# Residual finiteness for central extensions of lattices in
$\operatorname{PU}(n,1)$ and negatively curved projective varieties

Matthew Stover
Temple University
<EMAIL_ADDRESS>Domingo Toledo
University of Utah
<EMAIL_ADDRESS>
###### Abstract

We study residual finiteness for cyclic central extensions of cocompact
arithmetic lattices $\Gamma<\operatorname{PU}(n,1)$ simple type. We prove that
the preimage of $\Gamma$ in any connected cover of $\operatorname{PU}(n,1)$,
in particular the universal cover, is residually finite. This follows from a
more general theorem on residual finiteness of extensions whose characteristic
class is contained in the span in $H^{2}(\Gamma,\mathbb{Z})$ of the Poincaré
duals to totally geodesic divisors on the ball quotient
$\Gamma\backslash\mathbb{B}^{n}$. For $n\geq 4$, if $\Gamma$ is a congruence
lattice, we prove residual finiteness of the central extension associated with
any element of $H^{2}(\Gamma,\mathbb{Z})$.

Our main application is to existence of cyclic covers of ball quotients
branched over totally geodesic divisors. This gives examples of smooth
projective varieties admitting a metric of negative sectional curvature that
are not homotopy equivalent to a locally symmetric manifold. The existence of
such examples is new for all dimensions $n\geq 4$.

_To Herb Clemens in celebration of his many contributions to our understanding
and enjoyment of the topology of algebraic varieties_

##  1 Introduction

It is a classical result that if
$\Gamma<\operatorname{PSL}_{2}(\mathbb{R})\cong\operatorname{PU}(1,1)$ is a
lattice and $\widetilde{\Gamma}$ denotes its preimage in the universal
covering group
$\widetilde{\operatorname{PSL}}_{2}(\mathbb{R})\cong\widetilde{\operatorname{PU}}(1,1)$,
then $\widetilde{\Gamma}$ is residually finite. In fact, $\widetilde{\Gamma}$
is a linear group, i.e., it admits a faithful representation into
$\operatorname{GL}_{N}(\mathbb{R})$ for some $N$, even though its ambient Lie
group $\widetilde{\operatorname{PSL}}_{2}(\mathbb{R})$ is not a linear Lie
group. See [7, §IV.48] for an account of several closely-related perspectives
on how one can prove these results. In the other direction, following work of
Deligne [8], one can use the congruence subgroup property to build many
lattices in nonlinear Lie groups that are not linear or residually finite,
e.g., the preimage of $\operatorname{Sp}_{2n}(\mathbb{Z})$ in the universal
cover of $\operatorname{Sp}_{2n}(\mathbb{R})$ for $n\geq 2$.

These examples illustrate the interest in studying linearity of lattices in
nonlinear Lie groups. Particularly interesting are the universal covering
groups $\widetilde{\operatorname{PU}}(n,1)$ of $\operatorname{PU}(n,1)$,
$n\geq 2$. In recent work [21], we developed tools for proving residual
finiteness of preimages of lattices in $\operatorname{PU}(n,1)$ in
$\widetilde{\operatorname{PU}}(n,1)$, each inspired by a variant of the proof
for $\operatorname{PU}(1,1)$, and found the first examples of residually
finite lattices in $\widetilde{\operatorname{PU}}(2,1)$. One purpose of this
paper is to vastly improve upon the class of examples with this property.
Recall that arithmetic lattices in $\operatorname{PU}(n,1)$ can be classified
using hermitian forms on $\mathcal{D}^{r}$, where $\mathcal{D}$ is a certain
central simple division algebra with involution of second kind. In this paper
we consider those of _simple type_ , namely those constructed using hermitian
forms over number fields; see §3.1 for a precise description. We will prove:

######  Theorem 1.1.

Let $\Gamma<\operatorname{PU}(n,1)$ be a cocompact arithmetic lattice of
simple type. Then the preimage of $\Gamma$ in
$\widetilde{\operatorname{PU}}(n,1)$ is residually finite. In fact, the
preimage of $\Gamma$ in any connected cover of $\operatorname{PU}(n,1)$ is
residually finite.

This answers [21, Qu. 1] for these lattices. Moreover, the method of proof
completely answers [21, Qu. 2]. While completing this manuscript, we learned
that Richard Hill also very recently proved Theorem 1.1 using related but
ultimately quite different methods [11]. The proof of Theorem 1.1 is in §4,
where we prove the following much more general result about residual
finiteness of central extensions of lattices in $\operatorname{PU}(n,1)$.

######  Theorem 1.2.

Suppose that $\Gamma\backslash\mathbb{B}^{n}$ is a smooth compact ball
quotient with $\Gamma$ a congruence arithmetic lattice of simple type. Let
$\widetilde{\Gamma}$ be the central extension of $\Gamma$ by $\mathbb{Z}$ with
characteristic class $\phi\in H^{2}(\Gamma,\mathbb{Z})$.

  1. 1.

If $n\geq 4$, then $\widetilde{\Gamma}$ is residually finite.

  2. 2.

If $n<4$, then $\widetilde{\Gamma}$ is residually finite under the following
additional assumptions:

    1. (a)

If $n=3$, assume that $\phi\in
H^{1,1}(\Gamma\backslash\mathbb{B}^{3},\mathbb{C})\cap
H^{2}(\Gamma\backslash\mathbb{B}^{3},\mathbb{Z})$.

    2. (b)

If $n=2$, assume that $\phi$ is contained in the span of the Poincaré duals to
the totally geodesic divisors on $\Gamma\backslash\mathbb{B}^{2}$.

Moreover, if $\overline{\phi}\in H^{2}(\Gamma,\mathbb{Z}/d)$ is the reduction
modulo $d$ of any class $\phi$ in $H^{2}(\Gamma,\mathbb{Z})$ satisfying the
above hypotheses, then the central extension of $\Gamma$ by $\mathbb{Z}/d$
associated with $\overline{\phi}$ is residually finite.

We now describe why Theorem 1.1 is a consequence of Theorem 1.2. Residual
finiteness is a commensurability invariant, so it suffices to assume that
$\Gamma$ is a torsion-free congruence arithmetic lattice. Then, the canonical
class $c_{1}(\Gamma\backslash\mathbb{B}^{n})\in
H^{2}(\Gamma\backslash\mathbb{B}^{n},\mathbb{Z})$ is the characteristic class
associated with the preimage $\widetilde{\Gamma}$ of $\Gamma$ in
$\widetilde{\operatorname{PU}}(n,1)$, and similarly the reduction
$\widetilde{\Gamma}_{d}$ of $\widetilde{\Gamma}$ modulo $d$ is isomorphic to
the preimage of $\Gamma$ in the $d$-fold connected cover of
$\operatorname{PU}(n,1)$. Under our hypotheses,
$c_{1}(\Gamma\backslash\mathbb{B}^{n})$ is contained in the span of the
Poincaré duals to the totally geodesic divisors on
$\Gamma\backslash\mathbb{B}^{n}$; see Theorem 3.2. The proof uses
Kudla–Millson theory [15], following an analogous argument of Bergeron, Li,
Millson, and Moeglin for orthogonal Shimura varieties [3, Cor. 8.4]. Thus
$c_{1}(\Gamma\backslash\mathbb{B}^{n})$ satisfies the hypotheses of Theorem
1.2, and Theorem 1.1 follows.

The proof of Theorem 1.2 begins by using work of Bergeron, Millson, and
Moeglin [4] to show that for any class satisfying the hypotheses of the
theorem, there is a congruence subgroup $\Gamma^{\prime}\leq\Gamma$ so that
the pullback of $\phi$ to $\Gamma^{\prime}\backslash\mathbb{B}^{n}$ is in the
image of the cup product from
$\bigwedge\nolimits^{2}H^{1}(\Gamma^{\prime}\backslash\mathbb{B}^{n},\mathbb{C})$;
this is Theorem 3.3. Our previous work [21, Thm. 5.1], stated in slightly
different language as Theorem 2.16 below, then implies that the associated
central extension $\widetilde{\Gamma}$ of $\Gamma$ by $\mathbb{Z}$ has a two-
step nilpotent quotient that is injective on the center of
$\widetilde{\Gamma}$. The claims regarding residual finiteness of
$\widetilde{\Gamma}$ and $\widetilde{\Gamma}_{d}$ follow from results in [21,
§2] that we recall in §2.2 below.

######  Remark 1.3.

As in [21], we also obtain that every extension considered in Theorem 1.2 is
linear. Our proofs of residual finiteness proceed by constructing a nilpotent
quotient of the extension group that is injective on the center. Using
linearity of finitely generated nilpotent groups we easily get linearity of
the extension. We state only the weaker results because residual finiteness is
the property of interest to us for our applications.

Taking the special case in Theorem 1.2 where $\phi$ is the Poincaré dual to a
codimension one totally geodesic subvariety, we have the following.

######  Corollary 1.4.

Let $M=\Gamma\backslash\mathbb{B}^{n}$ be a closed complex hyperbolic
$n$-manifold with $\Gamma$ a congruence arithmetic lattice. If $D\subset M$ is
a smooth embedded codimension one totally geodesic subvariety,
$\mathcal{O}(D)$ is the line bundle over $M$ associated with $D$, and
$\mathcal{O}(D)^{\times}\subset\mathcal{O}(D)$ is the complement of the zero
section, then $\pi_{1}(\mathcal{O}(D)^{\times})$ is residually finite.

If $\Gamma<\operatorname{PU}(n,1)$ is an arithmetic lattice so that
$\Gamma\backslash\mathbb{B}^{n}$ contains an embedded codimension one totally
geodesic submanifold, then $\Gamma$ is of simple type. See Proposition 3.1,
which generalizes an argument from work of Möller and Toledo [16]. All known
nonarithmetic ball quotients of dimension $n\geq 2$ are well known to be
commensurable with quotients of the ball by _complex reflection groups_ ,
which implies that they contain codimension one totally geodesic subvarieties
(e.g., see [19, Thm. 1.3], where the proof works verbatim in all dimensions).
Thus residual finiteness of $\pi_{1}(\mathcal{O}(D)^{\times})$ is relevant,
and remains open, for nonarithmetic lattices. We note for contrast that
arithmetic ball quotients of simple type contain infinitely many distinct
immersed codimension one totally geodesic subvarieties, but recent work of
Bader–Fisher–Miller–Stover [1] and, independently, Baldi–Ullmo [2] proves that
nonarithmetic ball quotients only contain finitely many immersed codimension
one totally geodesic subvarieties (and, more generally, only finitely many of
any positive dimension that are maximal with respect to inclusion).

Our final application of Theorem 1.2 is to the existence of covers of ball
quotients branched over totally geodesic divisors, answering a question raised
to the second author by Gromov over $40$ years ago. Existence of these covers
allows us to prove the following in §5.

######  Theorem 1.5.

For all $n\geq 2$, there are $n$-dimensional smooth complex projective
varieties admitting a Kähler metric of strongly negative curvature, thus a
Riemannian metric of strictly negative sectional curvature, that are not
homotopy equivalent to a locally symmetric manifold.

Strongly negative curvature is in the sense defined by Siu [18, §2]. Examples
for $n=2$ are due to Mostow–Siu [17], and further examples were produced by
Zheng [23, 24]. Deraux gave examples for $n=3$ [9]. We contribute a wealth of
new examples for $n=2,3$ and the first examples for $n\geq 4$. Our examples
are cyclic branched covers of ball quotients branched over smooth (but
possibly disconnected) totally geodesic divisors. Existence of these branched
covers follows from Corollary 1.4. The fact that they admit a strongly
negatively curved Kähler metric is a theorem of Zheng [23, Thm. 1] that
generalizes the famous work of Mostow and Siu [17]. Zheng notes in [23, p. 135
& 151] that existence of branched covers was a primary obstruction to proving
Theorem 1.5 in higher dimensions.

Finally we must prove that these covers are not homotopy equivalent to a
locally symmetric space. This was shown for the previous examples by computing
Chern numbers. This becomes delicate in higher dimensions, so we provide a
proof of a different kind. Assuming our branched cover is homotopy equivalent
to a locally symmetric manifold, hyperbolicity of the fundamental group, work
of Carlson and Toledo [6], and Siu rigidity [18] allows us to conclude that
the manifold is in fact biholomorphic to a ball quotient. We then study the
normal bundle to the branch locus to derive a contradiction.

#### Acknowledgments

Profound thanks are due to Nicolas Bergeron, who explained to us how to apply
Kudla–Millson theory and his work to prove the results on cup products
necessary to apply our previous methods. In particular, the arguments for
Theorem 3.2 and Theorem 3.3 should be attributed to him. Stover was partially
supported by Grant Number DMS-1906088 from the National Science Foundation.

##  2 Residual finiteness, central extensions, and
branched covers

###  2.1 Basic objects

We review the connections among the following objects. Suppose:

  1. 1.

$X$ is a smooth aspherical manifold with $\Gamma=\pi_{1}(X)$;

  2. 2.

$p:L\to X$ is a complex line bundle over $X$, $L^{\times}$ is $L$ with its
zero section removed, and $\widetilde{\Gamma}=\pi_{1}(L^{\times})$;

  3. 3.

$s:X\to L$ is a section of $L$ transverse to the zero section.

The homotopy sequence of the fibration $p:L^{\times}\to X$ with fiber
$\mathbb{C}^{\times}$ gives a central extension

$0\longrightarrow\mathbb{Z}\longrightarrow\widetilde{\Gamma}\longrightarrow\Gamma\longrightarrow
1$

connecting the above fundamental groups. The zero set

$Z(s)=\\{x\in X:s(x)=0\\}$

of $s$ is a smooth real codimension two submanifold of $X$ with normal bundle
isomorphic to $L|_{Z(s)}$, where $L$ is viewed as an $\mathbb{R}^{2}$-bundle.

####  2.1.1 Group of central extensions

It is well known that for a given group $\Gamma$ and fixed Abelian group $A$
the isomorphism classes of central extensions

$E:\,0\longrightarrow
A\longrightarrow\widetilde{\Gamma}\longrightarrow\Gamma\longrightarrow 1$

form a group in natural bijection with $H^{2}(\Gamma,A)$, where the
correspondence is given by the characteristic class $\chi(E)$ of the extension
$E$; e.g., see [5, §VI.3]. This correspondence is functorial in both of the
variables. Given an injection ${\iota:\Lambda\to\Gamma}$, then the restriction
$\iota^{*}(E)$ of $E$ to $\Lambda$ has characteristic class
$\iota^{*}(\chi(E))$. Similarly, if $p:A\to B$ is surjective and $p_{*}(E)$
denotes the extension of $\Gamma$ obtained by dividing $A$ and
$\widetilde{\Gamma}$ by the kernel of $p$, then
$\chi(p_{*}(E))=p_{*}(\chi(E))$.

######  Remark 2.1.

Note that the word “extension” is used in two (closely related) meanings: the
exact sequence $E$ above, or just the middle group $\widetilde{\Gamma}$. We
will refer to $\widetilde{\Gamma}$ as _the group of the extension $E$_. Given
$\iota:\Lambda\to\Gamma$ as above, the group $\widetilde{\Lambda}$ of the
extension $\iota^{*}(E)$ is then the preimage of $\Lambda$ in
$\widetilde{\Gamma}$ for the natural projection $\widetilde{\Gamma}\to\Gamma$.

####  2.1.2 Group of line bundles

It is also well known that given a space $X$, the isomorphism classes of
complex line bundles over $X$ form a group under tensor product that is
isomorphic to $H^{2}(X,\mathbb{Z})$, with isomorphism given by the Chern class
$c_{1}(L)$. The isomorphism is natural with respect to smooth maps $f:Y\to X$,
namely $c_{1}(f^{*}L)=f^{*}(c_{1}(L))$.

####  2.1.3 Comparison of characteristic classes

Suppose that we are in the above situation: $X$ is a smooth aspherical
manifold and $\Gamma=\pi_{1}(X)$, hence $H^{*}(X)\cong H^{*}(\Gamma)$ (meaning
that there is an isomorphism of cohomology with any coefficients). In order to
compare the two characteristic classes defined above, we need a functorial
isomorphism between the two cohomologies. More precisely, we need a natural
isomorphism between cohomology of the complex $C^{*}(\Gamma,\mathbb{Z})$ of
Eilenberg–MacLane cochains on $\Gamma$ and the cohomology of a topologically
defined complex $C^{*}(X,\mathbb{Z})$ of cochains on $X$, where one could
choose either singular or Čech cochains on $X$. There is a well known method
for establishing a natural correspondence between the cohomologies of these
two complexes by embedding both in the bicomplex
$\bigoplus_{p,q}C^{p}(\Gamma,C^{q}(\widetilde{X},\mathbb{Z}))$ of
Eilenberg–MacLane cochains on $\Gamma$ with coefficients in the (singular or
Čech) cochains on the universal cover $\widetilde{X}$ with its action of
$\Gamma$. Standard arguments give natural isomorphisms of the two cohomologies
in question to that of the total complex of the bicomplex. The isomorphism
thus obtained will give a functorial isomorphism between
$H^{2}(\Gamma,\mathbb{Z})$ and $H^{2}(X,\mathbb{Z})$.

Thus, in our situation of a complex line bundle $L$ over a smooth aspherical
manifold $X$ we have three cohomology classes in $H^{2}(X,\mathbb{Z})\cong
H^{2}(\Gamma,\mathbb{Z})$:

  1. 1.

The characteristic class $\chi(E)\in H^{2}(\Gamma,\mathbb{Z})$ of the
extension

$E:\,0\longrightarrow\mathbb{Z}\longrightarrow\widetilde{\Gamma}\longrightarrow\Gamma\longrightarrow
1,$ (1)

where $\widetilde{\Gamma}=\pi_{1}(L^{\times})$.

  2. 2.

The Chern class $c_{1}(L)\in H^{2}(X,\mathbb{Z})$.

  3. 3.

The Poincaré dual $[Z(s)]^{\vee}$ of the homology class $[Z(s)]\in
H_{n-2}(X,\mathbb{Z})$ of the zero set $Z(s)$ of the section $s$.

######  Lemma 2.2.

The three cohomology classes defined in 1 - 3 above are equal, except possibly
up to a factor of $\pm 1$ arising from sign conventions.

####  2.1.4 The same considerations over projective varieties

Now suppose, in addition to the above assumptions, that $X$ is a smooth
projective variety. Let $D=D_{1}\cup\dots\cup D_{k}$, where
$D_{1},\dots,D_{k}$ are smooth and connected codimension one subvarieties of
$X$ that are pairwise disjoint, i.e., $D_{i}\cap D_{j}=\varnothing$ if $i\neq
j$. We will not distinguish the smooth codimension one subvariety $D$ from the
smooth divisor $D=D_{1}+\dots+D_{k}$.

######  Definition 2.3.

A pair $(X,D)$ consisting of a smooth projective variety $X$ and a divisor $D$
satisfying the above conditions will be called a _good pair_.

We can assign a holomorphic line bundle $\mathcal{O}(D)\to X$ to $D$ with the
property that it has a holomorphic section $s$ with zero set $Z(s)=D$ and
vanishing to first order on $D$, meaning that, in local coordinates on $X$ and
$\mathcal{O}(D)$, the differential $d_{x}s$ is nonzero for any $x\in D$. These
properties define $\mathcal{O}(D)$ uniquely, and its section $s$ uniquely up
to a nonzero multiplicative constant, i.e., we can replace $s$ by $\lambda s$
for any $\lambda\in\mathbb{C}^{\times}$. Transversality of $s$ implies that
$\mathcal{O}(D)|_{D}$ is isomorphic to the normal bundle $N(D)$ of $D$ in $X$.

Let $d>1$ be an integer and suppose that $\mathcal{O}(D)$ is divisible by $d$
in the group of holomorphic line bundles on $X$. That is, suppose that there
is a holomorphic line bundle $p:L\to X$ with
$L^{\otimes^{d}}\cong\mathcal{O}(D)$. We have a natural $d^{th}$ power map
$L\to L^{\otimes^{d}}$ defined by

$v\longmapsto
v^{d}=\overset{d~{}\textrm{factors}}{\overbrace{v\otimes\dots\otimes v}}.$

######  Definition 2.4.

Suppose $(X,D)$ is a good pair and $p:L\to X$ a line bundle with
$L^{\otimes^{d}}\cong\mathcal{O}(D)$ as above. Then the space $Y$ defined by

$Y=\left\\{v\in L:v^{d}=s(p(v))\right\\}$

will be called a _cyclic $d$-fold branched cover of $X$ branched along $D$_.

More precisely, $Y$ could be called the cyclic branched cover associated with
$L$. Note the following properties of $Y$:

######  Lemma 2.5.

Given a good pair $(X,D)$ and $L$ a line bundle so
$L^{\otimes^{d}}\cong\mathcal{O}(D)$, the cyclic branched cover $Y$ branched
along $D$ satisfies:

  1. 1.

As a subset of $L$, $Y$ is a smooth projective subvariety stable under the
multiplication maps $v\mapsto\zeta v$ where $\zeta$ is a primitive $d^{th}$
root of unity, which define an action of the cyclic group $\mathbb{Z}/d$ on
$Y$ with quotient $X$.

  2. 2.

The projection $p:Y\to X$ is $d$-to-$1$ over $X\smallsetminus D$ and
$1$-to-$1$ over $D$.

  3. 3.

If $\mathbf{O}$ denotes the zero section of $L$, then

$p|_{Y\smallsetminus\mathbf{O}}:Y\smallsetminus\mathbf{O}\to X\smallsetminus
D$

is a $\mathbb{Z}/d$ covering space.

###### Proof.

By writing local equations for $Y$ at each point $y\in Y\subset L$ it is clear
that $Y$ is a smooth subvariety. Since $L$ is a quasiprojective variety, so is
$Y$, and $Y$ is finite over $X$, hence compact, so $Y$ is projective. The
defining equation of $Y$ is invariant under the $\mathbb{Z}/d$-action, which
is free on $L\smallsetminus\mathbf{O}$ and fixes $\mathbf{O}$ point-wise. This
makes the remaining statements clear. ∎

For the existence of branched covers as in Definition 2.4 we have:

######  Proposition 2.6.

Suppose $(X,D)$ is a good pair and $d>1$ is an integer. With notation as in
§2.1.1, the following are equivalent:

  1. 1.

There exists a holomorphic line bundle $p:L\to X$ with
$L^{\otimes^{d}}\cong\mathcal{O}(D)$.

  2. 2.

There exists $\alpha\in H^{2}(X,\mathbb{Z})$ with
$c_{1}(\mathcal{O}(D))=d\cdot\alpha$.

  3. 3.

There exits $\gamma\in H^{2}(\Gamma,\mathbb{Z})$ with $\chi(E)=d\cdot\gamma$,
where $\Gamma=\pi_{1}(X)$,
${\widetilde{\Gamma}=\pi_{1}(\mathcal{O}(D)^{\times})}$, and $E$ is the
extension in Equation (1).

  4. 4.

There exists $z\in H_{2n-2}(X,\mathbb{Z})$ with $d\cdot z=[D]$, where
$n=\dim_{\mathbb{C}}(X)$ and $[D]$ is the homology class of $D$.

  5. 5.

The class $(r_{d})_{*}(c_{1}(\mathcal{O}(D)))\in H^{2}(X,\mathbb{Z}/d)$
vanishes, where

$(r_{d})_{*}:H^{2}(X,\mathbb{Z})\longrightarrow H^{2}(X,\mathbb{Z}/d)$

is the map induced by the coefficient homomorphism
${r_{d}:\mathbb{Z}\to\mathbb{Z}/d}$.

  6. 6.

The class $\chi((r_{d})_{*}(E))\in H^{2}(\Gamma,\mathbb{Z}/d)$ vanishes, where
$(r_{d})_{*}(E)$ is the extension

$0\longrightarrow\mathbb{Z}/d\longrightarrow\widetilde{\Gamma}_{d}\longrightarrow\Gamma\longrightarrow
1$

for $\widetilde{\Gamma}_{d}=\widetilde{\Gamma}/d\mathbb{Z}$ obtained from $E$
by dividing by the kernel of $r_{d}$.

If any one of these conditions holds, then there exists a cyclic $d$-fold
branched cover $Y\to X$ branched along $D$ (associated with $L$) as in
Definition 2.4.

###### Proof.

Clearly (1) implies (2) with $\alpha=c_{1}(L)$, and (2) and (4) are equivalent
by Poincaré duality and Lemma 2.2. Also (2) and (5) are seen to be equivalent
by looking at the Bockstein long exact sequence associated with the
coefficient sequence $0\to\mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}/d\to 0$. The
equivalence of (2) and (3), and similarly (5) and (6), follow from Lemma 2.2.
The only subtle point is to prove that (2) implies (1). We refer to [12, §7]
for a simple proof based on the exponential sequence
$0\to\mathbb{Z}\to\mathcal{O}\to\mathcal{O}^{*}\to 1$. ∎

###  2.2 Residual Finiteness

Recall that a finitely generated group $\Gamma$ is _residually finite_ if, for
every nontrivial $\gamma\in\Gamma$, there exists a homomorphism
$\rho:\Gamma\to F$ from $\Gamma$ to a finite group $F$ so that $\rho(\gamma)$
is nontrivial. This property is a commensurability invariant of groups. In
particular, if $\Lambda\leq\Gamma$ is a subgroup of finite index, then
$\Lambda$ is residually finite if and only if $\Gamma$ is residually finite.

We will be interested in residual finiteness for the groups
$\widetilde{\Gamma}$ of central extensions
$0\to\mathbb{Z}\to\widetilde{\Gamma}\to\Gamma\to 1$ of residually finite
groups $\Gamma$ by $\mathbb{Z}$. Throughout this section $q$ will denote the
projection $\widetilde{\Gamma}\to\Gamma$ and $\sigma$ will be the image of $1$
under the injection $\mathbb{Z}\to\widetilde{\Gamma}$, hence the infinite
cyclic group $\langle\sigma\rangle$ is the kernel of $q$.

Since $\Gamma$ is residually finite, to prove that $\widetilde{\Gamma}$ is
residually finite it suffices to prove that for every $i\neq 0$ there exists a
homomorphism from $\widetilde{\Gamma}$ to some finite group that is nontrivial
on $\sigma^{i}$. Indeed, for any $\gamma\in\widetilde{\Gamma}$ not in
$\langle\sigma\rangle$, $q(\gamma)$ is nontrivial and, by assumption, there
exists a homomorphism $\rho:\Gamma\to F$ for some finite group $F$ with
$\rho(q(\gamma))$ nontrivial. Thus $\rho\circ q:\widetilde{\Gamma}\to F$ is a
homomorphism that is nontrivial on $\gamma$. Observe that a slight extension
of this reasoning gives:

######  Lemma 2.7 (Lem. 2.4 [21]).

Let $E$ be a central extension of a residually finite group $\Gamma$ by
$\mathbb{Z}$ with group $\widetilde{\Gamma}$ as above, and let
$\sigma\in\widetilde{\Gamma}$ be the image of $1\in\mathbb{Z}$. If there is a
homomorphism $\phi:\widetilde{\Gamma}\to H$ where $H$ is residually finite and
$\phi$ is injective on $\langle\sigma\rangle$, then $\widetilde{\Gamma}$ is
residually finite.

For $d\in\mathbb{Z}$, $d>1$, we will also be interested in the reductions
modulo $d$ of the above extensions, namely:

$0\longrightarrow\mathbb{Z}/d\longrightarrow\widetilde{\Gamma}_{d}\longrightarrow\Gamma\longrightarrow
1,$ (2)

where $\widetilde{\Gamma}_{d}=\widetilde{\Gamma}/d\mathbb{Z}$. Similar
considerations apply with regard to residual finiteness of
$\widetilde{\Gamma}_{d}$. The following definition will turn out to be very
useful:

######  Definition 2.8.

Let $\Gamma$ be a residually finite group, $E$ be an extension of $\Gamma$ by
$\mathbb{Z}$ with group $\widetilde{\Gamma}$ as in Equation (1), and let
$\sigma\in\widetilde{\Gamma}$ be the image of $1\in\mathbb{Z}$. We say that
_$E$ satisfies condition $N$_ if there exists a nilpotent quotient
$\mathcal{N}$ of $\widetilde{\Gamma}$ under which the image of $\sigma$ has
infinite order. For a positive integer $k$ we say that _$E$ satisfies
condition $N_{k}$_ if there is a nilpotent quotient $\mathcal{N}$ of
$\widetilde{\Gamma}$ with step size at most $k$ in which the image of $\sigma$
has infinite order.

######  Lemma 2.9.

Let $E$ be a central extension of a residually finite group $\Gamma$ by
$\mathbb{Z}$ with group $\widetilde{\Gamma}$. If $E$ satisfies condition $N$,
then $\widetilde{\Gamma}$ is residually finite.

###### Proof.

Since finitely generated nilpotent groups are residually finite, this is an
immediate consequence of Lemma 2.7. ∎

First we need to know how residual finiteness behaves under the two basic
operations of restriction (or pullback) and reduction (push-forward) of
extensions defined in §2.1.1.

######  Lemma 2.10.

Let $\Gamma$ be a residually finite group and $E$ denote the central extension
$0\to\mathbb{Z}\to\widetilde{\Gamma}\to\Gamma\to 1$. If $d$ is an integer
$>1$, let $r_{d}:\mathbb{Z}\to\mathbb{Z}/d$ denote reduction mod $d$ and let
$E_{d}$ denote the central extension $(r_{d})_{*}(E)$.

  1. 1.

Let $\Lambda\leq\Gamma$ be a subgroup of finite index and
$\iota:\Lambda\to\Gamma$ be the inclusion map. If $\widetilde{\Lambda}$
denotes the group of the central extension $\iota^{*}(E)$, then
$\widetilde{\Lambda}$ is residually finite if and only if $\widetilde{\Gamma}$
is residually finite.

  2. 2.

If the group $\widetilde{\Gamma}_{d}=\widetilde{\Gamma}/(d\mathbb{Z})$ is
residually finite for all $d>1$, then $\widetilde{\Gamma}$ is residually
finite.

  3. 3.

Suppose $E$ satisfies condition N of Definition 2.8. Then the group
$\widetilde{\Gamma}_{d}$ of $(r_{d})_{*}(E)$ is residually finite for all
$d\geq 1$.

###### Proof.

  1. 1.

Since $\widetilde{\Lambda}$ is a finite index subgroup of
$\widetilde{\Gamma}$, this equivalence is a standard property of residual
finiteness.

  2. 2.

As explained before the statement of Lemma 2.7, to prove residual finiteness
of $\widetilde{\Gamma}$ we only need to separate nontrivial elements of the
kernel of $q$, i.e., the infinite cyclic group $\langle\sigma\rangle$, from
the identity. Consider an element $\sigma^{i}$,
$i\in\mathbb{Z}\smallsetminus\\{0\\}$, and choose any integer $d>1$ prime to
$i$. Then $(r_{d})_{*}(\sigma^{i})$ is nontrivial in $\widetilde{\Gamma}_{d}$.
Since $\widetilde{\Gamma}_{d}$ is residually finite, there is a homomorphism
$\phi:\widetilde{\Gamma}_{d}\to F$ for some finite group $F$ with
$\phi(\sigma^{i})$ nontrivial. Then $\phi\circ(r_{d}):\widetilde{\Gamma}\to F$
has $\phi\circ(r_{d})(\sigma^{i})$ nontrivial, hence $\widetilde{\Gamma}$ is
residually finite.

  3. 3.

The statement is more technical, so is the proof. See [21, Lem. 2.6].

∎

###  2.3 Application of condition $N$

We will need the following useful fact about residually finite
$\mathbb{Z}/d$-extensions.

######  Proposition 2.11.

Suppose the group $\widetilde{\Gamma}_{d}$ of an extension $E$ of a group
$\Gamma$ by $\mathbb{Z}/d$ as in Equation (2) is residually finite. Then there
is a finite index subgroup $\Lambda\leq\Gamma$ such that $\iota^{*}(E)$ is the
trivial extension, where $\iota:\Lambda\hookrightarrow\Gamma$ is the
inclusion. Equivalently, if $\alpha\in H^{2}(\Gamma,\mathbb{Z}/d)$, then there
exists $\iota:\Lambda\hookrightarrow\Gamma$ of finite index so that
$\iota^{*}(\alpha)\in H^{2}(\Lambda,\mathbb{Z}/d)$ vanishes.

###### Proof.

The first conclusion is a restatement of [21, Lem. 2.2]. The second statement
is the reformulation in terms of characteristic classes. ∎

######  Theorem 2.12.

Suppose the extension $E$ of $\Gamma$ by $\mathbb{Z}$ with group
$\widetilde{\Gamma}$ satisfies condition $N$ of Definition 2.8 and that
$\Gamma$ is residually finite. Let $\eta\in H^{2}(\Gamma,\mathbb{Z})$ denote
the characteristic class of $E$. Then for each integer $d>1$ there exists a
subgroup $\Lambda\leq\Gamma$ of finite index, depending on $d$, such that all
of the following equivalent conditions hold:

  1. 1.

The extension
$\iota^{*}((r_{d})_{*}(E)):0\to\mathbb{Z}/d\to\widetilde{\Lambda}_{d}\to\Lambda\to
1$ (that is, the restriction to $\Lambda$ of the reduction of $E$ modulo $d$),
is the trivial extension.

  2. 2.

The class $\iota^{*}((r_{d})_{*}(\eta))=(r_{d})_{*}(\iota^{*}(\eta))\in
H^{2}(\Lambda,\mathbb{Z}/d)$ is trivial.

  3. 3.

The restriction $\iota^{*}(\eta)$ is divisible by $d$ in
$H^{2}(\Lambda,\mathbb{Z})$, that is, there exists ${\alpha\in
H^{2}(\Lambda,\mathbb{Z})}$ with $\eta=d\cdot\alpha$.

###### Proof.

Since $E$ satisfies condition $N$, the third statement of Lemma 2.10 gives us
that $\widetilde{\Gamma}_{d}=\widetilde{\Gamma}/d\mathbb{Z}$ is residually
finite. Proposition 2.11 gives us the first statement, and the second
statement is the equivalent formulation in terms of characteristic classes. ∎

###  2.4 Existence of branched covers

Let $(X,D)$ be a good pair in the sense of Definition 2.3 and
$\Gamma=\pi_{1}(X)$. We want to construct cyclic $d$-fold branched covers,
branched along $D$, in the sense of Definition 2.4. Proposition 2.6 gives a
number of equivalent conditions for the existence of such a cover, for
example, when $c_{1}(\mathcal{O}(D))$ divisible by $d$ in
$H^{2}(\Gamma,\mathbb{Z})$.

Given $(X,D)$ and $d$, it may be very difficult to check this divisibility
condition. However, there are situations where the following weaker statement
is already interesting: within a given class of good pairs $\\{(X,D)\\}$ prove
that there exist many members for which the divisibility condition holds. The
class of pairs $(X,D)$ of interest to us is that of arithmetic ball quotients
$X$ of simple type and $D$ a totally geodesic divisor on $X$. By “many
members” we mean that any $(X,D)$ in the class has a finite unramified cover
$p:X^{\prime}\to X$ so that $(X^{\prime},p^{*}(D))$ satisfies the above
divisibility condition. Here $p^{*}(D)$ means the divisor on $X^{\prime}$ with
support $p^{-1}(D)$ such that all connected components have multiplicity one.
We record the following simple but necessary fact:

######  Lemma 2.13.

Suppose $(X,D)$ is a good pair and $p:X^{\prime}\to X$ is a finite unramified
cover. Then $(X^{\prime},p^{*}(D))$ is a good pair. If $s$ is the canonical
section of $\mathcal{O}(D)$, then $p^{*}(s)$ is the canonical section of
$\mathcal{O}(p^{*}(D))=p^{*}(\mathcal{O}(D))$.

###### Proof.

Smoothness of $p^{-1}(D)$ and vanishing to first order of $p^{*}(s)$ on
$p^{-1}(D)$ are both clear from the fact that $p:X^{\prime}\to X$ is locally a
biholomorphism. ∎

######  Theorem 2.14.

Let $(X,D)$ be a good pair, let $d>1$ be an integer, and let:

  1. 1.

$\Gamma=\pi_{1}(X)$ and $\widetilde{\Gamma}=\pi_{1}(\mathcal{O}(D)^{\times})$;

  2. 2.

$E:0\to\mathbb{Z}\to\widetilde{\Gamma}\to\Gamma\to 1$ be the extension given
by the homotopy sequence of the fibration
$\mathcal{O}(D)^{\times}\to\mathcal{O}(D)$.

Suppose that $E$ satisfies condition $N$ of Definition 2.8. Then there is a
finite unramified cover $p:X^{\prime}\to X$ so that $c_{1}(p^{*}(D))$ is
divisible by $d$ in $H^{2}(X^{\prime},\mathbb{Z})$.

###### Proof.

This is an immediate consequence of Theorem 2.12, by defining $X^{\prime}$ to
be the covering space of $X$ associated with the subgroup $\Lambda\leq\Gamma$.
∎

######  Corollary 2.15.

Let $(X,D)$ be a good pair and suppose the extension

$0\longrightarrow\mathbb{Z}\longrightarrow\pi_{1}(\mathcal{O}(D)^{\times})\longrightarrow\pi_{1}(\mathcal{O}(D))\longrightarrow
1$

satisfies condition $N$. Then there is a finite unramified cover
$p:X^{\prime}\to X$ such that there exists a cyclic $d$-fold cover $Y\to
X^{\prime}$ branched along $p^{*}(D)$.

Finally, we give a sufficient condition for certain extensions that occur very
frequently to satisfy condition $N$. More precisely, we look at a class of
extensions for which we can characterize those that satisfy the stronger
condition $N_{2}$ of Definition 2.8.

######  Theorem 2.16 (Thm. 5.1 [21]).

Let $X$ be a closed aspherical manifold with fundamental group $\Gamma$ and
let $L\to X$ be a complex line bundle with Chern class $\omega\in
H^{2}(X,\mathbb{Z})$. Equivalently, let
$\widetilde{\Gamma}=\pi_{1}(L^{\times})$ and let $E$ be the extension as in
Equation (1) with group $\widetilde{\Gamma}$ and characteristic class
$\omega\in H^{2}(\Gamma,\mathbb{Z})$. Assume that $\omega$ has infinite order
in $H^{2}(\Gamma,\mathbb{Z})$ and let $\omega_{\mathbb{Q}}$ denote its image
in $H^{2}(\Gamma,\mathbb{Q})$. Then, $E$ satisfies condition $N_{2}$ of
Definition 2.8 if and only if $\omega_{\mathbb{Q}}$ is in the image of the map

$c_{\mathbb{Q}}:\bigwedge\nolimits^{2}H^{1}(X,\mathbb{Q})\longrightarrow
H^{2}(X,\mathbb{Q})$

given by evaluation of the cup product.

######  Remark 2.17.

In the original posted version of [21] this theorem is stated and proved with
the extra hypothesis that cup product with $\omega$ defines an _injective_ map
$H^{1}(X,\mathbb{Q})\to H^{3}(X,\mathbb{Q})$. This hypothesis is not
necessary, and a careful look at the proof shows that it is not used. We
explain briefly what we expect to write in a revised version.

Appearing at the bottom right of Diagram (3) in the statement of Lemma 5.6 is
an isomorphism

$H^{2}(Y,\mathbb{Q})\overset{\sim}{\longleftarrow}H^{2}(X,\mathbb{Q})/\langle\omega\rangle.$

Simply replace this part of the diagram by an injection

$H^{2}(Y,\mathbb{Q})\ \ {\hookleftarrow}\ \
H^{2}(X,\mathbb{Q})/\langle\omega\rangle,$

leaving the rest of the diagram unchanged. This uses exactness of the Gysin
sequence before the appearance of the map $H^{1}(X)\to H^{3}(X)$ given by cup
product with $\omega$.

Then, keep Lemma 5.7 as it is, and in the paragraph _Proof of Theorem 5.1_
keep everything exactly the same. The crucial point is the equality
$\pi^{*}(I_{X})=I_{Y}$, which follows from the isomorphism

$\pi^{*}:H^{1}(X,\mathbb{Q})\overset{\sim}{\longrightarrow}H^{1}(Y,\mathbb{Q})$

and consequently the isomorphism

$\pi^{*}:\bigwedge\nolimits^{2}H^{1}(X,\mathbb{Q})\overset{\sim}{\longrightarrow}\bigwedge\nolimits^{2}H^{1}(Y,\mathbb{Q}).$

The rest of the proof is unchanged.

##  3 Arithmetic lattices in $\operatorname{PU}(n,1)$

###  3.1 Congruence arithmetic lattices of simple type

The arithmetic lattices under consideration in this paper are those of _simple
type_. These are constructed as follows, where we generally follow the
notation established in [4, §6]. Let $E/F$ be a totally imaginary quadratic
extension of a totally real field with $[F:\mathbb{Q}]=d\geq 2$, let
$\tau_{1},\dots,\tau_{d}:E\to\mathbb{C}$ be representatives for the complex
conjugate pairs of embeddings of $E$, and let $x\mapsto\overline{x}$ be the
Galois involution of $E/F$, which extends to complex conjugation under any
complex embedding of $E$.

Fix a nondegenerate hermitian vector space $V$ over $E$ of dimension $n+1$,
and let $V_{\tau_{j}}\cong\mathbb{C}^{n+1}$ be the completion of $V$ with
respect to the complex embedding $\tau_{j}$ of $E$. We assume that
$V_{\tau_{1}}$ has signature $(n,1)$ and that $V_{\tau_{j}}$ is definite for
$j\geq 2$. The assumption that $d\geq 2$ implies that $j\geq 2$, so there is
at least one completion where $V$ is a definite hermitian space. It follows
that $V$ is anisotropic.

The unitary group $\operatorname{U}(V)$ is a reductive algebraic group over
$F$, and let $G_{1}=\mathrm{Res}_{F/\mathbb{Q}}\operatorname{U}(V)$ be the
restriction of scalars from $F$ to $\mathbb{Q}$. The above assumptions on the
signature of $V$ at the various complex embeddings of $E$ imply that

$G_{1}(\mathbb{R})\cong\operatorname{U}(n,1)\times\prod_{j\geq
2}\operatorname{U}(n+1).$

We can identify the ball $\mathbb{B}^{n}$ with $G_{1}(\mathbb{R})/K_{\infty}$,
where

$K_{\infty}\cong\left(\operatorname{U}(n)\times\operatorname{U}(1)\right)\times\prod_{j\geq
2}\operatorname{U}(n+1)$

is a chosen maximal compact subgroup of $G_{1}(\mathbb{R})$.

Similarly, $G$ will denote the restriction of scalars from $F$ to $\mathbb{Q}$
of the group of unitary similitudes of $V$. If $\mathbb{A}_{\mathbb{Q}}$
denotes the adeles of $\mathbb{Q}$ and $\mathbb{A}_{\mathbb{Q}}^{f}$ the
finite adeles, for each prime $p$ choose an open compact subgroup
$K_{p}<G(\mathbb{Q}_{p})$, and set $K=\prod
K_{p}<G(\mathbb{A}_{\mathbb{Q}}^{f})$. We then obtain a Shimura variety

$S(K)=G(\mathbb{Q})\backslash\left(\mathbb{B}^{n}\times
G(\mathbb{A}_{\mathbb{Q}}^{f})\right)/K.$

As described in [4, §6.4], $S(K)$ is a finite disjoint union of connected
components $S(\Gamma_{j})=\Gamma_{j}\backslash\mathbb{B}^{n}$, where

$\Gamma_{j}<G_{\textrm{ad}}(\mathbb{R})\cong\operatorname{PU}(n,1)\times\prod_{j\geq
2}\operatorname{PU}(n+1)$

is a cocompact lattice. We will call lattices $\Gamma_{j}$ of this kind
_congruence arithmetic lattices of simple type_. If we choose $K$ sufficiently
small (e.g., neat) then each $\Gamma_{j}$ is torsion-free and the ball
quotient $S(\Gamma_{j})$ is both a manifold and a smooth projective variety.
From this point forward, all $K$ are assumed to be sufficiently small in this
sense.

###  3.2 Cohomology of Shimura varieties

In the notation from §3.1, if $K^{\prime}\leq
K<G(\mathbb{A}_{\mathbb{Q}}^{f})$, then there is a finite map
$S(K^{\prime})\to S(K)$ with pullback

$H^{*}(S(K),R)\longrightarrow H^{*}(S(K^{\prime}),R)$

for any coefficient ring $R$. This allows us to define

$H^{*}(\mathrm{Sh}(G),R)=\varinjlim_{K}H^{*}(S(K),R).$

See [4, §6.6] for further details. For each $K$, we also have the Chern form
$c_{1}(S(K))\in H^{2}(S(K),\mathbb{Z})$, which of course behaves well with
respect to pull-back. In particular, we obtain a limiting class
$\widetilde{c}_{1}\in H^{2}(\mathrm{Sh}(G),\mathbb{Z})$.

###  3.3 Totally geodesic hypersurfaces

We start with the following, which justifies restricting to lattices of simple
type when considering arithmetic ball quotients containing a totally geodesic
hypersurface. The case $n=2$ is found in [16, §1].

######  Proposition 3.1.

For $n\geq 2$, let $\Gamma<\operatorname{PU}(n,1)$ be an arithmetic lattice so
that $\Gamma\backslash\mathbb{B}^{n}$ contains a totally geodesic codimension
one subvariety. Then $\Gamma$ is of simple type.

###### Sketch.

Suppose $\Gamma<\operatorname{PU}(n,1)$ is arithmetic. Then there exists a
totally imaginary quadratic field $E$, a central simple division algebra
$\mathcal{D}$ over $E$ of degree $d$ with involution $\sigma$ of the second
kind, and a $\sigma$-hermitian form $h$ on $\mathcal{D}^{r}$ so that the
algebraic group $\mathcal{G}$ associated with $\Gamma$ is the projective
unitary group of $h$. Note that $n+1=dr$. The case where $\Gamma$ has simple
type is precisely the case $d=1$ (i.e., $\mathcal{D}=E$) and $r=n+1$.

If $\Gamma\backslash\mathbb{B}^{n}$ contains a totally geodesic subvariety of
the form $\Lambda\backslash\mathbb{B}^{n-1}$, it is well known that $\Lambda$
is also arithmetic. Then there is a totally imaginary subfield $E_{0}$ of $E$,
a subalgebra $\mathcal{D}_{0}\subseteq\mathcal{D}$ over $E_{0}$ of degree
$d_{0}$ dividing $d$, and an embedding of $\mathcal{D}_{0}^{r_{0}}$ in
$\mathcal{D}^{r}$ so that $d_{0}r_{0}=n$ and the restriction of $h$ to the
image of $\mathcal{D}_{0}^{r_{0}}$ is the algebraic group associated with
$\Lambda$. The fact that $\mathcal{G}$ is noncompact at exactly one place of
$E$ forces $E_{0}=E$. Now, $d_{0}$ divides $n$ and $n+1$, which forces
$d_{0}=1$. Then $r_{0}=n$, but $r_{0}\leq r\leq n+1$, so $r$ is $n$ or $n+1$.
However, $n\geq 2$ and $r$ divides $n+1$, so $r=n+1$, hence $d=1$, which
proves the proposition. ∎

One often classifies the immersed totally geodesic complex codimension one
subvarieties of arithmetic ball quotients $\Gamma\backslash\mathbb{B}^{n}$ by
$\Gamma$-orbits of lines $\ell\subset V$ so that the restriction of the
hermitian form to $\ell$ is positive under $\ell\hookrightarrow V_{\tau_{1}}$
or, dually, by codimension one subspaces $W\subset V$ so that $W_{\tau_{1}}$
has signature $(n-1,1)$. We instead follow the notation of Kudla–Millson [15,
§2 & 9] and Bergeron–Millson–Moeglin [4, §8], since it will be used for the
proof of our main results.

For each $d\times d$ hermitian matrix $\beta\in\operatorname{M}_{d}(E)$ that
is positive definite at $\tau_{1}$, Kudla and Millson define a _special cycle_
$C_{\beta}$. This is a locally finite cycle consisting of totally geodesic
subvarieties of (complex) codimension $d$ [15, p. 132]. Moreover, all immersed
totally geodesic subvarieties are contained in the support of some
$C_{\beta}$. See the above references for further technical details, since
they will not concern us in this paper. As pointed out to us by Nicolas
Bergeron, the following result has the exact same proof as [3, Cor. 8.4],
except that paper covers the orthogonal case. We give the argument for
completeness.

######  Theorem 3.2.

Let $X=\Gamma\backslash\mathbb{B}^{n}$ be a smooth compact ball quotient,
$n\geq 2$, with $\Gamma$ a congruence arithmetic lattice of simple type, and
let $c_{1}$ be the Chern form on $X$, considered as an element of
$H^{2}(X,\mathbb{C})$. Then $c_{1}$ is contained in the span of the collection
of all Poincaré duals to codimension one totally geodesic subvarieties of $X$.

###### Proof.

Suppose not. Then there exists $\eta\in
H^{2}(X,\mathbb{C})^{*}=H^{2(n-1)}(X,\mathbb{C})$ such that:

$\displaystyle\int_{X}\eta\wedge c_{1}$ $\displaystyle\neq 0$
$\displaystyle\int_{C}\eta$ $\displaystyle=0$ $\displaystyle\textrm{for all
totally geodesic}~{}C~{}\textrm{of codimension}~{}1$

In other words, the periods of $\eta$ over all codimension $1$ geodesic
subvarieties are zero.

Following, [15, p. 126], let $\mathcal{L}(t)$ be the space of hermitian
$1\times 1$ matrices of rank $t\leq 1$ over the ring of integers $\mathcal{O}$
of the totally imaginary field $E$ from the definition of the arithmetic
lattice $\Gamma$. Of course, $\mathcal{L}(1)=\mathcal{O}^{*}$ and
$\mathcal{L}(0)=\\{0\\}$. For each $\beta\in\mathcal{L}(t)$, we have the
totally geodesic cycle $C_{\beta}$ of codimension $t$ in $X$, where $C_{0}=X$
and for $\beta\in\mathcal{O}^{*}$ we have $C_{\beta}$ of codimension $1$.

For $\beta\in\mathcal{O}$ and $\zeta\in\mathbb{C}^{[F:\mathbb{Q}]}$, following
[15, p. 125], define

$e_{*}(\beta\zeta)=\prod_{j=1}^{[F:\mathbb{Q}]}\exp(\pi
i\operatorname{tr}_{\mathbb{C}/\mathbb{R}}(\tau_{j}(\beta))\zeta_{\tau_{j}}),$

where the product is over the places $\\{\tau_{j}\\}$ of $E$ and the
coordinates of $\zeta$ are labelled accordingly. Note that $e_{*}(0\zeta)=1$
for all $\zeta$. As on [15, p. 127] we then have that

$\displaystyle P(\zeta,\eta)$
$\displaystyle=\sum_{t=0}^{1}\sum_{\beta\in\mathcal{L}(t)}\left(\int_{C_{\beta}}\eta\wedge
c_{1}^{n-t}\right)e_{*}(\beta\zeta)$
$\displaystyle=e_{*}(0\zeta)\int_{X}\eta\wedge
c_{1}+\sum_{\beta\in\mathcal{O}^{*}}e_{*}(\beta\zeta)\int_{C_{\beta}}\eta$
$\displaystyle=\int_{X}\eta\wedge c_{1}$

is a constant function of $\zeta$ with value $\int_{X}\eta\wedge c_{1}\neq 0$,
since $\int_{C_{\beta}}\eta=0$ for all $\beta$ by hypothesis.

However, by [15, Thm. 3], the function $P(\zeta,\eta)$ of $\zeta$ is a
holomorphic modular form of weight $n+1$ for a suitable congruence subgroup of
the group of $\mathcal{O}$-points of a certain form of $\operatorname{U}(1,1)$
over $F$. Since a nonzero constant function is not a holomorphic modular form
of higher weight, this contradiction proves the theorem. ∎

###  3.4 The cup product and special cycles

The purpose of this section is to prove the following theorem, whose proof was
communicated to us by Nicolas Bergeron.

######  Theorem 3.3.

Let $X=\Gamma\backslash\mathbb{B}^{n}$ be a compact congruence arithmetic ball
quotient of simple type and $\mathrm{SC}^{1}(X)\subseteq H^{2}(X,\mathbb{C})$
be the span of the Poincaré duals to the totally geodesic codimension one
subvarieties of $X$. Then:

  1. 1.

For any class $\sigma\in\mathrm{SC}^{1}(X)$, there is a congruence cover
$p:X^{\prime}\to X$ so that $p^{*}(\sigma)\in\mathrm{SC}^{1}(X^{\prime})$ is
in the image of the cup product map

$c:\bigwedge\nolimits^{2}H^{1}(X^{\prime},\mathbb{C})\longrightarrow
H^{2}(X^{\prime},\mathbb{C}).$

  2. 2.

If $n\geq 3$, then for every $\phi\in H^{1,1}(X,\mathbb{C})$ we can find a
congruence cover $p:X^{\prime}\to X$ so that $p^{*}(\phi)$ is contained in the
image of $c$.

  3. 3.

If $n\geq 4$, then for all $\phi\in H^{2}(X,\mathbb{C})$ we can find a
congruence cover $p:X^{\prime}\to X$ so that $p^{*}(\phi)$ is in the image of
$c$.

###### Proof.

Let $G$ be the $\mathbb{Q}$-algebraic group associated with $X$ and $\Gamma$
as in §3.1, and suppose that $X$ is a connected component of the Shimura
variety $S(K)$. The theorem follows fairly directly from deep structural
results on cohomology arising from the theta correspondence; see [4, §7] for a
precise definition.

Taking $q=1$ and the appropriate values of $(a,b)$ in [4, Cor. 7.3], we see
that $H^{1}(X,\mathbb{C})$ is generated by the classes of theta lifts for all
$n\geq 2$, $H^{1,1}(X,\mathbb{C})$ is generated by theta lifts when $n\geq 3$,
and all of $H^{2}(X,\mathbb{C})$ is generated by the theta correspondence for
$n\geq 4$. For $n\geq 2$, Kudla and Millson proved that the Poincaré duals to
codimension $1$ geodesic subvarieties are contained in the subspace generated
by theta lifts and Poincaré duals to special cycles span the subspace
generated by theta lifts [13, 14, 15] (also see [4, p. 9]).

Now, suppose that $\sigma\in H^{2}(X,\mathbb{C})$ is contained in the subspace
generated by theta lifts. The above discussion implies that this is the case
for every class satisfying one of the three conditions from the statement of
the theorem. Let

$\widetilde{\sigma}\in
H^{*}(\mathrm{Sh}(G),\mathbb{C})=\varinjlim_{K}H(S(K),\mathbb{C})$

be the image of $\sigma$ in $H^{2}(\mathrm{Sh}(G),\mathbb{C})$ as in §3.2.
Using [4, Prop. 5.4 & 5.19], arguing almost verbatim as in the proof of [4,
Thm. 9.3] (the only change being the degrees of the forms under
consideration), we have that $\widetilde{\sigma}$ is in the image of the cup
product map

$\bigwedge\nolimits^{2}H^{1}(\mathrm{Sh(G)},\mathbb{C})\longrightarrow
H^{2}(\mathrm{Sh(G)},\mathbb{C}).$

This implies that there is a congruence cover $p:X^{\prime}\to X$ so that

$p^{*}(\sigma)\in\mathrm{Im}\left(\bigwedge\nolimits^{2}H^{1}(X^{\prime},\mathbb{C})\longrightarrow
H^{2}(X^{\prime},\mathbb{C})\right),$

which completes the proof of the theorem. ∎

######  Remark 3.4.

By the universal coefficients theorem, Theorem 3.3 is also true with
coefficients in any subfield of $\mathbb{C}$. In particular, it holds over
$\mathbb{Q}$.

Theorem 3.2 and Theorem 3.3 combine with Remark 3.4 to give the following.

######  Corollary 3.5.

Let $X$ be a smooth compact congruence arithmetic ball quotient of simple type
with dimension $n\geq 2$. Suppose that $\phi\in H^{2}(X,\mathbb{Z})$ is one of
the following classes:

  1. 1.

$c_{1}(X)$;

  2. 2.

$c_{1}(D)$ for $D\subset X$ a reduced effective divisor with support a union
of totally geodesic subvarieties, or any linear combination of such classes;

  3. 3.

any class in $H^{1,1}(X,\mathbb{Q})\cap H^{2}(X,\mathbb{Z})$ if $n\geq 3$;

  4. 4.

an arbitrary element of $H^{2}(X,\mathbb{Z})$ if $n\geq 4$.

Then there is a congruence cover $p:X^{\prime}\to X$ so that the image of
$p^{*}(\phi)$ in $H^{2}(X^{\prime},\mathbb{Q})$ is contained in the image of
the cup product map

$c_{\mathbb{Q}}:\bigwedge\nolimits^{2}H^{1}(X^{\prime},\mathbb{Q})\longrightarrow
H^{2}(X^{\prime},\mathbb{Q}).$

######  Remark 3.6.

Passage to a finite cover in Corollary 3.5 is necessary. For example, let $X$
be the Cartwright–Steger surface. Then $X$ is the quotient of $\mathbb{B}^{2}$
by a congruence arithmetic lattice of simple type (e.g., see [20, p. 90]).
Then $H^{1}(X,\mathbb{Z})\cong\mathbb{Z}^{2}$, so the image of the cup product
in $H^{2}(X,\mathbb{Z})$ is generated by the primitive element
$\alpha\wedge\beta$, where $\alpha$ and $\beta$ span $H^{1}(X,\mathbb{Z})$.
Since $c_{1}(X)^{2}\neq 0$ but $(\alpha\wedge\beta)^{2}=0$, we see that
$c_{1}(X)$ cannot be in the image of the cup product map. As shown by Dzambic
and Roulleau [10], the Stover surface, a $21$-fold étale cover of $X$ with
fundamental group a congruence arithmetic lattice, satisfies the conclusion of
Corollary 3.5.

##  4 Proofs of Theorems 1.1 and 1.2

We begin by proving Theorem 1.2, then use it to prove Theorem 1.1 as sketched
in the introduction.

###### Proof of Theorem 1.2.

Suppose that $\Gamma<\operatorname{PU}(n,1)$, $\phi\in
H^{2}(\Gamma\backslash\mathbb{B}^{n},\mathbb{Z})$, and $\widetilde{\Gamma}$
satisfy the hypotheses of the theorem. By §2.1.1 (in particular Remark 2.1)
and Lemma 2.2, if $\Gamma^{\prime}\leq\Gamma$ is a finite index subgroup, then
the preimage $\widetilde{\Gamma}^{\prime}$ of $\Gamma^{\prime}$ in
$\widetilde{\Gamma}$ is the central extension of $\Gamma^{\prime}$ with
characteristic class $p^{*}(\phi)$, where
$p:\Gamma^{\prime}\backslash\mathbb{B}^{n}\to\Gamma\backslash\mathbb{B}^{n}$
is the associated cover. Thus $\widetilde{\Gamma}^{\prime}$ is a finite index
subgroup of $\widetilde{\Gamma}$, and by Lemma 2.10 it suffices to prove that
$\widetilde{\Gamma}^{\prime}$ and the associated reductions
$\widetilde{\Gamma}_{d}^{\prime}$ of $\widetilde{\Gamma}^{\prime}$ modulo $d$
are residually finite.

By Corollary 3.5, we can choose $\Gamma^{\prime}$ to be a congruence subgroup
of $\Gamma$ such that

$p^{*}(\phi)\in\mathrm{Im}\left(\bigwedge\nolimits^{2}H^{1}(X^{\prime},\mathbb{Q})\longrightarrow
H^{2}(X^{\prime},\mathbb{Q})\right),$

where $X^{\prime}=\Gamma^{\prime}\backslash\mathbb{B}^{n}$. By Theorem 2.16,
the extension with group $\widetilde{\Gamma}^{\prime}$ satisfies condition
$N_{2}$. Since $\Gamma^{\prime}<\operatorname{PU}(n,1)$ is residually finite,
$\widetilde{\Gamma}^{\prime}$ is residually finite by Lemma 2.9 and
$\widetilde{\Gamma}_{d}$ is residually finite by Lemma 2.10. Thus
$\widetilde{\Gamma}$ and $\widetilde{\Gamma}_{d}$ are also residually finite,
and this completes the proof of the theorem. ∎

We now deduce Theorem 1.1 from Theorem 1.2.

###### Proof of Theorem 1.1.

Let $\Gamma<\operatorname{PU}(n,1)$ be a cocompact arithmetic lattice of
simple type and $\widetilde{\Gamma}$ its preimage in
$\widetilde{\operatorname{PU}}(n,1)$. If $n=1$, then it is classical that
$\widetilde{\Gamma}$ is residually finite; see the introduction to [21] for
discussion. If $n\geq 2$, first note that if $\Lambda$ is commensurable with
$\Gamma$, then the preimage $\widetilde{\Lambda}$ of $\Lambda$ to
$\widetilde{\operatorname{PU}}(n,1)$ is commensurable with
$\widetilde{\Gamma}$ (cf. Remark 2.1). Therefore $\widetilde{\Gamma}$ is
residually finite if and only if $\widetilde{\Lambda}$ is residually finite,
hence we can assume without loss of generality that $\Gamma$ is a torsion-free
congruence arithmetic lattice.

Now, $\widetilde{\Gamma}$ is the central extension of $\Gamma$ with
characteristic class $c_{1}(\Gamma\backslash\mathbb{B}^{n})$. Moreover, the
reduction $\widetilde{\Gamma}_{d}$ of $\widetilde{\Gamma}$ modulo $d$ is
isomorphic to the preimage of $\Gamma$ in the $d$-fold connected cover of
$\operatorname{PU}(n,1)$. By Theorem 3.2, we have that
$c_{1}(\Gamma\backslash\mathbb{B}^{n})$ is contained in the span of the
Poincaré duals to the totally geodesic cycles on
$\Gamma\backslash\mathbb{B}^{n}$, so Theorem 1.2 applies to give residual
finiteness of $\widetilde{\Gamma}$ and $\widetilde{\Gamma}_{d}$. This
completes the proof. ∎

##  5 Branched covers of ball quotients

We begin with the key application of our main results, which provides
existence of the branched covers used to prove Theorem 1.5.

######  Proposition 5.1.

Fix $d\geq 2$. Let $X=\Gamma\backslash\mathbb{B}^{n}$ be a smooth compact
complex hyperbolic $n$-manifold, $n\geq 2$, with $\Gamma$ a congruence
arithmetic lattice of simple type. Then we can pass to a congruence cover
$X^{\prime}=\Gamma^{\prime}\backslash\mathbb{B}^{n}$ so that $X^{\prime}$
contains a totally geodesic divisor $D^{\prime}$ such that:

  1. 1.

$(X^{\prime},D^{\prime})$ is a good pair in the sense of Definition 2.3, and
consequently

  2. 2.

there exists a smooth projective variety $Y$ and a cyclic branched cover
$f:Y\to X^{\prime}$ with branch divisor $D^{\prime}$ and branching of degree
$d$ along each irreducible component of $D^{\prime}$.

###### Proof.

As described in §3.3, it is well known that $X$ contains an _immersed_ totally
geodesic codimension one ball quotient $D$. It is a standard fact that we can
replace $X$ with a finite congruence cover and assume that $D$ is in fact
embedded, hence $(X,D)$ is a good pair in the sense of Definition 2.3.
Corollary 3.5 and Theorem 2.16 imply that condition $N_{2}$ is satisfied by
the central extension of $\Gamma$ by $\mathbb{Z}$ associated with
$\pi_{1}(\mathcal{O}(D)^{\times})$ as in §2.1. Therefore, by Corollary 2.15
there is a further finite cover $p:X^{\prime}\to X$ so that there exists a
cyclic $d$-fold branched cover $Y\to X^{\prime}$, branched over the totally
geodesic divisor $D^{\prime}=p^{*}(D)$, with $Y$ smooth. ∎

We will apply the following theorem of Zheng [23], where we restate Zheng’s
notion of a “good cover” with smooth totally geodesic branch divisor in our
language. His condition _negative definite complex curvature operator_ is the
same as Siu’s very strong negative curvature, which implies strong negative
curvature, which in turn implies negative sectional curvature. See [18, §2].

######  Theorem 5.2 (Thm. 1 [23]).

Let $f:Y\to X$ be a $d$-fold cyclic branched cover of a smooth compact ball
quotient $X$ with branch divisor $D$ a disjoint union of smooth totally
geodesic codimension one subvarieties and branching of order $d$ along each
component. Then $Y$ admits a Kähler metric with negative definite complex
curvature operator. In fact, the metric on $Y$ is strongly negative in the
sense of Siu, hence $Y$ is strongly rigid.

We now prove Theorem 1.5.

###### Proof of Theorem 1.5.

Let $Y$ be one of the smooth projective varieties provided by Proposition 5.1.
Then $Y$ admits a metric of strongly negative curvature by Theorem 5.2, so to
complete the proof we must show that $Y$ is not homotopy equivalent to a
locally symmetric manifold. Since $Y$ is a closed manifold admitting a
complete metric of negative curvature, it is well known that $\pi_{1}(Y)$
cannot contain a free abelian subgroup of rank $r\geq 2$. A cocompact lattice
in a semisimple Lie group of real rank $r$ contains a subgroup isomorphic to
$\mathbb{Z}^{r}$ by a theorem of Wolf [22, Thm. 4.2], hence we only need to
rule out the possibility that $\pi_{1}(Y)$ is isomorphic to a lattice in an
adjoint simple Lie group of real rank one.

Let $Z$ be a rank one locally symmetric space homotopy equivalent to $Y$. We
can exclude $Z$ being real, quaternionic, or Cayley hyperbolic by a result of
Carlson and Toledo [6, Cor. 3.3]: since $Y$ is a compact Kähler manifold, it
cannot be homotopy equivalent to a locally symmetric space $Z$ unless $Z$ is
locally Hermitian symmetric. This and real rank one implies that $Z$ is a ball
quotient. Considering cohomological dimension we see that $\dim(Y)=\dim(Z)$.

If $f:Y\to Z=\Lambda\backslash\mathbb{B}^{n}$ is a homotopy equivalence, and
$g:Z\to Y$ a homotopy inverse to $f$, since both $Y$ and $Z$ have strongly
negative curvature, Siu rigidity [18, Thm. 1] implies that we can assume that
both $f$ and $g$ are holomorphic maps. Thus $f\circ g$ and $g\circ f$ are
holomorphic self maps of $Z$ and $Y$, respectively, that are homotopic to the
identity, hence they are the identity. It follows that $f$ is a
biholomorphism, and $g=f^{-1}$. See also [18, Thm. 2]

By definition of a branched cover, $\mathbb{Z}/d$ acts on $Y$ by holomorphic
automorphisms with quotient the ball quotient $X^{\prime}$ from the statement
of Proposition 5.1. Let $\rho\in\operatorname{Aut}(Y)$ be a generator for this
action and $D^{\prime}$ be the branch locus, which we recall is a disjoint
union of smooth totally geodesic subvarieties of $X^{\prime}$. If
$\mathcal{D}$ denotes the preimage of $D^{\prime}$ on $Y$, then $\mathcal{D}$
is a divisor with support a disjoint union of smooth subvarieties of $Y$, each
isomorphic to a smooth totally geodesic subvariety $D_{i}\subset X^{\prime}$.
The fixed point set of $\rho$ is $\mathcal{D}$, and the cyclic group generated
by $\rho$ acts freely on $Y\smallsetminus\mathcal{D}$.

Consider the holomorphic automorphism

$\sigma=\left(f\circ\rho\circ f^{-1}\right):Z\to Z$

of period $d$ fixing the divisor $f(\mathcal{D})$. Since automorphisms of $Z$
are isometries for its Bergman metric and fixed sets of isometries are totally
geodesic, it follows that $f(\mathcal{D})$ is a totally geodesic divisor on
$Z$ and the support of $f(\mathcal{D})$ consists of a disjoint union of smooth
subvarieties of $Z$ isomorphic to the ball quotient $D_{i}$. In particular,
each $D_{i}$ is isometric to a smooth totally geodesic subvariety of both
$X^{\prime}$ and $Z$. Moreover, since $f$ is a biholomorphism, the normal
bundle to $D_{i}$ in $Z$ is isomorphic to the normal bundle of $D_{i}$ in $Y$.
We use this to derive a contradiction.

If $X$ is an $n$-dimensional ball quotient and $D_{i}$ a smooth totally
geodesic subvariety of codimension one, then the normal bundle to $D_{i}$ in
$X$ has first Chern class $\nu$ satisfying $n\nu=c_{1}(D_{i})$ in
$H^{2}(D_{i},\mathbb{Q})$. One way to see this is to use the principle of
proportionality in dual symmetric spaces: any relation between characteristic
classes with $\mathbb{Q}$-coefficients that holds in one element of a dual
pair must also hold in the other. For $\mathbb{P}^{n-1}\subset\mathbb{P}^{n}$
this equality is easily checked. Indeed, the normal bundle is the bundle
$\nu=\mathcal{O}(1)$ with ${c_{1}(\nu)=x}$, where $x$ is a positive generator
of $H^{2}(\mathbb{P}^{n-1},\mathbb{Z})$, while ${c_{1}(\mathbb{P}^{n-1})=nx}$,
thus the equivalent equality must also hold for a geodesic divisor in a ball
quotient.

In particular, the first Chern class of the normal bundle to $D_{i}$ is
completely independent of the ambient ball quotient. Applying this to
$X^{\prime}$ and considering the branched cover $Y\to X^{\prime}$, the normal
bundle to $D_{i}$ in $Y$ has first Chern class

$d\nu\in H^{2}(D_{i},\mathbb{Q}),$

hence the same holds for the normal bundle to $D_{i}$ in $Z$. This implies
that $c_{1}(D_{i})=dc_{1}(D_{i})$, but $d\neq 1$ and $c_{1}(D_{i})$ is
nontrivial, hence we arrive at a contradiction. This proves that $Y$ cannot be
homotopy equivalent to a ball quotient, which completes the proof of the
theorem. ∎

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# Multi-style Neural Radiance Field with AdaIN

Yu-Wen  Pao  An-Jie  Li

###### Abstract

In this work, we propose a novel pipeline that combines AdaIN and NeRF for the
task of stylized Novel View Synthesis111The implementation is available at
https://github.com/paoyw/Stylized-NeRF-with-AdaIN. Compared to previous works,
we make the following contributions: 1) We simplify the pipeline. 2) We extend
the capabilities of model to handle the multi-style task. 3) We modify the
model architecture to perform well on styles with strong brush strokes. 4) We
implement style interpolation on the multi-style model, allowing us to control
the style between any two styles and the style intensity between the stylized
output and the original scene, providing better control over the stylization
strength.

##  1 Introduction

Stylized Novel View Synthesis is a technique that takes real-world scene
photographs as input and generates stylized images of the same scene from
different viewpoints. This technology enables various applications in computer
graphics by creating virtual scenes with artistic styles based on real-world
environments.

Previous works have explored the combination of AdaIN and NeRF techniques for
stylized Novel View Synthesis. However, we found that previous architectures
were complex, and each model could only support a single style. We aimed to
design a simpler approach that achieves comparable results to previous works
while allowing the model to support multiple styles.

To achieve our objectives, we incorporate the style feature extracted by AdaIN
and the voxels information from NeRF into the final layers of the NeRF model
(which are Multilayer Perceptrons (MLP) used to predict color and density of
the voxel) . This integration allows us to stylize the original scene voxel-
wise and ultimately achieve the stylized rendering of the entire scene.

Through extensive experimentation, our proposed pipeline effectively stylizes
scenes and performs well when switching between different styles or training
with multiple styles. Additionally, we implemented style interpolation
functionality on our multi-style model, allowing us to blend two different
styles or control the degree of stylization through interpolation between the
stylized scene and the original scene.

##  2 Related Works

###  2.1 Image Style Transfer

In recent years, image style transfer has been a popular research topic in
computer vision and image processing. One line of work focuses on deep
learning-based methods. These approaches aim to learn representations that
capture the style and content of input images separately, enabling the style
transfer between different contents images. Gatys et al.[1] introduced a
neural network-based method, which utilized deep convolution neural networks
to separate the representations of the content and style images and then
recombine them to create visually appealing stylized outputs. Huang et al.[2]
introduced AdaIN which enables the style transfer process by dynamically
adjusting the mean and variance of features in a deep network, effectively
transferring the style statistics from a style image to a content image. Zhu
et al.[3] proposed an unsupervised image-to-image translation framework, which
leverages the power of generative adversarial networks to learn mapping
functions between two domains.

###  2.2 Novel View Synthesis

Novel view synthesis aims to generate visually consistent and plausible images
from unseen viewpoints using limited input information. In recent years, due
to the flourishing of deep learning, neural networks have brought new light to
this field. NeRF[4] uses MLPs to learn the mapping function between positions
and view directions to color and density.

###  2.3 Style-transfer with NeRF

Chiang et al.[5] is one of the pioneers to combine style transfer and NeRF.
Their framework consists of two networks, a NeRF to find the implicit
representation of the 3D scene and a hypernetwork to transfer the style.
What’s more, they use two-stage training and a patch sub- sampling approach
procedure to fix the problem of memory burden and training difficulties.

Huang et al.[6] combined NeRF and AdaIN using mutual learning. After training
the vanilla NeRF and AdaIN, it mutually trains the style module, its inputs,
which are learnable latent code, and the decoder of AdaIN with a combination
of four loss terms.

Zhang et al.[7] introduced a nearest neighbor-based loss to capture style
details while maintaining multi-view consistency. What’s more, they proposed
the deferred back-propagation method to make optimization of memory-intensive
NeRF using style losses defined on full-resolution rendered images possible.

However, most of the methods mentioned above ignore that density in 3D scenes
can be an important role in some styles. Also, they cannot transfer a single
scene to multiple styles using only one model.

##  3 Method

We propose a simple framework combining AdaIN and NeRF to achieve multi-style
transfer for novel view synthesis. To be more specific, we use a three-stage
training procedure. First, the 2D stylization training stage learns the
generation of stylized images by AdaIN. Secondly, the geometric training stage
learns the mapping function of position and view direction to opacity and
radiance color of a 3D scene by NeRF. Finally, the 3D scene stylization
training stage uses the camera parameters and its corresponding image after
stylization as inputs and takes the MLPs with part of pre-trained NeRF from
the previous stage as the network to learn the style transfer of a 3D scene.

###  3.1 2D Stylization Network

For 2D stylization network, we use AdaIN[2] as our method. First, AdaIN uses
pre-trained VGG[8] as encoder to extract the features of the content and style
images. Secondly, adaptive instance normalize layer aligns the channel-wised
means and variance of content image to those of style image. Finally, a CNN-
based decoder will decode the transformed features to stylized images. The
loss function to train the decoder
$\mathcal{L}=\mathcal{L_{c}}+\lambda\mathcal{L_{s}}$, where $\mathcal{L_{c}}$
is the content loss and $\mathcal{L_{s}}$ is the style loss. The content loss
$\mathcal{L_{c}}$ aligns the feature to reconstruct stylized images and the
feature of the stylized images. Style loss $L_{s}$ aligns the means and
variances of the features of style images and stylized images. We train an
AdaIN using MS-COCO[9] as our content images and 100 images mostly from
WikiArt as our style images.

###  3.2 NeRF

NeRF[4] uses MLPs to model the randiance color and opacity of a 3D scene. The
MLPs take 3D position $\textbf{x}\in\mathbb{R}^{3}$ and viewing direction
$\mathbf{b}\in\mathbb{R}^{2}$ as inputs. The outputs of MLPs are opacity
$\sigma(\mathbf{x})\in\mathbb{R}$ and radiance color
$c(\mathbf{x},\mathbf{d})\mathbb{R}^{3}$. The following integral tells us how
to obtain the single pixel on the image from the radiance color and opacity
along a ray.

$C(\mathbf{r})=\int_{t=0}^{\infty}\sigma(\mathbf{o}+t\mathbf{d},\mathbf{d})\exp{(-\int_{s=0}^{t}\sigma(\mathbf{o}+s\mathbf{d})ds)}$

To estimate this continuous integral, they uses quadrature. More specific,
they use a two-stage sampling strategy, a stratified sampling, which sample
one sample in evenly-spaced bins, and hierarchical volume sampling, which
tends to samples more points biased towards the parts with high density.

To improve the performance, NeRF also uses positional encoding $\gamma(\cdot)$
to map $\mathbf{x}$ and d to Fourier features.

$\gamma(x)=(\sin{(x)},\cos{(x)},\dots,\sin{(2^{L-1}x)},\cos{(2^{L-1}x)})$

We take the PyTorch implementation of NeRF from [10].

###  3.3 Multi-style NeRF

Figure 1: Multi-style NeRF

Given a set of N images $\\{\mathcal{I}_{n}\\}_{n=1}^{N}$ and their
corresponding camera poses $\\{(R_{n},t_{n})\\}_{n=1}^{N}$. We train the
vanilla NeRF using $\\{\mathcal{I}_{n}\\}_{n=1}^{N}$ and
$\\{(R_{n},t_{n})\\}_{n=1}^{N}$. Then, we stylize the images,
$\\{I_{n}\\}_{n=1}^{N}$, to stylized images, $\\{I_{n,m}\\}_{n=1,m=1}^{N,M}$,
with $M$ style images $\\{S_{m}\\}_{m=1}^{M}$ with pre-trained AdaIN. We take
the stylized images $\\{I_{n,m}\\}_{n=1,m=1}^{N,M}$ and their coresponding
camera poses $\\{(R_{n,m}=R_{n},t_{n,m}=t_{n})\\}_{n=1,m=1}^{N,M}$ as the new
dataset.

For model architecture, we take the top few layers of NeRF[4], which have been
trained with $\\{\mathcal{I}_{n}\\}_{n=1}^{N}$ and
$\\{(R_{n},t_{n})\\}_{n=1}^{N}$, and frozen their weight. For density, the
output feature of the frozen layers will be passed to a MLP $MLP_{\alpha}$ to
predict the density. For color prediction, we will first extract the feature
of the style images using a frozen VGG-based model like [2]. As mentioned in
[11, 12, 13, 2], feature statistics can carry the style information of an
image. Hence, we use another MLP $MLP_{style}$ to extract the feature of the
feature statistics extracted from the VGG-based model. In original NeRF[4],
the color prediction will take the view direction $\mathbf{d}$ as one of the
inputs. We follow the same concept by concatenating the output of the frozen
NeRF with view direction $\mathbf{d}$, passing it to a MLP $MLP_{view}$. Then,
we concatenate the output of $MLP_{style}$ and the output of $MLP_{view}$ and
pass it to another MLP $MLP_{rgb}$ to predict the color of a voxel.

To render a pixel from an image, we use the integral the same as the original
NeRF[4] with the same sampling strategy.

$\hat{C}(\mathbf{r},I_{style})=\sum_{i=1}^{N}T_{i}(1-exp(-\sigma_{i}\delta_{i}))\mathbf{c_{i}}\textit{,
where }T_{i}=exp(-\sum_{j=1}^{i-1}\sigma_{j}\delta_{j})$

For loss function, we use the same function as the original NeRF[4] and no
need for explicit loss function for style transfer as other 3D style transfer
method[5, 6, 7].

$\mathcal{L}=\sum_{i=1}^{N}\sum_{j=1}^{M}\sum_{\mathbf{r}\in\mathcal{R}_{i}}\left[\parallel\hat{C}(\mathbf{r},I_{i,j})-C(\mathbf{r},I_{i,j})\parallel\right]$

For the other hyper-parameters, we follow the same as the original NeRF[4].

#### Density Aware

To make the prediction of density vary between different style images, we
modify the framework to 2. First, we add a new MLP to transform the style
features captured by AdaIN to the features for the density prediction. Then,
we concatenate the output of this MLP with the output of the pre-trained NeRF.
Finally, we feed the concatenated features to a MLP to predict the density of
a voxel. The remain training procedure is the same as the one mentioned above.

Figure 2: Multi-style NeRF with Density Aware

#### Style Control and Style Interpolation

To do the style interpolation, we can easily interpolate the feature
statistics of the style image ${\mathcal{S}_{m}}_{m=1}^{M}$. To control the
balance of the content and style, we can train the multi-style NeRF by taking
the original content image $\mathcal{I}_{n}$ as another style. Then, we
directly interpolate the feature statistics of the original image and the
style image.

##  4 Results

We invite readers to watch our videos from our Github for better assessment of
3D stylization quality.

##  5 Conclusion

We present Multi-style NeRF, which is a novel pipeline for stylized novel view
synthesis that combines AdaIN and NeRF techniques. Our contributions include
simplifying the pipeline of the previous stylization of novel view synthesis,
extending its capabilities to handle multiple styles, modifying the model
architecture to handle styles with multiple styles, and implementing style
interpolation for better control over stylization strength.

## References

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