# Weak approximation for certain quadric surface bundles

Nick Rome<EMAIL_ADDRESS>University of Michigan
Department of Mathematics
East Hall, 530 Church Street, Ann Arbor, MI 48109

###### Abstract.

We investigate weak approximation for a class of biquadratic fourfolds inside
$\mathbb{P}^{2}\times\mathbb{P}^{3}$, some of which appear in the recent work
of Hassett–Pirutka–Tschinkel [13].

##  1\. Introduction

Let $k$ be a number field of degree $d$ and let $\Omega$ denote the set of
valuations on $k$. Given a smooth algebraic variety $X$ over $k$, we have the
following embeddings

$X(k)\hookrightarrow X(\mathbb{A}_{k})\subset\prod_{\nu\in\Omega}X(k_{\nu}),$

where the first map is the diagonal embedding of the rational points into the
adéles. When the set of rational points on $X$ is non-empty, one would like to
be able to discuss (either qualitatively or quantitatively) how the rational
points on $X$ are distributed. We say that the variety $X$ satisfies _weak
approximation_ if the image of $X(k)$ is dense in $X(\mathbb{A}_{k})$, under
the product topology. In his 1970 ICM talk, Manin observed that one can use
the Brauer group, ${\rm Br}(X)=H^{2}_{\acute{e}t}(X,\mathbb{G}_{m})$, to
define a set $X(\mathbb{A}_{k})^{{\rm Br}}$ with $X(k)\subset
X(\mathbb{A}_{k})^{{\rm Br}}\subset X(\mathbb{A}_{k})$, which can sometimes
obstruct weak approximation. Colliot-Thélène [2] conjectured that this
_Brauer–Manin obstruction_ is the only obstruction to weak approximation for
any smooth, projective, geometrically integral and rationally connected
variety. Our first result is to confirm Colliot-Thélène’s conjecture for a
particular class of fourfolds over $k$.

######  Theorem 1.1.

Let $k$ be a number field and $X/k$ the biprojective variety in
$\mathbb{P}^{2}\times\mathbb{P}^{3}$ defined by the equation

$xyt_{1}^{2}+xzt_{2}^{2}+yzt_{3}^{2}+F(x,y,z)t_{4}^{2}=0,$ (1.1)

where $F$ is a non-degenerate ternary quadratic form over $k$. Then the
Brauer–Manin obstruction is the only obstruction to weak approximation for any
smooth projective model of $X$.

There exist two general methods by which to prove theorems of this nature. The
first is the descent method, a generalisation of the classical descent theory
of elliptic curves via the use of universal torsors (c.f. [22]). The second is
the fibration method, in which one exploits the existence of a fibration
$f:X\rightarrow Y$ where the base $Y$ and the fibres of $f$ satisfy the
desired property. The varieties in Theorem 1.1 naturally admit the structure
of quadric surface bundles over $\mathbb{P}^{2}$ by the projection map

$\displaystyle X$ $\displaystyle\rightarrow\mathbb{P}^{2}$
$\displaystyle(x:y:z;\mathbf{t})$ $\displaystyle\mapsto(x:y:z),$

and so it is the latter method which will be relevant.

The earliest example of the use of a fibration to study local-global
principles in a family of varieties is due to Hasse in his proof of the local-
global principle for quadratic forms in 4 variables (see e.g. [18, Ch. IV,
Thm. 8]). These ideas were then generalised by Colliot-Thélène and Sansuc [5]
who replaced Hasse’s use of the prime number theorem in arithmetic
progressions with an evocation of Schinzel’s Hypothesis, which allowed them to
show conditionally that the Brauer–Manin obstruction is the only one for a
large class of conic bundles over $\mathbb{P}^{1}$. This was extended by
Colliot-Thélène and Swinnerton-Dyer [9](building on work of Serre [19],
Swinnerton-Dyer [23] and Salberger [16]) to establishing the conjecture
(conditionally on a variant of Schinzel’s Hypothesis) for pencils of
_generalised_ Severi–Brauer varieties (in the language of [9]), of which
quadric surfaces are an example. Skorobogatov [21](and subsequently Colliot-
Thélène–Skorobogatov [8]) was able to establish unconditionally that the
Brauer–Manin obstruction is the only one for quadric surface bundles over
$\mathbb{P}^{1}$ of rank $\leqslant 3$, by combining the fibration and descent
methods. Finally, if all of the degenerate fibres of a quadric surface bundle
over $\mathbb{P}^{1}$ are defined over $\mathbb{Q}$, then the Brauer–Manin
obstruction is unconditionally known to be the only one thanks to the work of
Browning–Mattheisen–Skorobogatov [1, Theorem 1.4].

Over higher dimensional bases, much less is known. Conditional on Schinzel’s
Hypothesis, Wittenberg [26, Corollaire 3.6] has shown that a fibration over
$\mathbb{P}^{n}$ into generalised Severi–Brauer varieties has the property
that the Brauer–Manin obstruction is the only one. Theorem 1.1 gives a rare
example of an unconditional proof that the Brauer–Manin obstruction is the
only one, without a specific restriction on the number (or degree) of the
degenerate fibres. We take this moment to point out that conic bundle
fibrations have a vast and illustrious history in the literature and that
fibrations over projective space into quadrics of dimension $\geqslant 3$
always satisfy weak approximation.

Our motivation for studying the particular class of varieties that we do comes
from a recent breakthrough by Hassett–Pirutka–Tschinkel [13] on classical
rationality problems in algebraic geometry. It was observed in that paper that
the Brauer group of certain quadric surface bundles can contain non-constant
elements, hence it is natural to wonder if these Brauer group elements might
obstruct weak approximation.

In particular, Hassett–Pirutka–Tschinkel were able to show for the first time
that rationality is not deformation invariant in families of complex fourfolds
by exhibiting a family for which many of the fibres were rational but a ‘very
general’ fibre was not even stably rational. This result was proven using the
specialisation method pioneered by Voisin [25] and later refined by Colliot-
Thélène–Pirutka [6] and Schreieder [17]. The upshot of this method is that if
one can find a special fibre $V_{b}$ in the family such that

  1. (1)

$H^{2}_{\text{nr}}(\mathbb{C}(V_{b}),\mathbb{Z}/2\mathbb{Z})$ is non-trivial,

  2. (2)

$V_{b}$ admits a desingularisation which is universally
$\text{CH}_{0}$-trivial,

then a very general fibre is not stably rational. In [13], these conditions
were shown to be satisfied by the fourfold $V_{b}$ defined by the equation

$xyt_{1}^{2}+xzt_{2}^{2}+yzt_{3}^{2}+F_{\text{HPT}}(x,y,z)t_{4}^{2}=0,$

where

$F_{\text{HPT}}(x,y,z)=x^{2}+y^{2}+z^{2}-2(xy+xz+yz).$ (1.2)

The unramified cohomology group (c.f. Section 3) used in the specialisation
method satisfies

$H^{2}_{\text{nr}}(\mathbb{C}(V_{b}),\mathbb{Z}/2\mathbb{Z})={\rm
Br}_{\mathbb{C}}(\widetilde{V_{b}})[2],$

where $\widetilde{V_{b}}$ is a desingularisation for $V_{b}$ (the group on the
right hand side is independent of the choice of desingularisation). This means
that Hassett–Pirutka–Tschinkel had to explicitly construct a non-trivial
element, specifically the quaternion algebra $(x/z,y/z)_{\mathbb{C}(V_{b})}$,
in the ($\mathbb{C}$-)Brauer group of $V_{b}$. The major novelty of this paper
is to use the techniques employed by Hassett–Pirutka–Tschinkel (and others in
the area) over _non algebraically closed_ fields in order to apply them, for
the first time, to the study of problems of a Diophantine nature.

The main result of this paper is to completely understand when weak
approximation does and does not hold for the varieties appearing in Theorem
1.1.

######  Theorem 1.2.

Let $k$ be a number field and $X/k$ the biprojective variety in
$\mathbb{P}^{3}\times\mathbb{P}^{2}$ defined by the equation

$xyt_{1}^{2}+xzt_{2}^{2}+yzt_{3}^{2}+F(x,y,z)t_{4}^{2}=0,$

where $F$ is a non-degenerate ternary quadratic form over $k$ such that

$\begin{cases}\text{The forms }F(0,y,z),F(x,0,z),\text{ and }F(x,y,0)\text{
are all squares},\\\ F(x,y,z)\not\in k(x,y,z)^{2}.\end{cases}$ (1.3)

If $F$ takes only positive values in all real embeddings, then $X$ satisfies
weak approximation. Otherwise, $X(k_{\nu})$ has two connected components for
$\nu$ real and weak approximation holds in one but may fail in the other.

It is possible to construct examples where weak approximation fails in one of
the components and examples where weak approximation is satisfied (c.f. the
final remark of the paper). We note that this provides new families of
examples of failure of weak approximation on rational varieties caused by a
transcendental Brauer group element. This can never occur for the more
familiar setting of pencils of conics but Harari has previously provided
examples of conic bundles over higher dimensional bases [12].

######  Remark.

The first condition is a little odd looking from a geometric point of view.
Perhaps more natural would be the condition that the conic $F(x,yz)=0$ is
tangent to each of the coordinate axis. However, as we shall see in the course
of the proof, this condition is not strong enough for fields $k$ which are not
algebraically closed. Tangency would allow us to deduce that $F(0,y,z)$ is
square in $k[y,z]$ modulo constants, however we really need for it to be a
square.

######  Remark.

The Hassett–Pirutka–Tschinkel example (1.2) certainly satisfies condition
(1.3).

######  Remark.

Our varieties are not smooth and hence we must clarify what we mean by weak
approximation in this setting. Following Colliot-Thélène–Xiu [10, Section 8],
we say that a singular variety $V$ satisfies weak approximation if for any
finite set of places of $k$, there exists a resolution of singularites
$\widetilde{V}\xrightarrow{\varphi}V$ such that the smooth locus of $V$ is
dense in the set $\prod_{\nu\in S}\varphi(\widetilde{V}(k_{\nu}))$. Note that
this definition is independent of the resolution of singularites chosen.

The second scenario in Theorem 1.2 where the rational points are restricted to
a connected component occurs often when ${\rm Br}(X)/{\rm Br}(k)$ is finite,
as is the case here, see for instance [24, §3] or [4, Prop 7.2].

### Acknowledgements

This paper formed part of the author’s thesis and he is grateful to his
supervisor Tim Browning for suggesting the problem and his continued guidance.
The author has greatly benefited from a number of useful discussions with
Julian Lyczak to whom he is indebted. The author would also like to thank
Christopher Frei, Tim Dokchitser, Wei Ho and, in particular, Diego Izquierdo
for their valuable comments on previous drafts. The author is grateful for the
suggestions of anonymous referees which have significantly reshaped the
exposition of the paper.

##  2\. Proof of Theorem 1.1

A systematic study of the arithmetic of quadric surface bundles over
$\mathbb{P}^{1}$ was conducted by Skorobogatov [21]. It was proved that if
there are very few closed points on $\mathbb{P}^{1}$ above which the fibre is
not geometrically integral then the Brauer–Manin obstruction is the only one
for the total space. Our goal is to bootstrap this result into a result about
our quadric surface bundles over $\mathbb{P}^{2}$, which themselves have very
few bad fibres. We start by gathering together some salient facts about the
geometry of quadric surface bundles.

######  Lemma 2.1 ([21, Corollary 4.1]).

Let $Z/k$ be a quadric surface bundle over $\mathbb{P}_{k}^{1}$ such that the
fibre above at most two closed geometric fibres is defined by a quadratic form
of rank $\leqslant 2$. Then, for any smooth proper model of $Z$, the
Brauer–Manin obstruction is the only obstruction to weak approximation and to
the existence of rational points.

######  Lemma 2.2 ([21, Lemma 3.1]).

Let $Z/k$ be a quadric surface bundle over $\mathbb{P}^{1}_{k}$ and let
$Q_{\eta}$ denote the quadratic form whose vanishing defines the generic
fibre. If $\text{det}(Q_{\eta})$ is not a square in $\overline{k}(\eta)$, then
$\operatorname{Pic}(Z_{\overline{k}(\eta)})$ is isomorphic to $\mathbb{Z}$ as
${\rm Gal}(\overline{k}/k)$ modules.

As discussed in the introduction, we will use the fibration method to
establish Theorem 1.1, specifically Harari’s fibration method with a smooth
section.

######  Theorem 2.3 ([11, Théorème 4.3.1]).

Let $V$ and $B$ be geometrically integral varieties over a field $k$ such that
$B$ satisfies weak approximation and there exists a dominant morphism
$V\xrightarrow{\pi}B$ which admits a section $s$. The Brauer–Manin obstruction
is the only one for any smooth projective of model of $V$ if the following are
satisfied:

  1. (1)

The generic fibre $V_{\eta}$ is a geometrically integral, geometrically
rational variety over $k(B)$ and $s$ defines a smooth point in $V_{\eta}$.

  2. (2)

For any smooth projective model $X$ of $V_{\eta}$, ${\rm
Br}(\overline{X}):={\rm Br}(X\times_{k(B)}\overline{k(B)})$ is trivial and
$\operatorname{Pic}(\overline{X})$ has no torsion.

  3. (3)

There exists a non-empty open $U\subset B$ such that $\forall b\in U$ the
Brauer–Manin obstruction is the only one for all smooth proper models of
$V_{b}$.

###### Proof of Theorem 1.1.

We will apply Theorem 2.3 to establish that the Brauer–Manin obstruction is
the only one for smooth proper models of $X$. We cannot apply this to the
natural fibration $X\rightarrow\mathbb{P}^{2}$, however, as we cannot
guarantee the existence of a smooth section. Instead we take an approach
inspired by the proof of [1, Thm 1.5]. We will consider the quasi-projective
variety obtained by dehomogenising the equation defining $X$. Setting $z=1$,
we have

$xyt_{1}^{2}+xt_{2}^{2}+yt_{3}^{2}+F(x,y,1)t_{4}^{2}=0,$

which defines a variety $V$ inside $\mathbb{A}^{2}\times\mathbb{P}^{3}$.
Again, the natural map to $\mathbb{A}^{2}$ does not necessarily have a smooth
section. Instead, we use the following map

$\displaystyle V$ $\displaystyle\xrightarrow{\psi}\mathbb{A}^{1}$
$\displaystyle(x,y;t_{1},\dots,t_{4})$ $\displaystyle\mapsto x.$

This map does admit a section given by

$x\mapsto(x,0;0,0,1,0).$

Each fibre of $\psi$, say $V_{x}$, is a threefold which admits the structure
of a quadric surface bundle over $\mathbb{A}^{1}$, with the map to
$\mathbb{A}^{1}$ given by projecting on the $y$-variable. When $x\neq 0$, the
fibre $V_{x}$ is a quadric surface bundle whose only essential singularity
(i.e. fibre defined by a quadratic form of rank $\leqslant 2$) occurs above
$y=0$. It follows, by Lemma 2.1, that for any projective model of a fibre
above the set $U=\\{x\neq 0\\}$ the Brauer–Manin obstruction is the only one.
To handle the second condition, we first observe that by Lemma 2.2, the Picard
group of the geometric generic fibre of $\psi$ is $\mathbb{Z}$. Hence, for any
smooth projective model $X$ of $V_{\eta}$, $\operatorname{Pic}(\overline{X})$
has no torsion. The Brauer group of $\overline{V_{\eta}}$ is isomorphic to
${\rm Br}(\overline{k(\mathbb{P}^{2})})$ (for instance, by Proposition 3.1).
Hence, for any smooth projective model $X$ of $V_{\eta}$, we have ${\rm
Br}(\overline{X})\simeq{\rm Br}(\overline{k(\mathbb{P}^{2})})$, which is
trivial (because $\overline{k(\mathbb{P}^{2})}$ is a $C_{1}$ field). Finally,
we note that the singular locus of the generic fibre of $\psi$ consists of
those $(y;\mathbf{t})$ where precisely two of the $t_{i}$ are 0, hence this
section does indeed correspond to a smooth point on the generic fibre.
Therefore, by Theorem 2.3, the Brauer–Manin obstruction is the only one for
any smooth projective model of $V$. Since $V$ and $X$ are birational, any
resolution of singularities for $X$ will be a smooth projective variety
birational to $V$ and hence the conclusion applies. ∎

##  3\. Computing the Brauer Group

Throughout this and all subsequent sections we fix a number field $k$, let
$X/k$ be the variety in the statement of Theorem 1.2 and $\widetilde{X}$ a
fixed desingularisation. If we hope to exhibit failures of weak approximation,
we need to find non-trivial elements in the Brauer group of $\widetilde{X}$.

Let $X_{\eta}$ be the generic fibre of the map $X\rightarrow\mathbb{P}^{2}$.
Specifically $X_{\eta}$ is a quadric surface over the function field
$K=k(\mathbb{P}^{2})$. We have the injection

${\rm Br}(X_{\eta})\hookrightarrow{\rm Br}(k(X))$

so we start by studying the Brauer group of the generic fibre $X_{\eta}$.

######  Proposition 3.1 ([7, Prop. 6.2.3]).

Let $K$ be a field with $\text{char}(K)\neq 2$ and $Q/K$ a smooth projective
quadric of dimension 1 or 2. Then the map ${\rm Br}(K)\rightarrow{\rm Br}(Q)$
is surjective. Moreover,

  1. (1)

Suppose Q is a conic. If $Q(K)\neq\varnothing$ then the map is an isomorphism.
If $Q(K)=\varnothing$, then the kernel of the map is isomorphic to
$\mathbb{Z}/2\mathbb{Z}$.

  2. (2)

Suppose $Q$ is a quadric surface. If the discriminant of $Q$ is a square in
$K$ and $Q(K)\neq\varnothing$ then the map is an isomorphism. Otherwise, the
kernel is isomorphic to $\mathbb{Z}/2\mathbb{Z}$.

Since $F(x,y,z)$ is non-square in $k(x,y,z)$, the map ${\rm
Br}(k(\mathbb{P}^{2}))\rightarrow{\rm Br}(X_{\eta})$ is an isomorphism and we
have the injection

${\rm Br}(k(\mathbb{P}^{2}))\xrightarrow{\cong}{\rm
Br}(X_{\eta})\hookrightarrow{\rm Br}(k(X))$

This means that non-trivial elements of ${\rm Br}(k(\mathbb{P}^{2}))$ are sent
to nontrivial elements of ${\rm Br}(k(X))$, which is isomorphic to ${\rm
Br}(k(\widetilde{X}))$ by birationality. Consider the quaternion algebra
$\alpha=(-x/z,-y/z)_{k(\mathbb{P}^{2})}\in{\rm Br}(k(\mathbb{P}^{2}))$, and
denote by $\beta$ its image in ${\rm Br}(k(X))$. We will work only in the open
chart $\\{z\neq 0\\}$ of $\mathbb{P}^{2}$, the computations in the charts
$\\{y\neq 0\\}$ and $\\{x\neq 0\\}$ following mutatis mutandis. Henceforth, we
write $\alpha$ as $(-x,-y)$ using the affine co-ordinates on this chart. The
quaternion algebra $(-x,-y)$ has nontrivial residues (e.g. along the axis
$\\{x=0\\}$) hence it is nontrivial in ${\rm Br}(k(\mathbb{P}^{2}))$.
Therefore, we conclude that its image under the map ${\rm
Br}(k(\mathbb{P}^{2}))\hookrightarrow{\rm Br}(k(\widetilde{X}))$ is
nontrivial.

We claim that in fact this image $\beta$ lies in the subgroup ${\rm
Br}(\widetilde{X})\subset{\rm Br}(k(X))$. The tool which we will use to
establish this is _unramified cohomology_. A large part of the appeal of this
approach is that one never needs to explicitly compute a smooth model of the
singular variety in order to study the Brauer group.

###  3.1. Unramified Cohomology

For the basic definitions of unramified cohomology groups, we refer the reader
to [14]. For smooth projective varieties, by the purity theorem, we can
compute the unramified cohomology by studying residues at codimension one
points. Indeed

$H^{i}_{nr}(k(V)/k,\mu_{n}^{\otimes j})=\bigcap_{x\in
V^{(1)}}\text{Ker}\left(H^{i}(k(V),\mu_{n}^{\otimes
j})\xrightarrow{\partial_{x}^{i}}H^{i-1}(\kappa(x),\mu_{n}^{\otimes
j-1})\right),$

with the intersection here running over DVR’s associated to codimension one
points $x$ where $\partial^{i}_{x}$ is the associated residue map and
$\kappa(x)$ the residue field. One important property of these groups is their
connection to torsion in the Brauer group.

######  Proposition 3.2 ([14, Prop 3.7]).

If $V$ is a smooth projective variety over $k$ then

$H^{2}_{nr}(k(V)/k,\mathbb{Z}/2\mathbb{Z})\simeq{\rm Br}(V)[2].$

Since quaternion algebras are 2 torsion elements in the Brauer group, it
suffices to prove that our Brauer class $\beta$ lies in
$H_{nr}^{2}(k(\widetilde{X})/k,\mathbb{Z}/2\mathbb{Z})$. Note that we can
compute the residues of quaternion algebras explicitly. Suppose that $a,b\in
k(V)^{*}$ and $\nu$ a rank one valuation on $k(V)$ with residue field
$\kappa(\nu)$. Then, the residue in ${\rm Br}(k(V))[2]$ of the cup product
$(a,b):=a\cup b\in H^{2}(k(V),\mathbb{Z}/2\mathbb{Z})$ is given by

$\partial^{2}_{\nu}(a,b)=(-1)^{\nu(a)\nu(b)}\overline{\frac{a^{\nu(b)}}{b^{\nu(a)}}},$
(3.1)

where $\overline{\frac{a^{\nu(b)}}{b^{\nu(a)}}}$ is the image of
$\frac{a^{\nu(b)}}{b^{\nu(a)}}$ in $\kappa(\nu)^{*}/(\kappa(\nu)^{*})^{2}$. We
will use this formula to compute the residue of $\alpha$ along codimension one
points in $\mathbb{P}^{2}$. This can be related to the residue of $\beta$
along codimension one points in $\widetilde{X}$ by the following compatibility
result.

######  Proposition 3.3 ([14, Prop 3.4]).

Let $A\subset B$ be discrete valuation rings with fields of fractions
$M\subset L$ respectively. Assume that $\text{char}(M)$ and $\text{char}(L)$
are both not 2. Let $\pi_{A}$ (resp. $\pi_{B}$) be a uniformising parameter of
$A$ (resp. $B$) and $\kappa(A)$ (resp. $\kappa(B)$) the residue field of $A$
(resp. $B$). Let $e$ be the valuation of $\pi_{A}$ in $B$. Then the following
diagram commutes

${H^{i}(L,\mathbb{Z}/2\mathbb{Z})}$${H^{i-1}(\kappa(B),\mathbb{Z}/2\mathbb{Z})}$${H^{i}(M,\mathbb{Z}/2\mathbb{Z})}$${H^{i-1}(\kappa(A),\mathbb{Z}/2\mathbb{Z})}$$\scriptstyle{\partial_{B}^{i}}$$\scriptstyle{\partial_{A}^{i}}$$\scriptstyle{Res_{M/L}}$$\scriptstyle{eRes_{\kappa(A)/\kappa(B)}}$

where $Res_{K/L}$ and $Res_{\kappa(A)/\kappa(B)}$ are the restriction maps in
Galois cohomology.

This allows us to compute the residue at a codimension 1 point of
$\widetilde{X}$ with codimension 1 image in $\mathbb{P}^{2}$ by studying the
residue along its image. However, codimension 1 points in $\widetilde{X}$
could also map down to the generic point of $\mathbb{P}^{2}$ or a closed
point. To handle some of these cases we will have to make use of the
following.

######  Proposition 3.4 ([14, Cor 3.12]).

Let $A$ be a local ring with field of fractions $F$, with $\text{char}(F)\neq
2$ and residue field $\kappa$. Let $Q/F$ be a quadric corresponding to the
vanishing of the quadratic form $q\simeq\langle 1,-a,-b,abd\rangle$. Let $v$
be a discrete valuation on $F(Q)$ with valuation ring $B$. Assume that there
is an injection $A\rightarrow B$ of local rings and that upto multiplication
by a square in $F$, the element $d$ is a unit in $A$ and a square in $\kappa$.
Let ${\alpha=(a,b)\in H^{2}(F,\mathbb{Z}/2\mathbb{Z})}$ and $\alpha^{\prime}$
its image in $H^{2}(F(Q),\mathbb{Z}/2\mathbb{Z})$. Then
$\partial_{v}^{2}(\alpha^{\prime})=0$.

Let $\nu:k(X)^{*}\rightarrow\mathbb{Z}$ be a valuation, $S$ the associated
valuation ring and $\kappa_{\nu}$ the residue field. If
$k(\mathbb{P}^{2})\subset S$ then $\nu(x)=\nu(y)=0$ hence
$(-x,-y)_{k(\mathbb{P}^{2})}$ has trivial residue. Otherwise, $S\cap
k(\mathbb{P}^{2})=R$ is a discrete valuation ring with field of fraction
$k(\mathbb{P}^{2})$. The image of Spec$(R)$ in $\mathbb{P}^{2}$ is either a
codimension one point or a closed point.

Codimension one. If the codimension one point $P$ is not the generic point of
a component of $\\{xy=0\\}$ then $(-x,-y)$ is unramified at $P$. The diagram
in Proposition 3.3 in this case is

${H^{2}(k(X),\mathbb{Z}/2\mathbb{Z})}$${H^{1}(\kappa_{\nu},\mathbb{Z}/2\mathbb{Z})}$${H^{2}(k(\mathbb{P}^{2}),\mathbb{Z}/2\mathbb{Z})}$${H^{1}(\kappa_{R},\mathbb{Z}/2\mathbb{Z}),}$$\scriptstyle{\partial_{\nu}^{2}}$$\scriptstyle{\partial_{P}^{2}}$

where $\kappa_{R}$ denotes the residue field of the DVR $R=S\cap
k(\mathbb{P}^{2})$. Since $\alpha$ is unramified at $P$, we have
$\partial_{P}^{2}(\alpha)=0$ and therefore, by the commutativity of the
diagram, we must have $\partial_{\nu}^{2}(\beta)=0$.

Otherwise, suppose that $P$ is the generic point of the line $\\{x=0\\}$. In
this case, the discriminant $d$ is given by $F(0,y,z)$ in the residue field,
which is square by (1.3). Furthermore, since the valuation of $F$ with respect
to $x$ is 0, it is indeed a unit in the associated local ring. Hence, by
Proposition 3.4, we conclude that the residue of $\beta$ along $v$ in $K(Q)$
is trivial. The same reasoning holds if $P$ is the generic point of
$\\{y=0\\}$.

Codimension two. If $P$ is a closed point of $\mathbb{P}^{2}_{k}$, then there
is a local homomorphism of local rings
$\mathscr{O}_{\mathbb{P}^{2},P}\rightarrow S$ which induces an embedding
$k\rightarrow\kappa_{\nu}$. If $P$ does not lie in $\\{xy=0\\}$, then $x$ and
$y$ are units in the local ring $\mathscr{O}_{\mathbb{P}^{2},P}$ and hence in
$S$. Therefore, $\partial_{\nu}^{2}(\beta)=0$. If $P$ lies on $\\{x=0\\}$ but
not $\\{y=0\\}$ then in the residue field $\kappa(P)$, the image of $-y$ in
$\kappa(P)$ lies in $(k^{*})^{2}$. Hence, it is a square in $S$. Therefore, we
have $\beta=0$ in $H^{2}(K(Q)_{\nu},\mathbb{Z}/2\mathbb{Z})$ and thus
$\partial_{\nu}(\beta)=0$. Finally, if $P$ is on both lines then we know that
$F(0,0,1)$ is a nonzero square in the residue field and thus by Hensel’s
Lemma, it is a square in the completion $k(X)_{\nu}$. We can apply Proposition
3.4 as above to conclude that $\partial^{2}_{\nu}(\beta)=0$.

######  Remark.

In the above proof we have made pivotal use of the conditions (1.3). One
intuitive justification for the above theorem is given by Abhyankar’s Lemma
(as observed by Colliot-Thélène [3, §3]). Namely, since the ramification locus
of the quaternion algebra $(-xy,-yz)_{k(\mathbb{P}^{2})}$ is contained within
the ramification locus of $X\rightarrow\mathbb{P}^{2}$, the ramification
cancels in $\widetilde{X}$ (see e.g. [20, p. 116] for an example of this
phenomenon). Concretely, in this case $-yz$ is not a square modulo $x$ in
$k(\mathbb{P}^{2})$ but _is_ a square modulo $x$ in $k(X)$. This explains why
the quaternion algebra $\alpha$ has nontrivial residues but these residues
vanish when you move from $\mathbb{P}^{2}$ to $X$.

###  3.2. The full Brauer group

The purpose of this section is to prove the following result.

######  Proposition 3.5.

Let $X$ the variety defined in Theorem 1.2. Then,

${\rm Br}(X)/{\rm Br}(k)\cong\mathbb{Z}/2\mathbb{Z},$

and is generated by $\beta$.

Proposition 3.1 lets us compare the Brauer group of the generic fibre with the
Brauer group of the function field $k(\mathbb{P}^{2})$. We shall also need the
following result in degree 1 (see e.g. [14, Thm 3.10], although the result is
originally due to Arason).

######  Lemma 3.6.

Let $Q$ be a quadric defined by the vanishing of a non-degenerate quadratic
form $q$ over $K$.

  1. (1)

If $q$ has dimension at least 3, then the natural map

$H^{1}(K,\mathbb{Z}/2\mathbb{Z})\rightarrow
H^{1}_{\text{nr}}(K(Q)/K,\mathbb{Z}/2\mathbb{Z})$

is injective.

  2. (2)

If $q$ has dimension 2, then the map

$H^{1}(K,\mathbb{Z}/2\mathbb{Z})\rightarrow
H^{1}_{\text{nr}}(K(Q)/K,\mathbb{Z}/2\mathbb{Z})$

has kernel isomorphic to $\mathbb{Z}/2\mathbb{Z}$ generated by the class of
the discriminant of $q$.

Our first step in the proof of Proposition 3.5 is to show that we can restrict
our attention to the 2-torsion part of the Brauer group.

######  Lemma 3.7.

The quotient ${\rm Br}(X)/{\rm Br}(k)$ is 2-torsion.

###### Proof.

We will prove this by exploiting the fibration $f:X\rightarrow\mathbb{P}^{2}$.
We make use of the following commutative diagram (c.f. [7, Eqn (10.1)]) coming
from the purity sequences for the two varieties

${0}$${{\rm Br}(X)}$${{\rm
Br}(X_{\eta})}$${\bigoplus\limits_{P}H^{1}(k(X_{P}),\mathbb{Q}/\mathbb{Z})}$${0}$${{\rm
Br}(k)}$${{\rm
Br}(K)}$${\bigoplus\limits_{P\in(\mathbb{P}^{2})^{(1)}}H^{1}(k(P),\mathbb{Q}/\mathbb{Z}).}$$\scriptstyle{f^{*}}$$\scriptstyle{\cong}$

Let $\gamma\in{\rm Br}(X)$. If $P$ is a codimension one point on
$\mathbb{P}^{2}$ which is not an irreducible component of the ramification
divisor $\\{xyz=0\\}$ then the residue $\partial_{P}(\gamma)=0$ (see for
instance [7, Prop 10.1.4]). Hence, the only points in the direct sum which we
need consider are $P\subset\\{xyz=0\\}$. If $P$ is such a point, by the
commutativity of the diagram, we have

$\partial_{P}(\gamma)\in\operatorname{Ker}[\text{res}_{k(X_{P})/k(P)}H^{1}(k(P),\mathbb{Q}/\mathbb{Z})\rightarrow
H^{1}(k(X_{P}),\mathbb{Q}/\mathbb{Z})].$

Note that by the nature of the fibration, the non-split fibres are given by
the vanishing of quadratic forms of rank two. Therefore, there exists a
quadratic extension $F_{P}$ over which the non-split fibres can be written as
a union of two transversal planes. In particular, this means

$\partial_{P}(\gamma)\in\operatorname{Ker}[\text{res}_{F_{P}/k(P)}H^{1}(k(P),\mathbb{Z}/2\mathbb{Z})\rightarrow
H^{1}(F_{P},\mathbb{Z}/2\mathbb{Z})].$

Since there are exactly 3 codimension 1 points on $\mathbb{P}^{2}$ over which
the fibre is non-split, we deduce the existence of a sequence ${\rm
Br}(k)\rightarrow{\rm
Br}(X)\rightarrow\left(\mathbb{Z}/2\mathbb{Z}\right)^{3}$, which gives the
claim. ∎

It remains to determine the group ${\rm Br}(X)[2]$. Our proof is based on [14,
Lemma 3.10]. We know that every non-trivial element of ${\rm Br}(X)[2]$ comes
from a class in $H^{2}(K,\mathbb{Z}/2\mathbb{Z})$ and we’ll aim to show that
if a class maps to something non-trivial in ${\rm Br}(X)[2]$ then its residue
along any codimension one point in $\mathbb{P}^{2}$ must be in the subgroup
generated by the residue of $\alpha$ along that codimension one point. From
this, we will then deduce that this element can differ from $\alpha$ at most
by a constant class.

###### Proof of Proposition 3.5.

Let $\gamma\in H^{2}(K,\mathbb{Z}/2\mathbb{Z})$ and
$D\in(\mathbb{P}^{2})^{(1)}$ such that $\delta_{D}(\gamma)\neq 0$. Let $q$ be
the quadratic form over $K$ which defines the quadric $Q$. The divisor
associated to the closed subscheme $q=0$ in $\mathbb{P}^{2}_{\mathscr{O}_{D}}$
is either integral or a union of two planes. Let $\nu$ be the valuation
attached either to the integral divisor or one of the two planes. Then
Proposition 3.3 gives the following diagram

${H^{2}(K(Q),\mathbb{Z}/2\mathbb{Z})}$${H^{1}(\kappa_{\nu},\mathbb{Z}/2\mathbb{Z})}$${H^{2}(K,\mathbb{Z}/2\mathbb{Z})}$${H^{1}(\kappa_{D},\mathbb{Z}/2\mathbb{Z}).}$$\scriptstyle{\partial_{\nu}}$$\scriptstyle{\partial_{D}}$$\scriptstyle{\text{Res}_{D/\nu}}$

The field $\kappa_{\nu}$ is the function field of a quadric over $\kappa_{D}$,
which we denote $\overline{q}$. If at most one of the coefficients of $q$ has
odd valuation with respect to $D$ then $\overline{q}=0$ is a smooth conic or
quadric surface. Hence, by Lemma 3.6 (1), the map $\text{Res}_{D/\nu}$ is
injective. Since $\gamma$ lands in $H^{2}(K(Q),\mathbb{Z}/2\mathbb{Z})$ the
map along the top arrow is trivial. Therefore, $\partial_{D}(\gamma)=0$.

This means at least 2 of the coefficients of $q$ must have odd valuation. The
only way this can happen is if $D$ is one of the components of $\\{xyz=0\\}$.
Suppose $D$ corresponds to the line $\\{x=0\\}$. Then, $\kappa(\nu)$ is the
function field of the quadric over $\kappa(D)$ defined by the equation

$\overline{yz}U^{2}+\overline{F(x,y,z)}V^{2}=0.$

By Lemma 3.6 (2), $\operatorname{Ker}\text{Res}_{D/\nu}$ is generated by the
class of the discriminant of $\overline{q}$ which is
$-\overline{yzF(x,y,z)}\in\kappa(D)^{*}/(\kappa(D)^{*})^{2}$. However,
$\overline{F(x,y,z)}$ is in $(\kappa(D)^{*})^{2}$, by conditions (1.3). Thus
the class is given by $-\overline{yz}$, which is precisely the residue of
$(-x/z,-y/z)$ along $D$ (by applying the formula (3.1)). The exact same
calculation holds for the other two lines.

Hence, if $\gamma$ is a non-trivial class in $H^{2}(K,\mathbb{Z}/2\mathbb{Z})$
giving rise to a non-constant element of ${\rm Br}(X)$ and $D$ is one of the
three coordinate axes, then
$\partial_{D}(\gamma)\in\\{0,\partial_{D}(\alpha)\\}$. It remains to rule out
the possibility that $\gamma$ is just ramified along one or two of the
coordinate axes (not all three like $\alpha$). However, $\gamma$ cannot just
be ramified along one axis because ${\rm Br}(\mathbb{A}^{2})$ is trivial. For
the same reason, $\gamma-\alpha$ cannot be ramified along just one axis which
means that $\gamma$ cannot be ramified along exactly two axes. Thus, $\gamma$
must be ramified along all three axes and thus must satisfy
$\partial_{D}(\gamma)=\partial_{D}(\alpha)$ for all codimension one points
$D$. Hence, it follows that the induced element in ${\rm Br}(X)$ can differ
from $\beta$ by at most constant classes. ∎

##  4\. The Brauer–Manin Obstruction set

In this final section, we explicitly compute the Brauer–Manin obstruction,
which combined with Theorem 1.1 finishes the proof of Theorem 1.2. Recall the
definition of the Brauer–Manin obstruction set (e.g. [15, Defn 8.2.5])

$X(\mathbb{A}_{k})^{{\rm Br}}=\\{(P_{\nu})_{\nu}\in
X(\mathbb{A}_{k}):\sum_{\nu}\text{inv}_{\nu}\text{ev}_{\gamma}(P)=0\in\mathbb{Q}/\mathbb{Z}\,\,\forall\gamma\in{\rm
Br}(X)\\}.$

Theorem 1.2 follows immediately from explicitly determining the image of the
evaluation maps.

######  Proposition 4.1.

Let $\beta\in{\rm Br}(X)$ be the non-trivial class from the previous section.
For every non-archimedean place $\nu$, $\text{inv}_{\nu}\text{ev}_{\beta}$ is
identically zero on all $k_{\nu}$ points.

Since the Brauer–Manin obstruction is the only obstruction to weak
approximation, and $\beta$ the only non-trivial Brauer class, the above shows
that the failure of weak approximation is completely determined by the
archimedean places.

###### Proof.

Proposition 4.1 will be proved by an explicit computation of the invariant
maps. Throughout, $(a,b)_{\nu}$ will refer to the Hilbert symbol associated to
the local field $k_{\nu}$. The evaluation of the class $\beta$ at a point
$P\in X(\mathbb{A}_{k})$ is equal to the evaluation of the pre-image
$\alpha\in{\rm Br}(k(\mathbb{P}^{2}))$ at the point
$B\in\mathbb{P}^{2}(\mathbb{A}_{k})$ below $P$. Fix a point
$P=(x_{\nu}:y_{\nu}:z_{\nu};\mathbf{t}_{\nu})_{\nu}\in X(\mathbb{A}_{k})$,
then

$\text{inv}_{\nu}\text{ev}_{\beta}(x_{\nu}:y_{\nu}:z_{\nu};\mathbf{t}_{\nu})=\frac{1}{2}\iff(-x_{\nu}y_{\nu},-y_{\nu}z_{\nu})_{\nu}=-1.$

Let $\mathfrak{p}$ a prime ideal in $\mathscr{O}_{k}$ and consider the conic

$x_{\mathfrak{p}}z_{\mathfrak{p}}U^{2}+y_{\mathfrak{p}}z_{\mathfrak{p}}V^{2}+W^{2}=0$

in $k_{\mathfrak{p}}$. By the definition of the Hilbert symbol,
$\text{inv}_{\mathfrak{p}}\text{ev}_{\beta}(x_{\mathfrak{p}}:y_{\mathfrak{p}}:z_{\mathfrak{p}};\mathbf{t}_{\mathfrak{p}})=0$
if and only if the conic has a $k_{\mathfrak{p}}$ solution. The vector
$(t_{1,\mathfrak{p}},t_{2,\mathfrak{p}},t_{3,\mathfrak{p}},t_{4,\mathfrak{p}})$
is a non-trivial solution to the quadratic form in four variables defined by
(1.1). The discriminant of this quadratic form is
$F(x_{\mathfrak{p}},y_{\mathfrak{p}},z_{\mathfrak{p}})$, which is a square
modulo $\mathfrak{p}$ and hence a square in $k_{\mathfrak{p}}$ by Hensel’s
lemma. Therefore, by the criteria for local solubility of quadratic forms [18,
Ch 6, Thm 6], we must have

$\prod_{i<j}(a_{i},a_{j})_{\mathfrak{p}}=(-1,-1)_{\mathfrak{p}},$

where $a_{i}$ are the coefficients of the quadratic form. In our case, after
simplifying the corresponding expression, we have

$(x_{\mathfrak{p}}y_{\mathfrak{p}},y_{\mathfrak{p}}z_{\mathfrak{p}})_{\mathfrak{p}}(x_{\mathfrak{p}}y_{\mathfrak{p}},x_{\mathfrak{p}}z_{\mathfrak{p}})_{\mathfrak{p}}(y_{\mathfrak{p}}z_{\mathfrak{p}},x_{\mathfrak{p}}z_{\mathfrak{p}})_{\mathfrak{p}}=(-1,-1)_{\mathfrak{p}}.$

Using the multiplicativity of the Hilbert symbol, we write

$(-x_{\mathfrak{p}}z_{\mathfrak{p}},-y_{\mathfrak{p}}z_{\mathfrak{p}})_{\mathfrak{p}}=(x_{\mathfrak{p}}z_{\mathfrak{p}},y_{\mathfrak{p}}z_{\mathfrak{p}})_{\mathfrak{p}}(-1,y_{\mathfrak{p}}z_{\mathfrak{p}})_{\mathfrak{p}}(x_{\mathfrak{p}}z_{\mathfrak{p}},-1)_{\mathfrak{p}}(-1,-1)_{\mathfrak{p}}.$

Combining these expressions, we see that

$(-x_{\mathfrak{p}}z_{\mathfrak{p}},-y_{\mathfrak{p}}z_{\mathfrak{p}})_{\mathfrak{p}}=(x_{\mathfrak{p}}z_{\mathfrak{p}},x_{\mathfrak{p}}y_{\mathfrak{p}})(x_{\mathfrak{p}}y_{\mathfrak{p}},y_{\mathfrak{p}}z_{\mathfrak{p}})(-1,y_{\mathfrak{p}}z_{\mathfrak{p}})(x_{\mathfrak{p}}z_{\mathfrak{p}},-1)=(x_{\mathfrak{p}}y_{\mathfrak{p}},-x_{\mathfrak{p}}y_{\mathfrak{p}})_{\mathfrak{p}}.$

This Hilbert symbol is certainly $+1$ because the associated conic
$x_{\mathfrak{p}}y_{\mathfrak{p}}U^{2}-x_{\mathfrak{p}}y_{\mathfrak{p}}V^{2}=W^{2}$
always has the non-trivial rational point $(1,1,0)$. ∎

######  Remark.

Informally speaking, the reason this result is true is similar to the reason
why $\beta$ lives in ${\rm Br}(X)$. Namely, if a prime divides $x$, then the
nature of the equation forces $-yz$ to be a square, which kills the invariant
map.

Finally, we will study the invariant map at archimedean places of $k$. If
$\nu$ is a complex place then the conic
$x_{\nu}z_{\nu}U^{2}+y_{\nu}z_{\nu}V^{2}+W^{2}=0$ always has points and thus
the invariant map is identically zero. The value of the invariant map at real
places is determined by the signs of $x,y$ and $z$.

######  Proposition 4.2.

Let $\nu$ be a real place of $k$ and let
$P_{\nu}=(x_{\nu}:y_{\nu}:z_{\nu};\mathbf{t}_{\nu})\in X(k_{\nu})$. If
$F(x_{\nu},y_{\nu},z_{\nu})>0$ then
$\text{inv}_{\nu}\text{ev}_{\beta}(P_{\nu})=+1$. Otherwise,

$\text{inv}_{\nu}\text{ev}_{beta}(P_{\nu})=\begin{cases}\frac{1}{2}\quad{}\text{
if }x,y,z\text{ all have the same sign in }k_{\nu},\\\ 0\quad{}\text{
otherwise.}\end{cases}$

###### Proof.

If $F(x_{\nu},y_{\nu},z_{\nu})>0$ then it is a square in $\mathbb{R}$ and we
may appeal to the local solubility criterion as in the proof of Proposition
4.1. The same Hilbert symbol computation shows that
$(-x_{\nu}z_{\nu},-y_{\nu}z_{\nu})_{\nu}=+1$. Otherwise, by the explicit
description of the real Hilbert symbol

$(-x_{\nu}z_{\nu},-y_{\nu}z_{\nu})_{\nu}=\begin{cases}-1\text{ if
}-x_{\nu}z_{\nu}<0\text{ and }-y_{\nu}z_{\nu}<0,\\\ +1\text{
otherwise.}\end{cases}$

This condition is equivalent to the one in the statement of the proposition. ∎

Together Propositions 4.1 and 4.2 completely describe the obstruction set and
hence the statement of Theorem 1.2 follows.

######  Remark.

Consider the Hassett–Pirutka–Tschinkel example over $\mathbb{Q}$ where
${F(x,y,z)=x^{2}+y^{2}+z^{2}-2(xy+xz+yz)}$. The point $P=(x,y,z;\mathbf{t})\in
X(\mathbb{Q})$ given by $P=(1,1,1;1,1,1,1)$ has the property that $F(x,y,z)<0$
and $x,y$ and $z$ share the same sign. Therefore weak approximation fails.
However, if one takes instead $F(x,y,z)=x^{2}+y^{2}+z^{2}+2(xy+xz+yz)$, then
the conditions (1.3) are met, and $F$ always takes positive values. Hence,
weak approximation holds for the resulting fourfold.

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# The rotational kinetic Sunyaev-Zeldovich contribution to the temperature
asymmetry toward the M31 halo

Noraiz Tahir1,2,∗; Francesco De Paolis1,2,3,†; Asghar Qadir4,‡;
Achille A. Nucita1,2,3,§
1 Department of Mathematics and Physics “E. De Giorgi”,
University of Salento, Via per Arnesano, I-73100 Lecce, Italy.
2 INFN, Sezione di Lecce, Via per Arnesano, 73100 Lecce, Italy.
3 INAF, Sezione di Lecce, Via per Arnesano, 73100 Lecce, Italy.
4 Abdus Salam School of Mathematics, G.C. University, Lahore, Pakistan.
∗<EMAIL_ADDRESS>
†<EMAIL_ADDRESS>
‡<EMAIL_ADDRESS>
§<EMAIL_ADDRESS>

###### Abstract

Temperature asymmetry in the cosmic microwave background (CMB) data by the
Planck satellite has been discovered and analyzed toward several nearby edge-
on spiral galaxies. It provides a way to probe galactic halo rotation, and to
constrain the baryon fraction in the galactic halos. The frequency
independence of the observed data provides a strong indication of the Doppler
shift nature of the effect, due to the galactic halo rotation. It was proposed
that this effect may arise from the emission of cold gas clouds populating the
galactic halos. However, in order to confirm this view, other effects that
might give rise to a temperature asymmetry in the CMB data, have to be
considered and studied in detail. The main aim of the present paper is to
estimate the contribution in the CMB temperature asymmetry data due to the
free-free emission by hot gas (particularly electrons) through the rotational
kinetic Sunyaev-Zeldovich (rkSZ) effect. We concentrate, in particular, on the
M31 galactic halo and compare the estimated values of the rkSZ induced
temperature asymmetry with those obtained by using the SMICA pipeline of the
Planck data release, already employed to project out the SZ sources and for
lensing studies. As an additional consistency check, we also verified that the
hot gas diffuse emission in the X-ray band does not exceed that detected in
the soft X-ray band by ROSAT observations. We note that our results clearly
show that the rkSZ effect gives only a minor contribution to the observed M31
halo temperature asymmetry by Planck data.

##  1 Introduction

To describe the Universe we need to know what is in it and where all its
components reside. This is a difficult task. For example, the “missing baryon
problem” is one of the unsolved issues in modern cosmology. We know that $\sim
5\%$ of the Universe is composed of baryons (Ade et al., 2016), but not all
these baryons have been detected as yet. What we can see is $\approx 60\%$ of
the total baryonic budget constituting the luminous parts of the galaxies
lying in what is called the Layman $\alpha$ forest, in the warm-hot
intergalactic gas (WHIG), in the the warm-hot intergalactic medium (WHIM) in
clusters of galaxies, in the circumgalactic medium (CGM), and in the form of
cold gas (Burkert, 2003; Cen e al., 1999; Li et al., 2018a; Katherine et al.,
1996; Gerhard et al., 1996). The remaining part of the baryonic budget is
still hidden. It has been suggested that galactic halos contain a non-
negligible fraction of these baryons (Cen et al., 2006; Fraser-McKelvie et
al., 2011; Nicastro et al., 2008; Zhang et al., 2012; Tahir et al., 2019a, b;
Richards et al., 2018), but in what form and how many baryons are present
there are still open questions.

The analysis of the cosmic microwave background (CMB) data of WMAP and Planck
for various nearby edge-on spiral galaxies showed the existence of a large-
scale (up to about 100 kpc in many cases) temperature asymmetry (which means
that one side turns out to be hotter than the other in the microwaves) with
respect to the considered galaxy’s rotational axis, with the important
characteristic of being almost frequency independent. This gave a strong
indication of a Doppler shift effect induced by the galactic halo rotation (De
Paolis et al., 2014, 2015; Gurzadyan et al., 2015; De Paolis et al., 2016;
Gurzadyan et al., 2018; De Paolis et al., 2019). De Paolis et al. (1995a)
proposed that galactic halos contain stable cold gas clouds in thermodynamic
equilibrium with the CMB and that these clouds contribute a non-negligible
fraction $f$ of the galactic halo dark matter. If these clouds are there, the
only way to detect them is through the Doppler shift effect induced by the
rotation of these clouds with the galactic halos (De Paolis et al., 1995b,
1998). The predicted Doppler shift was indeed detected by the CMB data, and
this data was then used to model the galactic halo rotation and estimate the
gas cloud contribution (Tahir et al., 2019a, b; Qadir et al., 2019). However,
the main problem with the existence of these clouds is their stability
(Padmanabhan, 1990), but the analysis by Qadir et al. (2019) indicated that
this equilibrium is possible and the clouds can be stable at the current CMB
temperature.

It is clear that showing that these clouds may exist does not prove that they
actually do exist and make up a substantial fraction of the missing baryons,
nor that they are responsible for the temperature asymmetry in the CMB data
observed toward several nearby galaxy halos. Indeed, before concluding that
this in fact occurs it is necessary to analyze and exclude other effects that
may contribute, even partially, to the observed asymmetry toward nearby
spirals. Many other effects, from a theoretical point of view, can contribute,
such as the rotational kinetic Sunyaev-Zeldovich (rkSZ) effect by hot gas
diffuse in the galactic halos (Matilla et al., 2020), synchrotron emission by
fast moving electrons (Dolag et al., 2000), free-free (or bremsstrahlung)
emission (Sun et al., 2010), and anomalous microwave emission (AME) from dust
grains in the halos (Leitch et al., 2013). The aim of the present paper is to
make a first step forward in this direction and consider in detail the
contribution of the rkSZ effect in the observed asymmetry in the Planck data
toward M31. The contributions of the other effects will be considered
elsewhere.

The manuscript is arranged as follows. In Section 2 we model the density
distribution of the hot gas in the M31 galaxy halo and its circular velocity.
We therefore estimate the total mass of the hot gas component. In order to
check the consistency of our adopted model we then estimate the diffuse X-ray
flux toward the M31 halo and compare it with the diffuse emission observed in
the soft X-ray band by the ROSAT satellite which is available (West et al.,
1997) and in other works in the literature. We further proceed to estimate the
expected temperature asymmetry values derived from our modeling of the hot
plasma in the M31 halo and compare them with the results obtained using the
SMICA pipeline of the Planck data. Finally, our main conclusions are presented
in Section 3.

##  2 Modeling of the rkSZ effect

The Sunyaev-Zeldovich (SZ) effect has become a cornerstone of modern cosmology
since it allows us to obtain a deeper understanding of several important
issues in astrophysics and cosmology. The SZ effect can be divided into two
types: the thermal SZ (thSZ) and the kinetic SZ (kSZ) effects. The first is
due to the inverse Compton scattering of the CMB photons off the hot ICM
(Zeldovich et al., 1996). The thSZ effect is redshift independent, so it is
used to detect clusters of galaxies at high redshifts where other
observational methods fail (Ade et al., 2015; Bleem et al., 2020; Hilton et
al., 2021), and to determine the Hubble constant (Birkinshaw, 1999; Reese et
al., 2002; Manson et al., 2001; Kozmanyan et al., 2019). To be precise, the
kSZ effect is due to the source’s peculiar motion where the hot plasma is
responsible for the inverse Compton scattering on CMB photons (Sunyaev et al.,
1980). This effect is often used to estimate the free electron distribution,
and to infer the circumgalactic (CG) peculiar velocity once the free electron
density distribution is given (Birkinshaw, 1999). The amplitude of the kSZ
signal is sensitive to the product of the gas density and bulk velocity
relative to the CMB frame, which is proportional only to the line-of-sight
(los) component of the electron gas velocity in galaxies and galactic halos
(Birkinshaw, 1999). However, the measurement of the kSZ effect, at the present
level of accuracy, is extremely difficult. More precise experiments in the
near future will enable us to study the kSZ effect in more detail. The early
detection of the kSZ effect was due to the proper motion of galaxy clusters.
Hand et al. (2012) analyzed Atacama Cosmology Telescope (ACT) and Baryon
Oscillation Spectroscopic Survey (BOSS) data using mean pairwise statistics,
which gave a new constraint to measure the kSZ effect. The same approach was
then used by De Bernardis et al. (2017) & Li et al. (2018b) for the case of
galaxies. The kSZ signal due to the proper motion of clusters has been
detected in stacked data (Lavaux et al., 2013; Schaan et al., 2016) and
through high-resolution imaging of various individual systems (Adam et al.,
2017). It has also been detected in cross-correlation analyses of projected
fields (Hill et al., 2016; Ferraro et al., 2016; Doré et al., 2004).

The kSZ effect induces two kinds of contributions: the first derives from the
motion of the source as a whole and appears as a monopole-like temperature
shift centered on the source. For example, this contribution has been used
frequently in the literature in order to trace the density profile of the
diffuse gas in some galaxy clusters (Baxter et al., 2019). The second
contribution derives from the internal motion of the hot gas within the
source, and it may have a complicated morphology (Baldi et al., 2018). In
particular, the bulk rotational motion of the gas is expected to give a
dipole-like kSZ signal if one side of the source is moving toward the
observer, while the other is moving away. This effect is referred to as the
rotational kSZ, or rkSZ (Baxter et al., 2019). There is some agreement that
the rkSZ effect may be particularly active on galaxy cluster scales (Cooray et
al., 2002; Chluba et al., 2002; Baldi et al., 2018), and in this case it may
give rise to temperature asymmetry on the order of $10\leavevmode\nobreak\
\mu$K. Recently, the rkSZ induced signal has been detected at about $3\sigma$
confidence level toward about $10^{4}$ massive galaxies of low redshift with
halo mass on the order of $10^{11}\leavevmode\nobreak\ M_{\odot}$, using the
Planck data (Matilla et al., 2020).

As a first approximation and to get an idea of the induced effect in the CMB
by the rkSZ effect, we consider the closest massive galaxy, M31. We assume the
M31 halo to be filled by a hot plasma. Therefore, the variation in the CMB
temperature induced by the rkSZ effect depends on the los integral of the hot
gas number density and on its circular velocity, and can be estimated through
the equation

$\frac{\Delta T}{T}({\bf
n})=\frac{\sigma_{T}}{c}\int_{los}n_{e}^{h}(r)\leavevmode\nobreak\ {\bf
n}\cdot{\bf v}\leavevmode\nobreak\ dl,$ (1)

where ${\bf n}$ is the unit vector that defines the point on the sky where the
CMB temperature is measured, $\sigma_{T}$ is the Thomson cross section, $c$ is
the speed of light, $n_{e}^{h}(r)$ is the hot electron number density
distribution profile, $r$ is associated with the three-dimensional radial
distance from the M31 center, and ${\bf v}$ is the hot gas velocity.
Therefore, eq. (1) can be rewritten as

$\frac{\Delta
T}{T}(R,\phi)=\frac{\sigma_{T}}{c}\int_{los}n_{e}^{h}(r)v_{c}(R)\cos\phi\sin
i\leavevmode\nobreak\ dl,$ (2)

where $v_{c}(R)$ is the circular velocity of the hot electron gas; $R$ is the
projected radial distance, that is the distance with respect to the M31
rotation axis; $\phi$ is the usual azimuthal angle; and $i=77.5^{\circ}$ is
the inclination angle of M31 (i.e., the angle between the normal to the M31
disk and the line of sight). The hot gas is assumed to be at a temperature
$\simeq 10^{6}$ K (West et al., 1997) and is taken to be responsible for the
inverse Compton scattering of the CMB photons that distort the background
radiation from its blackbody spectral shape. Here we note that the step from
eq. (1) to eq. (2) only requires a spherically symmetric hot gas distribution
moving along circular orbits of radius $R$ with velocity $v(R)$. In our model,
following Matilla et al. (2020), the hot electron gas number density
$n_{e}^{h}(r)$ is assumed to be given by

$n_{e}^{h}(r)=\frac{n_{\circ}^{h}}{\mu_{e}\left(\frac{r}{r_{c}}+\frac{3}{4}\right)\left(\frac{r}{r_{c}}+1\right)^{2}},$
(3)

where $n_{\circ}^{h}$ is the central hot electron number density,
$\mu_{e}=1.18$ is the mean atomic weight per electron, and $r_{c}$ is the M31
halo core radius. In our model we assume $n_{\circ}^{h}\simeq 3.4\times
10^{-2}\leavevmode\nobreak\ {\rm cm^{-3}}$ and $r_{c}\simeq 15$ kpc (West et
al., 1997). The number density profile of the hot electron gas in the M31
halo, obtained using eq. (3), is shown in Fig. 1.

Figure 1: Hot electron gas number density profile in the M31 halo up to a
galactocentric distance of about 100 kpc. The central value
$n^{h}_{\circ}\simeq 3.4\times 10^{-2}\,{\rm cm^{-3}}$ has been adopted.

Matilla et al. (2020) also give an explicit expression for the hot gas
circular velocity profile $v_{c}(R)$ expressed as

$v_{c}(R)=\sqrt{\alpha\left[\ln\left(1+\frac{R}{R_{vir}}\right)-\left(\frac{R}{R_{vir}}\right)\left(1+\frac{R}{R_{vir}}\right)^{-1}\right]},$
(4)

where $\alpha=(4\pi Gr_{c}^{2}m_{p}n_{\circ}^{h}R_{vir})/R$, $G$ is Newton’s
gravitational constant, $m_{p}=1.67\times 10^{-24}$ g is the proton mass, and
$R_{vir}$ is the M31 halo virial radius which is assumed to be $R_{vir}\simeq
200$ kpc (Tamm et al., 2012). The hot electron circular velocity radial
profile is shown in Fig. 2.

Figure 2: Hot gas circular velocity radial profile in the M31 halo up to a
galactocentric distance of about 100 kpc.

With the model above, the total hot gas mass in the M31 halo within $R$ can be
easily calculated as

$M_{h}(\leq R)=\int_{0}^{R}4\pi r^{2}m_{p}n_{e}^{h}(r)dr.$ (5)

In Fig. 3 we give the estimated hot gas mass in the M31 halo up to about 100
kpc. It can be seen that our simple model returns a value for the hot gas mass
in the M31 halo in agreement with that estimated in West et al. (1997). The
estimated hot gas mass in West et al. (1997) is $\simeq 7.0\times
10^{6}\leavevmode\nobreak\ M_{\odot}$ up to 20 kpc, while from our model we
get about $6.0\times 10^{6}\leavevmode\nobreak\ M_{\odot}$ within 20 kpc and
$\simeq 2.0\times 10^{7}\leavevmode\nobreak\ M_{\odot}$ up to about 100 kpc.

Figure 3: M31 halo hot gas mass within $R$ calculated through eq. (5).

Our next aim is to estimate, on the basis of the model above for the hot gas
electrons radial profile, the X-ray flux emitted by the hot electron gas in
the M31 halo in order to perform an additional consistency check of the model,
by verifying that the diffuse flux does not exceed the observational data in
the soft X-ray band available in the literature.

As done before, we assume that the hot diffuse gas in the halo of the M31
galaxy is at a temperature $T\simeq 10^{6}$ K (West et al., 1997). It should
then give rise to a thermal emission through the bremsstrahlung emission,
which should be responsible for a soft X-ray background. The emissivity of
this plasma can be described, as a function of the galactocentric distance $R$
and the emission frequency $\nu$, by the relation

$\displaystyle\epsilon_{ff}(R,\nu)=$ $\displaystyle 6.80\times
10^{-32}Z(n_{e}^{h}(R))^{2}T^{-0.5}\times$
$\displaystyle\exp\left(-\frac{h\nu}{k_{B}T}\right)G_{ff}(\nu)\leavevmode\nobreak\
{\rm erg\leavevmode\nobreak\ s^{-1}cm^{-3}\leavevmode\nobreak\ Hz^{-1}},$ (6)

where $Z$ is the gas metallicity, assumed to be $\simeq 0.074$ (Gilbert et
al., 2020); $h$ is Planck’s constant; $k_{B}$ is the Boltzmann constant; and
$G_{ff}$ is the Gaunt factor. In the relevant case of X-ray emission, $G_{ff}$
can be written as (Rosswog et al., 2011)

$G_{ff}(\nu)=\left(\frac{\sqrt{3}}{\pi}\right)\log\left[\frac{k_{B}T}{h\nu}\right].$
(7)

The total radiated power per unit volume by free-free emission from the hot
electron gas in the M31 halo can be thus estimated by substituting eq. (7) in
eq. (6) and integrating over the energy band which is assumed to be $0.5-2$
keV, which is the soft band of ROSAT. The total radiated power per unit volume
can be written as

$\displaystyle
P_{ff}(R)=\int_{\nu_{i}}^{\nu_{f}}\epsilon_{ff}(R,\nu)\leavevmode\nobreak\
d\nu=$ $\displaystyle\frac{4.57\times 10^{-26}}{(0.75+2.65\times
10^{-21}R)^{2}(1+2.65\times 10^{-21}R)^{4}}\leavevmode\nobreak\ {\rm
erg\leavevmode\nobreak\ s^{-1}\leavevmode\nobreak\ cm^{-3}}.$ (8)

Thus, the total luminosity emitted within the radius $R$ can be estimated by

$\Gamma_{ff}(\leq R)=\int_{A}P_{ff}(R)\leavevmode\nobreak\
dA\leavevmode\nobreak\ {\rm erg\leavevmode\nobreak\ s^{-1}},$ (9)

where $A$ is the aperture toward M31 halo within R. Similarly, the flux
density can be calculated by

$\Phi_{ff}(R)=\frac{\Gamma_{ff}(R)}{4\pi D^{2}}\leavevmode\nobreak\ {\rm
erg\leavevmode\nobreak\ s^{-1}\leavevmode\nobreak\ cm^{-2}},$ (10)

where $D\simeq 744$ kpc is the distance to the M31 center (Ribas et al.,
2005). In Table 2 of West et al. (1997), the diffuse soft X-ray flux for the
energy band $0.5-2$ keV toward the M31 halo is estimated. Assuming a hot gas
core radius $r_{c}=15$ kpc, they derive the luminosity of the spherical halo
$\approx 7.2\times 10^{39}\leavevmode\nobreak\ {\rm erg\leavevmode\nobreak\
s^{-1}}$ up to about 20 kpc. In order to check the consistency of our model,
we compare the X-ray luminosity toward the M31 halo obtained by using our
model results by using eq. (9) and then the flux density from eq. (10). Our
estimated luminosity $\Gamma_{ff}$ comes out to be $\approx 2.80\times
10^{40}\leavevmode\nobreak\ {\rm erg\leavevmode\nobreak\ s^{-1}}$ up to 20
kpc, in substantial agreement with that estimated from observations by West et
al. (1997) (see their Table 2). The corresponding X-ray flux $\Phi_{ff}$ turns
out to be $\approx 4.24\times 10^{-10}\leavevmode\nobreak\ {\rm
erg\leavevmode\nobreak\ s^{-1}\leavevmode\nobreak\ cm^{-2}}$. We can therefore
conclude that our model results are in good enough agreement with the
observations.

We are now in a position to estimate the temperature excess $\Delta T$ induced
by the rkSZ effect, through eq. (2). In Fig. 4 we give the maximum value of
the contribution from the rkSZ effect to the temperature variation of the CMB.
This result is obtained by using eq. (2) and scanning the major axis of the
M31 halo from left to right. As can be seen, the rkSZ-induced temperature
variation peaks at about 5 kpc from the M31 center and then slowly decreases
toward the external M31 halo region. Similarly, in Table 1 we give the maximum
value of the estimated temperature variation induced by the rkSZ effect
(central column) at various radial distances from the M31 center (specified in
the first column) and compare them with the detected excess given in the third
column. The values given in the third column correspond to the SMICA processed
data. Here we note that the SMICA data were chosen because they are less
contaminated than the other available Planck bands. It is new Planck data
product and has already been adopted, for example to search for SZ sources and
in lensing studies (Akrami et al., 2018; Aghanim et al., 2020). We also note
that the central column of Table 1 refers to the maximum value of the rkSZ-
induced temperature variation along the major axis of the M31 galactic halo
edge-on map, without including the instrument beam smoothing. As can be seen,
the estimated values of $\Delta T$ are much lower with respect to the values
observed by the Planck satellite. Hence, we can safely say that the
contribution induced by the rkSZ effect toward the M31 halo turns out to be
negligible.

Until now we have assumed that the hot gas temperature in the M31 halo is
$T\simeq 10^{6}$ K, and the hot electron central number density is
$n_{\circ}^{h}\simeq 3.4\times 10^{-2}\leavevmode\nobreak\ {\rm cm^{-3}}$. The
question arises of what changes in the values of $\Delta T$ induced by the
rkSZ effect by varying these parameters, and requiring that the diffuse X-ray
flux values estimated in Table 2 of West et al. (1997) are not violated. We
increased the value of the hot gas temperature to a few times $\sim 10^{7}$ K,
and decreased the value of the central number density to $\sim
10^{-4}-10^{-5}\leavevmode\nobreak\ {\rm cm^{-3}}$. As a result we find that
the estimated temperature variation induced by the rKSZ effect increased only
negligibly for each considered M31 halo radius. This makes our results robust
enough.

Table 1: Estimated maximum temperature variation (central column) induced by the rkSZ effect at various radial regions (indicated in the left column) along the major axis of the M31 halo. Equation (2) has been used and a hot gas temperature at $T\simeq 10^{6}$ K has been assumed. In the right column the SMICA observed temperature variation toward each region is also given. Region | Estimated | Observed
---|---|---
$R$ | $\Delta T_{est}$ | $\Delta T_{obs}$
kpc |  $\mu$ K |  $\mu$K
21.4 | $1.16\times 10^{-1}$ | 2.66
31.1 | $1.06\times 10^{-1}$ | 6.44
41.4 | $9.98\times 10^{-2}$ | 17.6
51.8 | $9.54\times 10^{-2}$ | 39.2
77.8 | $8.45\times 10^{-2}$ | 21.7
103.8 | $7.86\times 10^{-2}$ | 21.4
Figure 4: Maximum value of the rkSZ-induced temperature variation along the
major axis of the M31 galactic halo as a function of the M31 halo projected
radius $R$ is given.

##  3 Conclusions

Galactic halos are certainly much less studied with respect to galactic disks,
and there are still many unanswered questions related to their dynamics. The
Planck data of the CMB opened up a new window to probe the halo dynamics in
detail, and the rotation of disks and halos of various nearby edge-on spiral
galaxies has been investigated using these data. The independence of the CMB
temperature asymmetry of the frequency toward the M31 halo constitutes a
strong indication of its Doppler nature as a consequence of the M31 halo
rotation. What is not well understood is whether the true cause of the Doppler
shift is fully due to virial clouds populating the M31 halo or if there is a
non-negligible contribution from other effects, such as the rkSZ effect, free-
free emission by free electrons, synchrotron radiation by fast moving
electrons, and/or anomalous microwave emission by dust grains.

In the present paper we estimated in particular the contribution to the
observed temperature asymmetry toward the M31 halo by the rkSZ effect. We
started our analysis by constraining the hot gas parameters in the M31 halo.
Following West et al. (1997) we assumed that the central number density of the
hot electron gas is $\approx 3\times 10^{-2}\leavevmode\nobreak\ {\rm
cm^{-3}}$ and its temperature is $\approx 10^{6}$ K.

We then used eq. (3) to estimate the density distribution of this hot gas in
the M31 halo (see also Fig. 1). Similarly, the circular velocity $v_{c}(R)$ of
this hot gas was estimated by using eq. (4) and the resulting radial profile
was shown in Fig. 2. The obtained values for the hot gas circular velocity
results were found to be in the range $50-250$ km/s. We then estimated the
total mass distribution of this hot gas in the M31 halo, as shown in Fig. 3,
and found a total hot gas mass $\approx 1.9\times 10^{7}M_{\odot}$ up to about
100 kpc.

The next step was checking if the assumed parameters for the hot gas in the
M31 halo give results that are not in contradiction with observations. In
order to tackle this problem we considered the diffuse X-ray flux due to free-
free emission by hot electrons in the M31 halo, and as a result we estimated
theoretically the expected luminosity and flux in $0.5-2$ keV energy band. The
estimated luminosity was $\approx 2.8\times 10^{40}\leavevmode\nobreak\ {\rm
erg\leavevmode\nobreak\ s^{-1}}$ up to about 20 kpc, which is in rather good
agreement with that observed in the soft X-ray band by ROSAT (see Section 2).

The final step was estimating the expected temperature asymmetry, $\Delta T$,
induced by the rkSZ effect as a function of the projected radial distance $R$
in the M31 halo by using eq. 2. As is clear from Table 1, the contribution due
to the rkSZ effect can be considered negligible since it can produce an effect
certainly less than about $1\%$ of the temperature anisotropy detected in the
SMICA maps of the Planck satellite toward the M31 halo. This result makes the
findings in De Paolis et al. (2011, 2014), and in particular the
interpretation of the temperature anisotropy detected toward the M31 halo as
being due to a population of cold clouds rotating about a rotation axis, more
robust. It cannot be excluded, however, that future observations such as those
expected in the 220 GHz channel (which is the cross-over frequency of the tSZ
component where the kSZ signal is expected to be larger) of the Atacama
Cosmology Telescope (ACT) (see, e.g., Bleem et al. 2022), of the Simons
observatory (Hervías-Caimapo et al., 2022), of the CMB-S4 experiment (Galli et
al., 2022), and of CMB-HD (Sehgal et al., 2019), can bring more precise CMB
data allowing a mapping of the rotation of galactic halos using the rkSZ
effect.

## Acknowledgements

NT, FDP and AAN acknowledges the TAsP (Theoretical Astroparticle Physics) and
Euclid INFN projects. The anonymous referee is also acknowledget for the
useful and constructive report.

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generatorexamplerandom triplet$n$ channels$3$ channels$3$-channel loss(Gatys
et al., 2015)pretrainedpretrainedVGG-19VGG-19examplegenerated

albedo

normal

roughness

metalness

ambient

albedo

normal

roughness

metalness

ambient

Figure 1.  We train a material texture generator with multiple channels such
as _albedo_ , _normal_ , _roughness_ , _metalness_ and _ambient occlusion_ by
passing random channel triplets to the $3$-channel loss proposed by Gatys et
al. (2015).

# Passing Multi-Channel Material Textures to a 3-Channel Loss

Thomas Chambon  Unity Technologies ,  Eric Heitz  Unity Technologies and
Laurent Belcour  Unity Technologies

(2021)

###### Abstract.

Our objective is to compute a textural loss that can be used to train texture
generators with multiple material channels typically used for physically based
rendering such as _albedo_ , _normal_ , _roughness_ , _metalness_ , _ambient
occlusion_ , etc. Neural textural losses often build on top of the feature
spaces of pretrained convolutional neural networks. Unfortunately, these
pretrained models are only available for 3-channel RGB data and hence limit
neural textural losses to this format. To overcome this limitation, we show
that passing random triplets to a 3-channel loss provides a multi-channel loss
that can be used to generate high-quality material textures.

††submissionid: 1234††copyright: rightsretained††journalyear:
2021††conference: SIGGRAPH ’21 Talks; August 09-13, 2021; Virtual Event,
USA††booktitle: Special Interest Group on Computer Graphics and Interactive
Techniques Conference Talks (SIGGRAPH ’21 Talks), August 09-13, 2021††doi:
10.1145/3450623.3464685††isbn: 978-1-4503-8373-8/21/08††ccs: Computing
methodologies Rendering

##  1\. Introduction

A neural textural loss allows for generating textures by image optimization
(Gatys et al., 2015) or training generative models (Ulyanov et al., 2016).
Typically, the loss is computed from the statistics of the feature activations
in pretrained Convolutional Neural Networks (CNNs) such as VGG-19 (Simonyan
and Zisserman, 2015). These pretrained CNNs are mainly available for RGB
inputs, i.e. a $3$-channel formats. This is limiting for material textures
used in physically based rendering that have multiple channels such as
_albedo_ , _normal_ , _roughness_ , _metalness_ , _ambient occlusion_ , etc.
We thus investigate how a 3-channel loss can be applied to $n$-channel
textures.

Our first attempt was inspired by previous work that generates material
textures from RGB photographs examples (Aittala et al., 2016). They use a
differentiable renderer to light the material textures and create an RGB
render that can be passed to an RGB loss. The material textures can then be
optimized via gradient descent by backpropagating gradients through the
differential renderer. As shown in Figure 2-(a), we found this approach to be
unstable with non-diffuse materials, especially sharp speculars. Indeed, in
addition to texture synthesis, the optimizer also needs to solve a challenging
inverse rendering problem. Aittala et al. (2016) report using additional
priors and considerable engineering efforts.

Fortunately, we can take advantage of explicitly provided material channels
and avoid solving a difficult inverse rendering problem if we find a simpler
way to pass $n$ channels to a 3-channel loss. We tested different more or less
elaborated ideas such as computing a partial component analysis, training a
$n$-to-3 channel encoder, etc. In the end, we found that the best solution
consists of choosing random triplets in the $n$ channels. It provides a
surprisingly simple and well-founded approach with stable outcome shown in
Figure 2-(b).

| albedo | normal | rendered | albedo | normal | roughness | rendered
---|---|---|---|---|---|---|---
example |  |  |  |  |  |  |
(a) diff. render |  |  |  |  |  |  |
(b) ours |  |  |  |  |  |  |
| diffuse-only material | diffuse and rough-specular material
Figure 2.  (a) Passing a differentiable render to a 3-channel loss works with
diffuse materials but becomes unstable with rough speculars. (b) Our approach
is robust and stable.

##  2\. Our Multi-Channel Textural Loss

### The 3-channel loss.

We build upon the 3-channel textural loss introduced by Gatys et al. (2015).
They define the textural distance between two RGB images $I_{3}$ and
$\tilde{I}_{3}$ as the MSE between the Gram matrices of the activations
produced by the images in the $L$ layers of a pretrained CNN:

(1)
$\displaystyle\mathcal{L}_{3\text{-channel}}\left(I_{3},\tilde{I}_{3}\right)=\sum_{l=1}^{L}\frac{1}{N_{l}^{2}}\|G^{l}-\tilde{G}^{l}\|^{2},$

where $G^{l}$ and $\tilde{G}^{l}$ are the Gram matrices of the $N_{l}$ deep
features extracted from respectively $I$ and $\tilde{I}$ at layer $l$ in the
pretrained CNN. In our experiments, we use a pretrained VGG-19 (Simonyan and
Zisserman, 2015).

### Combining multiple 3-channel losses.

Accounting for more than 3 channels can be done by adding multiple 3-channel
losses applied on the different maps. For instance, in Figure 3-(a) we
optimize for the sum of two 3-channel losses computed for the _albedo_ and the
_roughness_ triplets separately. This produces a texture whose _albedo_ and
_normal_ look realistic separately but do not match together because the
correlations between the albedo and the normal have not been accounted for.

### Our $n$-channel loss.

In order to account for all the inter-channel correlations, we define the loss
between two $n$-channel images $I_{n}$ and $\tilde{I}_{n}$ as the expectation
of the 3-channel loss over _all_ the possible triplets:

(2)
$\displaystyle\mathcal{L}_{n\text{-channel}}\left(I_{n},\tilde{I}_{n}\right)=\mathbb{E}\left[\mathcal{L}_{3\text{-channel}}\left(\text{triplet}(I_{n}),\text{triplet}(\tilde{I}_{n})\right)\right],$

where $\text{triplet}\left(I_{n}\right)$ is a $3$-channel image whose channels
are chosen randomly among the $n$ channels of $I_{n}$. As shown in Figure
3-(b), $\mathcal{L}_{n\text{-channel}}$ preserves inter-channel correlations.
The generated textures have the same feature at the same places channel-wide.
The downside is a direct evaluation of $\mathcal{L}_{n\text{-channel}}$
requires evaluating $\mathcal{L}_{3\text{-channel}}$ for all possible triplets
and averaging the results. A material of $n$ channels hence requires $n^{3}$
different evaluations, which is untractable in practice.

### Stochastic evaluation.

To overcome this problem we proceed as shown in Figure 1. Instead of
evaluating $\mathcal{L}_{3\text{-channel}}$ on all the possible triplets, we
only evaluate it on a _single_ triplet that is randomized for each batch
during learning. In other words, we compute a stochastic estimate of Equation
(2):

(3)
$\displaystyle\hat{\mathcal{L}}_{n\text{-channel}}\left(I_{n},\tilde{I}_{n}\right)=\mathcal{L}_{3\text{-channel}}\left(\text{triplet}(I_{n}),\text{triplet}(\tilde{I}_{n})\right).$

This evaluation is fast, practical and does not change the optimum of the
$L^{2}$ minimization because the estimate is unbiased. Furthermore, it remains
robust because the randomness induced by the stochastic evaluation is similar
to the natural randomness of stochastic gradient descent and well-handled by
state-of-the-art optimizers.

##  3\. Training and results

We use our loss as a drop-in extension of a 3-channel loss to train generative
architectures of $n$ rather than $3$ channels. We train a mono-texture
(Ulyanov et al., 2016) and a multi-texture (Li et al., 2017) generators, which
we adapted to output $n$ channels. We use
$\hat{\mathcal{L}}_{n\text{-channel}}$ for sole loss function without further
priors or regularization terms and we train with the Adam optimizer. Note that
we use a vanilla implementation of the 3-channel loss of Gatys et al. (2015).
Several improvements to this loss have been published and implementing them
would directly benefit to our $n$-channel extension as well. Figure 1 shows a
result generated by our mono-texture generator. Figure 4 shows a result
generated by our multi-texture generator. These generative architectures are
capable of producing arbitrarily-large texture at inference time with
variation (no verbatim copying of the exemplar). Our supplemental material
shows further results with different sets of channels on various textures.

##  4\. Conclusion

We have proposed a simple approach to extend existing 3-channel textural
losses to $n$ channels. The main idea is to span all the possible inter-
channel correlations thanks to a stochastic evaluation that can be implemented
in a single line of code. With this approach we hope to bring a vast
literature of neural texture synthesis approaches to material texture
synthesis without further efforts.

✗

✓

albedoroughnessalbedoroughness(a)(b)$\mathcal{L}_{3\text{-channel}}$ on
albedo$+\mathcal{L}_{3\text{-channel}}$ on
roughness$\mathcal{L}_{n\text{-channel}}$ Figure 3.  (a) Optimizing multiple
$3$-channel losses on separate triplets produces textures that look realistic
independently but that are not correlated, i.e. the spatial features do not
match. (b) Our loss preserves inter-channel correlations since it optimizes
for all the possible channel combinations.  example 1generatedexample 2
---
Figure 4.  We train a multi-texture generator that allows for interpolating
material textures. We trained the same architecture for 16 textures that
includes these 2 examples.

## References

  * (1)
  * Aittala et al. (2016) Miika Aittala, Timo Aila, and Jaakko Lehtinen. 2016.  Reflectance Modeling by Neural Texture Synthesis.  _ACM Trans. Graph._ 35, 4, Article 65 (2016), 13 pages.
  * Gatys et al. (2015) Leon A. Gatys, Alexander S. Ecker, and Matthias Bethge. 2015.  Texture Synthesis Using Convolutional Neural Networks. In _Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 1_ _(NIPS’15)_. 262–270.
  * Li et al. (2017) Yijun Li, Chen Fang, Jimei Yang, Zhaowen Wang, Xin Lu, and Ming Hsuan Yang. 2017\.  Diversified texture synthesis with feed-forward networks. In _Proceedings - 30th IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2017_. 266–274.
  * Simonyan and Zisserman (2015) Karen Simonyan and Andrew Zisserman. 2015.  Very Deep Convolutional Networks for Large-Scale Image Recognition. In _International Conference on Learning Representations_.
  * Ulyanov et al. (2016) Dmitry Ulyanov, Vadim Lebedev, Andrea Vedaldi, and Victor Lempitsky. 2016.  Texture Networks: Feed-Forward Synthesis of Textures and Stylized Images. In _Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48_ _(ICML’16)_. 1349–1357.