# Products of curves as ball quotients

Matthew Stover111This material is based upon work supported by Grant Number
DMS-2203555 from the National Science Foundation.
Temple University
<EMAIL_ADDRESS>
###### Abstract

For any $g_{1},g_{2}\geq 0$, this paper shows that there is a cocompact
lattice $\Gamma<\operatorname{PU}(2,1)$ such that the ball quotient
$\Gamma\backslash\mathbb{B}^{2}$ is birational to a product $C_{1}\times
C_{2}$ of smooth projective curves $C_{j}$ of genus $g_{j}$. The only prior
examples were $\mathbb{P}^{1}\times\mathbb{P}^{1}$, due to Deligne–Mostow and
rediscovered by many others, and a lesser-known product of elliptic curves
whose existence follows from work of Hirzebruch. Combined with related new
examples, this answers the rational variant of a question of Gromov in the
positive for surfaces of Kodaira dimension $\kappa\leq 0$, namely that they
admit deformations $V^{\prime}$ such that there is a compact ball quotient
$\Gamma\backslash\mathbb{B}^{2}$ with a rational map
$\Gamma\backslash\mathbb{B}^{2}\dashrightarrow V^{\prime}$. Often the proof
gives the stronger conclusion that $V^{\prime}$ is birational to a ball
quotient orbifold. All examples are shown to be arithmetic, and even
arithmeticity of Hirzebruch’s example appears to be new.

##  1 Introduction

If $\Gamma<\operatorname{PU}(2,1)$ is a discrete subgroup acting cocompactly
on the unit ball $\mathbb{B}^{2}$ in $\mathbb{C}^{2}$, then the ball quotient
$\Gamma\backslash\mathbb{B}^{2}$ is isomorphic to a normal projective variety
$X$. A realization of a normal projective variety as a ball quotient connects
with many independently interesting problems, for example the existence of
special line arrangements on surfaces, the geography of Chern numbers of
surfaces, and existence of extremal orbifold Kähler–Eisenstein metrics; to
only scratch the surface, see for example [21, 20, 26, 11]. Concretely, Gromov
asked the following question [17, p. 187].

######  Question 1.1 (Gromov).

Does every irreducible (smooth?) algebraic $\mathbb{C}$-variety $V$ admit a
dominating regular (only rational?) map from some
$\Gamma\backslash\mathbb{B}^{n}$ to some deformation $V^{\prime}$ of $V$?

Benson Farb asked a similar question, namely whether smooth projective
surfaces are dominated by ball quotients. It turns out that many of the most
important explicit examples of $2$-dimensional ball quotient orbifolds,
constructed by Deligne and Mostow [9], are rational and hence birational to
$\mathbb{P}^{1}\times\mathbb{P}^{1}$. See [25] for a precise description of
the normal varieties associated with Deligne–Mostow lattices and [2, 7, 12]
(and references therein) for other perspective and more on similar examples.
Somewhat less well-known is Hirzebruch’s construction of a compact ball
quotient birational to a product of elliptic curves [21, §2]; technically,
Hirzebruch considers a high degree cover and this paper extracts the simpler
orbifold from his idea. This paper fully settles the rational variant of
Question 1.1 for products of curves, giving the first infinite family of
concrete examples for which Question 1.1 has a positive answer.

######  Theorem 1.2.

Given $g_{1},g_{2}\geq 0$, there exist smooth projective curves $C_{j}$ of
genus $g_{j}$ so that $C_{1}\times C_{2}$ is birational to a compact ball
quotient orbifold.

It is worth noting that the birational map from the ball quotient used to
prove Theorem 1.2 to $C_{1}\times C_{2}$ is always dominant. Smooth projective
surfaces of Kodaira dimension $-\infty$ are precisely the _ruled surfaces_ ,
namely those birational to $\mathbb{P}^{1}\times C$ for some smooth curve $C$
[4, Ex. VII.3]. Therefore, Theorem 1.2 has the following corollary.

######  Corollary 1.3.

The birational variant of Question 1.1 has a positive answer for smooth
projective surfaces $V$ of Kodaira dimension $\kappa(V)=-\infty$.
Specifically, there is a deformation $V^{\prime}$ of $V$ and a ball quotient
$\Gamma\backslash\mathbb{B}^{2}$ admitting a birational map
$\Gamma\backslash\mathbb{B}^{2}\dashrightarrow V^{\prime}$.

The proof of Theorem 1.2 consists of three primary steps. The first two are
carried out in §3 and §5, which construct examples for the cases
$g_{1}=g_{2}=2$ and $g_{1}=g_{2}=1$, respectively. The ball quotient in §3 is
birational to $C\times C$, where $C$ is the unique smooth projective curve of
genus $2$ admitting an automorphism of order $5$, first classified by Wiman
[37]. The quotient in §5 was essentially constructed by Hirzebruch, though the
example in [21, §2] is technically a large degree cover of the orbifold
presented in §5. The proof that each example is a ball quotient uses the
orbifold Bogomolov–Miyaoka–Yau inequality, which is recalled in §2.

The third step in the proof of Theorem 1.2 is to handle both products of
higher-genus curves and ruled surfaces of genus $g\geq 2$, which are
constructed by taking appropriate quotients and covers of the two primary
examples. These are constructed in four stages in §7.

Smooth projective surfaces of Kodaira dimension zero are also related to
products of curves, and it was known that many can be realized as _toroidal
compactifications_ of noncompact finite volume ball quotients. See for
examples [22, 24, 15, 16]. In §8, more new examples of ball quotient orbifolds
inspired by twists on the noncompact examples prove the following result.

######  Theorem 1.4.

Let $V$ be a smooth complex projective surface of Kodaira dimension
$\kappa(V)\leq 0$. Then there is a compact ball quotient
$\Gamma\backslash\mathbb{B}^{n}$ and a rational map
$f:\Gamma\backslash\mathbb{B}^{2}\dashrightarrow V^{\prime}$ to a deformation
$V^{\prime}$ of $V$.

The proofs of Theorems 1.2 and 1.4 give even more precise results. For many
cases the key step in the argument is proving that a blowup $Y$ of the
appropriate $V^{\prime}$ is the underlying analytic space for a ball quotient
orbifold, i.e., $Y$ admits a complete orbifold metric of constant
biholomorphic sectional curvature $-1$. Specifically, combining this paper
with previous work of Deligne–Mostow proves the existence of compact ball
quotient orbifold structures on:

  * $\star$

del Pezzo surfaces of degree at least five

  * $\star$

the blowup of a product $C_{1}\times C_{2}$ of curves $C_{j}$ of genus
$g_{j}\geq 2$ at $3(g_{1}-1)(g_{2}-1)$ points

  * $\star$

the blowup of an abelian surface at any finite number of points

  * $\star$

blowups of all bielliptic surfaces except one class (the one associated with
$\mathbb{Z}/6$) at sufficiently many points

The del Pezzo cases are covered by Deligne–Mostow orbifolds [25, Thm. 4.1].
For products of higher-genus curves, see Remark 7.2. For the remaining cases,
note that if $Y$ is the underlying analytic space for a ball quotient
orbifold, then any étale covering of $Y$ also naturally a ball quotient
orbifold. The example in §5 is an abelian surface blown up at one point, and
the proof of Theorem 1.4 does the same for blowups of bielliptic surfaces.
Appropriate étale covers give examples blown up at more points. The proof of
Theorem 1.4 only constructs ball quotient orbifold structures on _singular_ K3
and Enriques surfaces, and these are the only cases in Kodaira dimension zero
where this paper does not answer Question 1.1 using a dominant holomorphic
map.

An example of a ball quotient orbifold structure on a smooth K3 surface was
constructed in unpublished work of Naruki [32]; see also [23, §4.4]. Combining
this example with other examples from this paper leads to the following
result, proved at the conclusion of §8, which completely answers Farb’s
variant of Question 1.1 in Kodaira dimension zero.

######  Theorem 1.5.

Every minimal smooth projective surface of Kodaira dimension zero is smoothly
dominated by a compact ball quotient.

As for Question 1.1 in higher Kodaira dimension, general type surfaces seem
out of reach at the moment. However the Kodaira dimension one case seems quite
reasonable. Surfaces of Kodaira dimension one all admit elliptic fibrations
[4, Prop. IX.2], Theorem 1.2 covers trivial fibrations, and Theorem 1.4 adds
bielliptic surfaces. Thus the essential step in answering Question 1.1 for
surfaces Kodaira dimension one is to answer the following question asked by
Benson Farb.

######  Question 1.6 (Farb).

Which elliptic fibrations are dominated by a ball quotient?

For Kodaira dimension one, examples were constructed in the thesis of Livne
[28] (see also [10, Ch. 16] and [30], which does not cite Livne), but little
else is known beyond the trivial higher-genus fibrations constructed here. The
most interesting subcase is perhaps that of Shioda modular surfaces [33],
which includes Livne’s examples.

Lastly, this paper studies arithmeticity of the examples used to prove the
main results. Unfortunately, the examples used to prove the results in this
paper do not provide new nonarithmetic lattices in $\operatorname{PU}(2,1)$.

######  Theorem 1.7.

All ball quotients used to prove Theorems 1.2, 1.4, and 1.5 are arithmetic.

The example in §3 is shown to be arithmetic in §4. It appears that
arithmeticity of Hirzebruch’s example, proved in §6, is new. As an aside,
similar techniques prove arithmeticity of a related example of Holzapfel [22]
that was claimed to be arithmetic with details to appear in a later paper that
does not seem to be available; see Remark 9.3. The examples from §8 are shown
to be arithmetic in §9. All proofs show by a geometric argument that the
orbifolds are commensurable with arithmetic Deligne–Mostow orbifolds. Since
all other examples in §7 are commensurable with the two primary examples,
Theorem 1.7 follows. See the end of §9 for the precise catalogue of results
that prove Theorem 1.7. For a new explicit construction of a nonarithmetic
orbifold (commensurable with a previously-known example), see Remarks 8.2 and
9.2.

In fact, in light of Question 1.1, arithmeticity can be considered a feature
of this paper. A precise enumeration of smooth projective surfaces of general
type seems hopeless at the current moment. However, arithmetic lattices in
$\operatorname{PU}(2,1)$ can be enumerated and precisely classified by
increasing volume, with only finitely many of a given volume bound [5, Thm.
A]. Thus a positive answer to the following question could provide an
intriguing conceptual step forward toward classification of surfaces of
general type.

######  Question 1.8.

Let $V$ be a smooth projective surface of general type that is not a smooth
ball quotient. Is there a deformation $V^{\prime}$ of $V$ and a cocompact
arithmetic lattice $\Gamma<\operatorname{PU}(2,1)$ so that $V^{\prime}$ is
birational to the underlying analytic space for
$\Gamma\backslash\mathbb{B}^{2}$?

One could also include quotients of the bidisk
$\mathbb{B}^{1}\times\mathbb{B}^{1}$ in Question 1.8, since bidisk quotients
have already proven useful in constructing general type surfaces (e.g., see
[3]). Lattices acting on the bidisk are also enumerable, since they are either
a product of Fuchsian groups or are arithmetic by the Margulis arithmeticity
theorem [29, p. 4], so this addition would add little complication towards a
goal of enumerating general type surfaces.

### Acknowledgements

I thank Benson Farb for stimulating conversations about the contents of this
paper and for pointing out Naruki’s K3 ball quotient [32].

##  2 Ball quotient pairs

Let $\mathbb{B}^{2}$ be the unit ball in $\mathbb{C}^{2}$ with its Bergman
metric and $\operatorname{PU}(2,1)$ its group of holomorphic isometries. A
smooth projective surface $X$ is a smooth ball quotient if there is a torsion-
free discrete group $\Gamma<\operatorname{PU}(2,1)$ so that
$\Gamma\backslash\mathbb{B}^{2}$ is biholomorphic to $X$. Yau famously proved
that $X$ is a smooth ball quotient if and only if the canonical bundle $K_{X}$
is ample and the Chern numbers of $X$ satisfy $c_{1}^{2}(X)=3c_{2}(X)$, i.e.,
$K_{X}^{2}=3e(X)$, where $e(X)$ is the topological Euler characteristic and
$K_{X}^{2}$ is the self-intersection of $K_{X}$. Over time this result has
been generalized in many ways. This paper requires a very special case of the
main results in [26, 27].

######  Theorem 2.1.

Suppose $Y$ is a smooth projective surface, and $D_{1},\dots,D_{n}$ are
smooth, reduced, irreducible curves on $Y$. Given integers
$r_{1},\dots,r_{n}$, consider the $\mathbb{Q}$-divisor

$\mathcal{D}=\sum_{j=1}^{n}\left(1-\frac{1}{r_{j}}\right)D_{j}$

on $Y$. Then there is a lattice $\Gamma<\operatorname{PU}(2,1)$ so that the
$\Gamma\backslash\mathbb{B}^{2}$ is an orbifold with underlying analytic space
biholomorphic to $Y$ and orbifold locus $\bigcup D_{j}$ where $D_{j}$ has
orbifold weight $r_{j}$ if and only if $K_{Y}+\mathcal{D}$ is ample and the
relative Chern numbers of the pair $(Y,\mathcal{D})$ satisfy
$c_{1}^{2}(Y,\mathcal{D})=3c_{2}(Y,\mathcal{D})$.

When the conditions of Theorem 2.1 are satisfied $(Y,\mathcal{D})$ will be
called a _ball quotient pair_. The relative Chern numbers in the statement of
the theorem are simply

$\displaystyle c_{1}^{2}(Y,\mathcal{D})$
$\displaystyle=(K_{Y}+\mathcal{D})^{2}$ (1) $\displaystyle
c_{2}(Y,\mathcal{D})$ $\displaystyle=e^{\mathrm{orb}}(Y,\mathcal{D})$ (2)

where $e^{\mathrm{orb}}$ is the orbifold Euler characteristic of $Y$ with
$\mathcal{D}$ is considered as the orbifold locus with appropriate weights.
This is subtle when curves meet, since one must know local groups at crossing
points. When the curves $D_{j}$ are disjoint there are no crossing points with
local groups to determine, hence

$c_{2}(Y,\mathcal{D})=e(Y)-\sum_{j=1}^{n}\left(1-\frac{1}{r_{j}}\right)e(D_{j})$
(3)

in that special case. The primary example in §5 will have normal crossings,
but is already known to be a ball quotient, and the examples in §8 will
require the normal crossings variant in the exact same way: the local group at
a smooth point where two curves of weights $r_{j}$ and $r_{k}$ meet
transversally is simply $\mathbb{Z}/r_{j}\times\mathbb{Z}/r_{k}$.

##  3 The first example

To begin, consider the smooth plane curve $C^{\circ}$ of genus $2$ with
equation

$y^{2}=x^{5}+1$ (4)

in coordinates $(x,y)$ on $\mathbb{C}^{2}$. If $\xi$ denotes a primitive
$5^{th}$ root of unity, then

$\xi(x,y)=(\xi x,y)$ (5)

defines an order $5$ automorphism of $C^{\circ}$. The two fixed points of
$\xi$ are $a_{j}=(0,(-1)^{j})$ for $j=1,2$. Projection $(x,y)\mapsto y$
realizes $C^{\circ}$ as the five-fold Galois branched cover of $\mathbb{C}$
branched over $\pm 1$ with group $\mathbb{Z}/5=\langle\xi\rangle$.

Let $C$ denote the unique smooth projective curve birational to $C^{\circ}$,
which can be realized as the five-fold regular cover of $\mathbb{P}^{1}$
branched over $\\{\pm 1,\infty\\}$. The preimage of $\infty$ is a point
$a_{0}\in C$ so that $C\smallsetminus\\{a_{0}\\}$ is isomorphic to
$C^{\circ}$, and the action of $\xi$ extends to an action of $\mathbb{Z}/5$ on
$C$ with fixed points precisely at the three points $\\{a_{0},a_{1},a_{2}\\}$.
It is originally due to Wiman in 1895 [37], that $C$ is the unique smooth
projective curve of genus $2$ admitting an automorphism of order $5$; see [19]
for an alternate treatment through Fuchsian groups.

Set $X=C\times C$, and let $C_{j}$ be the graph of $\xi^{j}$ for $0\leq j\leq
4$. Since $\xi^{k-j}$ has the same fixed points as $\xi$ for any $j\neq k$,

$C_{j}\cap C_{k}=\\{x_{0},x_{1},x_{2}\\}$

where $x_{j}=(a_{j},a_{j})$. Let $\pi:Y\to X$ be the blowup of $X$ at
$\\{x_{0},x_{1},x_{2}\\}$, $E_{j}$ denote the exceptional divisor associated
with $x_{j}$, and $D_{j}$ be the proper transform of $C_{j}$ to $Y$. See
Figure 1 for a depiction of this simple arrangement of curves on $Y$ and [4,
§II.1] for basic properties of blowups that will be used in what follows.

$E_{1}$$E_{0}$$E_{2}$$D_{2}$$D_{1}$$D_{0}$$D_{3}$$D_{4}$ Figure 1: The
arrangement of lines on $Y$

Let $C_{h}$ and $C_{v}$ denote a general horizontal and vertical divisor of
$X$, respectively. The canonical divisor $K_{X}$ of $X$ is then
$2(C_{h}+C_{v})$ by [18, II Ex. 8.3] and the fact that the canonical divisor
on a smooth projective curve of genus $g$ has degree $2g-2$. If $D_{h}$ and
$D_{v}$ are the proper transforms of $C_{h}$ and $C_{v}$ to $Y$, then

$K_{Y}=2(D_{h}+D_{v})+\sum E_{j}.$ (6)

Note that $K_{Y}^{2}=5$. Since $C_{j}\cdot C_{h}=C_{j}\cdot C_{v}=1$ for all
$j$ and $D_{j}\cdot E_{k}=1$ for all $j,k$, it follows that $K_{Y}\cdot
D_{j}=7$, and thus $D_{j}^{2}=-5$ by adjunction [4, I.15]. These calculations
are now used to prove the following result, which provides the first of the
two examples bootstrapped to prove Theorem 1.2.

######  Theorem 3.1.

Let $Y$ be the surface and $\\{D_{j}\\}$ be the curves of genus $2$ on $Y$
described previously in this section, and consider the $\mathbb{Q}$-divisor

$\mathcal{D}=\sum_{j=0}^{4}\left(1-\frac{1}{5}\right)D_{j}$ (7)

on $Y$. Then $(Y,\mathcal{D})$ is a ball quotient pair.

###### Proof.

Define $\mathcal{L}=K_{Y}+\mathcal{D}$. To apply Theorem 2.1, the first task
is to compute the Chern numbers $c_{1}^{2}(Y,\mathcal{D})$ and
$c_{2}(Y,\mathcal{D})$. From earlier in this section, $K_{Y}^{2}=5$,
$K_{Y}\cdot D_{j}=7$, and $D_{j}^{2}=-5$ for each $0\leq j\leq 4$, hence it
follows that

$c_{1}^{2}(Y,\mathcal{D})=45.$ (8)

Since the curves $D_{j}$ are disjoint of genus $2$ and $c_{2}(Y)=7$, Equation
(3) gives

$c_{2}(Y)-\sum_{j=0}^{4}\left(1-\frac{1}{5}\right)(-2)=15.$ (9)

Thus $c_{1}^{2}(Y,\mathcal{D})=3c_{2}(Y,\mathcal{D})$ as required.

It remains to show that $\mathcal{L}$ is ample. By the Nakai–Moishezon
criterion [18, Thm. V.1.10], since $\mathcal{L}^{2}>0$ it suffices to show
that $\mathcal{L}\cdot A>0$ for every irreducible curve $A$ on $Y$. As $Y$ is
the blowup of the minimal surface $X$ of general type, $K_{X}$ is ample and so
$K_{Y}\cdot A>0$ for any irreducible curve $A$ on $Y$ that is not one of the
three exceptional curves. Since distinct curves have nonnegative intersection
number, $\mathcal{L}\cdot A>0$ when $A$ is not one of the three exceptional
curves or one of the curves $D_{j}$. It therefore suffices to show that
$\mathcal{L}\cdot A>0$ when $A$ is either an exceptional curve or one of the
curves $D_{j}$. Direct computation gives

$\mathcal{L}\cdot E_{j}=\mathcal{L}\cdot D_{j}=3$

for all relevant $j$, and this completes the proof. ∎

##  4 Arithmeticity of the example in §3

Let $(Y,\mathcal{D})$ be the ball quotient pair in Theorem 3.1 and
$\Gamma\backslash\mathbb{B}^{2}$ be the associated ball quotient orbifold.
Continuing with the notation of §3, consider the action of
$G=(\mathbb{Z}/5)^{2}$ on $X=C\times C$ given by
$(z,w)\mapsto(\xi^{r}z,\xi^{s}w)$. Since the points $x_{j}=(a_{j},a_{j})\in X$
are global fixed points of the action, where $a_{j}\in\\{\pm 1,\infty\\}$ are
the fixed points of $\xi$, the action of $G$ lifts to an action on the blowup
$Y$ stabilizing the three exceptional curves $E_{j}$. Let $Z$ be the quotient
$Y/G$.

######  Lemma 4.1.

The surface $Z$ is isomorphic to $\mathbb{P}^{2}$ blown up in the vertices of
the complete quadrangle.

###### Proof.

Recall that the complete quadrangle in $\mathbb{P}^{2}$ consists of the six
lines where either two coordinates are equal or one is equal to zero; see [25,
Fig. I(vi)] or Figure 3 below for the standard depiction of the quadrangle.
Since $C/\langle\xi\rangle$ is isomorphic to $\mathbb{P}^{1}$, $X/G$ is
isomorphic to $\mathbb{P}^{1}\times\mathbb{P}^{1}$. For each point $y$ on an
exceptional curve $E_{j}$, one checks directly in coordinates that its
stabilizer $\operatorname{Stab}_{G}(y)$ is generated by complex reflections,
hence $Y/G$ is a smooth surface by the Chevalley–Shephard–Todd theorem [6,
§V.5.5, Th. 4] and $E_{j}$ maps to a smooth curve $\widehat{E}_{j}$ on $Z$
isomorphic to $\mathbb{P}^{1}$. Moreover, contraction of
$\\{\widehat{E}_{j}\\}$ is a map $Z\to\mathbb{P}^{1}\times\mathbb{P}^{1}$, and
therefore $\widehat{E}_{j}$ is a $(-1)$-curve; see [1, §III.4]. This proves
that $Z$ is isomorphic to $\mathbb{P}^{1}\times\mathbb{P}^{1}$ blown up at
three points on the diagonal, and it is a classical exercise that this surface
is isomorphic to $\mathbb{P}^{2}$ blown up at the four vertices of the
complete quadrangle. ∎

Moreover, there is a $\mathbb{Q}$-divisor $\mathcal{E}$ on $Z$ so that
$(Z,\mathcal{E})$ is a ball quotient pair and the associated ball quotient
orbifold $\Lambda\backslash\mathbb{B}^{2}$ is isomorphic to
$(\Gamma\backslash\mathbb{B}^{2})/G$. To prove this, let
$\operatorname{Aut}_{\mathcal{D}}(Y)$ denote the automorphisms of $Y$
preserving $\mathcal{D}$, in that sense that it not only preserves the support
of $\mathcal{D}$ but also preserves the weights of divisors. By the definition
of an orbifold cover, the quotient $Y/\operatorname{Aut}_{\mathcal{D}}(Y)$ is
again a ball quotient and the induced map is an orbifold covering.

######  Lemma 4.2.

The action of $G$ on $Y$ induces an action on
$\Gamma\backslash\mathbb{B}^{2}$, hence the quotient
$(\Gamma\backslash\mathbb{B}^{2})/G$ is a ball quotient
$\Lambda\backslash\mathbb{B}^{2}$ for a lattice
$\Lambda<\operatorname{PU}(2,1)$ containing $\Gamma$ as a normal subgroup with
quotient $G$. The orbifold $\Lambda\backslash\mathbb{B}^{2}$ arises from a
ball quotient pair $(Z,\mathcal{E})$ for some $\mathbb{Q}$-divisor
$\mathcal{E}$ on $Z$.

###### Proof.

Recall that the curves $D_{j}$ on $Y$ are the proper transforms of the graphs
of $\xi^{j}$, hence are isomorphic to $C$. The action of $G$ cyclically
permutes these curves, and the stabilizer of each $D_{j}$ is isomorphic to
$\mathbb{Z}/5$ acting on $C$ by the usual action. This alone suffices to imply
that $G$ acts on $\Gamma\backslash\mathbb{B}^{2}$, since the $G$-action on $Y$
preserves the orbifold locus. All statements in the lemma are restatements of
this fact and hence follow directly. ∎

The next goal is to explicitly determine the $\mathbb{Q}$-divisor
$\mathcal{E}$ on $Z$ and hence (equivalently) describe the orbifold structure
on $Z$ as $\Lambda\backslash\mathbb{B}^{2}$. The diagonal subgroup of $G$ acts
on an exceptional curve $E_{j}$ as a complex reflection through $E_{j}$ and
the quotient of $G$ by the diagonal subgroup acts on $E_{j}$ as $y\mapsto
y^{5}$. The remaining curves stabilized by $G$ are the proper transforms
$H_{j},V_{j}$ ($j\in\\{0,1,2\\}$) of the horizontal and vertical curves on $X$
with fixed coordinate $a_{j}$. The action of $G$ on any of these curves
$H_{j},V_{j}$ is generated by a complex reflection of order $5$ and the usual
action of $\mathbb{Z}/5$ on $C$.

See Figure 2 for a depiction of the arrangement of curves on $Z$ defined by
the images of the curves $D_{j}$, $E_{j}$, $H_{j}$, and $V_{j}$. The branch
divisor for the quotient map $\pi$ is the union of the $E_{j}$, $H_{j}$, and
$V_{j}$, and the image of the curves $D_{j}$ is the proper transform to $Z$ of
the diagonal of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ with respect to the
natural blowup map $Z\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ commuting with the
action of $G$.

$\pi(H_{1})$$\pi(H_{0})$$\pi(H_{2})$$\pi(V_{2})$$\pi(V_{0})$$\pi(V_{1})$$\pi(\\{D_{j}\\})$$\pi(E_{2})$$\pi(E_{0})$$\pi(E_{1})$
Figure 2: The arrangement of curves on $Z$

To understand the ball quotient orbifold structure on $Z$, first recall that
the orbifold structure on $Y$ consists of the five curves $D_{j}$, each with
orbifold weight $5$. The curves $D_{j}$ are not in the branch locus of $\pi$,
so the image on $Z$ has orbifold weight $5$ on
$\Lambda\backslash\mathbb{B}^{2}$. The remaining curves in the arrangement are
not in the orbifold locus on $Y$, but $\pi$ branches to order $5$ along each,
so it becomes a curve of orbifold weight $5$ in the ball quotient structure on
$Z$. In other words, the orbifold locus on $Z$ consists of precisely the
curves in the arrangement in Figure 2, each with weight $5$. Said yet one more
way, if

$\mathcal{E}=\left(1-\frac{1}{5}\right)\left(\pi(\\{D_{j}\\})+\sum_{j=0}^{2}\left(\pi(E_{j})+\pi(H_{j})+\pi(V_{j})\right)\right)$

then $(Z,\mathcal{E})$ is the ball quotient pair associated with
$\Lambda\backslash\mathbb{B}^{2}$ and the action of $G$ on $Y$ induces an
orbifold covering
$\Gamma\backslash\mathbb{B}^{2}\to\Lambda\backslash\mathbb{B}^{2}$.

Using the standard identification of the arrangement in Figure 2 with the
blowup of the complete quadrangle in $\mathbb{P}^{2}$ at its vertices, this
implies that $(Z,\mathcal{E})$ can be realized as the arrangement shown in
[25, Fig. I(i)]. Factoring in the orbifold weight of each curve, this is the
Deligne–Mostow quotient with weights $(2,2,2,2,2)/5$ by [25, Thm. 4.1], which
is arithmetic [9, p. 86]. Note that $(Z,\mathcal{E})$ is associated with the
quotient of $\mathbb{B}^{2}$ by the ‘pure’ Deligne–Mostow lattice, not its
extension by the action of the symmetric group $S_{6}$ on the weights. This
completes the proof of the following result.

######  Theorem 4.3.

The ball quotient orbifold constructed in §3 is arithmetic.

##  5 The second example

Let $T$ be the elliptic curve $\mathbb{Z}[\zeta]\backslash\mathbb{C}$, where
$\zeta=e^{2\pi i/6}$, and consider the abelian surface $T\times T$. In
coordinates $(z,w)$ on $T\times T$, let $T_{j}$ be the graph of multiplication
by $j\in\mathbb{Z}[\zeta]$. As noted in [21, §1], the curves $T_{0}$,
$T_{\infty}$ (i.e., $z=0$), $T_{1}$, and $T_{\zeta}$ are embedded elliptic
curves that mutually intersect at the origin $(0,0)$ and otherwise in no other
point.

Let $Y$ be the blowup of $T\times T$ at $(0,0)$, $L$ be the exceptional curve,
$D_{j}$ denote the proper transform of $T_{j}$ to $Y$, and define

$\mathcal{D}=\\!\left(1-\frac{1}{3}\right)\\!\left(L+\sum D_{j}\right)\\!.$

The following proposition is essentially equivalent to the result proved in
[21, §2], and hence covers the case $g_{1}=g_{2}=1$ in Theorem 1.2. A proof
based on §2 is provided for completeness.

######  Proposition 5.1.

The pair $(Y,\mathcal{D})$ defined in this section is a ball quotient pair.

###### Proof.

Note that $K_{Y}$ is numerically equivalent to the exceptional curve $L$
(e.g., see [4, §II.1]). Since the $D_{j}$ are disjoint, each $D_{j}$ has self-
intersection $-1$, and $L\cdot D_{j}=1$ for all $j$ it follows that:

$\displaystyle c_{1}^{2}(Y,\mathcal{D})$ $\displaystyle=(L+\mathcal{D})^{2}$
$\displaystyle=\\!\left(\frac{5}{3}L+\frac{2}{3}\sum D_{j}\right)^{2}$
$\displaystyle=\frac{13}{3}$ $\displaystyle c_{2}(Y,\mathcal{D})$
$\displaystyle=e(Y)-\left(1-\frac{1}{3}\right)e\\!\left(L\smallsetminus\\{4\textrm{pts}\\}\right)$
$\displaystyle\quad-\sum\left(1-\frac{1}{3}\right)e\\!\left(D_{j}\smallsetminus\\{1\textrm{pt}\\}\right)-\left(1-\frac{1}{9}\right)e\\!\left(\\{4\textrm{pts}\\}\right)$
$\displaystyle=1-\frac{2}{3}(-2)-\frac{8}{3}(-1)-\frac{32}{9}$
$\displaystyle=\frac{13}{9}$

Thus $c_{1}^{2}(Y,\mathcal{D})=3c_{2}(Y,\mathcal{D})$.

It remains to prove that $\mathcal{L}=K_{Y}+\mathcal{D}$ is ample, which
follows the same script as the proof of Theorem 3.1. Notice that $\mathcal{L}$
is an effective linear combination of $L$ and the $D_{j}$. Any curve on $Y$
not $L$ or some $D_{j}$ meets some curve in the support of $\mathcal{L}$.
Indeed, suppose that $C$ does not meet $D_{0}$ or $D_{\infty}$. Then the image
of $C$ in $T\times T$ must meet either $C_{0}$ or $C_{\infty}$, which means
that $C$ meets $L$. It follows that any such $C$ meets some curve in the
support of $\mathcal{D}$, and thus $\mathcal{L}\cdot C>0$. Finally, the
computations

$\displaystyle L\cdot\mathcal{L}$
$\displaystyle=\frac{5}{3}(-1)+\frac{8}{3}=1$ $\displaystyle
D_{j}\cdot\mathcal{L}$ $\displaystyle=\frac{5}{3}+\frac{2}{3}(-1)=1$

then suffice by the Nakai–Moishezon criterion [18, Thm. V.1.10] to imply that
$\mathcal{L}$ is ample. Thus $(Y,\mathcal{D})$ is a ball quotient pair. ∎

######  Remark 5.2.

The manifold $W_{3}$ in [21, §2] is a degree $3^{8}$ cover of the orbifold
given by Proposition 5.1.

######  Remark 5.3.

The examples provided by Theorem 3.1 and Proposition 5.1 are closely related.
Indeed, both are constructed using the unique smooth projective curve of genus
$g\in\\{2,1\\}$ admitting an automorphism of order $2g+1$; in fact, both admit
one of order $4g+2$, and they are the unique curves of their genus achieving
this maximum. A systematic search found no obvious analogous higher genus ball
quotient pair.

##  6 Arithmeticity of the example in §5

Let $(Y,\mathcal{D})$ be as in §5, $\Gamma\backslash\mathbb{B}^{2}$ be the
associated ball quotient, and retain all other notation from that section. As
in §4, $\Gamma$ is shown to be arithmetic is by proving that a certain
quotient of $\Gamma\backslash\mathbb{B}^{2}$ is an arithmetic Deligne–Mostow
orbifold. The argument this time is somewhat more complicated, since the
quotient is a singular rational surface that is not as simple to identify.
While there may be an identification of the quotient relying on more classical
techniques, this section uses an auxiliary ball quotient, the noncompact
manifold constructed by Hirzebruch in the same paper [21, p. 356], which
admits an action by the same finite group. This argument has the added bonus
of giving a natural explanation for a certain group of order $72$, previously
found only computationally, relating Hirzebruch’s noncompact example to a
Picard modular surface that is also a Deligne–Mostow orbifold [35, Appendix].

Let $\mu_{6}=\langle\zeta\rangle$ be the group of $6^{th}$ roots of unity, and
$F$ be the subgroup of $\operatorname{GL}_{2}(\mu_{6})$ generated by:

$\displaystyle\alpha$ $\displaystyle=\begin{pmatrix}1&\zeta^{2}\\\
1&-1\end{pmatrix}$ $\displaystyle\beta$ $\displaystyle=\begin{pmatrix}1&0\\\
1&\zeta^{2}\end{pmatrix}$

There is a natural action of $F$ on $T\times T$ fixing the origin, and hence
an action on the blowup $Y$. The action of $\alpha$ on the curves $T_{j}$ is:

$\displaystyle\alpha(z,0)$ $\displaystyle=(z,z)$ $\displaystyle\alpha(z,z)$
$\displaystyle=(-\zeta z,0)$ $\displaystyle\alpha(0,z)$
$\displaystyle=(\zeta^{2}z,-z)$ $\displaystyle\alpha(z,\zeta z)$
$\displaystyle=(0,-\zeta^{2}z)$

Noting that $(\zeta^{2}z,-z)=(w,\zeta w)$ for $w=\zeta^{2}z$, $\alpha$
permutes the $T_{j}$ by the permutation $(1\,3)(2\,4)$ for the ordering
$\\{T_{0},T_{\infty},T_{1},T_{\zeta}\\}$. Moreover, $\alpha$ has order $12$
with

$\alpha^{2}=\begin{pmatrix}\zeta&0\\\
0&\zeta\end{pmatrix}=\zeta\operatorname{Id}$

central in $F$. Further calculations show that $\beta$ acts on the $T_{j}$ as
$(1\,3\,4)$, $\beta^{3}$ is the identity on $T\times T$, $(\alpha\beta)$ acts
as $(2\,4\,3)$ on the $\\{T_{j}\\}$, and
$(\alpha\beta)^{3}=-\operatorname{Id}$ on $T\times T$. Therefore the
restriction to permutations of the $\\{T_{j}\\}$ defines a homomorphism of $F$
onto the alternating group $A_{4}$ with kernel $\zeta\operatorname{Id}$.

######  Remark 6.1.

This will not be used in what follows, but one can deduce an abstract
presentation

$F=\\!\left\langle\alpha,\beta\ |\
\alpha^{12},\beta^{3},[\alpha^{2},\beta],(\alpha\beta)^{3}=\alpha^{6}\right\rangle$

with $\langle\alpha^{2}\rangle\cong\mathbb{Z}/6$ central and
$F/\langle\alpha^{2}\rangle\cong A_{4}$. In other words, $F$ is a central
extension of $A_{4}$ by $\mathbb{Z}/6$.

The induced action of $F$ on $Y$ also permutes the proper transforms $D_{j}$
of the curves $T_{j}$ and the action on the exceptional curve $L$ is by the
linear action of the given matrix on lines through the origin of $T\times T$.
Exactly as in Lemma 4.2, the action of $F$ on $Y$ preserves the orbifold
structure for the pair $(Y,\mathcal{D})$, hence
$F\leq\operatorname{Aut}_{\mathcal{D}}(Y)$ and the following lemma holds.

######  Lemma 6.2.

The action of $F$ on $Y$ induces an action on
$\Gamma\backslash\mathbb{B}^{2}$. Therefore there is a lattice
$\Lambda<\operatorname{PU}(2,1)$ containing $\Gamma$ as a normal subgroup of
index $|F|=72$ so that
$\Lambda\backslash\mathbb{B}^{2}=(\Gamma\backslash\mathbb{B}^{2})/F$.

The goal is now to determine the space $Y/F$ and prove that $\Lambda$ is an
arithmetic Deligne–Mostow lattice. The first step is to show that the quotient
is isomorphic to a scheme-theoretic blowup of the weighted projective plane
$\mathbb{P}(1,2,3)$ by first proving that $(Y\smallsetminus\bigcup D_{j})/F$
is isomorphic to the complement of a certain point in $\mathbb{P}(1,2,3)$.
This open variety is the underlying space for a _different_ Deligne–Mostow
orbifold.

Consider the divisor $\mathcal{D}_{\infty}=\sum D_{j}$ on $Y$, which fits into
§2 in the more general setting of weight $r_{j}=\infty$ that allows for cusps.
It is now well-established in the literature that $(Y,\mathcal{D}_{\infty})$
is a ball quotient pair in this sense, i.e., that
$Y^{\circ}=Y\smallsetminus\bigcup D_{j}$ is the quotient
$\Delta\backslash\mathbb{B}^{2}$ of $\mathbb{B}^{2}$ by an arithmetic lattice
$\Delta<\operatorname{PU}(2,1)$. For example, see Example 1 in §6 and §7 of
[15]. The exact same logic that was applied to $Y$ and
$\Gamma\backslash\mathbb{B}^{2}$ gives the following lemma.

######  Lemma 6.3.

The action of $F$ on $Y^{\circ}$ induces an action on
$\Delta\backslash\mathbb{B}^{2}$. Therefore there is a lattice
$\widehat{\Delta}<\operatorname{PU}(2,1)$ containing $\Delta$ as a normal
subgroup of index $|F|=72$ so that
$\widehat{\Delta}\backslash\mathbb{B}^{2}=(\Delta\backslash\mathbb{B}^{2})/F$.

The identification of $Y^{\circ}/F$ with $\mathbb{P}(1,2,3)$ minus a point
will come from the following proposition, which identifies the group
$\widehat{\Delta}$.

######  Proposition 6.4.

The group $\widehat{\Delta}$ in Lemma 6.3 is the Deligne–Mostow lattice with
weights $(5,4,1,1,1)/6$.

###### Proof.

It is known that $\Delta$ is commensurable with the Picard modular group
$\operatorname{PU}(2,1;\mathbb{Z}[\zeta])$ [15, §7]. A direct calculation
shows that $\Delta\backslash\mathbb{B}^{2}$ has Euler characteristic $1$,
hence $\widehat{\Delta}\backslash\mathbb{B}^{2}$ has orbifold Euler
characteristic $1/72$, which is the smallest possible orbifold Euler
characteristic of a noncompact arithmetic quotient of $\mathbb{B}^{2}$ by [34,
Thm. 1.1] and Chern–Gauss–Bonnet. It is already known that $\Gamma$ is a
normal subgroup with index $72$ in the Deligne–Mostow lattice with weights
$(5,4,1,1,1)/6$ [35, Appendix], and this must be $\widehat{\Delta}$ by Mostow
rigidity. ∎

######  Remark 6.5.

The weights $(5,4,1,1,1)/6$ only satisfy condition $\Sigma\mathrm{INT}$ of
Deligne and Mostow, hence there is no ambiguity of ‘pure’ lattice versus the
extension by the $S_{3}$ action on weights.

$z_{235}$$z_{245}$$z_{234}$$S_{3}$$y_{\infty}$$\times$$\mathrm{A}_{2}$$\times$$\mathrm{A}_{1}$
Figure 3: The map
$(\mathbb{P}^{2}\smallsetminus\\{z_{234},z_{235},z_{245}\\})\to(\mathbb{P}(1,2,3)\smallsetminus\\{y_{\infty}\\})$

######  Corollary 6.6.

The quotient $Y^{\circ}/F$ is the complement of the point $y_{\infty}$ in
$\mathbb{P}(1,2,3)$ indicated in Figure 3.

###### Proof.

Following Theorem 4.1(vi), (5.1)(vi), and Figure I(vi) in [25],
$\widehat{\Delta}\backslash\mathbb{B}^{2}$ has underlying space the quotient
$(\mathbb{P}^{2}\smallsetminus\\{z_{234},z_{235},z_{245}\\})/S_{3}$, where
$z_{ijk}$ is the vertex of the complete quadrangle in $\mathbb{P}^{2}$
indicated in Figure 3 and $S_{3}$ acts by coordinate permutations. The
quotient of $\mathbb{P}^{2}$ by $S_{3}$ is classically known to be
$\mathbb{P}(1,2,3)$ (e.g., see [10, Ch. 11]), where the image of the complete
quadrangle and the singularities of type $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$
are depicted in Figure 3. This proves the lemma. ∎

The above information is now used to prove the following result.

######  Proposition 6.7.

The space $\Lambda\backslash\mathbb{B}^{2}$ in Lemma 6.2 is the Deligne–Mostow
quotient with weights $(14,13,3,3,3)/18$, which has underlying analytic space
the scheme theoretic blowup of $\mathbb{P}(1,2,3)$ at $y_{\infty}$.

###### Proof.

The complement in $\Gamma\backslash\mathbb{B}^{2}$ of the four curves $D_{j}$
in the orbifold locus is isomorphic to $Y^{\circ}$ as a complex manifold. The
action of $F$ on the $D_{j}$ is transitive, so they map to a single curve
$\widehat{D}$ in the orbifold $\Lambda\backslash\mathbb{B}^{2}$ in Lemma 6.2.
By Corollary 6.6, the quotient

$\left(\Gamma\backslash\mathbb{B}^{2}\smallsetminus\bigcup D_{j}\right)\\!/F$

is isomorphic to $\mathbb{P}(1,2,3)\smallsetminus\\{y_{\infty}\\}$, and hence
$\Lambda\backslash\mathbb{B}^{2}$ has underlying space obtained by replacing
$y_{\infty}$ with the curve $\widehat{D}$. Thus must be made precise.

The stabilizer $F_{j}$ in $F$ of a curve $D_{j}$ is isomorphic to
$\mathbb{Z}/6\times\mathbb{Z}/3$ generated by the center
$\langle\zeta\operatorname{Id}\rangle$ along with a lift of the appropriate
$3$-cycle in $\mathrm{A}_{4}$. That $3$-cycle acts on $D_{j}$ by $\zeta^{2}$,
so there is a $\mathbb{Z}/3$ subgroup $R_{j}<F_{j}$ that acts trivially on
$D_{j}$, and $F_{j}/R_{j}$ is the usual action of $\zeta$ on $D_{j}$. In other
words, the image of $D_{j}$ on $\Lambda\backslash\mathbb{B}^{2}$ is isomorphic
to $T/\langle\zeta\rangle\cong\mathbb{P}^{1}$. The orbifold weight of $D_{j}$
on $\Gamma\backslash\mathbb{B}^{2}$ is $3$, and $F_{j}$ has subgroup of order
$3$ acting as a complex reflection through $D_{j}$, so $\widehat{D}$ has
orbifold weight $9$ on $\Lambda\backslash\mathbb{B}^{2}$.

The only fixed point of $\zeta$ on $T$ is the origin, which on $D_{j}$ is its
point of intersection with the exceptional curve $L$. The action of $\zeta$ on
$L$ is trivial, hence $\zeta$ locally acts as a complex reflection at
$D_{j}\cap L$, and thus this fixed point projects to a smooth point of
$\Lambda\backslash\mathbb{B}^{2}$ by the Chevalley–Shephard–Todd theorem [6,
§V.5.5, Th. 4]. The action of $-1=\zeta^{3}$ on $D_{j}$ has three additional
fixed points at nontrivial $2$-torsion elements of $T$, where the action on
$Y$ is locally by $-\operatorname{Id}$. These three points are permuted by the
action of $\zeta^{2}$, hence they determine a single $\mathrm{A}_{1}$
singularity on $Y/F$ that lies on $\widehat{D}$. The two additional fixed
points of $\zeta^{2}$, which are nontrivial $3$-torsion points identified by
the action of $\zeta^{3}$, have stabilizer $(\mathbb{Z}/3)^{2}<F_{j}$
generated by complex reflections, hence they descend to a smooth point of
$Y/F$.

In conclusion, $\widehat{D}$ is isomorphic to $\mathbb{P}^{1}$, passes through
an $\mathrm{A}_{1}$ singularity of $Y/F$, and has orbifold weight $9$ for the
ball quotient structure on $\Lambda\backslash\mathbb{B}^{2}$. Contraction of
$\widehat{D}$ is a map to $\mathbb{P}(1,2,3)$. The universal property of
blowing up [18, Prop. II.7.14] implies that this map factors through the
scheme theoretic blowup of $\mathbb{P}(1,2,3)$ at $y_{\infty}$, but
$\widehat{D}$ is reduced and irreducible, hence $Y/F$ must be isomorphic to
the blowup. See Figure 4 and compare with ${}_{2}R$ in the figure on page 113
of [10]. It remains to determine the rest of the orbifold locus of
$\Lambda\backslash\mathbb{B}^{2}$, i.e., the image on $Y/F$ of the orbifold
locus of $\Gamma\backslash\mathbb{B}^{2}$ and any additional ramification of
the projection.

$18$$3$$9$$\times$$\mathrm{A}_{2}$$\times$$\mathrm{A}_{1}$$\times$$\mathrm{A}_{1}$
Figure 4: The quotient $Y/F$ with orbifold weights

The only other curve in the orbifold locus of $\Gamma\backslash\mathbb{B}^{2}$
is the exceptional curve $L$, which has orbifold weight $3$ and is stabilized
by $F$. The central $\langle\zeta\operatorname{Id}\rangle<F$ acts by a complex
reflection through $L$, since scalar matrices act trivial on lines through the
origin. The action of $F/\langle\zeta\operatorname{Id}\rangle\cong A_{4}$ is
by the standard action of $A_{4}$ on $\mathbb{P}^{1}$. Thus $L$ projects to a
smooth rational curve on $L/F$ with orbifold weight $18$. One can see either
through analysis of the action of $F$ or by noticing that $L$ maps to the
curve of orbifold weight $6$ for the interpretation of $Y^{\circ}/F$ as the
$(5,4,1,1,1)/6$ Deligne–Mostow orbifold that the image of $L$ is the curve in
Figure 4 passing through the second $\mathrm{A}_{1}$ singularity of $Y/F$.

Finally, any additional ramification of $Y\to Y/F$ arises from complex
reflections in $F$. Every element of $F$ is conjugate to $\zeta^{r}\alpha$ or
$\zeta^{r}\beta$ for some $r$, thus the only complex reflections in $F$ are
the conjugates of $\beta$. Computing the eigenvectors of $\beta$, the curve
$T_{\beta}$ on $T\times T$ fixed by $\beta$ is the embedded $T$ with
parametrization

$z\longmapsto\\!\left((1-\zeta^{2})z,z\right)\\!.$

The full stabilizer of $T_{\beta}$ in $F$ also contains the center, which
again acts on $T_{\beta}$ as multiplication by $\zeta$. Either similar
analysis as with $\widehat{D}$ and $L$ or analogy with the $(5,4,1,1,1)/6$
Deligne–Mostow orbifold implies that the image of $T_{\beta}$ on $Y/F$ is the
singular curve in Figure 4 with orbifold weight $3$ on
$\Lambda\backslash\mathbb{B}^{2}$.

Therefore Figure 4 depicts all the orbifold locus on
$\Lambda\backslash\mathbb{B}^{2}$. Following [10, Ch. 11] and [25, Thm.
4.1(ii)], this is precisely the Deligne–Mostow orbifold with weights
$(14,13,3,3,3)/18$. This completes the proof of the proposition. ∎

Since the Deligne–Mostow orbifold with weights $(14,13,3,3,3)/18$ is
arithmetic [31, p. 102], the main result of this section now follows
immediately from Proposition 6.7.

######  Theorem 6.8.

The ball quotient orbifold constructed in §5 is arithmetic.

##  7 Remaining cases for Theorem 1.2

The results in §3 and §5 provide the cases $g_{1}=g_{2}=2$ and $g_{1}=g_{2}=1$
in the proof of Theorem 1.2, respectively. The case $g_{1}=g_{2}=0$ is handled
by Deligne–Mostow orbifolds (e.g., see the proof of Theorem 4.3). This section
bootstraps from those examples to complete the proof of Theorem 1.2.

######  Proposition 7.1.

Given $g_{1},g_{2}\geq 2$, there exist smooth projective curves $C_{j}$ of
genus $g_{j}$ so that $C_{1}\times C_{2}$ is birational to a ball quotient.

###### Proof.

Let $\Gamma<\operatorname{PU}(2,1)$ be the lattice determined by the example
in Theorem 3.1 and retain all notation from §3. Forgetting the orbifold
structure on $\Gamma\backslash\mathbb{B}^{2}$ induces an exact sequence

$1\longrightarrow
R\longrightarrow\Gamma\longrightarrow\pi_{1}(Y)\cong\pi_{1}(C)\times\pi_{1}(C)\longrightarrow
1$

where $R$ is the normal closure in $\Gamma$ of the torsion elements associated
with the orbifold locus.

Genus two curves admit étale covers by a curve of any given genus greater than
two, so given $g_{1},g_{2}\geq 2$ there exist curves $C_{j}$ of genus $g_{j}$
and étale covers $C_{j}\to C$. In particular, there is an étale cover
$C_{1}\times C_{2}\to C\times C$. This determines a finite index subgroup
$\Delta^{\prime}<\pi_{1}(Y)$ and a cover $Y^{\prime}\to Y$ with $Y^{\prime}$
birational to $C_{1}\times C_{2}$. The pullback of $\Delta^{\prime}$ to
$\Gamma$ under the surjection is a finite index subgroup $\Gamma^{\prime}$
with the property that $\Gamma^{\prime}\backslash\mathbb{B}^{2}$ is an
orbifold with underlying analytic space $Y^{\prime}$. This proves the
proposition. ∎

######  Remark 7.2.

The surface birational to $C_{1}\times C_{2}$ that is biholomorphic to a ball
quotient in the proof of Proposition 7.1 is the blowup of $C_{1}\times C_{2}$
at $3(g_{1}-1)(g_{2}-1)$ points.

The next proposition handles higher-genus ruled surfaces in a similar fashion
to Proposition 7.3, again building off of the example from §3.

######  Proposition 7.3.

Given $g_{2}\geq 2$, there exists a smooth projective curve $C_{2}$ of genus
$g_{2}$ so that $\mathbb{P}^{1}\times C_{2}$ is birational to a ball quotient.

###### Proof.

Let $C\times C$ be as in Theorem 3.1, and consider the action of
$\mathbb{Z}/5$ on the first coordinate by $\xi$. The quotient
$C/\langle\xi\rangle$ is isomorphic to $\mathbb{P}^{1}$. Since $\xi$ fixes the
three points $a_{j}\in\\{\pm 1,\infty\\}$, it fixes the points
$x_{j}=(a_{j},a_{j})$ on the product. This implies that there is an induced
action of $\xi$ on the blowup surface $Y$. Moreover, $\xi$ cyclically permutes
the graphs $C_{j}$ of $\xi^{j}$, and thus the action on $Y$ permutes the
curves $D_{j}$ that form the orbifold locus of
$\Gamma\backslash\mathbb{B}^{2}$ for the ball quotient structure on $Y$.

If $\operatorname{Aut}_{\mathcal{D}}(Y)$ denotes automorphisms of $Y$
preserving $\mathcal{D}$, recall from the discussion around Lemma 4.2 that the
quotient $Y/\operatorname{Aut}_{\mathcal{D}}(Y)$ is again a ball quotient.
This implies that the action of $\xi$ induces an action of $\mathbb{Z}/5$ on
$\Gamma\backslash\mathbb{B}^{2}$, meaning that there is a lattice
$\Gamma_{\xi}<\operatorname{PU}(2,1)$ containing $\Gamma$ as a normal subgroup
of index $5$ so that the orbifold covering
$\Gamma\backslash\mathbb{B}^{2}\to\Gamma_{\xi}\backslash\mathbb{B}^{2}$ is
induced by this action.

Now, the complement $Y^{\circ}=Y\smallsetminus\bigcup E_{j}$ of the
exceptional curves in $Y$ is isomorphic to $C\times C$ minus the three points
$x_{j}$. Thus $Y^{\circ}/(\mathbb{Z}/5)$ is isomorphic to
$\mathbb{P}^{1}\times C=(C\times C)/\langle\xi\rangle$ minus the three points
$(\infty,a_{0})$, $(1,a_{1})$, and $(-1,a_{2})$. In other words, the
underlying orbifold for $\Gamma_{\xi}\backslash\mathbb{B}^{2}$ is birational
to $\mathbb{P}^{1}\times C$. This proves the proposition for the case
$g_{2}=2$. The case $g_{2}>2$ is now proved using étale covers of $C$ exactly
as in Proposition 7.1. ∎

######  Remark 7.4.

The surfaces in the proof of Proposition 7.3 biholomorphic to a ball quotient
are singular surfaces birational to a smooth ruled surface. On the other hand,
it follows from the proof that for all $g\geq 2$ there is a curve $C_{g}$ of
genus $g$ such that the smooth surface $\mathbb{P}^{1}\times C_{g}$ is
holomorphically dominated by a ball quotient.

The following lemma will be used to construct examples for the case $g_{1}=1$
and $g_{2}\geq 2$.

######  Lemma 7.5.

The curve $C$ from §3 has an étale cover $C^{\prime}$ that admits a
holomorphic mapping onto an elliptic curve $E^{\prime}$.

###### Proof.

The fact that $C$ is the cyclic cover of $\mathbb{P}^{1}$ branched over three
points to order five means that the uniformization of $C$ as a hyperbolic
$2$-manifold $\Sigma\backslash\mathbb{H}^{2}$ realizes $\Sigma$ as an index
five subgroup of the triangle group $\Delta(5,5,5)$, which itself is an index
six subgroup of $\Delta(2,3,10)$. From the presentation

$\Delta(2,3,10)=\\!\left\langle a,b\ |\ a^{2},b^{3},(ab)^{10}\right\rangle$

one can show that the subgroup $\Lambda$ generated by the words
$(ab)^{2}ab^{-1}aba$ and $abab^{-1}a(ba)^{2}$ has index ten with quotient
$\Lambda\backslash\mathbb{H}^{2}$ having genus zero and one cone point of
order three. Thus the quotient is isomorphic to an elliptic curve
$E^{\prime}$. Letting $C^{\prime}$ be the curve with fundamental group
$\Sigma\cap\Lambda$ that completes the diagram

${(\Sigma\cap\Lambda)\backslash\mathbb{H}^{2}=C^{\prime}}$${\Sigma\backslash\mathbb{H}^{2}=C}$${\Lambda\backslash\mathbb{H}^{2}\cong
E^{\prime}}$${\Delta(5,5,5)\backslash\mathbb{H}^{2}\cong\mathbb{P}^{1}}$${\Delta(2,3,10)\backslash\mathbb{H}^{2}\cong\mathbb{P}^{1}}$

gives the desired cover. ∎

######  Proposition 7.6.

Let $C$ be the curve with genus $2$ from §3 and $C^{\prime}\to E^{\prime}$ be
the map from an étale cover of $C$ to an elliptic curve $E^{\prime}$ provided
by Lemma 7.5. For all $g\geq 2$ there is an étale cover $C_{g}$ of $C$ and a
compact ball quotient $\Gamma_{g}\backslash\mathbb{B}^{2}$ that admits a
dominant rational map onto $E^{\prime}\times C_{g}$.

###### Proof.

From the proof of Proposition 7.1, there is a ball quotient
$\Gamma_{g}\backslash\mathbb{B}^{2}$ with underlying analytic space isomorphic
to the blowup of $C^{\prime}\times C_{g}$ at some collection of points, where
$C_{g}$ is any étale cover of $C$ with genus $g$. Contracting the exceptional
curves an applying the mapping $C^{\prime}\to E^{\prime}$ on the first
coordinate proves the proposition. ∎

The remaining example required to prove Theorem 1.2 is constructed using the
example from §5.

######  Proposition 7.7.

There is a compact ball quotient $\Gamma^{\prime}\backslash\mathbb{B}^{2}$
whose underlying analytic space is biholomorphic to a ruled surface of genus
one.

###### Proof.

The proof will employ the example and notation from §6. In particular, let
$\Gamma\backslash\mathbb{B}^{2}$ be the example from §5 and consider the
automorphism

$\beta=\begin{pmatrix}1&0\\\ 1&\zeta^{2}\end{pmatrix}$

of $T\times T$ with order $3$. This acts on $T\times T$ as a complex
reflection stabilizing the curve $T_{\beta}$ parametrized by

$z\longmapsto\\!\left((1-\zeta^{2})z\,,\,z\right)\\!.$

The induced action on the blowup $Y$ is again by a complex reflection of order
$3$ (the multiplicative order of $\zeta^{2}$) through the proper transform
$D_{\beta}$ of $T_{\beta}$ to $Y$.

Note that $\beta$ acts trivially on the first factor of $T\times T$ and for a
fixed $z_{0}$ acts on the vertical curve $(z_{0},w)$ by
$w\mapsto\zeta(z_{0}-w)$, which has quotient $\mathbb{P}^{1}$. Thus $(T\times
T)/\langle\beta\rangle$ is simply $T\times\mathbb{P}^{1}$, and this realizes
$T\times T$ as the $3$-fold branched cover of $T\times\mathbb{P}^{1}$ branched
over the graph of multiplication by $\zeta^{2}$. It was shown in §6 that
$\beta$ acts on $\Gamma\backslash\mathbb{B}^{2}$, and it follows exactly as in
the proof of Proposition 7.3 that
$(\Gamma\backslash\mathbb{B}^{2})/\langle\beta\rangle$ is a ball quotient with
underlying analytic space birational to $\mathbb{P}^{1}\times T$. This proves
the proposition. ∎

The results in this section now prove Theorem 1.2.

###### Proof of Theorem 1.2.

Combine Propositions 7.1, 7.3, 7.6, and 7.7. ∎

##  8 Surfaces with $\kappa(V)=0$

The goal of this section is to prove Theorem 1.4. By Corollary 1.3, it
suffices to consider surfaces of Kodaira dimension $0$. The case of an abelian
surface is covered by the example in §5. By [4, Thm. VIII.2], the remaining
smooth projective surfaces of Kodaira dimension zero are:

  * $\star$

K3 surfaces, which define one connected deformation space [1, Thm. 22.3]

  * $\star$

Enriques surfaces, which form one connected deformation space [1, Thm. 18.5]

  * $\star$

bielliptic surfaces, which fall into seven connected deformation spaces [4,
List VI.20], [36, Rem. 2]

Bielliptic surfaces are surfaces of the form $(E_{\lambda}\times E_{\tau})/G$,
where $E_{\lambda}$ and $E_{\tau}$ are elliptic curves and $G$ is a finite
group acting freely on the product. By the Bagnera–de Franchis theorem [4,
List VI.20],

$G\in\\!\left\\{\mathbb{Z}/2,(\mathbb{Z}/2)^{2},\mathbb{Z}/4,\mathbb{Z}/4\oplus\mathbb{Z}/2,\mathbb{Z}/3,(\mathbb{Z}/3)^{2},\mathbb{Z}/6\right\\}\\!,$

where each group determines exactly one connected deformation space of
surfaces [36, Rem. 2]. As described in more detail below, each deformation
space of surfaces of Kodaira dimension zero contains an element somehow
closely related to a product of elliptic curves, and this connection will be
used to prove Theorem 1.4.

A second abelian example is required to cover all bielliptic surfaces. This is
a compact variant on a noncompact example due to Holzapfel [22] also
considered in [16]. The example presented here is analogous to the example in
§5 alongside Hirzebruch’s noncompact ball quotient from the same paper (which
appears in the proof of Proposition 6.7) based on the abelian surface defined
using the Eisenstein integers. Consider the elliptic curve
$E=\mathbb{C}/\mathbb{Z}[i]$ defined by the Gaussian integers and the curves

$\displaystyle C_{0}$ $\displaystyle=(z,z)$ $\displaystyle H_{0}$
$\displaystyle=\\!\left(\\!z\,,\,\frac{1}{2}\\!\right)$ $\displaystyle C_{1}$
$\displaystyle=(z,-z)$ $\displaystyle H_{1}$
$\displaystyle=\\!\left(\\!z\,,\,\frac{i}{2}\\!\right)$ $\displaystyle C_{2}$
$\displaystyle=(z,iz)$ $\displaystyle V_{0}$
$\displaystyle=\\!\left(\\!\frac{1}{2}\,,\,z\\!\right)$ $\displaystyle C_{3}$
$\displaystyle=(z,-iz)$ $\displaystyle V_{1}$
$\displaystyle=\\!\left(\\!\frac{i}{2}\,,\,z\\!\right)$

on $E\times E$. These eight curves intersect in six points, with exactly four
of the curves passing through each point and exactly four intersection points
on each $C_{j}$ and two on each $H_{j}$ and $V_{j}$. See [16, §3] for the
coordinates of the intersection points.

$D_{2}$$D_{3}$$D_{1}$$D_{0}$$A_{0}$$B_{0}$$A_{1}$$B_{1}$$E_{0}$$E_{4}$$E_{3}$$E_{2}$$E_{5}$$E_{6}$
Figure 5: The blown-up Gaussian abelian surface

Let $Y$ be the blowup of $E\times E$ at these six points, $D_{j}$ (resp.
$A_{j}$, $B_{j}$) be the proper transform of $C_{j}$ (resp. $H_{j}$, $V_{j}$)
and $E_{j}$ be the exceptional curves, ordered as indicated in Figure 5. Then
consider the $\mathbb{Q}$-divisor

$\mathcal{D}=\\!\left(1-\frac{1}{2}\right)\\!\sum
E_{j}+\\!\left(1-\frac{1}{4}\right)\\!\left(\sum A_{j}+\sum B_{j}+\sum
D_{j}\right)$ (10)

on $Y$. Note that the curves $A_{j}$, $B_{j}$, and $D_{j}$ are all disjoint,
as are the six exceptional curves, and each $E_{j}$ meets exactly four curves.
Then

$\displaystyle c_{1}^{2}(Y,\mathcal{D})$
$\displaystyle=6(-1)\\!\left(2-\frac{1}{2}\right)^{2}\\!$
$\displaystyle+4(-2)\\!\left(1-\frac{1}{4}\right)^{2}\\!+4(-4)\\!\left(1-\frac{1}{4}\right)^{2}\\!$
$\displaystyle+2(6)(4)\\!\left(2-\frac{1}{2}\right)\\!\left(1-\frac{1}{4}\right)$
$\displaystyle=27$

since $K_{Y}$ is numerically equivalent to $\sum E_{j}$, $E_{j}^{2}=-1$,
$A_{j}^{2}=B_{j}^{2}=-2$, and $D_{j}^{2}=-4$. As for $c_{2}(Y,\mathcal{D})$,
$c_{2}(Y)=6$ and every normal crossing of two curves has local group
$\mathbb{Z}/4\times\mathbb{Z}/2$, thus weight $8$, and so

$\displaystyle c_{2}(Y,\mathcal{D})$ $\displaystyle=6$
$\displaystyle-6(-2)\\!\left(1-\frac{1}{2}\right)\\!-4(-2)\\!\left(1-\frac{1}{4}\right)$
$\displaystyle-4(-4)\\!\left(1-\frac{1}{4}\right)\\!-24\\!\left(1-\frac{1}{8}\right)$
$\displaystyle=9$

and hence $c_{1}^{2}(Y,\mathcal{D})=3c_{2}(Y,\mathcal{D})$. The proof that
$K_{Y}+\mathcal{D}$ is ample is exactly the same as in §5: one only needs to
check that $K_{Y}+\mathcal{D}$ pairs positively with the curves in the
arrangement under the intersection pairing. This proves the following result.

######  Proposition 8.1.

Let $Y$ be the blown up abelian surface defined in this section and
$\mathcal{D}$ the $\mathbb{Q}$-divisor on $Y$ defined in Equation (10). Then
$(Y,\mathcal{D})$ is a ball quotient pair.

######  Remark 8.2.

Giving each curve in the support of $\mathcal{D}$ orbifold weight $3$ also
defines a ball quotient pair. This alternate example is actually
nonarithmetic; see Remark 9.2 below.

The example from Proposition 8.1 and the example from §5 will now be used to
cover to other surfaces of Kodaira dimension zero and prove Theorem 1.4.

###### Proof of Theorem 1.4.

The proof proceeds by constructing an example from each deformation space.
Note that the abelian surface case is handled by §5 and Proposition 8.1.

First, let $(Y,\mathcal{D})$ be the ball quotient pair from Proposition 8.1
and $\Gamma\backslash\mathbb{B}^{2}$ the associated ball quotient orbifold.
This will be used to produce a K3 surface, an Enriques surface, and all but
three types of bielliptic surfaces.

_K3 and Enriques surfaces._ The union of the curves $C_{j}$, $H_{j}$, and
$V_{j}$ on $E\times E$ used to define $\mathcal{D}$ is stable under
multiplication by $i\operatorname{Id}$ in such a way that orbifold weights are
preserved, which implies that it induces an action of $\mathbb{Z}/4$ on
$\Gamma\backslash\mathbb{B}^{2}$. If $\Delta$ is the associated lattice in
$\operatorname{PU}(2,1)$ containing $\Gamma$ as an index four subgroup, then
$\Delta\backslash\mathbb{B}^{2}$ is birational to $(E\times E)/\langle
i\operatorname{Id}\rangle$. Let $\Lambda<\Delta$ be the intermediate subgroup
associated with $\langle-\operatorname{Id}\rangle$. Then $(E\times
E)/\langle-\operatorname{Id}\rangle$ is a Kummer variety [1, V.16], and thus
$\Lambda\backslash\mathbb{B}^{2}$ is birational to a K3 surface. Moreover, the
induced action of $\mathbb{Z}/2$ on $\Lambda\backslash\mathbb{B}^{2}$ is free
away from the singular points of the Kummer surface, hence
$\Delta\backslash\mathbb{B}^{2}$ is birational to an Enriques surface [4,
Prop. VIII.17].

The bielliptic examples constructed from the pair $(Y,\mathcal{D})$ from
Proposition 8.1 are all compact analogues of noncompact examples in [16, §3].

_A $\mathbb{Z}/2$ bielliptic surface._ Consider the automorphism

$\psi_{2}(z,w)=\\!\left(z+\frac{1+i}{2}\,,\,-w+\frac{1+i}{2}\right)$

of $E\times E$ with order two. This automorphism permutes the curves $C_{j}$,
$H_{j}$, and $V_{j}$ and hence permutes their intersection points and induces
an automorphism $\widehat{\psi}_{2}$ of $Y$. Moreover, the action on $Y$ fixes
the collection of curves with the same orbifold weight on
$\Gamma\backslash\mathbb{B}^{2}$, thus there is a lattice $\Lambda_{2}$
containing $\Gamma$ with index two so that
$\Lambda_{2}\backslash\mathbb{B}^{2}$ has underlying analytic space
$Y/\langle\widehat{\psi}_{2}\rangle$. Conjugating $\psi_{2}$ by translation by
$(1+i)/4$ in the second coordinate shows that it is equivalent to the
automorphism

$\phi_{2}(z,w)=\\!\left(z+\frac{1+i}{2}\,,\,-w\right),$

which is the standard automorphism defining a $\mathbb{Z}/2$ bielliptic
quotient of $E\times E$. Thus $\Lambda_{2}\backslash\mathbb{B}^{2}$ is
birational to a $\mathbb{Z}/2$ bielliptic surface.

_A $\mathbb{Z}/4$ bielliptic surface._ Consider the $4$-fold étale cover
$E^{\prime}\times E$ of $E\times E$ associated with the subgroup of
$\mathbb{Z}[i]$ generated by $4$ and $i$. This induces a ball quotient
orbifold $\Gamma^{\prime}\backslash\mathbb{B}^{2}$ covering
$\Gamma\backslash\mathbb{B}^{2}$ with underlying analytic space the blowup
$Y^{\prime}$ of $E^{\prime}\times E$ at $24$ points and orbifold locus the
preimage $\mathcal{D}^{\prime}$ of $\mathcal{D}$. The preimage
$C_{j}^{\prime}$ of each curve $C_{j}$ in $E^{\prime}\times E$ is irreducible,
and can be described in coordinates in exactly the same way by considering
$C_{j}^{\prime}$ as the image in $E^{\prime}\times E$ of the graph of
multiplication by $i^{j}$ in $\mathbb{C}^{2}$. The curves $H_{j}$ similarly
have irreducible preimage $H_{j}^{\prime}$ given as the image of the
appropriate horizontal $\mathbb{C}$ in $\mathbb{C}^{2}$, but the curves
$V_{j}$ each lift to four irreducible curves whose coordinates are of the form
$(\tau,z)$ where $\tau$ is one of the four preimages of $1/2,i/2\in E$ in
$E^{\prime}$. Thus the support of $\mathcal{D}^{\prime}$ consists of fourteen
curves.

The point $1\in E^{\prime}$ is a $4$-torsion point that generates the group of
deck transformations for the covering map to $E$. Therefore

$\psi_{4}(z,w)=(z+1,iw)$

is an automorphism of $E^{\prime}\times E$ with quotient a $\mathbb{Z}/4$
bielliptic surface. Direct check shows that $\psi_{4}$ preserves
$\mathcal{D}^{\prime}$ and hence induces an automorphism of
$\Gamma^{\prime}\backslash\mathbb{B}^{2}$ with quotient birational to a
$\mathbb{Z}/4$ bielliptic surface.

_A $(\mathbb{Z}/2)^{2}$ and a $\mathbb{Z}/4\times\mathbb{Z}/2$ bielliptic
surface._ Let $\Gamma_{4}\backslash\mathbb{B}^{2}$ be the ball quotient in
this section that is birational to a $\mathbb{Z}/4$ bielliptic surface. As
shown at the end of the proof of [16, Thm. 1.4], there are étale covers of a
$\mathbb{Z}/4$ bielliptic surface by both $(\mathbb{Z}/2)^{2}$ and
$\mathbb{Z}/4\times\mathbb{Z}/2$ bielliptic surfaces. This induces orbifold
covers of $\Gamma_{4}\backslash\mathbb{B}^{2}$ with underlying analytic space
birational to surfaces of the desired kind, hence completing these two cases.

The remainder of the proof relies on the example from §5, which was
constructed using the Eisenstein integers $\mathbb{Z}[\zeta]$, where $\zeta$
is a primitive $6^{th}$ root of unity. Recall that $T$ is the elliptic curve
$\mathbb{Z}[\zeta]\backslash\mathbb{C}$. The arguments are completely
analogous to the previous ones, except now are closely related to the
bielliptic ball quotients in [15, §6] instead of those found in [16]. While
these examples are commensurable with the example in §5, it is easier to
describe them explicitly instead of giving various group actions in
coordinates. It will be convenient to let $\rho=\zeta^{2}$ be a primitive
$3^{rd}$ root of unity, and recall that $\rho-1$ divides $3$.

_A $\mathbb{Z}/3$ bielliptic surface._ This example is analogous to [15,
§6.3]. The automorphism

$\phi_{3}(z,w)=\\!\left(z+\frac{\rho-1}{3}\,,\,\rho w\right)$

of the abelian surface $T\times T$ from §5 has order three and quotient a
$\mathbb{Z}/3$ bielliptic surface $Z_{3}$. First, notice that the image
$C_{0}^{\prime}$ in $Z_{3}$ of the diagonal $C_{0}$ in $T\times T$ has genus
one and a unique singular point of order three. Indeed, $\phi_{3}(C_{0})\cap
C_{0}$ consists of the $\phi_{3}$-orbit of the $3$-torsion point $(1/3,1/3)$,
which therefore maps to a singular point of $C_{0}^{\prime}$. If $C_{1}$ is
the horizontal curve $(z,1/3)$ on $T\times T$, then $\phi_{3}(C_{1})$ does not
meet $C_{1}$, hence the image $C_{1}^{\prime}$ is a smooth curve of genus one
passing through the singular point of $C_{0}^{\prime}$.

Let $Y_{3}$ be the blowup of $Z_{3}$ at the singular point of $C_{0}^{\prime}$
with exceptional curve $E$ and $D_{j}$ be the proper transform of the image of
$C_{j}^{\prime}$, which is a smooth curve on $Y_{3}$. Then $D_{0}^{2}=-1$ and
$D_{1}^{2}=-3$ by adjunction [4, I.15], since $E$ is numerically equivalent to
the canonical divisor, $D_{0}\cdot E=1$, and $D_{1}\cdot E=3$. Then

$\mathcal{D}_{3}=\\!\left(1-\frac{1}{3}\right)\\!(E+D_{0}+D_{1})$

has $c_{1}^{2}(Y_{3},\mathcal{D}_{3})=13/3$ and
$c_{2}(Y_{3},\mathcal{D}_{3})=13/9$. A direct check analogous to all other
cases in this paper shows that $K_{Y_{3}}+\mathcal{D}$ is ample, and thus
$(Y_{3},\mathcal{D}_{3})$ is a ball quotient pair by Theorem 2.1.

_A $(\mathbb{Z}/3)^{2}$ bielliptic surface._ This example is analogous to [15,
§6.4]. Consider the $3$-fold cover $T^{\prime}$ of $T$ associated with the
index $3$ subgroup $(\rho-1)\mathbb{Z}[\zeta]$ of $\mathbb{Z}[\zeta]$. Then
the $3$-torsion subgroup of $T^{\prime}$ is generated by $1$ and $(\rho-1)/3$.
The automorphisms

$\displaystyle\psi_{a}(z,w)$ $\displaystyle=(z+1,\rho w)$
$\displaystyle\psi_{b}(z,w)=\\!\left(z+\frac{\rho-1}{3},w+\frac{\rho-1}{3}\right)$

of $T^{\prime}\times T$ commute, and the quotient by $T^{\prime}\times T$ by
the group they generate is a $(\mathbb{Z}/3)^{2}$ bielliptic surface $Z_{9}$.
As in the previous example, if $C_{0}$ is the image of the diagonal in
$\mathbb{C}^{2}$ in $T^{\prime}\times T$ and $C_{0}^{\prime}$ the image of
$C_{0}$ in $Z_{9}$, then $C_{0}^{\prime}$ has genus zero and one singular
point of order $3$. Again, let $C_{1}^{\prime}$ be the horizontal curve on
$Z_{9}$ through the singular point of $C_{0}^{\prime}$.

Let $Y_{9}$ be the blowup of $Z_{9}$ at the singular point of $C_{0}^{\prime}$
with exceptional curve $E$ and $D_{j}$ be the proper transform of
$C_{j}^{\prime}$ to $Y_{9}$. Then the triple $E,D_{0},D_{1}$ has exactly the
same intersection properties as the curves in the previous $\mathbb{Z}/3$
bielliptic example. It therefore follows by the exact same calculations that
if

$\mathcal{D}_{9}=\\!\left(1-\frac{1}{3}\right)\\!(E+D_{0}+D_{1})$

then $(Y_{9},\mathcal{D}_{9})$ is a ball quotient pair.

_A $\mathbb{Z}/6$ bielliptic surface._ To complete the proof of the theorem,
it remains to produce a lattice $\Gamma<\operatorname{PU}(2,1)$ so that
$\Gamma\backslash\mathbb{B}^{2}$ admits a dominant rational map to a
$\mathbb{Z}/6$ bielliptic surface. Let $\Gamma$ be a lattice so that the
underlying analytic space $Y_{3}$ is the blowup of a $\mathbb{Z}/3$ bielliptic
surface $Z_{3}$ at a single point. Any $\mathbb{Z}/6$ bielliptic surface is
the quotient of a $\mathbb{Z}/3$ bielliptic by a free action of
$\mathbb{Z}/2$, and every $\mathbb{Z}/3$ bielliptic admits such a free action.
In particular, there is a $2$-to-$1$ étale map from $Z_{3}$ to some bielliptic
surface $Z_{6}$. The composition

$\Gamma\backslash\mathbb{B}^{2}\longrightarrow Y_{3}\longrightarrow
Z_{3}\longrightarrow Z_{6}$

is dominant and rational. This completes the proof of the theorem. ∎

######  Remark 8.3.

Holzapfel noticed that the complement in $Y$ of the curves $A_{j},B_{j},D_{j}$
is a noncompact ball quotient [22], and that one could take a quotient by the
action of $-\operatorname{Id}$ to get a ball quotient with compactification
birational to a K3 surface. Kasparian and Kotzev noticed that the further
quotient by $i\operatorname{Id}$ gives a ball quotient with compactification
birational to an Enriques surface [24, Thm. 9]. The K3 and Enriques cases in
the proof of Theorem 1.4 are nothing more than applying the exact same
argument to the compact ball quotient orbifold from Proposition 8.1.

######  Remark 8.4.

Careful analysis shows that the $\mathbb{Z}/6$ bielliptic surface in the proof
of Theorem 1.4 does not inherit a ball quotient orbifold structure, and this
is the only case in Kodaira dimension zero where the proof does not provide a
ball quotient orbifold structure on a surface birational to the desired smooth
minimal model. It would be interesting to settle this case like the others by
an explicit ball quotient orbifold structure.

######  Remark 8.5.

If one uses toroidal compactifications instead of compact ball quotients,
almost every case in Theorem 1.4 was in the literature already. The case of an
abelian surface is covered by the examples in §5, either by the compact
example or the auxiliary example (also due to Hirzebruch [21]) that admits a
smooth toroidal compactification by the four elliptic curves $D_{j}$. See [13]
for a plethora of other related (albeit commensurable) examples. See Remark
8.3 for the K3 and Enriques cases. Arithmetic lattices in
$\operatorname{PU}(2,1)$ with smooth toroidal compactification a bielliptic
surface associated with $G$ either $\mathbb{Z}/3$ or $(\mathbb{Z}/3)^{2}$ were
constructed in [15, 14] and examples for the groups $\mathbb{Z}/2$,
$(\mathbb{Z}/2)^{2}$, $\mathbb{Z}/4$, $\mathbb{Z}/4\oplus\mathbb{Z}/2$ were
constructed in [16]. In other words, the only surface of Kodaira dimension
zero not already constructed in the literature in the more general setting of
toroidal compactifications is a bielliptic surface associated with the group
$\mathbb{Z}/6$.

The final task in this section is to prove Theorem 1.5.

###### Proof of Theorem 1.5.

The main example in §5 is a ball quotient orbifold with underlying space the
blowup of an abelian surface at a single point. This smoothly dominates any
abelian surface. Similarly, the proof of Theorem 1.4 constructs ball quotient
orbifolds with underlying space the blowup of a bielliptic surface, and these
examples suffice to show that any bielliptic surface is smoothly dominated by
a ball quotient. Thus only K3 and Enriques surfaces remain.

Naruki [32] constructed a K3 ball quotient as follows. See Ch. VIII and §V.7
of [1] for basic information about the structure of K3 surfaces and elliptic
fibrations used in what follows. There exists a K3 surface admitting an
elliptic fibration $X\to\mathbb{P}^{1}$ with the following properties:

  * $\star$

There are exactly three singular fibers of type $I_{7}$ and exactly three of
type $I_{1}$.

  * $\star$

The fibration admits seven distinct disjoint sections for which each section
intersects exactly one irreducible component of each $I_{7}$ singular fibers
and each curve on each $I_{7}$ singular fiber meets exactly one section.

Let $A_{j,k}$ denote the $k^{th}$ curve in the $j^{th}$ singular fiber of type
$I_{7}$ and $B_{k}$ be the $k^{th}$ section (for any ordering). Then
$A_{j,k}^{2}=B_{k}^{2}=-2$, and any curve meets exactly three other curves.

Consider the $\mathbb{Q}$-divisor

$\mathcal{C}=\\!\left(1-\frac{1}{7}\right)\\!\sum_{k=1}^{7}\\!\left(B_{k}+\sum_{j=1}^{3}A_{j,k}\right)$

on $X$. Since K3 surfaces have trivial canonical bundle and $c_{2}(X)=24$,

$\displaystyle c_{1}^{2}(X,\mathcal{C})$
$\displaystyle=\\!\left(\frac{6}{7}\right)^{2}\\!\left(28(-2)+2\\!\left(\frac{28}{2}(3)\right)\\!\right)$
$\displaystyle=\frac{144}{7}$ $\displaystyle c_{2}(X,\mathcal{C})$
$\displaystyle=24-(28)\\!\left(\frac{6}{7}\right)\\!(-1)-(42)\\!\left(\frac{48}{49}\right)\\!$
$\displaystyle=\frac{48}{7}$

hence $c_{1}^{2}(X,\mathcal{C})=3c_{2}(X,\mathcal{C})$. Moreover,
$\mathcal{C}$ is ample. Indeed, any reduced irreducible curve not in the
support of $\mathcal{C}$ must meet either the $I_{7}$ singular fiber or some
section. Then each curve in the support of $\mathcal{C}$ meets $\mathcal{C}$
with intersection number $(6/7)(-2+3)=6/7$, so $\mathcal{C}$ is ample by the
Nakai–Moishezon criterion [18, Thm. V.1.10]. Thus $(X,\mathcal{C})$ is a ball
quotient pair, and it is arithmetic by [32, Thm. 1].

Since all K3 surfaces are diffeomorphic [1, Cor. VII.8.6], all K3 surfaces are
dominated by $X$, and therefore all K3 surfaces are smoothly dominated by a
ball quotient. Since every Enriques surface has universal cover a K3 surface,
all Enriques surfaces are thus also smoothly dominated by a ball quotient.
This completes the proof of the theorem. ∎

##  9 Arithmeticity of the examples in §8

To prove Theorem 1.7, what remains is to prove arithmeticity of the examples
from §8 that are not arithmetic by construction.

######  Theorem 9.1.

The ball quotient pair $(Y,\mathcal{D})$ from Proposition 8.1 is arithmetic.

###### Proof.

Recall that $E$ is the elliptic curve associated with the Gaussian integers.
Consider the action of $F=(\mathbb{Z}/4)^{2}$ on $E\times E$ generated by
complex multiplication by $i$ on each factor. First, note that $F$ stabilizes
the set of points blown up to obtain $Y$. Indeed, the six points are

$\displaystyle(0,0)$ $\displaystyle\left(\frac{1}{2},\frac{1}{2}\right)$
$\displaystyle\left(\frac{i}{2},\frac{i}{2}\right)$
$\displaystyle\left(\frac{1+i}{2},\frac{1+i}{2}\right)$
$\displaystyle\left(\frac{1}{2},\frac{i}{2}\right)$
$\displaystyle\left(\frac{i}{2},\frac{1}{2}\right)$

where $(0,0)$ and $((1+i)/2,(1+i)/2)$ are fixed by all of $F$ and the other
four are permuted with stabilizer $(\mathbb{Z}/2)^{2}$.

The remaining points of $E\times E$ fixed by some nontrivial element of $F$
are of the form $(\tau_{1},\tau_{2})$ for $\tau_{j}$ a $2$-torsion point, and
each stabilizer is either $\mathbb{Z}/4\times\mathbb{Z}/2$ or
$\mathbb{Z}/2\times\mathbb{Z}/4$. Finally, note that $F$ permutes the curves
$C_{j}$, each having stabilizer $\mathbb{Z}/4$ acting as complex
multiplication. In particular, the induced action on $Y$ permutes curves in
the support of $\mathcal{D}$ with the same orbifold weight, hence $Z=Y/F$ is a
ball quotient orbifold with respect to the appropriate $\mathbb{Q}$-divisor
$\mathcal{E}$ with support the image in $Z$ of the orbifold locus for
$(Y,\mathcal{D})$ and any additional curves fixed by the action of $F$.

Since $(E\times E)/F$ is $\mathbb{P}^{1}\times\mathbb{P}^{1}$ and the six
exceptional curves of $Y$ map to three exceptional curves of $Z$, it follows
that $Z$ is the blowup of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ on three points
on the diagonal. As mentioned in the proof of Lemma 4.1, this means that $Z$
is isomorphic to the blowup of $\mathbb{P}^{2}$ at the vertices of the
complete quadrangle. Moreover, the curves in $Y$ stabilized by some subgroup
of $F$ are mapped to the lines in the quadrangle.

It remains to compute orbifold weights of the fixed curves and compare the
orbifold structure on $Z$ with Deligne–Mostow orbifolds. First, the four
curves $D_{j}$ are identified by $F$, and the action of the stabilizer is by
complex multiplication of order four. In particular, the image of all the
$D_{j}$ in $Z$ is a single curve of orbifold weight four. The points $(0,0)$
and $((1+i)/2,(1+i)/2)$ have stabilizer $F$, where the diagonal subgroup of
$F$ therefore acts as a complex reflection of order four through the
associated exceptional curves, which are $E_{5}$ and $E_{6}$ in Figure 5.
Since each $E_{j}$ has orbifold weight two, the image on $Z$ has orbifold
weight eight. The remaining blowup points $(1/2,1/2)$, $(1/2,i/2)$,
$(i/2,1/2)$, $(i/2,i/2)$ are permuted by $F$ and have stabilizer
$(\mathbb{Z}/2)^{2}$, so the diagonal subgroup acts as a complex reflection of
order two through $E_{j}$ for $1\leq j\leq 4$, and hence the image in $Z$ is a
single curve of orbifold weight four.

The curves $A_{j}$ (resp. $B_{j}$) coming from horizontal and vertical curves
are exchanged by one direct $\mathbb{Z}/4$ factor of $F$ with $\mathbb{Z}/2$
stabilizer acting as a complex reflection, and the other direct $\mathbb{Z}/4$
factor of $F$ acts by complex multiplication. Since the $A_{j}$ and $B_{j}$
have orbifold weight four, their images on $Z$ determine two curves in total,
each with orbifold weight eight. Similarly, the horizontal curves $(z,0)$ and
$(z,(1+i)/2)$ and their vertical analogues also are fixed by $F$ with
stabilizer containing an order four complex reflection; despite not being part
of the orbifold locus for $Y$, these determine four more curves of orbifold
weight four on $Z$. These are all the fixed curves for $F$, and hence
determine the orbifold locus on $Z$.

From the intersection patters of all these curves, one obtains the arrangement
of curves on $Z$ depicted in Figure 6. Note that this line arrangement on $Z$
is precisely the arrangement given by the blowup of the complete quadrangle on
$\mathbb{P}^{2}$ at its vertices. To see this, first notice that every curve
in the arrangement has self-intersection $-1$. Then, blowing down the four
curves of orbifold weight eight gives $\mathbb{P}^{2}$, and the image of the
arrangement is combinatorially the complete quadrangle. It is a classic
theorem in projective geometry that every combinatorial complete quadrangle is
projectively equivalent to the complete quadrangle [8, 6.13], hence the claim.
Then [25, Thm. 4.1(i)] identifies the pair $(Z,\mathcal{E})$ as the
Deligne–Mostow lattice with weights $(4,3,3,3,3)/8$, which is arithmetic [9,
p. 86]. This completes the proof of the theorem.

$8$$4$$4$$4$$8$$4$$4$$8$$4$$8$ Figure 6: The orbifold locus on $Z$ with
weights indicated

∎

######  Remark 9.2.

Applying the same argument to the example mentioned in Remark 8.2 gives the
Deligne–Mostow orbifold with weights $(6,5,5,4,4)/12$, which is
_nonarithmetic_ [9, p. 86].

######  Remark 9.3.

One can also use the techniques from this section to prove that Holzapfel’s
noncompact example in [22] is commensurable with the Picard modular group over
the Gaussian integers, and hence is arithmetic. Holzapfel claimed in [22] that
this result would appear in a later paper, but that paper does not seem to be
available.

The only remaining examples from §8 that are not immediately arithmetic by
virtue of having an arithmetic orbifold cover by construction are the
$\mathbb{Z}/3$ and $(\mathbb{Z}/3)^{2}$ bielliptic examples contained in the
proof of Theorem 1.4. By Theorem 6.8, it suffices to find an explicit orbifold
commonly covered by the example from §5 and the two bielliptic examples.
Notation from the construction of each is used significantly in what follows.

It ends up being natural to start with the $(\mathbb{Z}/3)^{2}$ example, since
it has the least symmetry. Recall that this is constructed by a blowup $Y_{9}$
of the quotient $Z_{9}$ of $T^{\prime}\times T$ by the group $F_{9}$ generated
by the automorphisms

$\displaystyle\psi_{a}(z,w)$ $\displaystyle=(z+1,\rho w)$
$\displaystyle\psi_{b}(z,w)=\\!\left(z+\frac{\rho-1}{3},w+\frac{\rho-1}{3}\right)$

with $\rho=\zeta^{2}$. Since $\tau=(\rho-1)/3\in T$ is one of the three fixed
points of $\rho$ acting on $T$, the map $r(z,w)=(z,\rho w)$ commutes with
$\psi_{a}$ and $\psi_{b}$. The first lemma gives a birational model for the
eventual common orbifold quotient.

######  Lemma 9.4.

The automorphism $r$ of $T^{\prime}\times T$ induces a nontrivial order three
automorphism $r_{9}$ of $Z_{9}$ with quotient isomorphic to the quotient
$B_{0}$ of $T\times\mathbb{P}^{1}$ by the order three automorphism
$(z,y)\mapsto(z+\tau,\rho y)$.

###### Proof.

That $r$ induces an automorphism of $Z_{9}$ follows from the fact that $r$
commutes with the generators $\psi_{a},\psi_{b}$ of $F_{9}$. Since $r$ has
fixed points, and thus is not in the deck group $F_{9}$, the induced
automorphism of $Z_{9}$ is nontrivial.

The goal is then to understand the quotient of $T^{\prime}\times T$ by the
$(\mathbb{Z}/3)^{3}$ group of automorphisms generated by $\psi_{a}$,
$\psi_{b}$, and $r$, which can be alternately generated by $r$ along with
$r^{-1}\circ\psi_{a}$ and $\psi_{b}$, which both act by translations. The
quotient of $T^{\prime}\times T$ by $r^{-1}\circ\psi_{a}$ is $T\times T$ by
definition and the transformation of $T\times T$ induced by $\psi_{b}$ is
diagonal translation by $\tau$. Note that the quotient of $T$ is by the action
of translation by $\tau$ is
$(\rho-1)^{-1}\mathbb{Z}[\zeta]\backslash\mathbb{C}$, which is homothetic and
thus isomorphic to $T$. The action on $T\times T$ induced by the action of $r$
is the action by $\rho$ on the second coordinate, so the quotient is
$T\times\mathbb{P}^{1}$. Thus there is a commutative diagram

${T^{\prime}\times T}$${Z_{9}}$${T\times
T}$${T\times\mathbb{P}^{1}}$${A=(T\times
T)/\langle(\tau,\tau)\rangle}$${B_{0}}$${T\times T}$${T\times\mathbb{P}^{1}}$

where the bottom $T\times T$ is the quotient of $T\times T$ by the
$(\mathbb{Z}/3)^{2}$ generated by the action of $\tau$ on each individual
coordinate, $T^{\prime}\times T\to Z_{9}$ is the covering projection, and
every other right-moving arrow is the action induced by $r$.

All that remains is to understand the automorphism of $T\times\mathbb{P}^{1}$
induced by translation by $(\tau,\tau)$ on $T\times T$. Since $r$ only changes
the second coordinate, the action on the first factor is evidently by $\tau$.
A point $y\in\mathbb{P}^{1}$ with preimage $z\in T$ is fixed under the induced
action if and only if $\rho z=z+k\tau$ for some $k\in\\{0,1,2\\}$, i.e.,
$(\rho-1)z=k\tau$. This implies the induced automorphism of $\mathbb{P}^{1}$
is nontrivial of order three, and hence is multiplication by $\rho$ in the
appropriate coordinates. This proves the lemma. ∎

The next goal is to see that $r_{9}$ induces an automorphism of the ball
quotient associated with the pair $(Y_{9},\mathcal{D}_{9})$.

######  Lemma 9.5.

The action of $r_{9}$ on $Z_{9}$ induces an action on the blowup $Y_{9}$. Let
$B$ denote the normal surface $Y_{9}/\langle\widehat{r}_{9}\rangle$ completing
the commutative diagram

${Z_{9}}$${Y_{9}}$${B_{0}}$${B}$

with birational horizontal arrows. Then there is a $\mathbb{Q}$-divisor
$\mathcal{E}$ on $B$ so that $(B,\mathcal{E})$ is a ball quotient pair with
$(Y_{9},\mathcal{D}_{9})$ as an orbifold cover.

###### Proof.

The first thing to prove is that $r_{9}$ induces an automorphism of the blowup
$Y_{9}$. Let $D$ denote the image in $T^{\prime}\times T$ of the diagonal of
$\mathbb{C}^{2}$, which projects to the curve $C_{0}^{\prime}$ in $Z_{9}$.
Notice that $\psi_{b}$ is diagonal translation and thus fixes $D$ and that
$r(D)=\psi_{a}(D)$ since $\rho z=\rho(z+1)$ on $T$. This implies that $r$
preserves the $F_{9}$-orbit of $D$ and thus $r_{9}$ stabilizes
$C_{0}^{\prime}$. To fix $C_{0}^{\prime}$, $r_{9}$ must fix its unique triple
point, and thus $r_{9}$ acts on the blowup $Y_{9}$ by a transformation
$\widehat{r}_{9}$ that fixes the proper transform $C_{0}$ of $C_{0}^{\prime}$.

Note that the fixed point $(0,0)\in D\subset T^{\prime}\times T$ of $r$ maps
to the singular point of $C_{0}^{\prime}$. The derivative of $r$ at $(0,0)$ is
not a scalar multiple of the identity, which implies that the action of
$\widehat{r}_{9}$ on the exceptional divisor $E$ of $Y_{9}$ is nontrivial,
i.e., $\widehat{r}_{9}$ does not act by a complex reflection through the
curve. On the other hand, $r$ does act as a complex reflection through the
lift to $T^{\prime}\times T$ of the horizontal curve $C_{1}^{\prime}$ through
the singular point of $C_{0}^{\prime}$, namely the curve with coordinates
$(z,0)$, hence $\widehat{r}_{9}$ also acts as a complex reflection through the
proper transform $C_{1}$ of $C_{1}^{\prime}$ to $Y_{9}$. Thus
$\widehat{r}_{9}$ fixes each curve in the orbifold locus for the pair
$(Y_{9},\mathcal{D}_{9})$ and the quotient $B$ of $Y_{9}$ by
$\langle\widehat{r}_{9}\rangle$ is a ball quotient orbifold. ∎

The next lemma completely describes the $\mathbb{Q}$-divisor for the pair
$(B,\mathcal{E})$.

######  Lemma 9.6.

The curves in the orbifold locus for the pair $(B,\mathcal{E})$ are:

  * $\star$

the image $\widehat{E}$ of the exceptional curve $E$ of $Y_{9}$, with orbifold
weight $3$

  * $\star$

the image $\widehat{C}_{0}$ of the curve $C_{0}$ on $Y_{9}$, with orbifold
weight $3$

  * $\star$

the image $\widehat{C}_{1}$ of the horizontal curve $C_{1}$ on $Y_{9}$, with
orbifold weight $9$

###### Proof.

From the discussion before the statement of the lemma, it suffices to show
that $C_{1}$ is the only curve on $Y_{9}$ on which $\widehat{r}_{9}$ has a
fixed point. Indeed, the given curves were shown to be contained in the
support of $\mathcal{E}$ with the given weight, so it suffices to show that
$\widehat{r}_{9}$ does not have either an isolated fixed point or another
fixed curve on which it acts by a complex reflection. Suppose that $(z,w)\in
T^{\prime}\times T$ projects to a fixed point of $\widehat{r}_{9}$. Then there
is an element $g$ of the covering group for $Z_{9}$ so that $g(z,w)=(z,\rho
w)$. In other words, there are $k,\ell\in\\{0,1,2\\}$ so that

$(z,\rho w)=\\!\left(z+k+\ell\frac{\rho-1}{3}\,,\,\rho^{k}w+\ell\tau\right)$

on $T^{\prime}\times T$. Then evidently $k=\ell=0$, and thus $\rho w=w$. In
other words, the only such curves are $(z,0)$, $(z,\tau)$, and $(z,2\tau)$,
which all map to the horizontal curve $C_{1}^{\prime}$ with proper transform
$C_{1}$. This proves the lemma. ∎

######  Proposition 9.7.

The ball quotient pair $(B,\mathcal{E})$ described by Lemma 9.6 is also a
quotient of the $\mathbb{Z}/3$ bielliptic example from §8, hence the
$\mathbb{Z}/3$ and $(\mathbb{Z}/3)^{2}$ bielliptic ball quotient orbifolds are
commensurable.

###### Proof.

The $\mathbb{Z}/3$ bielliptic example is constructed using the automorphism

$\phi_{3}(z,w)=(z+\tau,\rho w)$

of $T\times T$, which commutes with the automorphism $t(z,w)=(z+\tau,w+\tau)$
of order three. Moreover, $t$ permutes the three points
$(\rho^{j}/3,\rho^{j}/3)$, $j\in\\{0,1,2\\}$, that project to the triple point
on the curve $C_{0}^{\prime}$ on the bielliptic surface $Z_{3}$. It follows
that $t$ induces an automorphism $\widehat{r}_{3}$ of the blowup $Y_{3}$ of
$Z_{3}$.

To compute the action of $\widehat{r}_{3}$ on the exceptional divisor of
$Y_{3}$, one can consider the derivative of $\phi_{3}^{-1}\circ t$ fixing the
point $(1/3,1/3)$ on $T\times T$. This is the automorphism

$(z,w)\mapsto(z,\rho^{2}w+\tau),$

which has nonscalar derivative at $(1/3,1/3)$. This implies that the action of
$\widehat{r}_{3}$ stabilizing the exceptional curve is not by a complex
reflection.

Any other fixed point of $\widehat{r}_{3}$ is determined by a fixed point of
$\phi_{3}^{j}\circ t$ for some $j\in\\{0,1,2\\}$. The equation

$(z,w)=\\!\left(z+\tau+j\tau,\rho^{j}w+\tau\right),$

implies that $z=z+(1+j)\tau$, so $j=2$. Then $\rho^{2}w+\tau=w$, which has
only the three solutions $w=\rho^{k}/3$ for $k\in\\{0,1,2\\}$. In other words,
the only fixed points of $\widehat{r}_{3}$ on $Z_{3}$ are the horizontal curve
$C_{1}$ through the exceptional divisor, and $\widehat{r}_{3}$ acts on this
curve as a complex reflection.

As in many previous cases, to show that $\widehat{r}_{3}$ acts on the ball
quotient pair $(Y_{3},\mathcal{D}_{3})$ it only suffices to show that
$\widehat{r}_{3}$ stabilizes the curve $C_{0}$ on $Y_{3}$ determined by the
diagonal of $T\times T$. However, $t$ stabilizes the diagonal of $T\times T$
by definition, so $\widehat{r}_{3}$ indeed stabilizes $C_{1}$ and thus fixes
all the curves in the support of $\mathcal{D}_{3}$.

Therefore the proof of the proposition will be complete once it is shown that
the quotient ball quotient pair is the pair $(B,\mathcal{E})$ from Lemma 9.6
and that $\mathcal{D}_{3}$ maps to $\mathcal{E}$ under the quotient map, with
$C_{1}$ mapping to the curve of orbifold weight $9$. One of the equivalent
definitions of $B_{0}$ given in the proof of Lemma 9.4 is by taking the
quotient of $T\times T$ first by the diagonal action of $(\tau,\tau)$ (i.e.,
the transformation $t$) followed by the action of $\rho$ on the second
coordinate, which is precisely the induced action of $\phi_{3}$ on the
quotient $(T\times T)/\langle(\tau,\tau)\rangle$. Moreover, this description
implies that $\mathcal{D}_{3}$ indeed maps to $\mathcal{E}$ with the
appropriate orbifold weights. Thus $(Y_{3},\mathcal{D}_{3})$ and
$(Y_{9},\mathcal{D}_{9})$ are commensurable. ∎

######  Theorem 9.8.

The $\mathbb{Z}/3$ and $(\mathbb{Z}/3)^{2}$ bielliptic ball quotients
constructed in the proof of Theorem 1.4 are arithmetic.

###### Proof.

Since Proposition 9.7 shows that both bielliptic examples are commensurable
with the ball quotient pair $(B,\mathcal{E})$ from Lemma 9.6, it suffices to
prove arithmeticity of that example. The proof will show that the example from
§5 admits a $\mathbb{Z}/3$ action with quotient $(B,\mathcal{E})$.
Accordingly, this proof freely draws from the notation established in §5 and
§6.

Consider the automorphism $b=\rho^{2}\beta$ of $T\times T$, which has matrix

$\begin{pmatrix}\rho^{2}&0\\\ \rho^{2}&1\end{pmatrix}$

and eigenvectors $(1-\rho,1)$ and $(0,1)$ with respective eigenvalues
$\rho^{2}$ and $1$. It is already known from the proof of Proposition 6.7 that
$b$ acts as an automorphism of the ball quotient pair $(Y,\mathcal{D})$. It
acts nontrivially (i.e., not as a complex reflection) on the exceptional
curve, cyclically permutes the three curves $(z,0)$, $(z,z)$, and $(z,\zeta
z)$, and acts on the curve $(0,z)$ as a complex reflection. The only other
curve fixed by $b$ is the proper transform of the eigenline $((1-\rho)z,z)$,
where $b$ acts as complex multiplication by $\rho^{2}$, in particular not by a
complex reflection. It follows that the orbifold locus for the quotient of $Y$
by $\langle b\rangle$ has the same combinatorics and orbifold weights as
$\mathcal{E}$. It remains to prove that $Y/\langle b\rangle$ is isomorphic to
$B$ and that the orbifold locus $\mathcal{D}$ appropriately maps to
$\mathcal{E}$ under the projection.

Decompose $T\times T$ in terms of the two eigenlines $((1-\rho)x,x)$ and
$(0,y)$ of $b$. This decomposition is nonunique. Specifically, every point
$(z,w)$ can be written in coordinates as

$(z,w)=((1-\rho)x,x)+(0,y)$

where $y$ is uniquely defined and $x$ is only defined up to translation by
$\langle\tau\rangle$. In these coordinates, the action of $b$ is by
$(x,y)\mapsto(\rho^{2}x,y)$. More concretely, consider the self-covering of
$T\times T$ by itself induced by $\tau$ acting on the first coordinate. Then
$b$ is the automorphism of the quotient induced by $\rho^{2}$ acting on the
first coordinate of the cover. The quotient of the cover $T\times T$ by the
lift of $b$ is then $\mathbb{P}^{1}\times T$, and the further quotient by the
automorphism induced by the action of $\tau$ is isomorphic to $B_{0}$ by
construction of $B_{0}$. Considering the curve stabilizer calculations for the
$b$-action, it follows that the quotient orbifold is indeed the pair
$(B,\mathcal{E})$. This proves the theorem. ∎

This finally completes all the pieces required to prove Theorem 1.7.

###### Proof of Theorem 1.7.

By construction, all ball quotient orbifolds used to prove Theorems 1.2 and
1.4 are explicitly commensurable with the orbifolds shown to be arithmetic in
Theorems 4.3, 6.8, 9.1, and 9.8. ∎

## References

  * [1] W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven.  Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Springer-Verlag, second edition, 2004.
  * [2] G. Barthel, F. Hirzebruch, and T. Höfer.  Geradenkonfigurationen und Algebraische Flächen.  Aspects of Mathematics, D4. Friedr. Vieweg & Sohn, 1987.
  * [3] I. Bauer, F. Catanese, and R. Pignatelli.  Surfaces of general type with geometric genus zero: a survey.  In Complex and differential geometry, volume 8 of Springer Proc. Math., pages 1–48. Springer, 2011.
  * [4] A. Beauville.  Complex algebraic surfaces, volume 68 of London Mathematical Society Lecture Note Series.  Cambridge University Press, 1983.
  * [5] A. Borel and G. Prasad.  Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups.  Inst. Hautes Études Sci. Publ. Math., (69):119–171, 1989.
  * [6] N. Bourbaki.  Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines.  Actualités Scientifiques et Industrielles, No. 1337. Hermann, Paris, 1968.
  * [7] W. Couwenberg, G. Heckman, and E. Looijenga.  Geometric structures on the complement of a projective arrangement.  Publ. Math. Inst. Hautes Études Sci., (101):69–161, 2005.
  * [8] H. S. M. Coxeter.  Projective geometry.  Springer-Verlag, 1987.  Second edition.
  * [9] P. Deligne and G. D. Mostow.  Monodromy of hypergeometric functions and nonlattice integral monodromy.  Inst. Hautes Études Sci. Publ. Math., (63):5–89, 1986.
  * [10] P. Deligne and G. D. Mostow.  Commensurabilities among lattices in ${\rm PU}(1,n)$, volume 132 of Annals of Mathematics Studies.  Princeton University Press, 1993.
  * [11] Y. Deng and B. Cadorel.  A characterization of complex quasi-projective manifolds uniformized by unit balls.  Math. Ann., 384(3-4):1833–1881, 2022.
  * [12] M. Deraux.  Non-arithmetic lattices and the Klein quartic.  J. Reine Angew. Math., 754:253–279, 2019.
  * [13] L. F. Di Cerbo and M. Stover.  Multiple realizations of varieties as ball quotient compactifications.  Michigan Math. J., 65(2):441–447, 2016.
  * [14] L. F. Di Cerbo and M. Stover.  Bielliptic ball quotient compactifications and lattices in $\rm PU(2,1)$ with finitely generated commutator subgroup.  Ann. Inst. Fourier (Grenoble), 67(1):315–328, 2017.
  * [15] L. F. Di Cerbo and M. Stover.  Classification and arithmeticity of toroidal compactifications with $3\overline{c}_{2}=\overline{c}_{1}^{2}=3$.  Geom. Topol., 22(4):2465–2510, 2018.
  * [16] L. F. Di Cerbo and M. Stover.  Punctured spheres in complex hyperbolic surfaces and bielliptic ball quotient compactifications.  Trans. Amer. Math. Soc., 372(7):4627–4646, 2019.
  * [17] M. Gromov.  Super stable Kählerian horseshoe?  In Essays in mathematics and its applications, pages 151–229. Springer, 2012.
  * [18] R. Hartshorne.  Algebraic geometry.  Graduate Texts in Mathematics, No. 52. Springer-Verlag, 1977.
  * [19] W. J. Harvey.  Cyclic groups of automorphisms of a compact Riemann surface.  Quart. J. Math. Oxford Ser. (2), 17:86–97, 1966.
  * [20] F. Hirzebruch.  Arrangements of lines and algebraic surfaces.  In Arithmetic and geometry, Vol. II, volume 36 of Progr. Math., pages 113–140. Birkhäuser, 1983.
  * [21] F. Hirzebruch.  Chern numbers of algebraic surfaces: an example.  Math. Ann., 266(3):351–356, 1984.
  * [22] R.-P. Holzapfel.  Complex hyperbolic surfaces of abelian type.  Serdica Math. J., 30(2-3):207–238, 2004.
  * [23] B. Hunt.  Complex manifold geography in dimension $2$ and $3$.  J. Differential Geom., 30(1):51–153, 1989.
  * [24] A. Kasparian and B. Kotzev.  Weak form of Holzapfel’s conjecture.  J. Geom. Symmetry Phys., 19:29–42, 2010.
  * [25] F. C. Kirwan, R. Lee, and S. H. Weintraub.  Quotients of the complex ball by discrete groups.  Pacific J. Math., 130(1):115–141, 1987.
  * [26] R. Kobayashi.  Einstein-Kähler $V$-metrics on open Satake $V$-surfaces with isolated quotient singularities.  Math. Ann., 272(3):385–398, 1985.
  * [27] R. Kobayashi, S. Nakamura, and F. Sakai.  A numerical characterization of ball quotients for normal surfaces with branch loci.  Proc. Japan Acad. Ser. A Math. Sci., 65(7):238–241, 1989.
  * [28] R. A. Livne.  On certain covers of the universal elliptic curve.  ProQuest LLC, 1981.  Thesis (Ph.D.)–Harvard University.
  * [29] G. A. Margulis.  Discrete subgroups of semisimple Lie groups, volume 17 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3).  Springer-Verlag, 1991.
  * [30] A. Momot.  On modular ball-quotient surfaces of Kodaira dimension one.  ISRN Geom., Art. ID 214853, 2011.
  * [31] G. D. Mostow.  Generalized Picard lattices arising from half-integral conditions.  Inst. Hautes Études Sci. Publ. Math., (63):91–106, 1986.
  * [32] I. Naruki.  On a K3 surface which is a ball quotient.  Preprint, Max Planck Inst., Bonn, 1986.
  * [33] T. Shioda.  On elliptic modular surfaces.  J. Math. Soc. Japan, 24:20–59, 1972.
  * [34] M. Stover.  Volumes of Picard modular surfaces.  Proc. Amer. Math. Soc., 139(9):3045–3056, 2011.
  * [35] M. Stover and G. Urzúa.  Rigid surfaces arbitrarily close to the Bogomolov-Miyaoka-Yau line.  Amer. J. Math., 144(6):1783–1804, 2022.
  * [36] T. Suwa.  On hyperelliptic surfaces.  J. Fac. Sci. Univ. Tokyo Sect. I, 16:469–476 (1970), 1969.
  * [37] A. Wiman.  Ueber die hyperelliptischen Curven und diejenigen vom Geschlechte $p=3$, welche eindeutige Transformationen in sich zulassen.  Stockh. Akad. Bihang, 21(1):23p, 1895.