbb{S}\ |\ u>\tilde{u}\\}.$

The smallest $\tilde{u}$ such that $\Gamma_{\infty}\backslash
B_{\infty}(\tilde{u})$ injects into $X$ will be called the height of the
$q_{\infty}$ and will be denoted by $u_{\infty}.$ The complex ball quotient
$X$ has a finitely many cusps and they are in one-to-one correspondence with
the equivalence classes of parabolic fixed points of $\Gamma.$ For a cusp
$q_{i}$, there exists a $g\in\operatorname{PU}(Q)$ translating $q_{i}$ to
$q_{\infty}.$ The horoball based at cusp $q_{i}$ with height $\tilde{u}$ is
defined to be the image of the horoball based at cusp $q_{\infty}$ with height
$\tilde{u}$ translated by $g:$

$B_{i}(\tilde{u})=g\cdot B_{\infty}(\tilde{u})$

Accordingly, the height of cusp $q_{i}$ is the smallest $\tilde{u}$ such that
$\Gamma_{i}\backslash B_{i}(\tilde{u})$ injects into $X,$ where $\Gamma_{i}$
is the parabolic stabilizer of $q_{i}.$ We will denote the height of $q_{i}$
by $u_{i}.$ Thanks to the Parker’s generalization of Shimuzu’s lemma [Par98],
$B_{i}(\tilde{u})$ must inject into $X$ for a large enough $\tilde{u}.$ Note
that the height coordinate $u$ on $\mathbb{S}$ is invariant under the action
of Heisenberg rotations $\operatorname{U}(n-1)$ and Heisenberg translations
$\mathfrak{N}$ and hence the horoball is invariant with respect to the action
of these groups.

Let $t_{i}$ be the shortest vertical translation in the parabolic stabilizer
of $q_{i}$. The number $d_{i}=t_{i}/u_{i}$ is called the depth of cusp $q_{i}$
which is invariant under conjugating $\Gamma.$

######  Definition 2.3.

([BT18, Definition 3.7.]) The uniform depth of the cusps of $X$ is the largest
$d$ satisfying the following properties:

  1. 1.

for every $i$, $d\leq d_{i}$ (this gives that $\Gamma_{i}\backslash
B_{i}(t_{i}/d)$ injects into $X$).

  2. 2.

all $B_{i}(t_{i}/d)$ are disjoint.

##  3 Systole and depth of cusps

Let $X=\Gamma\backslash\mathbb{B}^{n},$ where
$\Gamma\subset\operatorname{PU}(n,1)$ is a torsion-free lattice whose
parabolic stabilizers are unipotent. The main goal of this section is to prove
Theorem 3.9, where we show that the systole $\operatorname{sys}(X)$ bounds the
uniform depth of cusps $d$ of $X$ from below.

To see the relation between the systole and depth of cusps, we first prove
that the length of a hyperbolic element in $\Gamma$ only depends on its non-
unit eigenvalues:

######  Proposition 3.1.

Suppose $h\in\Gamma$ is a hyperbolic element whose non-unit eigenvalues are
$re^{i\theta}$ and $r^{-1}e^{i\theta}$. Then,

$\ell(h)=2\cosh^{-1}\Big{(}\frac{1}{4}\cdot(r+\frac{1}{r})^{2}\Big{)}.$

###### Proof.

Since $h$ is hyperbolic, it fixes two distinct points $x_{1}$ and $x_{2}$ on
the boundary $\partial\mathbb{S}.$ As $\operatorname{PU}(Q)$ acts doubly
transitive on the boundary, there exists $P\in\operatorname{PU}(Q)$ such that
$P(x_{1})=q_{0}$ and $P(x_{2})=q_{\infty}.$ Now we can write

$\displaystyle d(x,hx)=d(Px,PhP^{-1}Px)=d(x^{\prime},PhP^{-1}x^{\prime}),$

where $x^{\prime}=Px.$ Suppose $x^{\prime}=(\zeta_{1},v_{1},u_{1}),$ and
$PhP^{-1}x^{\prime}=(\zeta_{2},v_{2},u_{2}).$ Since $PhP^{-1}$ fixes both
$q_{0}$ and $q_{\infty},$ it follows from Lemma 2.1 that for a complex number
$a$ and $A\in\operatorname{U}(n-1)$

$\displaystyle PhP^{-1}x^{\prime}=\begin{bmatrix}a&0&0\\\ 0&A&0\\\
0&0&1/\bar{a}\end{bmatrix}\begin{bmatrix}\frac{1}{2}(-||\zeta_{1}||^{2}-u_{1}+iv_{1})\\\
\zeta_{1}\\\
1\end{bmatrix}=\begin{bmatrix}\frac{a}{2}(-||\zeta_{1}||^{2}-u_{1}+iv_{1})\\\
A\zeta_{1}\\\ 1/\bar{a}\end{bmatrix}.$

This gives that $\zeta_{2}=\bar{a}A\zeta_{1}$ and

$\displaystyle\frac{1}{2}(-||\zeta_{2}||^{2}-u_{2}+iv_{2})=\frac{|a|^{2}}{2}(-||\zeta_{1}||^{2}-u_{1}+iv_{1}).$
(8)

Therefore, $u_{2}=|a|^{2}u_{1}$ and $v_{2}=|a|^{2}v_{1}.$ Note that as
conjugation does not change the eigenvalues, we have that $|a|^{2}=r^{2}\
\textit{or}\ \frac{1}{r^{2}}.$ On the other hand, the inequality 4 yields that

$\displaystyle d\big{(}x^{\prime},PhP^{-1}x^{\prime}\big{)}$
$\displaystyle\geq
2\cosh^{-1}\Big{(}\dfrac{|u_{1}+u_{2}|^{2}}{4u_{1}u_{2}}\Big{)}$
$\displaystyle\geq
2\cosh^{-1}\Big{(}\frac{1}{4}\cdot(r+\frac{1}{r})^{2}\Big{)}.$

Since this lower bound is realized on $\zeta_{1}=0,v=0,$ we can conclude the
desired equality. ∎

Consider $\gamma\in\Gamma$ and the corresponding element in the matrix group
$g_{\gamma}\in\operatorname{PU}(Q).$ When we talk about
$|\operatorname{tr}(\gamma)|,$ we mean $|\operatorname{tr}(g_{\gamma})|.$
Proposition 3.1 tells us if for every hyperbolic element $\gamma\in\Gamma$ we
have that $|\operatorname{tr(\gamma)}|$ is sufficiently large, then systole
$\operatorname{sys}(X)$ will be large, i.e., the systole
$\operatorname{sys}(X)$ can be estimated just be the trace of the hyperbolic
elements. In particular, the systole goes to infinity in a cofinal tower of
any ball quotient.

Consider the set associated to $\Gamma$

$S_{\Gamma}:=\big{\\{}\gamma\in\Gamma\big{|}|\operatorname{tr}(\gamma)|>n+1\big{\\}}.$

It follows from the classification of isometries that if $\gamma\in\Gamma$ has
$|\operatorname{tr}(\gamma)|>n+1,$ then $\gamma$ must be hyperbolic. Hence,
all elements of $S_{\Gamma}$ are hyperbolic. We associate the number

$\lambda_{\Gamma}:=\displaystyle\inf_{\gamma\in
S_{\Gamma}}|\operatorname{tr}(\gamma)|,$

to $\Gamma$ which will play a role as an intermediate quantity to relate the
systole of $X$ to the depth of cusps of $X$. Specifically, we can see how
$\operatorname{sys}(X)$ bounds $\lambda_{\Gamma}$ from below:

######  Proposition 3.2.

The following inequality holds:

$\displaystyle\lambda_{\Gamma}\geq 1-n+\sqrt{2}e^{\operatorname{sys}(X)/4}.$

###### Proof.

Consider $\gamma\in S_{\Gamma}.$ Let $re^{i\theta}$ and $r^{-1}e^{i\theta}$ be
eigenvalues of $\gamma$ which are not units. As $\operatorname{sys}(X)$ is the
lengtg of the shortest geodesic, Proposition 3.1 implies that

$r+r^{-1}\geq\sqrt{4\cosh\big{(}\frac{1}{2}\operatorname{sys}(X)\big{)}}.$

Since the other $n-1$ eigenvalues of $\gamma$ have norm $1,$ and
$\cosh(x)>\frac{1}{2}e^{x},$ by the triangle inequality we can conclude the
desired inequality.

∎

We prove a lemma which will help us to see the relation between the height of
cusp $q_{\infty}$ in terms of the trace of the hyperbolic elements in
$\Gamma:$

######  Lemma 3.3.

Suppose $\gamma\in\Gamma\setminus\Gamma_{\infty}$ written in from 2. For every
$z\in\mathbb{B}^{n}$ the following inequality holds:

$u(z)u(\gamma\cdot z)\leq\big{|}\dfrac{2}{c}\big{|}^{2}.$

###### Proof.

There are unique Heisenberg transformations $h_{1},h_{2}$ such that
$h_{1}(0)=\gamma(\infty)$ and $h_{2}^{-1}(0)=\gamma^{-1}(\infty).$ Consider
$\stackrel{{\scriptstyle\sim}}{{\smash{\gamma}\rule{0.0pt}{4.73611pt}}}=h_{1}^{-1}\gamma
h_{2}^{-1}$ and note that as the Heisenberg translations are stabilizers of
the $u-$coordinate, we have that $u(\gamma
z)=u(\stackrel{{\scriptstyle\sim}}{{\smash{\gamma}\rule{0.0pt}{4.73611pt}}}z).$
Since $\stackrel{{\scriptstyle\sim}}{{\smash{\gamma}\rule{0.0pt}{4.73611pt}}}$
swaps $\infty$ and $0,$ Lemma 2.1 tells us that
$\stackrel{{\scriptstyle\sim}}{{\smash{\gamma}\rule{0.0pt}{4.73611pt}}}$ must
have the form:

$\stackrel{{\scriptstyle\sim}}{{\smash{\gamma}\rule{0.0pt}{4.73611pt}}}:(\zeta,v,u)\longrightarrow\Big{(}\dfrac{A\zeta
r_{\gamma}^{2}}{||\zeta||^{2}+u-iv},\dfrac{-vr_{\gamma}^{4}}{\big{|}||\zeta||^{2}+u-iv\big{|}^{2}},\dfrac{ur_{\gamma}^{4}}{\big{|}||\zeta||^{2}+u-iv\big{|}^{2}}\Big{)}$

where $A\in U(n-1)$ and $r_{\gamma}=\sqrt{\dfrac{2}{|c|}}.$ This gives that

$u(z)u(\gamma\cdot
z)=u(z)u(\stackrel{{\scriptstyle\sim}}{{\smash{\gamma}\rule{0.0pt}{4.73611pt}}}\cdot
z)=\dfrac{u^{2}}{|||\zeta||^{2}+u-iv|^{2}}\cdot\big{|}\dfrac{2}{c}\big{|}^{2}\leq\big{|}\dfrac{2}{c}\big{|}^{2}.$

∎

Let $c_{m}$ be the minimal value of $|c|$ among all
$\gamma\in\Gamma\setminus\Gamma_{\infty}$ written in the form 2:

$\displaystyle\gamma=\begin{bmatrix}a&\tau^{*}&b\\\ \alpha&A&\beta\\\
c&\delta^{*}&e\end{bmatrix}.$ (9)

Note that as $\gamma\not\in\Gamma_{\infty},$ the $c$ entry cannot be zero (see
Remark 2.2). Lemma 3.3 implies that:

######  Proposition 3.4.

The horoball $B_{\infty}(\frac{2}{c_{m}})$ injects into $X,$ i.e., the height
of the cusp $q_{\infty}$ is at least $2/c_{m}$ and therefore the depth of
$q_{\infty}$ is at least $\frac{c_{m}\cdot t_{\infty}}{2}.$

We recall a lemma from Parker’s version of Shimizu’s lemma which tells us how
the $c$ entry shows up in the trace of (some of) hyperbolic elements:

######  Lemma 3.5.

(see [Par98, Lemma 2.6]) Let $g_{\infty}=(0,t_{\infty}),$ be the vertical
translation centered at $q_{\infty}$ and $h$ be an element of
$\operatorname{PU}(n,1)$ written in the form 2. Then,

$\operatorname{tr}[g_{\infty},h]=n+1+|\frac{tc}{2}|^{2}.$

###### Proof.

The matrix representations of $g_{\infty},$ $h$ and $h^{-1}$ are given by:

$g_{\infty}=\begin{bmatrix}1&0&i\frac{t}{2}\\\ 0&I&0\\\ 0&0&1\end{bmatrix},\
h=\begin{bmatrix}a&\tau^{*}&b\\\ \alpha&A&\beta\\\
c&\delta^{*}&e\end{bmatrix}\
h^{-1}=\begin{bmatrix}\bar{e}&\beta^{*}&\bar{b}\\\ \delta&A^{*}&\tau\\\
\bar{c}&\alpha^{*}&\bar{a}\end{bmatrix},$

To find $\operatorname{tr}[g_{\infty},h]$, note that

$g_{\infty}h_{0}=\begin{bmatrix}a+\frac{i}{2}ct&\tau^{*}+\frac{i}{2}t\delta^{*}&b+\frac{i}{2}te\\\
\alpha&A&\beta\\\ c&\delta^{*}&e\end{bmatrix},\
g_{\infty}^{-1}h_{0}^{-1}=\begin{bmatrix}\bar{e}-\frac{i}{2}\bar{c}t&\beta^{*}-\frac{i}{2}t\alpha^{*}&\bar{b}-\frac{i}{2}t\bar{a}\\\
\delta&A^{*}&\tau\\\ \bar{c}&\alpha^{*}&\bar{a}\end{bmatrix}$

Therefore, using the relation given by 2 it follows that

$\displaystyle\operatorname{tr}[g_{\infty},h]$
$\displaystyle=a\bar{e}+\frac{1}{4}|ct|^{2}+\frac{it}{2}(c\bar{e}-\bar{c}a)+\delta\tau^{*}+\frac{i}{2}t|\delta|^{2}+b\bar{c}+\frac{i}{2}te\bar{c}$
$\displaystyle\hskip
11.38092pt+\alpha\beta^{*}-\frac{i}{2}t|\alpha|^{2}+AA^{*}+\beta\alpha^{*}+c\bar{b}-\frac{i}{2}t\bar{a}c+\delta^{*}\tau+e\bar{a}$
$\displaystyle=n+1+\frac{1}{4}|ct|^{2}$

∎

Now, we can show the relation between the quantity $\lambda_{\Gamma}$ and
depth of each cusps of $X:$

######  Proposition 3.6.

The depth of each cusp of $X=\Gamma\backslash\mathbb{B}^{n}$ is at least
$\sqrt{\lambda_{\Gamma}-n-1}.$

###### Proof.

Since $\lambda_{\Gamma}$ is invariant under the conjugation by an element of
$\operatorname{PU}(n,1)$, it is sufficient to prove the lemma for the cusp
$q_{\infty}.$ Let $g_{\infty}=(0,t_{\infty})$ be the shortest vertical
translation in $\Gamma_{\infty}.$ Suppose that $h\in\Gamma$ is an element
which does not fix $q_{\infty}$ and written in the form 2. It follows from
Lemma 3.5 that

$\operatorname{tr}[g_{\infty},h]=n+1+\big{|}\dfrac{t_{\infty}c}{2}\big{|}^{2}.$

Since $h$ does not fix $\infty,$ we have that $c\neq 0,$ and therefore
$[g_{\infty},h]\in S_{\lambda}.$ This implies that

$\big{|}\dfrac{t_{\infty}c}{2}\big{|}\geq\sqrt{\lambda_{\Gamma}-n-1}.$

Since this inequality holds for every $h\in\Gamma\setminus\Gamma_{\infty},$ we
can conclude that

$\big{|}\dfrac{t_{\infty}c_{m}}{2}\big{|}\geq\sqrt{\lambda_{\Gamma}-n-1}.$

Hence, Proposition 3.4 implies

$d_{\infty}\geq\sqrt{\lambda_{\Gamma}-n-1},$

where $d_{\infty}$ is the depth of cusp $q_{\infty}.$ ∎

To pass from the individual depth of cups to the uniform depth of cusps we
will use this lemma:

######  Lemma 3.7.

([Par98, Lemma 2.5]) Let $B_{0}(\tilde{u}_{0})$ be the horoball of height
$\tilde{u}_{0}$ based at $q_{0}$, and let $B_{\infty}(\tilde{u}_{\infty})$ be
the horoball of height $\tilde{u}_{\infty}$ based at $q_{\infty}.$ These two
horoballs are disjoint if and only if

$\tilde{u}_{0}\cdot\tilde{u}_{\infty}\geq 4.$

######  Proposition 3.8.

Let $d$ be the uniform depth of cusps of $X$. Then,

$d\geq\operatorname{min}\\{(\lambda_{\Gamma}-n-1)^{\frac{1}{4}},(\lambda_{\Gamma}-n-1)^{\frac{1}{2}}\\}.$

###### Proof.

By Proposition 3.6, we only need to show that if
$d>(\lambda_{\Gamma}-n-1)^{\frac{1}{4}},$ then the horoballs are disjoint. Let
$g_{0}=(0,t_{0})$ be the shortest vertical translation based at $q_{0}$ and
$g_{\infty}=(0,t_{\infty})$ be the shortest vertical translation based at
$q_{\infty}.$ We write $g_{\infty}$ and $g_{0}$ in the form 7:

$\displaystyle g_{0}=\begin{bmatrix}1&0&0\\\ 0&I&0\\\
-it_{0}/2&0&0\end{bmatrix},\ \ \ \
g_{\infty}=\begin{bmatrix}1&0&-it_{\infty}/2\\\ 0&I&0\\\ 0&0&0\end{bmatrix}.$

Lemma 3.5 implies that
$\operatorname{tr}[g_{\infty},g_{0}]=n+1+\big{|}\frac{t_{0}t_{\infty}}{4}\big{|}^{2}.$
Therefore, $[g_{\infty},g_{0}]\in S_{\lambda}$ and it follows that

$t_{0}t_{\infty}\geq 4\sqrt{\lambda_{\Gamma}-n-1}.$

Consider $\tilde{u}_{0}:=\frac{t_{0}}{(\lambda_{\Gamma}-n-1)^{\frac{1}{4}}}$
and
$\tilde{u}_{\infty}:=\frac{t_{\infty}}{(\lambda_{\Gamma}-n-1)^{\frac{1}{4}}}.$
The inequality above implies that

$\tilde{u}_{0}\cdot\tilde{u}_{\infty}\geq 4,$

and therefore it follows from Lemma 3.7 that the horoballs are disjoint. ∎

Combining Proposition 3.2 with Proposition 3.8, we finally conclude that the
systole bounds the uniform depth from below:

######  Theorem 3.9.

Let $d$ be the uniform depth of cusps of $X.$ Then,

$\displaystyle
d\geq\operatorname{min}\\{\big{(}-2n+s^{\prime}\big{)}^{\frac{1}{4}},\big{(}-2n+s^{\prime}\big{)}^{\frac{1}{2}}\\},$

where $s^{\prime}=\sqrt{2}e^{\frac{1}{4}\operatorname{sys}(X)}.$

Direct computation gives the following Corollary which will be used later to
bound the uniform depth of cusps in terms of $\operatorname{sys}(X)$:

######  Corollary 3.10.

If $\operatorname{sys}(X)\geq 4\ln\big{(}5n+(4\pi)^{4}\big{)},$ then

$\displaystyle d>e^{\operatorname{sys}(X)/16}>4\pi.$ (10)

##  4 Thin-thick decomposition

In this section, we introduce a version of thin-thick decomposition relative
to the systole. This is not the same as Margulis’ thin-thick decomposition as
it depends on the lattice $\Gamma.$ The main goal of this section is to prove
Theorem 4.4 which tells us that the thin part of $X$ does not have any
subvariety.

Let $q_{i}$ be a cusp of $X$ with unipotent stabilizer $\Gamma_{i}.$ Fix
$\epsilon>0.$ Consider the set

$\stackrel{{\scriptstyle\sim}}{{\smash{U}\rule{0.0pt}{4.73611pt}}}_{i,\epsilon}=\\{x\in\mathbb{B}^{n}|\exists
g\in\Gamma_{i},d(x,g\cdot x)<\epsilon\\}.$

We define the $\epsilon$-thin neighborhood around the cusp $q_{i}$ as the set
$U_{i,\epsilon}:=\Gamma\backslash\stackrel{{\scriptstyle\sim}}{{\smash{U}\rule{0.0pt}{4.73611pt}}}_{i,\epsilon}.$
Also, we fix $\rho=\operatorname{sys}(X)/2$ and define the thin part of $X$ as
union of all $\rho$-thin neighborhood around cusps of $X:$

$X_{thin}:=\displaystyle\cup_{i=1}^{k}{U_{i,\rho}}.$

The following Proposition shows that $X_{thin}$ is actually disjoint union of
the $\rho$-thin neighborhood around cusps:

######  Proposition 4.1.

If $\epsilon<sys(X)/2$, then $U_{i,\epsilon}\cap U_{j,\epsilon}=\varnothing$
for $i\neq j.$

###### Proof.

For the sake of the contradiction assume that $x\in U_{i,\epsilon}\cap
U_{j,\epsilon}.$ It means that there exist $\gamma_{1}\in\Gamma_{i}$ and
$\gamma_{2}\in\Gamma_{j}$ such that
$d(\tilde{x},\gamma_{1}\cdot\tilde{x})<\epsilon$ and
$d(\tilde{x},\gamma_{2}\cdot\tilde{x})<\epsilon,$ where
$\tilde{x}\in\mathbb{B}^{n}$ is a fiber of $x.$ This in particular implies
that $d(\tilde{x},\gamma_{1}^{-1}\cdot\tilde{x})<\epsilon.$

Suppose $\gamma_{1}=(\tau,t)$ is a Heisenberg translation based at
$q_{\infty}$ and $\gamma_{2}=(\sigma,s)$ is a Heisenberg translation based at
$q_{0}.$ We represent $\gamma_{1}$ and $\gamma_{2}$ by the matrices
$g_{\infty},g_{0}\in\operatorname{PU}(Q)$ respectively, where

$\displaystyle g_{\infty}=\begin{bmatrix}1&-\tau^{*}&-(|\tau|+it)/2\\\
0&I_{n-1}&\tau\\\ 0&0&1\\\ \end{bmatrix},\ \ \ \ g_{0}=\begin{bmatrix}1&0&0\\\
\sigma&I_{n-1}&0\\\ -(|\tau|+is)/2&-\sigma^{*}&1\\\ \end{bmatrix}.$

Note that $\gamma_{1}^{-1}=(-\tau,-t)$ corresponds to $g^{-1}_{\infty}.$ We
can write:

$\displaystyle\big{|}\operatorname{tr}(g_{\infty}g_{0})\big{|}+\big{|}\operatorname{tr}(g^{-1}_{\infty}g_{0})\big{|}\geq\Big{|}\operatorname{tr}\big{(}(g_{\infty}+g_{\infty}^{-1})g_{0}\big{)}\Big{|}$
$\displaystyle=\Big{|}\operatorname{tr}\Big{(}\begin{bmatrix}2&0&-|\tau|^{2}\\\
0&2I_{n-1}&0\\\ 0&0&2\\\ \end{bmatrix}\begin{bmatrix}1&0&0\\\
\sigma&I_{n-1}&0\\\ (-|\sigma|^{2}+is)/2&-\sigma^{*}&1\\\
\end{bmatrix}\Big{)}\Big{|}$
$\displaystyle=\big{|}2(n+1)+\frac{1}{2}|\tau|^{2}(|\sigma|^{2}-is)\big{|}$
$\displaystyle\geq 2(n+1)+\frac{1}{2}|\tau|^{2}|\sigma|^{2}.$

Hence, either $|\operatorname{tr}(g_{\infty}g_{0})|\geq
n+1+\frac{1}{4}|\tau|^{2}|\sigma|^{2}$ or
$|\operatorname{tr}(g^{-1}_{\infty}g_{0})|\geq
n+1+\frac{1}{4}|\tau|^{2}|\sigma|^{2}$ and therefore either
$\gamma_{1}\gamma_{2}$ or $\gamma_{1}^{-1}\gamma_{2}$ must be hyperbolic. But
this implies that either
$d(\tilde{x},\gamma_{1}\gamma_{2}\cdot\tilde{x})\geq\operatorname{sys}(X)$ or
$d(\tilde{x},\gamma_{1}^{-1}\gamma_{2}\cdot\tilde{x})\geq\operatorname{sys}(X),$
which is a contradiction because
$d(\tilde{x},\gamma_{2}\cdot\tilde{x})<\operatorname{sys}(X)/2,$
$d(\tilde{x},\gamma_{1}\cdot\tilde{x})<\operatorname{sys}(X)/2,$ and
$d(\tilde{x},\gamma_{1}^{-1}\cdot\tilde{x})<\operatorname{sys}(X)/2.$ ∎

We define the thick part of $X$ as the complement of the thin part:

$X_{thick}:=X\backslash\displaystyle\cup_{i=1}^{k}{U_{i,\rho}}.$

Since every point in a thin part of $X$ has a displacement less than
$\operatorname{sys}(X)/2,$ the following Proposition tells us that
$X_{thcik}\neq\varnothing.$

######  Proposition 4.2.

There exists $x\in X$ such that

$\operatorname{inj_{x}(X)}\geq\operatorname{sys}(X)/2.$

###### Proof.

Assume for the sake of contradiction that for all $x\in X$ we have
$\operatorname{inj}_{x}(X)<\operatorname{sys}(X)/2.$ Therefore, the thin part
of $X$ covers all $X.$ However, this is not possible because the thin part of
$X$ is disjoint union of open sets by Proposition 4.1 and $X$ is connected.

∎

Now we show that the monodromy of $\rho$-thin part of $X$ around each cusp is
in the stabilizer of that cusp:

######  Lemma 4.3.

Suppose that $\epsilon\leq\operatorname{sys}(X)/2.$ Let $U^{\prime}_{i}$ be a
connected component of $U_{i,\epsilon}$ and $\iota:U^{\prime}_{i}\to X$ be the
identity map. Then, $\iota(\pi_{1}(U_{i}^{\prime}))$ is a subgroup of
$\Gamma_{i}.$

###### Proof.

As $\epsilon<\operatorname{sys}(X)/2,$ Proposition 4.1 implies that
$U_{i,\epsilon}$s are disjoint. Fix $x\in U^{\prime}_{i,\epsilon}$ and let
$\gamma:[0,1]\to X$ be a loop at $x$ which is representative of a class in
$\iota(\pi_{1}(U^{\prime},x)).$ Let $\tilde{x}$ be a lift of $x$ to the
universal cover $\mathbb{B}^{n}.$ As $\gamma$ is fully contained in
$U^{\prime}_{i,\epsilon},$ it lifts to a loop
$\tilde{\gamma}(t):[0,1]\to\tilde{U}_{i,\epsilon}$ which starts at
$\tilde{x}.$ Therefore, $\tilde{y}:=\gamma\cdot\tilde{x}=\tilde{\gamma}(1)$ is
in $\tilde{U}_{i,\epsilon}$ which means that
$d(\tilde{x},\tilde{y})<\epsilon.$ Let $\gamma^{\prime}\in\Gamma_{i}$ such
that $d(\tilde{x},\gamma^{\prime}\cdot\tilde{x})<\epsilon.$ By homogeneity, we
have

$d(\tilde{y},\gamma\gamma^{\prime}\gamma^{-1}\cdot\tilde{y})=d(\gamma\cdot\tilde{x},\gamma\gamma^{\prime}\gamma^{-1}\gamma\cdot\tilde{x})=d(\tilde{x},\gamma^{\prime}\cdot\tilde{x})<\epsilon.$

Since $\gamma\gamma^{\prime}\gamma^{-1}$ fixes $\gamma(q_{i})$ and
$\tilde{y}\in U_{i},$ we get that $\gamma(q_{i})=q_{i}$ and therefore
$\gamma\in\Gamma_{i}.$

∎

Finally, we show that every subvariety of $X$ intersects with $X_{thick}$,
that is, every subvariety of $X$ contains a point whose injectivity radius in
$X$ is larger than $\operatorname{sys}(X)/2:$

######  Theorem 4.4.

Every subvariety of $\overline{X}$ either intersects with $X_{thick}$ or fully
contained in $D.$

###### Proof.

For the sake of the contradiction assume that there exists a connected
subvariety $V$ which is fully contained in $X_{thin}.$ Without loss of
generality we can assume that $V$ is fully contained in a connected component
of a thin part around the cusp $q_{\infty}.$ Consider the function $-u$ which
is a plurisubharmonic function on the Siegel domain $\mathbb{S}$ and invariant
under the stabilizer $\Gamma_{\infty}$(see [BT18]). Therefore, it follows from
Lemma 4.3 that $-u$ is a well-defined function on every component of the thin
part around $q_{\infty}.$ Hence, $-u$ is a well-defined plurisubharmonic
function on $V.$ Notice that if a plurisubharmonic function achieves its
maximum, it has to be constant. Since $V$ is compact, $-u$ must be constant on
$V.$ However, it is not possible because the Kähler form on $X$ is induced
from $-2i\partial\bar{\partial}\log(u)$(see [BT18, Lemma 2.1]) and if $-u$
were constant, the volume of $V$ would be zero.

∎

##  5 Volumes of Subvarieties

In this section, we prove Theorem A, and Theorem B. We first restate the Hwang
and To’s theorem in the following way:

######  Theorem 5.1.

([HT02, Theorem 1.1]) Take $x\in X$ with injectivity radius
$r=\operatorname{inj}_{x}(X).$ Let $B(x,r)$ be Bergman ball of radius $r$
centered at $x.$ Suppose $V$ is an $m$-dimensional subvariety of $X$ passing
through $x.$ Then, the following inequality holds:

$\displaystyle\operatorname{vol_{X}(V\cap B(x,r))}$
$\displaystyle\geq\frac{(4\pi)^{m}}{m!}\sinh^{2m}(r)\cdot\operatorname{mult}_{x}(V).$
(11)

Hwang and To generalized the above-mentioned theorem for a general Hermitian
symmetric domain in [HT00, HT02]. Note that for non-compact $X$, the
injectivity radius escapes to zero as we get closer to the cusps. But, Theorem
4.4 tells us that every subvariety of $X$ has a point with injectivity as
large as $\operatorname{sys}(X)/2$. Hence, we will get the following theorem:

######  Theorem 5.2.

Let $V$ be an $m$-dimensional subvariety of $\overline{X}$ which is not
contained in $D.$ Then,

$\displaystyle\operatorname{vol}_{X}(V)$
$\displaystyle\geq\frac{(4\pi)^{m}}{m!}\sinh^{2m}\big{(}\operatorname{sys}(X)/2\big{)}.$
(12)

###### Proof.

Theorem 4.4 implies that $V\cap X_{thick}\neq\varnothing.$ By Proposition 4.2
there exits a point $x\in V$ with
$\operatorname{inj}_{x}(X)\geq\operatorname{sys}(X)/2.$ Now, Theorem 5.1
gives:

$\operatorname{vol}_{X}(V)\geq\frac{(4\pi)^{m}}{m!}\sinh^{2m}\big{(}\operatorname{sys}(X)/2\big{)}.$

∎

######  Corollary 5.3.

With the same notation as Theorem 5.2, we have that

$\displaystyle(K_{\overline{X}}+D)^{m}\cdot V$
$\displaystyle\geq(n+1)^{m}\sinh^{2m}\big{(}\operatorname{sys}(X)/2\big{)}.$

###### Proof.

Theorem 5.2 together with 6 gives:

$\displaystyle(K_{\overline{X}}+D)^{m}\cdot V$
$\displaystyle=\big{(}\frac{n+1}{4\pi}\big{)}^{m}m!\operatorname{vol}_{X}(V)\textit{\
(by \ref{ChernForm})}$
$\displaystyle\geq(n+1)^{m}\sinh^{2m}\big{(}\operatorname{sys}(X)/2\big{)}\textit{\
(by Theorem \ref{Bound Hyperbolic Volume})}$

∎

We recall Bakker and Tsimerman’s theorem which tells us that the uniform depth
of cusps of $X$ bounds the intersection numbers of $K_{\overline{X}}$ with
subvarieties of $\overline{X}$ which is not contained in $D.$

######  Theorem 5.4.

([BT18, Corrolary 3.8]) Suppose $d$ is the uniform depth of cusps. Then,

$K_{\overline{X}}+(1-\lambda)D$

is ample for $\lambda\in(0,d(n+1)/4\pi).$

Now, putting together Theorem 5.2, Theorem 5.4 and what we proved for the
uniform depth of cusps, Theorem 3.9, yields a lower bound for the degree of
$K_{\overline{X}}$ on $V$ in terms of $\operatorname{sys}(X):$

######  Theorem 5.5.

Let $V$ be an $m$-dimensional subvariety of $\overline{X}$ which is not
contained in $D.$ If $\operatorname{sys}(X)\geq
4\ln\big{(}5n+(4\pi)^{4}\big{)},$ then

$\displaystyle K_{\overline{X}}^{m}\cdot V$
$\displaystyle>(\frac{n}{4\pi})^{m}\cdot e^{m\operatorname{sys}(X)/16}.$ (13)

###### Proof.

We deal separately with the following two cases:

  1. 1.

$V\cap D=\varnothing:$ In this case we have $K_{X}^{m}\cdot
V=(K_{\overline{X}}+D)^{m}\cdot V$ and from Corollary 5.3 we get that

$\displaystyle K_{X}^{m}\cdot V$
$\displaystyle\geq(n+1)^{m}\sinh^{2m}\big{(}\operatorname{sys}(X)/2\big{)}$
(14) $\displaystyle\geq(\frac{n}{4\pi})^{m}e^{m\operatorname{sys}(X)}\textit{\
by the bound on the systole.}$ (15)

  2. 2.

$V\cap D\neq\varnothing:$ Since $\operatorname{sys}(X)\geq
4\ln\big{(}5n+(4\pi)^{4}\big{)}),$ 10 gives that the uniform depth of cusp is
at least $4\pi$ and it follows from Bakker-Tsimerman’s theorem (Theorem 5.4)
that $K_{\overline{X}}$ is ample. In particular, this implies that
$K_{\overline{X}|D}$ is ample. On the other hand, we know that the conormal
bundle $-D_{|D}$ is ample. Therefore, for every $i>1,$ we have

$\displaystyle K_{\overline{X}}^{m-i}\cdot(-D)^{i}\cdot
V=-(K_{\overline{X}|D})^{m-i}\cdot(-D_{|D})^{i-1}\cdot V_{|D}<0.$ (16)

By Bakker-Tsimerman’s theorem, Theorem 5.4, we get that

$\Big{(}K_{\overline{X}}-\big{(}(n+1)d/4\pi-1\big{)}D\Big{)}^{m}\cdot V\geq
0.$

Expanding this and combining with 16 gives:

$\displaystyle K_{\overline{X}}^{m}\cdot V$
$\displaystyle\geq\big{(}(n+1)d/4\pi-1\big{)}^{m}\cdot-(-D)^{m}\cdot V$
$\displaystyle\geq\big{(}(n+1)d/4\pi-1\big{)}^{m}\textit{(by ampleness of
$-D$)}$ $\displaystyle\geq\big{(}(n+1)d/4\pi-1\big{)}^{m}$
$\displaystyle\geq\Big{(}nd/4\pi\Big{)}^{m}\textit{ (by \ref{depth Bounds
2})}$ $\displaystyle\geq\big{(}n/4\pi\big{)}^{m}\cdot
e^{m\operatorname{sys}(X)/16}\textit{ (by \ref{depth Bounds 2})}.$ (17)

Combining 14 and 17 gives that:

$\displaystyle K_{\overline{X}}^{m}\cdot V>(\frac{n}{4\pi})^{m}\cdot
e^{m\operatorname{sys}(X)/16}.$

∎

The volume of a line bundle $L$ on an $m$-dimensional projective variety $V$
is defined as the non-negative real number

$\operatorname{vol}_{V}(L):=\limsup\limits_{b\rightarrow\infty}\dfrac{h^{0}(V,bL)}{b^{m}/m!},$

which measures the positivity of $L$ from the point of view of birational
geometry. If $L$ is a nef line bundle on $V$, then $vol_{V}(L)=L^{n}.$ Let
$V^{\prime}$ be a smooth variety birational to $V$ with a canonical bundle
$K_{V^{\prime}}.$ The canonical volume of the subvariety $V$ is

$\widetilde{\operatorname{vol}}_{V}:=\limsup\limits_{b\rightarrow\infty}\dfrac{h^{0}(V^{\prime},bK_{V^{\prime}})}{b^{m}/m!},$

which does not depend on the choice of $V^{\prime}.$

To prove the bound on the canonical volume for a subvariety which does not
intersect the boundary $D,$ we will use the following lemma proved by
Brunebarbe:

######  Lemma 5.6.

[Bru20a, Proposition 3.2] Let $Y$ be a Kähler manifold with Kähler form
$\omega$. Assume that its holomorphic bisectional curvature is non-positive
and that its holomorphic sectional curvature is bounded from above by $-c<0.$
Then,

$\frac{1}{c}\operatorname{Ric}_{\omega}\geq\omega.$

######  Theorem 5.7.

Let $V$ be an $m$-dimensional subvariety of $\overline{X}$ which is not
contained in $D.$ If $\operatorname{sys(X)}\geq
4\ln\big{(}5n+(8\pi)^{4}\big{)},$ then

$\displaystyle\widetilde{\operatorname{vol}}_{V}>(\frac{m}{4\pi})^{m}e^{m\operatorname{sys}(X)/16}.$
(18)

###### Proof.

Let $V^{\prime}\longrightarrow V$ be a desingularization such that the set-
theoretic preimage of the boundary divisor $D^{\prime}$ is a normal crossing
divisor. We consider two cases:

  1. 1.

$V\cap D=\varnothing:$ Let $\omega$ be the Kähler form induced by the Bergman
metric on $V^{\prime}.$ Since the holomorphic sectional curvature of the
Bergman metric is $-1$ and the curvature decreases on a subvariety, we can
apply Lemma 5.6 to get that $K_{V^{\prime}}-K_{\overline{X}|V^{\prime}}$ is
nef on $V^{\prime}$ and therefore

$\widetilde{\operatorname{vol}}_{V}=\operatorname{vol}_{V^{\prime}}(K_{V^{\prime}})\geq\operatorname{vol}_{V^{\prime}}(K_{\overline{X}|V^{\prime}}).$

Since the uniform depth of cusp is sufficiently large, by Theorem 5.4 we get
that $K_{\overline{X}}$ is ample and therefore

$\operatorname{vol}_{V^{\prime}}(K_{\overline{X}|V^{\prime}})=K_{\overline{X}|V^{\prime}}^{m}=K_{\overline{X}}^{m}\cdot
V=\operatorname{deg}_{\overline{X}}(V).$

We can conclude the desired inequality for this case by Theorem 5.5.

  2. 2.

$V\cap D\neq\varnothing:$ Since $\operatorname{sys}(X)\geq
4\ln\big{(}5n+(4\pi)^{4}\big{)}),$ 10 gives that the uniform depth of cusp is
at least $4\pi.$ By [Mem22, Theorem A] we get that twisted cotangent bundle
$\Omega^{1}_{V^{\prime}}\langle-rD^{\prime}\rangle$ is ample for every
$r\in(0,d/2\pi).$ Taking determinate gives that
$K_{V^{\prime}}+(1-mr)D^{\prime}$ is ample for every $r\in(0,d/2\pi).$
Therefore, with the bound on the depth we get that $K_{V^{\prime}}$ is ample.
On the other hand, we have that the bundle $-D^{\prime}_{|D^{\prime}}$ is
ample. Therefore, for every $i>1,$ we have

$\displaystyle K_{V^{\prime}}^{m-i}\cdot(-D^{\prime})^{i}\cdot
V^{\prime}=-(K_{V^{\prime}|D})^{m-i}\cdot(-D^{\prime}_{|D^{\prime}})^{i-1}\cdot
V_{|D^{\prime}}<0.$ (19)

Set $r^{\prime}=d/2\pi-\epsilon,$ for a small $\epsilon>0.$ Expanding
$\Big{(}K_{V^{\prime}}+(1-mr^{\prime})D^{\prime}\Big{)}^{m}\geq 0$ and using
19 gives that

$\displaystyle K_{V^{\prime}}^{m}$
$\displaystyle\geq\big{(}mr^{\prime}-1\big{)}^{m}(-D^{\prime}_{|D^{\prime}})^{m}$
$\displaystyle\geq\big{(}mr^{\prime}-1\big{)}^{m}\textit{ \ (by the ampleness
of $-D^{\prime}_{|D^{\prime}}$)}$ $\displaystyle\geq(md/4\pi)^{m}$
$\displaystyle\geq(\frac{m}{4\pi})^{m}e^{m\operatorname{sys}(X)/16}\textit{
(by \ref{depth Bounds 2})}.$

By the ampleness of $K_{V^{\prime}}$ we get
$\widetilde{\operatorname{vol}}_{V}=\operatorname{vol}_{V^{\prime}}(K_{V^{\prime}})=K_{V^{\prime}}^{m}$
and hence the claim follows.

∎

##  6 Effective global generation and very ampleness

In this section, we prove Corollary C, Corollary D based on the bound we found
for $\operatorname{deg}_{\overline{X}}(V)$ in Theorem 5.5. First, we analyze
the problem on the boundary divisor $D.$

###  6.1 Base-point freeness and very ampleness on $D.$

In this subsection, we prove that if the uniform depth of cusps is
sufficiently large, then $2K_{\overline{X}}$ does not have a base point on
$D,$ and moreover $3K_{\overline{X}}$ can separate any two points, and any
tangent direction on $D.$ We first prove that the restricted bundles on the
boundary satisfy these properties. Consider the decomposition of the boundary
divisor $D$ to the connected components $D=\sqcup_{i=1}^{k}D_{i}.$ Due to
[Mok12], we know that each $D_{i}$ is an abelian variety with ample conormal
bundle $O_{D_{i}}(-D_{i}).$

######  Lemma 6.1.

The line bundle $2K_{\overline{X}|D_{i}}$ is base-point free and
$3K_{\overline{X}|D_{i}}$ is very ample for every $i.$

###### Proof.

The adjunction formula gives that $K_{\overline{X}|D_{i}}\cong-D_{i|D_{i}}.$
As the conormal bundle is ample and $D_{i}$ is an abelian variety,
$-2D_{i|D_{i}}$ is base-point free and $-3D_{i|D_{i}}$ is very ample (see
[Ohb87]).

∎

In the next two lemmas, we see how we can lift the sections from the
restricted bundle to $\overline{X}.$ The base locus of a line bundle $L$ on
$\overline{X}$ will be denoted by $\operatorname{Bs}(L).$

######  Lemma 6.2.

Suppose that the uniform depth of cusps is larger than $4\pi,$ Then, the
following hold:

  1. 1.

$\operatorname{Bs}\big{(}2K_{\overline{X}}\big{)}\cap D=\varnothing$

  2. 2.

For any two points on different components of $D,$ there exists a global
section of $2K_{\overline{X}}$ which separate them.

###### Proof.

Let $L$ be $2K_{\overline{X}}.$

  1. 1.

By Lemma 6.1, $L_{|D}$ is base-point free and therefore it is enough to show
that we can lift the global section from $D$ to $\overline{X}$, that is,
$H^{0}(\overline{X},L)\longrightarrow H^{0}(D,L_{|D})$ is surjective. Consider
the following exact sequence on $\overline{X}:$

$0\longrightarrow L-D\longrightarrow L\longrightarrow L_{|D}\longrightarrow 0$

Writing the long exact sequence we can see that it is sufficient to show
$H^{1}(\overline{X},L-D)=0.$ As $L-D=K_{\overline{X}}+(K_{\overline{X}}-D),$
if the uniform depth is sufficiently large, then by Theorem 5.4
$K_{\overline{X}}-D$ is ample. Therefore, the vanishing of
$H^{1}(\overline{X},L-D)$ follows from Kodaira’s vanishing theorem.

  2. 2.

Suppose that we want to separate $x\in D_{i}$ and $y\in D_{j}$ with $i\neq j.$
It is sufficient to find a global section of $L-D_{i}$ which does not vanish
at $y.$ To this end, we can repeat the argument of the first part, but instead
of using Theorem 5.4, we should use [BT18, Proposition 3.6].

∎

######  Lemma 6.3.

If the uniform depth of cusps is larger than $2\pi,$ then $3K_{\overline{X}}$
can separate any two points, and any tangent direction on every connect
component of $D.$

###### Proof.

By Lemma 6.2 and Lemma 6.1, it is enough to show that we can lift the sections
from the boundary, i.e.,

$H^{0}(\overline{X},3K_{\overline{X}})\longrightarrow
H^{0}(D,3K_{\overline{X}|D})\longrightarrow 0.$

Hence, it is enough to show that $H^{1}(\overline{X},3K_{\overline{X}}-D)=0.$
Since $d>2\pi,$ it follows from Theorem 5.4 that $2K_{\overline{X}}-D$ is
ample. Therefore, by Kodaira’s vanishing theorem we get that
$H^{1}(\overline{X},3K_{\overline{X}}-D)=0.$ ∎

###  6.2 Global generation and very ampleness on $\overline{X}$

In this subsection, we see how we can conclude effective global generation and
effective very ampleness results by using Theorem 5.5. We first recall the
famous theorem of Angehrn and Siu on pointwise base point freeness:

######  Theorem 6.4.

[AS95, Theorem 0.1] Let $Y$ be a smooth projective variety of dimension $n,$
and let $L$ be an ample line bundle on $Y.$ Fix a point $y\in Y,$ and assume
that

$\displaystyle L^{m}\cdot V>\big{(}\frac{n(n+1)}{2}\big{)}^{m}$ (20)

for every subvariety $V$ of dimension $m$ passing through $y.$ Then, $K_{Y}+L$
has a section that does not vanish at $y.$

Combining Angehrn and Siu’s result with our Theorem 5.5 gives that if
$\operatorname{sys}(X)$ is sufficiently large relative to $n,$ then
$2K_{\overline{X}}$ is globally generated:

######  Theorem 6.5.

If $\operatorname{sys(X)}\geq 20\ln\big{(}5n+(4\pi)^{4}\big{)},$ then
$2K_{\overline{X}}$ is globally generated.

###### Proof.

Using 10 we get that $d>4\pi.$ Therefore, by Lemma 6.2, $2K_{\overline{X}}$
does not have any base-point on $D.$ On the other hand, Theorem 5.5 implies
that for every $m$-dimensional subvariety $V\subset\overline{X}$ which is not
contained in $D,$ we have

$\displaystyle K_{\overline{X}}^{m}\cdot V$
$\displaystyle\geq(\frac{n}{4\pi})^{m}\cdot e^{m\operatorname{sys}(X)/16}$
$\displaystyle\geq n^{m}\big{(}5n+(4\pi)^{4}\big{)}^{m}\textit{(by the bound
on $\operatorname{sys}(X)$)}$
$\displaystyle>\big{(}\frac{n(n+1)}{2}\big{)}^{m}.$

Therefore, Theorem 6.4 implies that for every point $x\in\overline{X}\setminus
D,$ there is a section of $2K_{\overline{X}}$ which does not vanish at $x.$
Hence, $2K_{\overline{X}}$ is globally generated. ∎

Now, we prove a proposition which will be used to show that
$2K_{\overline{X}}$ can separate a point in $X$ from a point in $D:$

######  Proposition 6.6.

If $\operatorname{sys(X)}\geq 20\ln\big{(}5n+(8\pi)^{4}\big{)},$ then for
every $x\in X$ there exits $s\in H^{0}(\overline{X},2K_{\overline{X}}-D)$ such
that $s$ does not vanish at $x.$

###### Proof.

Since $\operatorname{sys(X)}\geq 20\ln\big{(}5n+(8\pi)^{4}\big{)},$ the
uniform depth of cusps is larger than $4\pi$(see 10) and therefore by Bakker-
Tsimerman’s result, Theorem 5.4, $K_{\overline{X}}+(1-\lambda)D$ is ample for
$\lambda\in(0,\frac{(n+1)d}{4\pi}).$ On the other hand, as
$K_{\overline{X}|D}\cong-D_{D}$ and $-D_{|D}$ is ample, for every subvariety
$V$ of dimension $m$ and every $1\leq i\leq m$ we have

$\displaystyle(K_{\overline{X}}-2D)^{m-i}(-D)^{i}\cdot
V_{|D}=-(K_{\overline{X}}-2D)_{|D}^{i}(-D_{|D})^{j-1}\cdot
V=-3^{i}(-D_{|D})^{n-1}\cdot V_{|D}\leq 0.$ (21)

Expanding $(K_{\overline{X}}-(1-\frac{(n+1)d}{4\pi})D)^{m}\cdot V\geq 0$ and
using 21 we get:

$\displaystyle(K_{\overline{X}}-2D)^{m}\cdot V$
$\displaystyle\geq(\frac{(n+1)d}{4\pi}+1)^{m}(-D_{|D})^{m-1}\cdot V_{|D}$
$\displaystyle\geq(\frac{(n+1)d}{4\pi})^{m}\textit{(by the ampleness of
$-D_{|D}$)}$ $\displaystyle\geq\big{(}\frac{n+1}{4\pi}\big{)}^{m}\cdot
e^{m\operatorname{sys}(X)/16}\textit{ \ (by \ref{depth Bounds 2})}$
$\displaystyle>(n+1)^{m}n^{m}\textit{ \ (by the bound on
$\operatorname{sys}(X)$)}$

Hence, Theorem 6.4 gives that $2K_{\overline{X}}-D$ has a global section which
does not vanish at $x.$

∎

We recall the result of Ein-Lazarsfeld-Nakamay on the pointwise separation of
jets:

######  Theorem 6.7.

([ELN96, Theorem 4.4]) Let $Y$ be a smooth projective variety of dimension $n$
and let $L$ be an ample line bundle on $Y$ satisfying $L^{n}>(n+s)^{n}.$ Let
$b$ be a non-negative number such that $K_{Y}+bL$ is nef. Suppose that $m_{0}$
is a positive integer such that $m_{0}L$ is free. Then, for any point $y\in Y$
either

(a) $K_{Y}+L$ separates $s$-jets at $y$, or

(b) there exists a dimension $m$ subvariety $V$ containing $y$ and satisfying

$\displaystyle\operatorname{deg}_{L}(V)\leq\Big{(}b+m_{0}\cdot
m+\frac{n!}{(n-m)!}\Big{)}^{n-m}(n+s)^{n}$ (22)

Let $Y$ be a smooth projective variety and $L$ be a nef line bundle on $Y.$
Fix a point $y\in Y.$ The Seshadri constant of $L$ at $y$ is the real number

$\displaystyle\varepsilon(L,y)=\operatorname{inf}\frac{L\cdot
C}{\operatorname{mult}_{y}(C)},$

where the infimum is taken over all curves $C$ passing through $y.$

Plugging in our Theorem 5.5 and Theorem 6.5 to the result of Ein-Lazarsfeld-
Nakamay allows us to separates $s$-jets of $2K_{\overline{X}}$ on $X$ if
$\operatorname{sys}(X)$ is sufficiently large with respect to $n$ and $s:$

######  Theorem 6.8.

Let $s$ be a positive integer. Suppose that

$\operatorname{sys}(X)\geq
20\operatorname{max}\\{n\ln\big{(}(1+2n+n!)(n+s)\big{)},\ln\big{(}5n+(8\pi)^{4}\big{)}\\}.$

Then for every $x\in X,$ the line bundle $2K_{\overline{X}}$ separates
$s$-jets at $x.$ In particular, for every $x$ we have that

$\epsilon(K_{\overline{X}},x)\geq s/2.$

###### Proof.

Since $\operatorname{sys}(X)>20\ln\big{(}5n+(8\pi)^{4}\big{)},$ Theorem 6.5
implies that $2K_{\overline{X}}$ is globally generated. Also, as
$\operatorname{sys}(X)\geq 20\ln(n+s),$ Theorem 5.5 implies that

$\displaystyle K_{\overline{X}}^{n}>(n+s)^{n}.$

Note that plugging in the lower bounds on $\operatorname{sys}(X)$ in Theorem
5.5 gives that for a subvariety $V$ of dimension $m$ which does not contained
in $D,$ the following inequality holds:

$K_{\overline{X}}^{m}\cdot V\geq
n^{m}(1+2n+n!)^{n}(n+s)^{n}\geq(b+2m+\frac{n!}{(n-m))!})^{n-m}(n+s)^{n}.$

Now, applying Theorem 6.7 to $L=K_{\overline{X}},m_{0}=2$ and $b=1$ gives that
$2K_{\overline{X}}$ separates $s$-jest at every $x\in X.$

Combining the separation of jets with [BRH+09, Proposition 2.2.5 ] gives that
$\epsilon(2K_{\overline{X}},x)\geq s.$ Since
$\epsilon(2K_{\overline{X}},x)=2\epsilon(K_{\overline{X}},x),$ we get the
desired inequality. ∎

We recall a result of Kollar which tells us that a line can separate two
points if the degree of every subvariety passing through either of the points
with respect to the line bundle is sufficiently large relative to the
dimension of the ambient space:

######  Theorem 6.9.

([Kol97, Theorem 5.9]) Let $L$ be a nef and big divisor on a smooth projective
variety $Y$. Let $x_{1},x_{2}$ be closed points and assume that there are
positive numbers $c(k)$ with the following properties:

  1. 1.

If $V\subset Y$ is an irreducible $m$-dimensional subvariety which contains
$x_{1}$ or $x_{2}$ then

$L^{m}\cdot V>c(m)^{m}.$

  2. 2.

The numbers $c(k)$ satisfy the inequality

$\displaystyle\sum_{k=1}^{\operatorname{dim}(x)}\sqrt[k]{2}\frac{k}{c(k)}\leq
1.$

Then, $K_{Y}+L$ separates $x_{1}$ and $x_{2}.$

######  Definition 6.10.

([Tak93]) Let $L$ be a line bundle on a smooth projective variety $Y$ and $D$
be a divisor on $Y.$ The line bundle $L$ is said to be very ample modulo $D$
if the rational map $\Phi_{L}:Y\dashrightarrow\mathbb{P}(H^{0}(Y,O_{Y}(L))$ is
an embedding of $Y\setminus D.$

Note that Theorem 6.5 says that the rational map
$\Phi_{2K_{\overline{X}}}:Y\dashrightarrow\mathbb{P}(H^{0}(Y,O_{Y}(2K_{\overline{X}}))$
is globally defined map on $Y.$ Moreover, the following theorem gives that
this map is in particular injective on $X$ and can separate any two tangent
directions at whole $\overline{X}:$

######  Theorem 6.11.

Suppose that

$\operatorname{sys}(X)\geq
20\operatorname{max}\\{n\ln\big{(}(1+2n+n!)(n+1)\big{)},\ln\big{(}5n+(8\pi)^{4}\big{)}\\}.$

Then the map
$\Phi_{2K_{\overline{X}}}:\overline{X}\to\mathbb{P}\big{(}H^{0}(\overline{X},2K_{\overline{X}})\big{)}$
satisfies the following properties:

  1. 1.

If $\phi_{2K_{\overline{X}}}(x_{1})=\phi_{2K_{\overline{X}}}(x_{2})$ for some
$x_{1},x_{2}\in\overline{X},$ then $x_{1},x_{2}\in D_{i},$ where $D_{i}$ is
some connected component of $D.$

  2. 2.

$\Phi_{2K_{\overline{X}}}$ separates tangent directions at every $x\in X.$

###### Proof.

Separation of points: Note that by Lemma 6.2 if
$\phi_{2K_{\overline{X}}}(x_{1})=\phi_{2K_{\overline{X}}}(x_{2})$ and
$x_{1},x_{2}\in D,$ then they both lie on the same component of $D.$ Hence, we
only need to deal with the following two cases:

  1. 1.

$x_{1},x_{2}\in X:$ Let $V\subset\overline{X}$ be a subvariety of dimension
$m$ which passes through either $x_{1}$ or $x_{2}.$ Fix
$c=ne^{\operatorname{sys}(X)/20}.$ By Theorem 5.5 we have that

$\displaystyle K_{\overline{X}}^{m}\cdot V\geq c^{m}.$

Therefore, by Kollar’s Theorem (6.9) we can separate any two points
$x_{1},x_{2}\in X.$

  2. 2.

$x_{1}\in X,x_{2}\in D:$ By Proposition 6.6, there is a section $s\in
H^{0}(\overline{X},2K_{\overline{X}}-D)$ which does not vanish at $x_{1}.$
Therefore, as $2K_{\overline{X}}-D$ is a subbundle of $2K_{\overline{X}},$ we
get a section of $2K_{\overline{X}}$ which does not vanish at $x_{1},$ but
vanishes on $D$ and in particular at $x_{2}.$

Separation of tangent directions: For $x\in X,$ the separation of tangent
direction follows from Theorem 6.8 when $s=1.$

∎

In particular, Theorem 6.11 implies that $2K_{\overline{X}}$ is very ample
modulo $D.$

######  Theorem 6.12.

With the same assumption on $\operatorname{sys}(X)$ as Theorem 6.11,
$3K_{\overline{X}}$ is very ample.

###### Proof.

By Theorem 6.11, we only need to show that $3K_{\overline{X}}$ can separate
any two points and any tangent direction on any connected component of $D,$
which follows from Lemma 6.3. ∎

Putting all of these together, we get:

######  Corollary 6.13.

Suppose that

$\operatorname{sys}(X)\geq
20\operatorname{max}\\{n\ln\big{(}(1+2n+n!)(n+1)\big{)},\ln\big{(}5n+(8\pi)^{4}\big{)}\\}.$

Then, the following hold

  1. 1.

$2K_{\overline{X}}$ is globally generated and very ample modulo $D.$

  2. 2.

$3K_{\overline{X}}$ is very ample.

###### Proof.

The global generation of $2K_{\overline{X}}$ follows from Theorem 6.5. The
very ampleness modulo $D$ follows from Theorem 6.11. The very ampleness of
$3K_{\overline{X}}$ follows from Theorem 6.12. ∎

##  7 Seshadri Constant

The goal of this section is to study the relation between the Seshadri
constant and the systole of $X$ and in particular we prove Corollary E in this
section.

In addition to what we obtained on Theorem 6.8 about the Seshadri constant
$\epsilon(2K_{\overline{X}},x)$ for $x\in X,$ we can obtain the following
result which has smaller bound on $\operatorname{sys}(X):$

######  Corollary 7.1.

Suppose that $\operatorname{sys}(X)\geq 20\ln\big{(}5n+(8\pi)^{4}\big{)}.$ Let

$E:=\\{x\in X|\epsilon(K_{\overline{X}},x)<e^{\operatorname{sys}(X)/20}\\}.$

Then, $E$ satisfies the following properties:

  1. 1.

$E\cap X_{thick}=\varnothing.$

  2. 2.

$E$ does not contain any positive dimensional subvariety.

  3. 3.

$E$ is contained in a Zariski closed proper subset of $X.$

###### Proof.

  1. 1.

Fix $x\in X_{thick}.$ Let $C\subset\overline{X}$ be a curve passing through
$x.$ Since $x\in X_{thick}$ we have
$\operatorname{inj}_{x}(X)\geq\operatorname{sys}(X)/2.$ On the other hand, the
bound on the systole gives that $d\geq 8\pi,$ therefore by Theorem 5.4
$K_{\overline{X}}-D$ is ample. We can write:

$\displaystyle 2K_{\overline{X}}\cdot C$
$\displaystyle\geq(K_{\overline{X}}+D)\cdot C\textit{\ (By ampleness of
$K_{\overline{X}}-D$)\ }$
$\displaystyle\geq\frac{n+1}{4\pi}\operatorname{vol}_{X}(C)$
$\displaystyle\geq\frac{n+1}{4\pi}\sinh^{2}\big{(}\operatorname{sys}(X)/2\big{)}\cdot\operatorname{mult}_{x}(C)\textit{\
(by \ref{Hwanginequilty})}.$

Therefore,

$\epsilon(x,K_{\overline{X}})\geq\frac{n+1}{8\pi}\sinh^{2}\big{(}\operatorname{sys}(X)/2\big{)}>e^{\operatorname{sys}(X)/20},$

and this gives the first property.

  2. 2.

Combining (i) with Theorem 4.4 we conclude that $E$ does not have any positive
dimensional subvariety.

  3. 3.

Note that Theorem 5.5 implies that for every $m$-dimensional subvariety
$V\not\subset D,$ we have

$(K_{\overline{X}}^{m}\cdot
V)^{\frac{1}{m}}\geq\frac{n+1}{4\pi}e^{\operatorname{sys}(X)/16}.$

Putting this in [EKL95, Theorem 3.1] gives

$\displaystyle\epsilon(K_{\overline{X}},x)\geq\frac{1}{4\pi}e^{\operatorname{sys}(X)/16}>e^{\operatorname{sys}(X)/20}.$
(23)

for all $x\in\overline{X}$ off the union of countably many proper subvarieties
of $\overline{X}.$ On the other hand as $K_{\overline{X}}$ is ample by using
[EKL95, Lemma 1.4] we can conclude that the inequality 23 holds on Zarisiky
open set, i.e, $E$ is contained in a proper subvariety of $\overline{X}.$

∎

Consider the decomposition of the boundary divisor $D$ to the connected
components $D=\sqcup_{i=1}^{k}D_{i}.$ Due to [Mok12], we know that each
$D_{i}$ is an abelian variety with ample conormal bundle $O_{D_{i}}(-D_{i}).$
The adjunction formula gives that $K_{\overline{X}|D_{i}}$ is isomorphic to
the conormal bundle $O_{D_{i}}(-D_{i}).$ Suppose that
$D_{i}=\Lambda_{i}\backslash W_{i},$ where $W_{i}\cong\mathbb{C}^{n-1}$ is a
complex vector space of dimension $n-1,$ and
$\Lambda_{i}\cong\mathbb{Z}^{n-1}$ is a lattice in $W_{i}.$ It is classical
that every ample line bundle on $D_{i}$ determines a positive definite
Hermitian form on $W_{i}.$ Suppose $H_{i}$ is the positive definite Hermitian
form determined by $K_{\overline{X}|D_{i}}$ on $W_{i}.$ The real part

$B_{i}=\operatorname{Re}(H_{i})$

defines a Euclidean inner product on $W_{i}$ (see [Laz17, sec 5.3.A] for more
details). Let $l_{i}$ be the length of the shortest vector of $\Lambda_{i}$
with respect to $B_{i}.$ We define the systole of the boundary as

$\operatorname{sys}(D):=\operatorname{min}_{i=1}^{k}l_{i}.$

The following lemma gives a lower bound for the Seshadri constant of
$K_{\overline{X}|D}$ in terms of the systole of the boundary:

######  Lemma 7.2.

Let $x$ be a point on a connected component of the boundary, $D_{i}.$ Then,

$\epsilon(K_{\overline{X}|D_{i}},x)\geq\frac{\pi}{4}\cdot\operatorname{sys}(D)^{2}.$

###### Proof.

This follows from [Laz17, Theorem 5.3.6]. ∎

Combining this lemma with the previous results gives that if the systole of
$\overline{X}$ and $\operatorname{sys}(D)$ are sufficiently large, then the
Seshadri constant $\epsilon(K_{\overline{X}},x)$ is large and in particular
$2K_{\overline{X}}$ is very ample:

######  Corollary 7.3.

Suppose that $\operatorname{sys}(D)>2\sqrt{2n/\pi}$ and that

$\operatorname{sys}(X)\geq
20\operatorname{max}\\{n\ln\big{(}5n(1+2n+n!)\big{)},\ln\big{(}5n+(8\pi)^{4}\big{)}\\}.$

Then, for every $x\in\overline{X}$ we have

$\epsilon(K_{\overline{X}},x)\geq 2n,$

and in particular $2K_{\overline{X}}$ is very ample.

###### Proof.

Let $C\subset\overline{X}$ be a connected curve passing through a point
$x\in\overline{X}.$ We consider there cases:

  1. 1.

$x\in D$ and $C$ fully contained in a $D:$ Let $D_{i}$ be the connected
component of $D$ which contains $x.$ Lemma 7.2 implies that

$K_{\overline{X}}\cdot C=K_{\overline{X}|D_{i}}\cdot
C\geq\frac{\pi}{4}\operatorname{sys}(D)^{2}\cdot\operatorname{mult}_{x}(C)\geq
2n\cdot\operatorname{mult}_{x}(C).$

  2. 2.

$x\in D$ and $C$ is not contained in $D:$ Plugging in the bound on the systole
in Theorem 3.9 gives that the uniform depth of cusps $d$ is at least $8\pi.$
By the theorem of Bakker-Tsimerman, Theorem 5.4, the line bundle
$K_{\overline{X}}+(1-\lambda)D$ is ample for $\lambda\in(0,(n+1)d/4\pi)$.
Hence, we can write

$\displaystyle K_{\overline{X}}\cdot C$
$\displaystyle\geq(\frac{(n+1)d}{4\pi}-1)D\cdot C$
$\displaystyle\geq\frac{nd}{4\pi}\operatorname{mult}_{x}(C)$
$\displaystyle\geq 2n\operatorname{mult}_{x}(C).$

  3. 3.

$x\in X:$ For this case we will use Theorem 6.8. Plugging in $s=2n$ to this
theorem gives that:

$\displaystyle K_{\overline{X}}\cdot C\geq 2n\cdot\operatorname{mult}_{x}(C).$

Hence, for every $x\in X$ we get that $\epsilon(K_{\overline{X}},x)\geq 2n.$
Combining this with Demailly’s theorem [Dam92, Proposition 6.8).] implies that
$2K_{\overline{X}}$ is very ample.

∎

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