N_{m}}},\det\|\cdot\|_{m\phi}\right)=O(N_{m})$

when $\phi$ is a model metric. ∎

###### Proof of Theorem 9.2.

By definition, we have

$\sup
V_{\|\cdot\|_{m}}(m\phi)=\frac{\|\Psi_{m}(\omega_{m})\|_{(m\phi)^{\boxtimes
N_{m}}}}{\det\|\omega_{m}\|_{m}}$

for any choice of generator $\omega_{m}\in\det H^{0}(mL)$. By Lemma 9.7, it
follows that

$\log\delta_{m,\psi}(\phi)=\frac{1}{mN_{m}}\operatorname{vol}\left(\|\cdot\|_{m},\|\cdot\|_{m\phi}\right)+o(1)=\frac{1}{mN_{m}}\operatorname{vol}\left(\|\cdot\|_{m\psi},\|\cdot\|_{m\phi}\right)+o(1)$

and we thus see that Theorem 9.2 is indeed equivalent to Theorem 8.5. ∎

####  9.2. Equidistribution of Fekete points

We assume in this section that one of the following holds:

  * (a)

$K$ is Archimedean;

  * (b)

$K$ is non-Archimedean, either trivially or discretely valued, and of residue
characteristic zero.

By [BB10, BFJ15, BG+16, BJ18], the semipositive envelope $P(\phi)$ of each
continuous metric $\phi\in\mathcal{C}^{0}(L)$ is then continuous, and for any
auxiliary continuous sempositive metric $\psi$, the operator $\phi\mapsto
E(P(\phi),\psi)$ is differentiable on $\mathcal{C}^{0}(L)$, with directional
derivatives given by

$\frac{d}{dt}_{t=0}E(P(\phi+tu),\psi)=\int_{X^{\mathrm{an}}}u\,\operatorname{MA}(P(\phi)).$
(9.5)

Following the strategy of [BBW11], itself inspired by a variational argument
due to Szpiro-Ullmo-Zhang [SUZ97], we will then use (9.5) to infer from
Theorem 9.2 an equidistribution property for Fekete configurations.

######  Definition 9.10.

Let $\phi\in\mathcal{C}^{0}(L)$ be a continuous metric on $L$. A _Fekete
configuration_ for $\phi$ is a point $P\in(X^{N})^{\mathrm{an}}$ such that

$\sup_{(X^{N})^{\mathrm{an}}}|\Psi(\omega)|_{\phi^{\boxtimes
N}}=|\Psi(\omega)|_{\phi^{\boxtimes N}}(P)$

for some, hence any, generator $\omega\in\det H^{0}(L)$.

######  Theorem 9.11.

Assume that the ground field $K$ satisfies (a) or (b) above. For each $m\gg
1$, pick a Fekete configuration $P_{m}\in(X^{N_{m}})^{\mathrm{an}}$ for
$m\phi$. Then $P_{m}$ equidistributes to $\operatorname{MA}(P(\phi))$ as
$m\to\infty$.

The statement means that

$\int_{X^{\mathrm{an}}}u\,\delta_{P_{m}}\to\int_{X^{\mathrm{an}}}u\,\operatorname{MA}(P(\phi))$

for each $u\in\mathcal{C}^{0}(X^{\mathrm{an}})$, where we associate to
$P\in(X^{N})^{\mathrm{an}}$ the measure $\delta_{P}$ on $X^{\mathrm{an}}$
obtained by averaging over the image of $P$ in $(X^{\mathrm{an}})^{N}$.

###### Proof.

Fix an auxiliary continuous semipositive metric $\psi$ on $L$, and pick for
each $m$ a generator $\omega_{m}\in\det H^{0}(mL)$. To say that $P_{m}$ is a
Fekete configuration for $m\phi$ means that
$|\Psi_{m}(\omega_{m})|_{(m\phi)^{\boxtimes
N_{m}}}\in\mathcal{C}^{0}((X^{N_{m}})^{\mathrm{an}})$ is maximized (and, in
particular, nonzero) at $P_{m}$. In terms of

$F_{m}(\phi):=-\frac{1}{mN_{m}}\log\left(\frac{|\Psi_{m}(\omega_{m})|_{(m\phi)^{\boxtimes
N_{m}}}(P_{m})}{\det\|om_{m}\|_{m\psi}}\right),$

$P_{m}$ is Fekete for $m\phi$ iff $F_{m}(\phi)=-\log\delta_{m,\psi}(\phi)$.
For each $u\in\mathcal{C}^{0}(X^{\mathrm{an}})$, observe that

$F_{m}(\phi+u)=F_{m}(\phi)+\int_{X^{\mathrm{an}}}u\,\delta_{P_{m}}.$

Since $F_{m}(\phi+u)\geq-\log\delta_{m,\psi}(\phi)$, Theorem 9.2 yields

$\liminf_{m\to\infty}F_{m}(\phi+u)\geq E(P(\phi+u),\psi)$

and

$\lim_{m\to\infty}F_{m}(\phi)=E(P(\phi),\psi).$

Replacing $u$ with $tu$ for $0<t\ll 1$, we infer

$t\liminf_{m}\int_{X^{\mathrm{an}}}u\,\delta_{P_{m}}=\liminf_{m}\left(F_{m}(\phi+tu)-F_{m}(\phi+tu)\right)\geq
E(P(\phi+tu),\psi)-E(P(\phi),\psi),$

and hence

$\liminf_{m}\int_{X^{\mathrm{an}}}u\,\delta_{P_{m}}\geq\frac{d}{dt}_{t=0}E(P(\phi+tu),\psi)=\int_{X^{\mathrm{an}}}u\,\operatorname{MA}(P(\phi)).$

Applying this to $-u$, we conclude as desired

$\lim_{m}\int_{X^{\mathrm{an}}}u\,\delta_{P_{m}}=\int_{X^{\mathrm{an}}}u\,\operatorname{MA}(P(\phi)).$

∎

####  9.3. The pullback formula

We consider in this section a _polarized endomorphism_ $f$ of $(X,L)$, i.e. a
morphism $f:X\to X$ together with the data of an isomorphism $f^{*}L\simeq dL$
for some positive integer $d>1$. Since $f^{*}L$ is ample and $f$ is
necessarily proper, $(f^{*}L)^{n}=d^{n}(L^{n})$ implies that $f$ is finite
(and flat, as $X$ is assumed to be smooth), of degree $d^{n}$. By Theorem
A.15, we thus have a canonical isomorphism

$\langle f^{*}L^{n+1}\rangle\simeq d^{n}\langle L^{n+1}\rangle,$

which combines with the given isomorphism $f^{*}L\simeq dL$ to yield

$d^{n+1}\langle L^{n+1}\rangle\simeq d^{n}\langle L^{n+1}\rangle.$

This defines a canonical section

$R_{f}\in d^{n}(d-1)\langle L^{n+1}\rangle,$ (9.6)

which we call the _resultant section_ (see Corollary 9.17 below for the choice
of terminology).

On the other hand, the map $\mathcal{C}^{0}(L)\to\mathcal{C}^{0}(L)$ defined
by $\phi\mapsto d^{-1}f^{*}\phi$, being $1/d$-Lipschitz continuous, admits a
unique fixed point $\phi_{f}$, the _equilibrium metric_ of $f$. For any choice
of Fubini-Study metric $\phi$, the metrics $d^{-j}(f^{j})^{*}\phi$ are Fubini-
Study as well, and they converge uniformly to $\phi_{f}$, which is thus
asymptotically Fubini-Study.

######  Example 9.12.

For any $d\geq 2$, the equilibrium metric of the polarized endomorphism $f$ of
$(\mathbb{P}^{n},\mathcal{O}(1))$ induced by
$(x_{0},\dots,x_{n})\mapsto(x_{0}^{d}:\dots:x_{n}^{d})$ is
$\phi_{f}=\max_{i}\log|x_{i}|$.

######  Lemma 9.13.

The resultant $R_{f}\in d^{n}(d-1)\langle L^{n+1}\rangle$ has norm $1$ with
respect to the induced metric $d^{n}(d-1)\langle\phi_{f}^{n+1}\rangle$.

###### Proof.

For any continuous, psh-regularizable metric $\phi$ on $L$, the isomorphism
$\langle f^{*}L^{n+1}\rangle\simeq d^{n}\langle L^{n+1}\rangle$ is an isometry
with respect to $\langle f^{*}\phi^{n+1}\rangle$ and
$\langle\phi^{n+1}\rangle$. By definition of $\phi_{f}$, the isomorphism
$f^{*}L\simeq dL$ is an isometry with respect to $f^{*}\phi_{f}$ and
$d\phi_{f}$. It follows that the induced isomorphism $d^{n}(d-1)\langle
L^{n+1}\rangle\simeq K$ is an isometry with respect to
$d^{n}(d-1)\langle\phi_{f}^{n+1}\rangle$ and the canonical metric on $K$,
hence the result. ∎

We now get the following pull-back formula for the transfinite diameter, which
generalizes [DMR], [BB10, §6.3] in view of Corollary 9.17 below.

######  Theorem 9.14.

Let $\phi,\psi$ be continuous metrics on $L$, and assume that
$P(\phi),P(\psi)$ are continuous. Then

$\delta_{\infty,\psi}(d^{-1}f^{*}\phi)=c_{f}(\psi)\delta_{\infty,\psi}(\phi)^{1/d}$

with $c_{f}(\psi)>0$ such that

$\log c_{f}(\psi)=-\frac{1}{V(n+1)d^{n+1}}\log|R_{f}|_{d^{n}(d-1)\langle
P(\psi)^{n+1}\rangle}.$

###### Proof.

By Lemma 9.15, we have $P(d^{-1}f^{*}\phi)=d^{-1}f^{*}P(\phi)$. Since
$d^{-1}f^{*}\phi_{f}=\phi_{f}$, Theorem 9.2 and the coycle formula for $E$
give

$\log\delta_{\infty,\psi}(d^{-1}f^{*}\phi)=E\left(P(\psi),P(d^{-1}f^{*}\phi)\right)\\\
=E\left(P(\psi),\phi_{f}\right)+E\left(d^{-1}f^{*}\phi_{f},d^{-1}f^{*}P(\phi)\right)\\\
=E\left(P(\psi),\phi_{f}\right)+d^{-1}E\left(\phi_{f},P(\phi)\right)\\\
=(1-d^{-1})E\left(P(\psi),\phi_{f}\right)+d^{-1}E\left(P(\psi),P(\phi)\right)\\\
=(1-d^{-1})E\left(P(\psi),\phi_{f}\right)+\log\delta_{\infty,\psi}(\phi)^{1/d}.$

On the other hand, Lemma 9.13 yields

$\log|R_{f}|_{d^{n}(d-1)\langle
P(\psi)^{n+1}\rangle}=d^{n}(d-1)\left(\langle\phi_{f}^{n+1}\rangle-\langle
P(\psi)^{n+1}\rangle\right),$

and hence

$\frac{1}{Vd^{n+1}}\log|R_{f}|_{d^{n}(d-1)\langle
P(\psi)^{n+1}\rangle}=(n+1)(1-d^{-1})E\left(\phi_{f},P(\psi)\right).$

The result follows. ∎

######  Lemma 9.15.

Let $f:Y\to X$ be a finite, flat morphism of reduced projective $K$-schemes.
Suppose $L$ is a line bundle on $X$, and $\phi$ any bounded metric on
$L^{\mathrm{an}}$. Then $P(f^{*}\phi)=f^{*}P(\phi)$, and in particular if
$P(\phi)$ is continuous, so is $P(f^{*}\phi)$.

In the Archimedean case, this is a special case of [BB10, Proposition 2.9].

###### Proof.

Denote the degree of $f$ by $e$. We will show that any asymptotically Fubini-
Study metric $\psi$ on $f^{*}L$ with $\psi\leq f^{*}\phi$ admits a majoration
$\psi\leq f^{*}\varphi\leq f^{*}\phi$, where $\varphi$ is asymptotically
Fubini-Study.

If $K$ is non-Archimedean non-trivially valued, it follows from Proposition
5.9 and $1$-Lipschitz continuity of $f^{*}$ that
$f^{*}\operatorname{\overline{\mathcal{FS}}}(L)\subseteq\operatorname{\overline{\mathcal{FS}}}(f^{*}L)$.
The corresponding statement in the trivially valued case follows from this one
and considerations analogues to the proof in Proposition 6.16.

We thus have the inequality $f^{*}P(\phi)\leq P(f^{*}\phi)$ and since $P$ is
non-decreasing we find that

$\psi\leq f^{*}\varphi\leq f^{*}P(\phi)\leq P(f^{*}\phi)$

which implies the statement by taking supremum over all
$\psi=P_{m}(f^{*}\phi).$

Recall that there is a natural metric $N(\varphi)$ on the norm of a line
bundle with metric $\varphi$ (cf. (7.5)). If we are given an asymptotically
Fubini-Study metric $\psi\leq f^{*}\phi$, then $N(\psi)\leq
N(f^{*}\phi)=e\phi$, and $f^{*}\frac{1}{e}N(\psi)\leq f^{*}\phi$. By Lemma
7.8, $f^{*}\frac{1}{e}N(\psi)$ is also asymptotically Fubini-Study. For any
positive integer $k$, define inductively
$A_{k}\psi=\sup\\{f^{*}\frac{1}{e}N(A_{k-1}\psi),\psi\\}$, where we set
$A_{0}(\psi)=\psi$. It is an asymptotically Fubini-Study metric by Proposition
6.16. By using an auxiliary bounded reference metric on $L$, e.g. $\phi$, we
may identify our metrics with functions. Consider the fiberwise supremum

$v(y)=\sup_{x^{\prime}\in f^{-1}(y)}\psi(x^{\prime}).$

We want to show that $A^{k}\psi\to f^{\ast}v$ uniformly as $k\to\infty$. By we
have construction

$0\leq(f^{*}v-A_{k}\psi(x)\leq v(f(x))-\left[\frac{1}{e}\sum_{x^{\prime}\in
f^{-1}(f(x)}-{m_{x^{\prime}}}{A_{k-1}\psi}(x^{\prime})\right].$

Here $[f^{-1}(y)]=\sum m_{x^{\prime}}[x^{\prime}]$ denotes the fundamental
cycle of $f^{-1}(y)$. Since for any $y\in Y^{\mathrm{an}}$, there is always at
least one $x\in f^{-1}(y)$ such that ${A^{k-1}\psi}(x)=v(y)$, it follows that
any bound $M_{k}$ of the above can be choosen so that
$M_{k}\leq\left(1-\frac{1}{e}\right)M_{k-1}\leq\left(1-\frac{1}{e}\right)^{k}M_{0}$.
We conclude that $A^{k}\psi\to f^{*}v$ uniformly so that $f^{*}v$ is
asymptotically Fubini-Study and bounded below by $\psi$, above by $f^{*}\phi.$
This construction is independent of the implicit auxiliary metric used.

∎

####  9.4. The case toric varieties

We now illustrate the previous pull-back formula in the toric case. We thus
assume that $(X,L)$ is a smooth projective polarized toric variety with
respect to a split torus $T\simeq(\mathbb{G}_{m,K})^{n}$ with character
lattice $M=\operatorname{Hom}(T,\mathbb{G}_{m,K})$, which thus corresponds to
a Delzant polytope $\Delta\subset M_{\mathbb{R}}$.

For each integer $d\geq 2$, multiplication by $d$ on $M$ induces a polarized
endomorphism $m_{d}$ of $(X,L)$, defining a an equilibrium metric $\phi_{d}$
on $L$ and a resultant section $R_{d}\in d^{n}(d-1)\langle L^{n+1}\rangle$.
The next lemma describes the moduli space of polarized endomorphisms of degree
$d$ of $(X,L)$.

######  Lemma 9.16.

Set $N=\dim H^{0}(X,L)=\\#(M\cap\Delta)$. The space of polarized morphisms of
degree $d$ of $(X,L)$ is parametrized by a Zariski open subset of

$\mathbb{P}\left(\operatorname{Hom}\left(H^{0}(X,L),H^{0}(X,dL)\right)\right)\simeq\mathbb{P}\left(H^{0}(X,dL)^{N}\right)$

whose complement $Z$ has codimension $N-n$. In particular, $Z$ has codimension
greater than $1$ unless $(X,L)\simeq(\mathbb{P}^{n},\mathcal{O}(1))$.

###### Proof.

Since $X$ is smooth, $R(X,L)$ is generated in degree one, the data of a
polarized endomorphism of $(X,L)$ of degree $d$ is equivalent to that of a
linear map $H^{0}(X,L)\to H^{0}(X,dL)$ whose image is basepoint free. As a
result, the space of polarized endomorphisms of degree $d$ is isomorphic to
the complement in $\mathbb{P}\left(H^{0}(X,dL)^{N}\right)$ of the projection
$Z$ of the incidence variety

$I=\left\\{\left([s_{1}:\dots:s_{N}],x\right)\in\mathbb{P}\left(H^{0}(X,dL)^{N}\right)\times
X,s_{1}(x)=\dots=s_{N}(x)=0\right\\}$

Since $dL$ is basepoint free, the elements of $H^{0}(X,dL)$ vanishing at a
given closed point $x\in X$ is a hyperplane, and it follows that $\dim
I=n-1+N(N_{d}-1)$. We claim that the restriction to $I$ of the first
projection $H^{0}(X,dL)^{N}\times X\to H^{0}(X,dL)^{N}$ is generically finite,
which will imply $\operatorname{codim}Z=NN_{d}-1-\dim I=N-n$. Indeed, if
$s_{1},\dots,s_{n}\in H^{0}(X,dL)$ is a regular sequence of sections, the
fiber of $I$ over the $N$-tuple $(s_{1},\ldots,s_{1},s_{2},\ldots,s_{n})\in
H^{0}(X,dL)^{N}$ is finite.

The last point of the lemma follows from the embedding
$X\hookrightarrow\mathbb{P}H^{0}(X,L)$, which is an isomorphism iff $N-n=1$. ∎

In the case $N=n-1$, i.e. $(X,L)\simeq(\mathbb{P}^{n},\mathcal{O}(1))$, any
polarized morphism of degree $d$ is given by
$z\mapsto[f_{0}(z):\ldots:f_{n}(z)]$ for homogenous polynomials
$f_{0},\ldots,f_{n}$, of degree $d$, without common zeros, and the locus
$Z\subset\mathbb{P}\left(H^{0}(X,L)^{n+1}\right)$ is an irreducible divisor of
degree $(n+1)d^{n}$. As a result, it is defined by a unique polynomial
$\operatorname{Res}(f_{0},\ldots,f_{n})$ of degree $(n+1)d^{n}$ in the
coefficients of the $f_{i}$ and normalized by
$\operatorname{Res}(x_{0}^{d},\ldots,x_{n}^{d})=1$, cf. [GKZ94, Ch. 13].

######  Corollary 9.17.

Let $f$ be a polarized morphism of degree $d$ of the smooth polarized toric
variety $(X,L)$.

  * •

If $(X,L)\simeq(\mathbb{P}^{n},\mathcal{O}(1))$, then
$R_{f}=\operatorname{Res}(f)R_{d}$;

  * •

if not, then $R_{f}=R_{d}$.

###### Proof.

If $(X,L)\simeq(\mathbb{P}^{n},\mathcal{O}(1))$, we can restrict along the
hyperplane determined by $x_{n}=0$ and inductively compare how the Deligne
products and the resultants change. In the case of the resultant, the
transformation is described by the Poisson formula [GKZ94, Ch. 13, Theorem
1.2], and the Deligne products transform accordingly. This shows they are
equal up to some constant, and the constant is equal to 1 by evaluating at the
polarized endomorphism $m_{d}$.

If $(X,L)$ is not isomorphic to $(\mathbb{P}^{n},\mathcal{O}(1))$, the
previous lemma shows that the space of polarized degree $d$ endomorphisms of
$(X,L)$ is isomorphic to an open subset
$U\subset\mathbb{P}\left(H^{0}(X,dL)^{N}\right)$ whose complement $Z$ has
codimension at least $2$. The map $f\mapsto R_{f}/R_{d}$ defines a morphism
$U\to\mathbb{G}_{m}$, which is thus constant by normality and properness of
$\mathbb{P}\left(H^{0}(X,dL)^{N}\right)$, and hence equal to $1$ by evaluating
at $f=m_{d}$. ∎

######  Remark 9.18.

It follows from Lemma 9.13 that the constant $c_{f}(\psi)$ in Theorem 9.14 can
be expressed through $E(\phi,\phi_{can})$. This is also the case for the
arithmetic height of a toric variety over the integers and one finds that

$\log|R_{f}|_{\phi}=\log|R_{f}/R_{d}|+d^{n}(d-1)h(X_{\Delta},\overline{\mathcal{L}}_{\phi}).$

In the case $\phi$ is given by the usual Fubini-Study metric on
$\mathbb{P}^{n}$, we have
$h(\mathbb{P}^{n},\mathcal{O}(1))=\sum_{i=1}^{n}\sum_{j=1}^{i}\frac{1}{2j}.$
For an exhaustive discussion on computations of arithmetic heights of toric
varieties we refer the reader to [BPS14].

###  Appendix A Determinant of cohomology and Deligne pairings

The goal of this Appendix is to discuss (a slight generalization of) results
of Knudsen-Mumford [KM76], Deligne [Del], Elkik [Elk89], Munoz-Garcia [MG00]
and Ducrot [Duc05], which provide a rough Riemann-Roch theorem for the
determinant of cohomology.

####  A.1. Discussion of the results

For a projective scheme $X$ over a field $K$, the _determinant of cohomology_
of a line bundle $L$ is the line (i.e. one-dimensional $K$-vector space)

$\lambda(L):=\sum_{i=0}^{n}(-1)^{i}\det H^{i}(X,L),$

where we use additive notation for tensor products of lines. If $\pi:X\to Y$
is now a flat projective morphism of locally noetherian schemes, it was shown
by Knudsen and Mumford in [KM76] that the fiberwise determinant of cohomology
of a line bundle $L$ on $X$ glues together to define a line bundle
$\lambda_{X/Y}(L)$ on $Y$. Indeed, the derived direct image $R\pi_{*}L$ is a
perfect complex, i.e. there exists a bounded complex $E^{\bullet}$ of vector
bundles on $Y$ with $R^{q}\pi_{*}L$ as $q$-th cohomology sheaf, and the
determinant of cohomology of $L$ can then be described as

$\lambda_{X/Y}(L)=\sum_{i}(-1)^{i}\det E_{i}.$

Denoting by $n$ the relative dimension of $\pi$, the main result in F.
Ducrot’s paper [Duc05] implies that the functor
$\lambda_{X/Y}:\mathcal{P}(X)\to\mathcal{P}(Y)$ so defined between the Picard
categories of line bundles on $X$ and $Y$ admits a unique polynomial structure
of degree $n+1$ compatible with base change and restriction to a relative
Cartier divisor (see §A.8 for a precise statement). This result recovers in
one stroke the construction of Deligne pairings [Elk90, MG00] and the Knudsen-
Mumford expansion [KM76]. Indeed, it implies that the $(n+1)$-st iterated
difference

$\langle
L_{0},\dots,L_{n}\rangle_{X/Y}:=\sum_{I\subset\\{0,\dots,n\\}}(-1)^{n+1-|I|}\lambda_{X/Y}\left(\sum_{i\in
I}L_{i}\right)$

defines a multi-additive symmetric functor
$\mathcal{P}(X)^{n+1}\to\mathcal{P}(Y)$, the _Deligne pairing_ , and that we
have for each $L\in\mathcal{P}(X)$ an expansion

$\lambda_{X/Y}(mL)=\sum_{i=0}^{n+1}{m+i\choose i}M_{i}$

with line bundles $M_{i}$ on $Y$ such that $M_{n+1}=\langle
L^{n+1}\rangle_{X/Y}$. Whenever a Grothendieck-Riemann-Roch theorem is
available, we infer

$c_{1}\left(\langle
L_{0},\dots,L_{n}\rangle_{X/Y}\right)=\pi_{*}(c_{1}(L_{0})\cdot\dots\cdot
c_{1}(L_{n})\rangle,$

so that Deligne pairings lift the natural push-forward operation on the right-
hand side to the level of line bundles.

In the main body of the present paper, a version of these results in the
possibly non-noetherian setting of models over the valuation ring of a
complete non-Archimedean field is required, and the purpose of this appendix
is to summarize the results leading to this generalization. In order to keep
the treatment uniform, we shall say in this appendix that a scheme $X$ is
_admissible_ if either

  * (i)

$X$ is locally noetherian, or

  * (ii)

$X$ is locally finitely presented over the valuation ring $K^{\circ}$ of some
complete non-Archimedean field $K$.

####  A.2. Polynomial maps

In order to motivate the definitions in §A.3, we briefly recall some
background on polynomial maps and difference calculus. It is well-known that a
map $f:\mathbb{Z}\to\mathbb{Z}$ is polynomial of degree (at most) $n$ if and
only if it admits an expansion

$f(m)=\sum_{i=0}^{n}{m+i\choose i}b_{i}$

with coefficients $b_{i}\in\mathbb{Z}$. More generally, a map $f:A\to B$
between commutative groups is said to be _polynomial of degree $n$_ if for any
given $x_{1},\dots,x_{r}\in A$ we have an expansion

$f(m_{1}x_{1}+\dots+m_{r}x_{r})=\sum_{0\leq i_{1},\dots,i_{r}\leq
n}{m_{1}+i_{1}\choose i_{1}}\dots m_{r}+i_{r}\choose i_{r}b_{i_{1}\dots
i_{r}}$

for all $m_{i}\in\mathbb{Z}$, with coefficients $b_{i_{1}\dots i_{r}}\in B$.

Polynomiality can be characterized in terms of difference calculus. For a map
$f:\mathbb{Z}\to B$, define the difference $\Delta f:\mathbb{Z}\to B$ by

$(\Delta f)(m):=f(m+1)-f(m).$

Then

$\Delta{m+i\choose i}={m+i-1\choose i-1},$

which can be used to show by induction on $n$ that $f:\mathbb{Z}\to B$ is a
polynomial map of degree $n$ if and only if $\Delta^{n+1}f=0$. For a map
$f:\mathbb{Z}^{r}\to B$, one can introduce partial difference operators
$\Delta_{i}$, and $f$ is polynomial of degree $n$ if and only if
$\Delta^{\alpha}f=0$ for all $\alpha\in\mathbb{N}^{r}$ with $|\alpha|=n+1$,
where we have set
$\Delta^{\alpha}=D_{1}^{\alpha_{1}}\dots\Delta_{r}^{\alpha_{r}}$ for each
multiindex $\alpha$.

Consider now a map $f:A\to B$ between commutative groups. Mimicking
differential calculus, one defines the _difference at $x$_ $\delta_{x}f:A\to
B$ of a map $f:A\to B$ by setting

$(\delta_{x}f)(y)=f(x+y)-f(x),$

and the _$k$ -th iterated difference_ $\delta^{k}_{x}f:A^{k}\to B$ by

$(\delta^{k}_{x}f)(x_{1},\dots,x_{k}):=\delta_{x}\left(y\mapsto(\delta^{k-1}_{y}f)(x_{2},\dots,x_{k})\right)(x_{1})$
$=(\delta^{k-1}_{x+x_{1}}f)(x_{2},\dots,x_{k})-(\delta^{k-1}_{x}f)(x_{2},\dots,x_{k}).$

The map $\delta^{k}_{x}f:A^{k}\to B$ so defined is symmetric, as follows from
the explicit expression

$(\delta^{k}_{x}f)(x_{1},\dots,x_{k})=\sum_{I\subset\\{1,\dots,k\\}}(-1)^{k-|I|}f\left(x+\sum_{i\in
I}x_{i}\right).$ (A.1)

Given $x_{1},\dots,x_{r}\in A$, the map $g:\mathbb{Z}^{r}\to B$ defined by
$g(m_{1},\dots,m_{r})=f(\sum_{i}m_{i}x_{i})$ satisfies

$(\Delta^{\alpha}g)(m_{1},\dots,m_{r})=(\delta^{|\alpha|}_{\sum_{i}m_{i}x_{i}}f)(x_{1}^{\alpha_{1}},\dots,x_{r}^{\alpha_{r}})$
(A.2)

for all $\alpha\in\mathbb{N}^{r}$. Using this, we conclude that $f:A\to B$ is
polynomial of degree $n$ if and only if $\delta^{n+1}_{x}f=0$ for all $x\in
A$.

It is in fact enough to check this condition for $x=0$. Indeed, the operator
$\delta^{k}:=\delta^{k}_{0}$ determines all $\delta^{k}_{x}$ by

$(\delta^{k}_{x}f)(x_{1},\dots)=(\delta^{k}f)(x+x_{1},\dots)-(\delta^{k}f)(x,\dots).$
(A.3)

It further satisfies

$(\delta^{k+1}f)(x_{1},y_{1},\dots)=(\delta^{k}f)(x_{1}+y_{1},\dots)-(\delta^{k}f)(x_{1},\dots)-(\delta^{k}f)(y_{1},\dots),$

and we thus see that $f$ is polynomial of degree $n$ if and only if
$\delta^{n}f$ is multi-additive. Note also that $\delta^{k}$ can be understood
as a polarization operator, in the sense that $\delta^{k}(L(x^{k}))=k!L$ for
any symmetric multi-additive map $L:A^{k}\to B$; we can thus view the multi-
additive map $\delta^{n}f:A^{n}\to B$ associated to a polynomial map $f:A\to
B$ of degree $n$ as the polarization of its degree $n$ part.

######  Example A.1.

Let $X$ be an $n$-dimensional projective scheme over a field $K$, with
structure morphism $\pi:X\to\operatorname{Spec}K$, and pick a line bundle $L$
on $X$. The _Euler characteristic_

$\chi(L):=\sum_{i=0}^{n}(-1)^{i}\dim_{K}H^{i}(X,L)$

only depends on the class of $L$ in the Picard group $\operatorname{Pic}(X)$,
and Snapper’s theorem implies that $\chi:\operatorname{Pic}(X)\to\mathbb{Z}$
is a polynomial map of degree $n$, with degree $n$ polarization given by the
intersection pairing, i.e.

$(\delta^{n}\chi)(L_{1},\dots,L_{n})=(L_{1}\cdot\dots\cdot
L_{n}):=\deg\pi_{*}\left(c_{1}(L_{1})\cdot\cdots\cdot
c_{1}(L_{n})\cdot[X]\right).$

For later use, we finally note:

######  Lemma A.2.

Let $f:A\to B$ is a polynomial map of degree $n$, and set for all $0\leq i\leq
n$

$f_{n,i}(x):=(\delta^{i}f)(x^{i})-(\delta^{i+1}f)(x^{i+1})+...+(-1)^{n-i}(\delta^{n}f)(x^{n}).$

Then $f(mx)=\sum_{i=0}^{n}{m+i\choose i}f_{n,i}(x)$ for all $x\in A$ and
$m\in\mathbb{Z}$.

###### Proof.

Since $f$ is polynomial of degree $n$, we have

$g(m):=f(mx)=\sum_{i=0}^{n}{m+i\choose i}b_{i}$

for some $b_{i}\in B$. Since $\Delta_{0}^{k}{m+i\choose i}=1$ for $k\leq i$
and $0$ otherwise, (A.2) yields

$(\delta^{k}f)(x^{k})=\Delta_{0}^{k}g=\sum_{i\geq k}b_{i},$

and hence $b_{i}=f_{n,i}(x)$. ∎

####  A.3. Polynomial functors

A _commutative Picard category_ is a ’stacky version’ of a commutative group.
It is defined as a category $\mathcal{P}$ in which all arrows are
isomorphisms, together with an additivity functor and functorial associativity
and commutativity isomorphisms satisfying the expected compatibility
conditions, and such that for any object $x$ the endofunctors $y\mapsto x+y$
and $y\mapsto y+x$ are autoequivalences.

These axioms imply the existence of a neutral object $0$ and of an inverse
$-x$ for each object $x$, both unique up to unique isomorphism. The sum
$\sum_{i\in I}x_{i}$ of a finite family $(x_{i})_{i\in I}$ of objects in
$\mathcal{P}$ is well-defined up to unique isomorphism, and satisfies the
expected associativity rules. A commutative Picard category $\mathcal{P}$ is
_strictly commutative_ if the commutativity isomorphism induces the identity
on $x+x$ for each $x$, in which case $x$ and $-x$ can be contracted within a
sum without raising any sign issue.

In practice for us, $\mathcal{P}$ will be the category of line bundles or
$\mathbb{Q}$-line bundles on a given scheme, and isomorphisms between them,
both of which are strictly commutative Picard categories. Note that a
commutative group can also be viewed as a strictly commutative Picard
category.

In what follows, $\mathcal{P}$ and $\mathcal{Q}$ are stricly commutative
Picard categories. An _additive functor_ $F:\mathcal{P}\to\mathcal{Q}$ is a
functor equipped with a functorial additivity isomorphism $F(x+y)\simeq
F(x)+F(y)$ which is _commutative_ , expressed by the commutativity of the
induced diagram

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
22.39813pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{F(x+y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
46.39813pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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46.39813pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{F(x)+F(y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
76.5949pt\raise-29.5pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{$\textstyle{F(y+x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
46.39813pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise
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46.39813pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{F(y)+F(x)}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

and _associative_ , i.e. the commutativity of the diagram

$\textstyle{F((x+y)+z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F(x+y)+F(z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(F(x)+F(y)\right)+F(z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F(x+(y+z))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
F(x)+F(y+z)}$$\textstyle{F(x)+\left(F(y)+F(z)\right).}$

These conditions then yield a consistent system of functorial additivity
isomorphisms $F(\sum_{i\in I}x_{i})\simeq\sum_{i\in I}F(x_{i})$ for all finite
families $(x_{i})_{i\in I}$ in $\mathcal{P}$.

A _multi-additive functor_ $F:\mathcal{P}^{n}\to\mathcal{Q}$ is defined as a
functor equipped with functorial commutative and associative additivity data
in each variable, such that expanding out sums in the variables does not
depend on the order the operation is performed. A _symmetric functor_
$F:\mathcal{P}^{n}\to\mathcal{Q}$ is a functor equipped with symmetry
isomorphisms

$F(x_{\sigma(1)},\dots,x_{\sigma(n)})\simeq F(x_{1},\dots,x_{n})$

for each permutation $\sigma\in\mathfrak{S}_{r}$, compatible with the group
law on $\mathfrak{S}_{r}$, and a _symmetric, multi-additive_ functor has both
structures, with the expected compatibility condition.

Define the _$k$ -th iterated difference_ at an object $x$ in $\mathcal{P}$ of
a functor $F:\mathcal{P}\to\mathcal{Q}$ as the symmetric functor
$\delta^{k}_{x}F:\mathcal{P}^{k}\to\mathcal{Q}$ defined by setting

$(\delta^{k}_{x}F)(x_{1},\dots,x_{k}):=\sum_{I\subset\\{1,\dots,k\\}}(-1)^{k-|I|}F\left(x+\sum_{j\in
I}x_{j}\right),$

For $x=0$, we simply set $\delta^{k}:=\delta^{k}_{0}$. Recalling from §A.2
that a map $f$ between commutative groups is polynomial of degree $n$ if and
only if $\delta^{n}f$ is multi-additive, we introduce:

######  Definition A.3.

A _polynomial structure of degree $n$_ on $F$ is defined as a structure of
multi-additive functor on $\delta^{n}F$, compatible with its canonical
symmetry.

Ducrot introduces in [Duc05, Definition 1.6.1] the notion of _$k$ -cube
structure_ on $F$. By [Duc05, Proposition 1.9], an $(n+1)$-cube structure on
$F$ induces a polynomial structure of degree $n$ on $F$ (and the converse is
probably true as well, by [Duc05, 1.5.1, (d)]). Ducrot’s terminology comes
from the following well-known result.

######  Example A.4.

If $L$ is a line bundle on an abelian variety $A$, the _theorem of the cube_
asserts that for any variety $S$, the functor $F_{L}:A(S)\to\mathcal{P}(S)$
defined by $F_{L}(x):=x^{*}L$ admits a $3$-cube structure. It is thus
quadratic in our sense, i.e. $(x,y)\mapsto(x+y)^{*}L-x^{*}L-y^{*}L+0^{*}L$ is
biadditive. Further, the whole structure is compatible with base change.

In analogy with Lemma A.2, we have:

######  Lemma A.5.

Suppose that $F:\mathcal{P}\to\mathcal{Q}$ admits a polynomial structure of
degree $n$, and define for $0\leq i\leq n$ a functor
$F_{n,i}:\mathcal{P}^{i}\to\mathcal{Q}$ by setting

$F_{n,i}(x)=(\delta^{i}f)(x^{i})-(\delta^{i+1}f)(x^{i+1})+...+(-1)^{n-i}(\delta^{n}f)(x^{n}).$

For all $x\in\mathcal{P}$ and $m\in\mathbb{Z}$, we then have canonical
functorial isomorphisms

$F(mx)\simeq\sum_{i=0}^{n}{m+i\choose i}F_{n,i}(x).$

###### Proof.

Define $g:\mathbb{Z}\to\mathcal{Q}$ by $g(m):=F(mx)-\sum_{i=0}^{n}{m+i\choose
i}F_{n,i}(x)$. Since

$\Delta_{m}^{k}F(mx)=\sum_{i=0}^{k}(-1)^{k-i}{k\choose
i}F((m+ix)=(\delta^{k}_{mx}F)(x^{k}),$

and $\Delta^{k}_{0}{m+i\choose i}=1$ for $k\leq i$ and $0$ otherwise, we have
canonical isomorphisms $(\Delta^{k}g)(0)\simeq 0$ for $k=0,\dots,n$. Further,

$(\Delta^{n-1}g)(m+1)-(\Delta^{n-1}g)(m)=(\Delta^{n}g)(m)=(\delta^{n}_{mx}F)(x^{n})-(\delta^{n}F)(x^{n})$
$\simeq(\delta^{n}F)((m+1)x,x,\dots,x)-(\delta^{n}F)(mx,x,\dots,x)-(\delta^{n}F)(x,\dots,x)\simeq
0,$

for all $m\in\mathbb{Z}$, by multiadditivity of $\delta^{n}F$. Summing up
these relations, we get

$(\Delta^{n-1}g)(m)\simeq(\Delta^{n-1}g)(0)\simeq 0,$

and iterating the argument finally yields $g(m)\simeq 0$ for all $m$. ∎

####  A.4. The determinant of a perfect complex

Let $X$ be a scheme, and $E$ be a vector bundle on $X$, i.e. a finite locally
free $\mathcal{O}_{X}$-module. One denotes by $\det
E=\Lambda^{\operatorname{rk}E}E$ the determinant line. If

$0\to E^{\prime}\to E\to E^{\prime\prime}\to 0$ (A.4)

is an exact sequence of vector bundles, then there is a canonical isomorphism

$\det E\simeq\det E^{\prime}+\det E^{\prime\prime},$ (A.5)

where $+$ denotes the tensor product of line bundles, in additive notation.
However, given two vector bundles $E,F$, the isomorphism

$\det E+\det F\simeq\det(E\oplus F)\simeq\det(F\oplus E)\simeq\det F+\det E$

induced by the canonical isomorphism $E\oplus F\simeq F\oplus E$ coincides
with the canonical commutativity isomorphism only up to a factor
$(-1)^{(\operatorname{rk}E)(\operatorname{rk}F)}$.

To deal with this sign issue, one introduces the graded determinant functor
$E\mapsto(\det E,\operatorname{rk}E)$ with values in the Picard category of
graded line bundles and modifies the commutativity isomorphism as to satisfy
the Koszul rule of signs indicated. For the purpose of the present paper, it
will however be enough to view $\det E$ as an object in the Picard category
$\mathcal{P}(X)_{\mathbb{Q}}$ of $\mathbb{Q}$-line bundles on $X$, and we can
thus ignore the previous sign issue.

Next, recall that a complex $F^{\bullet}$ of $\mathcal{O}_{X}$-modules is

  * (i)

_pseudo-coherent_ if it is locally quasi-isomorphic to a bounded-above complex
of vector bundles;

  * (ii)

_of locally finite Tor-dimension_ if it is locally quasi-isomorphic to a
bounded complex of flat $\mathcal{O}_{X}$-modules;

  * (iii)

_perfect_ if it is locally quasi-isomorphic to a bounded complex of vector
bundles.

By [Stacks, Tag 08CQ], $F^{\bullet}$ is perfect if and only if it is pseudo-
coherent and of locally finite Tor-dimension. In [KM76, Theorem 2], Knudsen
and Mumford showed that setting

$\det E^{\bullet}:=\sum_{i}(-1)^{i}\det E^{i}$

for each bounded complex of vector bundles $E^{\bullet}$ gives rise to a
functor $F^{\bullet}\mapsto\det F^{\bullet}$ from the category of perfect
complexes $F^{\bullet}$ on $X$ and quasi-isomorphims between them to
$\mathcal{P}(X)$. This functor commutes with base change (i.e. we have a
canonical functorial isomorphism $\det Lf^{*}F^{\bullet}\simeq
g^{*}F^{\bullet}$ for any morphism $f:X^{\prime}\to X$), it is additive with
respect to short exact sequences of complexes in the sense of (A.5), and
satisfies a couple of other properties which then uniquely characterize it up
to unique isomorphism.

####  A.5. The direct image theorem

As mentioned in §A.1, we shall say for convenience that a scheme $X$ is
_admissible_ if it is is either locally noetherian (’noetherian case’) or
locally of finite presentation over the valuation ring $K^{\circ}$ of some
complete non-Archimedean field $K$ (’valuative case’). Obviously, any scheme
$X$ locally finitely presented over an admissible scheme is admissible.
Further, any scheme $X$ which is flat and locally of finite type (in
particular, proper) over a valuation ring $K^{\circ}$ as above is locally
finitely presented over $K^{\circ}$ (by [Nag66, Theorem 3’] or [RG71, Théorème
3.4.6], cf. Lemma 4.5), and hence is admissible.

Recall that an $\mathcal{O}_{X}$-module $F$ on a scheme $X$ is _coherent_ if
it is locally finitely presented, and for each open $U\subset X$, each locally
finite submodule of $F|_{U}$ is locally finitely presented as well. Crucially,
the structure sheaf $\mathcal{O}_{X}$ of any admissible scheme $X$ is
coherent, by [Ull, Corollary 1.8]. The following direct image theorem is of
course well-known in the noetherian case, and follows from the work of Kiehl
[Kie72] in the valuative case, as explained in [Ull, Theorem 3.5, Proposition
3.6] (note that an admissible scheme admits an open cover by stably coherent
rings, by [Ull, Example 3.3]).

######  Theorem A.6.

Let $Y$ be an admissible scheme and $\pi:X\to Y$ be a proper, finitely
presented morphism. For each coherent $\mathcal{O}_{X}$-module $F$,
$R^{q}\pi_{*}F$ is coherent for all $q\in\mathbb{N}$ (and zero for $q$ large
enough, locally on $Y$).

If we further assume that $\pi$ is projective, $Y$ is quasicompact and $L$ is
a $\pi$-ample line bundle on $X$, then the usual Serre vanishing theorem is
satisfied, i.e. for all $m\gg 1$ $F(mL):=F\otimes L^{m}$ is $\pi$-globally
generated and satisfies $R^{q}\pi_{*}F(mL)=0$ for $q\geq 1$.

######  Corollary A.7.

If $\pi:X\to Y$ is a flat, proper, finitely presented morphism with $Y$
admissible, then the derived direct image $R\pi_{*}E$ is perfect for each
vector bundle $E$ on $X$.

###### Proof.

Since $R^{q}\pi_{*}E=0$ for $q\gg 1$, $R\pi_{*}E$ is quasi-isomorphic to a
bounded-above complex. As $\mathcal{O}_{Y}$ is coherent, [Ull, Lemma 3.2] thus
says that $R\pi_{*}E$ is pseudo-coherent if and only it has coherent
cohomology, which is indeed the case by Theorem A.6. It thus remains to see
that $R\pi_{*}E$ has locally finite Tor-dimension, which can be done verbatim
as in [Stacks, Tag 08EV] (whose proof only uses the noetherianity assumption
to get pseudo-coherence). ∎

####  A.6. Regular sequences

Let $\pi:X\to Y$ be a flat, locally finitely presented morphism of schemes,
and let $s$ be a global section of a line bundle $L$ on $X$, defining a closed
subscheme $D\subset X$. Recall that $s$ is _$\pi$ -regular at $x\in X$_ if $D$
is a relative Cartier divisor at $x$, i.e. $s$ is a nonzerodivisor in
$\mathcal{O}_{X,x}$, and $D$ is $\pi$-flat at $x$. If this holds for all $x\in
X$, then $s$ is simply called $\pi$-regular. By [EGA, IV.11.3.7], $s$ is
$\pi$-regular at $x$ if and only the restriction of $s$ to the fiber through
$x$ is not a zerodivisor at $x$, and the set of $x\in X$ at which this holds
is open.

More generally, a sequence of sections $(s_{1},\dots,s_{p})$ of line bundles
$L_{1},\dots,L_{p}$ with zero schemes $D_{1},\dots,D_{p}$ is _$\pi$ -regular_
if

$s_{1},s_{2}|_{D_{1}},\dots,s_{p}|_{D_{1}\cap\dots\cap D_{p-1}}$

are $\pi$-regular.

######  Lemma A.8.

Let $\pi:X\to Y$ be flat, proper, finitely presented morphism of schemes, and
$s$ be a global section of a line bundle $L$ on $X$. Pick $y\in Y$, and assume
that $s$ is nonzero at each associated point of $X_{y}$. Then $s$ is
relatively regular over an open neighborhood of $y$.

###### Proof.

Denote by $U\subset X$ the open set of points at which $s$ is relatively
regular. Since $X_{y}$ is noetherian (being of finite type over a field), the
assumption implies that $s|_{X_{y}}$ is a nonzerodivisor at each $x\in X_{y}$,
and hence that $X_{y}\subset U$ by the above results. As $\pi$ is closed, it
follows that $\pi^{-1}(V)\subset U$ for some open neighborhood $V$ of $y$. ∎

As a consequence, we then have the following useful existence result for
relatively regular sections.

######  Proposition A.9.

Let $\pi:X\to Y$ be a flat, projective, finitely presented morphism with $Y$
admissible, and let $L$ be a $\pi$-ample line bundle on $X$. Then $mL$ admits
a relatively regular section locally over $Y$ for all $m\gg 1$.

###### Proof.

By coherence of $\pi_{*}(mL)$ and Lemma A.8, it is enough to show that for
each closed point $y\in Y$ and all $m\gg 1$, there exists
$s\in\pi_{*}(mL)_{y}$ such that $s|_{X_{y}}$ is nonzero at all associated
points of $X_{y}$. By ampleness of $L|_{X_{y}}$, for each $m\gg 1$ we may find
a section in $H^{0}(X_{y},mL_{y})$ not vanishing at the associated points of
$X_{y}$ (finitely many, since $X_{y}$ is noetherian). If we denote by
$\mathfrak{a}\subset\mathcal{O}_{X}$ the ideal sheaf of $X_{y}$, Theorem A.6
implies that $R^{1}\pi_{*}(\mathfrak{a}(mL))=0$ for all $m\gg 1$. The
restriction map $\pi_{*}(mL)_{y}\to H^{0}(X_{y},mL_{y})$ is thus surjective
for $m\gg 1$, hence the result. ∎

####  A.7. The determinant of cohomology

Let $\pi:X\to Y$ be a flat, proper, finitely presented morphism with $Y$
admissible. Thanks to Corollary A.7, we can introduce:

######  Definition A.10.

The _determinant of cohomology_ is the functor
$\lambda_{X/Y}:\mathcal{P}(X)\to\mathcal{P}(Y)_{\mathbb{Q}}$ that takes a line
bundle $L$ on $X$ to the $\mathbb{Q}$-line bundle

$\lambda_{X/Y}(L):=\det R\pi_{*}L.$

If $L$ is $\pi$-acyclic, i.e. $R^{q}\pi_{*}L=0$ for $q>0$, then $\pi_{*}L$ is
a vector bundle, and $\lambda_{X/Y}(L)$ is simply given as the ’naive’
determinant $\lambda_{X/Y}(L)=\det\pi_{*}L$. This holds in particular when
$n=0$ (i.e. $\pi$ finite flat), or for large enough multiples of a $\pi$-ample
line bundle.

######  Proposition A.11.

The determinant of cohomology satisfies the following two compatibility
properties.

  * (i)

It commutes with base change: for any morphism $f:Y^{\prime}\to Y$ with
$Y^{\prime}$ admissible, let

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{\pi^{\prime}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

be the corresponding Cartesian square. Then we have a canonical functorial
isomorphism

$\lambda_{X^{\prime}/Y^{\prime}}(g^{*}L)\simeq f^{*}\lambda_{X/Y}(L).$

  * (ii)

If $D$ is an effective relative Cartier divisor $D$ on $X$, then we have a
canonical functorial isomorphism

$\lambda_{X/Y}(L)-\lambda_{X/Y}(L-D)\simeq\lambda_{D/Y}(L|_{D}).$ (A.6)

###### Proof.

Since $L$ is flat over $Y$, we have
$R\pi^{\prime}_{*}{L^{\prime}}=Lf^{*}R\pi_{*}L$, cf. [Stacks, Tag 0A1D]. On
the other hand, the determinant functor satisfies $\det Lf^{*}=f^{*}\det$,
hence (i). If $D$ is an effective relative Cartier divisor, the restriction
exact sequence $0\to L(-D)\to L\to L|_{D}\to 0$ induces an exact sequence of
perfect complexes $0\to R\pi_{*}L(-D)\to R\pi_{*}L\to R(\pi|_{D})_{*}L|_{D}\to
0$, and (ii) follows by additivity of $\det$ in exact sequences. ∎

####  A.8. Deligne pairings and Knudsen-Mumford expansion

In this section, we fix a flat, projective, finitely presented morphism
$\pi:X\to Y$ of constant relative dimension $n$, with $Y$ admissible.

When $n=0$, $\pi:X\to Y$ is a finite flat morphism, and the determinant of
cohomology $\lambda_{X/Y}$ provides a canonical and functorial construction of
the norm of a line bundle [EGA, II.6.5] (compare for instance [Fer98,
Proposition 3.3]). To see this, recall first that the norm
$N_{X/Y}(f)\in\mathcal{O}_{Y}$ of $f\in\pi_{*}\mathcal{O}_{X}$ is defined as
the determinant of the endomorphism of $\pi_{*}\mathcal{O}_{X}$ defined by
multiplication by $f$, yielding a multiplicative map

$N_{X/Y}:\pi_{*}\mathcal{O}_{X}\to\mathcal{O}_{Y}.$ (A.7)

Now define a functor $N_{X/Y}:\mathcal{P}(X)\to\mathcal{P}(Y)$ by setting

$N_{X/Y}(L)=(\det\pi_{*}L)-(\det\pi_{*}\mathcal{O}_{X})$
$=\lambda_{X/Y}(L)-\lambda_{X/Y}(\mathcal{O}_{X})=(\delta\lambda_{X/Y})(L),$

######  Lemma A.12.

For each line bundle $L$ on $X$, $N_{X/Y}(L)$ coincides with the norm of $L$
as defined in [EGA, II.6.5].

###### Proof.

Observe that if $u\in H^{0}(X,\mathcal{O}_{X}^{*})$ is a unit and $L$ is a
line bundle on $X$, multiplication by $u$ defines an isomorphism $L\simeq L$,
whose induced isomorphism $\det\pi_{*}L\simeq\det\pi_{*}L$, and hence also
$N_{X/Y}(L)\simeq N_{X/Y}(L)$, are both given by multiplication by
$N_{X/Y}(u)$.

By [EGA, II.6.1.12.1], $L$ is trivial in a neighborhood of each fiber of
$\pi$, and $Y$ therefore admits an open cover $(Y_{i})$ with
$L|_{X_{i}}\simeq\mathcal{O}_{X_{i}}$ on $X_{i}:=\pi^{-1}(Y_{i})$. Set
$Y_{ij}=Y_{i}\cap Y_{j}$, $X_{ij}=X_{i}\cap X_{j}=\pi^{-1}(Y_{ij})$, and
denote by $u_{ij}\in H^{0}(X_{ij},\mathcal{O}_{X_{ij}}^{*})$ the corresponding
cocycle. The transition isomorphism $\mathcal{O}_{X_{ij}}\simeq
L|_{X_{ij}}\simeq\mathcal{O}_{X_{ij}}$ is given by multiplication by $u_{ij}$.
By the above observation, applying the functor $N_{X/Y}$ yields an isomorphism
$\mathcal{O}_{Y_{ij}}\simeq N_{X/Y}(L)|_{Y_{ij}}\simeq\mathcal{O}_{Y_{ij}}$
given by multiplication by $N_{X_{ij}/Y_{ij}}(u_{ij})$, which precisely means
that $N_{X/Y}(L)$ coincides with the norm of $L$ as defined in [EGA, II.6.5].
∎

Arguing as in [Duc05, 4.1.1], we next prove:

######  Proposition A.13.

There is a unique way to assign to each finite flat morphism $\pi:X\to Y$ with
$Y$ admissible an additivity structure on the norm functor
$N_{X/Y}:\mathcal{P}(X)\to\mathcal{P}(Y)$ that is compatible with base change
and such that the additivity isomorphisms

$N_{X/Y}(L+\mathcal{O}_{X})\simeq N_{X/Y}(L)+N_{X/Y}(\mathcal{O}_{X})$

are the canonical ones given by $N_{X/Y}(\mathcal{O}_{X})=\mathcal{O}_{Y}$.

###### Proof.

Pick two line bundles $L,L^{\prime}$ on $X$ and a cover $(Y_{i})$ of $Y$ with
trivializations $L|_{X_{i}}\simeq\mathcal{O}_{X_{i}}$,
$L^{\prime}|_{X_{i}}\simeq\mathcal{O}_{X_{i}}$. Given an additivity
isomorphism

$N_{X/Y}(L+L^{\prime})\simeq N_{X/Y}(L)+N_{X/Y}(L^{\prime})$

with the desired properties, the induced isomorphisms

$N_{X_{i}/Y_{i}}(\mathcal{O}_{X_{i}}+\mathcal{O}_{X_{i}})\simeq
N_{X_{i}/Y_{i}}(\mathcal{O}_{X_{i}})+N_{X_{i}/Y_{i}}(\mathcal{O}_{X_{i}})$

are necessarily equal to the canonical ones obtained by multiplicativity of
(A.7), which proves uniqueness. To establish existence, it is then enough to
argue locally on $Y$, since compatibility on overlaps will follow from
uniqueness, and the result is then straighforward, using again that any line
bundle on $X$ is trivial locally over $Y$. ∎

In the terminology of §A.3, Proposition A.13 says that the functor
$\lambda_{X/Y}$ is polynomial of degree $1$ when $n=0$. This is generalized by
the next result due to F. Ducrot [Duc05, Theorem 4.2]. h

######  Theorem A.14.

[Duc05, Theorem 4.2] There exists a unique way to assign to each flat,
projective, finitely presented morphism $\pi:X\to Y$ of relative dimension $n$
with $Y$ admissible a polynomial structure of degree $n+1$ on
$\lambda_{X/Y}:\mathcal{P}(X)\to\mathcal{P}(Y)_{\mathbb{Q}}$ with the
following properties:

  * (i)

it commutes with base change;

  * (ii)

it coincides with the above one when $n=0$;

  * (iii)

for any relative effective divisor $D$ on $X$, the polynomial structures on
$\lambda_{X/Y}$ and $\lambda_{D/Y}$ are compatible with the canonical
restriction isomorphism

$\lambda_{X/Y}(L)-\lambda_{X/Y}(L-D)\simeq\lambda_{D/Y}(L|_{D}).$

More precisely, [Duc05, Theorem 4.2] proves the existence of a canonical
$(n+2)$-cube structure on $\lambda_{X/Y}$, which yields (and is probably
equivalent to) a polynomial structure of degree $n+1$ on $\lambda_{X/Y}$, as
discussed in §A.3. Stricty speaking, the proof of Theorem A.14 assumes $X$ and
$Y$ to be locally noetherian, but all the arguments apply to the valuative
case as well, once Corollary A.7 is available. Since a polynomial structure of
degree $n+1$ on $\lambda_{X/Y}$ is by definition a multi-additive structure on
the $(n+1)$-st difference

$(\delta^{n+1}\lambda_{X/Y})(L_{0},\dots,L_{n})=\sum_{I\subset\\{0,\dots,n\\}}(-1)^{n+1-|I|}\lambda_{X/Y}\left(\sum_{i\in
I}L_{i}\right),$

and we can thus define the _Deligne pairing_ as the functor
$\mathcal{P}(X)^{n+1}\to\mathcal{P}(Y)_{\mathbb{Q}}$ that takes line bundles
$L_{0},\dots,L_{n}$ on $X$ to the $\mathbb{Q}$-line bundle

$\langle
L_{0},\dots,L_{n}\rangle_{X/Y}:=(\delta^{n+1}\lambda_{X/Y})(L_{0},\dots,L_{n}).$
(A.8)

######  Theorem A.15.

The Deligne pairing satisfies the following properties.

  * (i)

it is multi-additive, symmetric, and commutes with base change;

  * (ii)

for each relative effective Cartier divisor $D$ on $X$, we have canonical
multi-additive functorial isomorphisms

$\langle\mathcal{O}_{X}(D),L_{1},\dots,L_{n}\rangle_{X/Y}\simeq\langle
L_{1}|_{D},\dots,L_{n}|_{D}\rangle_{D/Y};$

  * (iii)

if $f:X^{\prime}\to X$ is a finite flat of degree $e$, then we have canonical
functorial isomorphisms

$\langle f^{*}L_{0},\ldots,f^{*}L_{n}\rangle_{X^{\prime}/Y}\simeq e\langle
L_{0},\ldots,L_{n}\rangle_{X/Y}.$

###### Proof.

(i) follows directly from Theorem A.14. Given $D$ as in (ii), taking the
$n$-th iterated difference of the restriction isomorphism

$\lambda_{D/Y}(L|_{D})\simeq\lambda_{X/Y}(L)-\lambda_{X/Y}(L-D)$

yields

$\langle
L_{1}|_{D},\dots,L_{n}|_{D}\rangle_{D/Y}=(\delta^{n}\lambda_{D/Y})(L_{1}|_{D},\dots,L_{n}|_{D})$
$\simeq(\delta^{n}\lambda_{X/Y})(L_{1},\dots,L_{n})-(\delta^{n}_{-D}\lambda_{X/Y})(L_{1},\dots,L_{n})\simeq-(\delta^{n+1}\lambda_{X/Y})(-D,L_{1},\dots,L_{n}),$

which is isomorphic to

$(\delta^{n+1}\lambda_{X/Y})(D,L_{1},\dots,L_{n})=\langle
D,L_{1},\dots,L_{n}\rangle_{X/Y},$

by multiadditivity of $\delta^{n+1}\lambda_{X/Y}$. Finally, let
$f:X^{\prime}\to X$ be finite and flat of degree $e$, so that
$E:=f_{*}\mathcal{O}_{X^{\prime}}$ is a rank $e$ vector bundle. By the
projection formula we have

$R(\pi\circ f)_{*}(f^{*}L)=R\pi_{*}Rf_{*}(f^{*}L)=R\pi_{*}(L\otimes E),$

and we get (iii) thanks to [Duc05, Proposition 4.7.1], which yields a
canonical isomorphism between the $(n+1)$-st difference of $L\mapsto\det
R\pi_{*}(L\otimes E)$ and $e\,\delta^{n+1}\lambda_{X/Y}$. ∎

By Lemma A.5, we finally get the following generalization of [KM76, Theorem
4].

######  Corollary A.16.

For each $0\leq i\leq n+1$, define a functor
$F_{n+1,i}:\mathcal{P}(X)^{i}\to\mathcal{P}(Y)_{\mathbb{Q}}$ by

$F_{n+1,i}(L):=\sum_{j=i}^{n+1}(-1)^{j-i}\left(\delta^{j}\lambda_{X/Y}\right)(L^{j}),$

For each line bundle $L$ on $X$ and $m\in\mathbb{Z}$, we then have functorial
isomorphisms

$\lambda_{X/Y}(mL)\simeq\sum_{i=0}^{n+1}{m+i\choose
i}F_{n+1,i}(L)=\frac{m^{n+1}}{(n+1)!}\langle L^{n+1}\rangle_{X/Y}+O(m^{n}),$

compatible with base change.

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# Distance Duality Test: The Evolution of Radio Sources Mimics a Nonexpanding
Universe

Pengfei Li Humboldt fellow.  Leibniz-Institute for Astrophysics, An der
Sternwarte 16, 14482 Potsdam, Germany

###### Abstract

Distance duality relation (DDR) marks a fundamental difference between
expanding and nonexpanding Universes, as an expanding metric causes angular
diameter distance smaller than luminosity distance by an extra factor of
$(1+z)$. Here we report a test of this relation using two independent samples
of ultracompact radio sources observed at 2.29 GHz and 5.0 GHz. The test with
radio sources involves only geometry, so it is independent of cosmological
models. Since the observed radio luminosities systematically increase with
redshift, we do not assume a constant source size. Instead, we start with
assuming the intensive property, luminosity density, does not evolve with
redshift and then infer its evolution from the resultant DDR. We make the same
assumption for both samples, and find it results in the same angular size-
redshift relation. Interestingly, the resultant DDR is fully consistent with a
nonexpanding Universe. Imposing the DDR predicted by the expanding Universe,
we infer the radio luminosity density evolves as $\rho_{L}\propto(1+z)^{3}$.
However, the perfect agreement with a nonexpanding Universe under the
assumption of constant luminosity densities poses a conspiracy and fine-tuning
problem: the size and luminosity density of ultracompact radio sources evolve
in the way that precisely mimics a nonexpanding Universe.

Cosmology (343); Observational cosmology (1146); Active galaxies (17);
Distance measure (395); Galaxy distances (590)

##  1 Introduction

The prevailing cosmology states that the Universe is uniformly expanding
through a growing scale factor in the metric, which makes a unique prediction
between angular diameter distance $D_{A}$ and luminosity distance $D_{L}$:
$D_{L}=(1+z)^{2}D_{A}$. This relation does not depend on specific cosmological
models, so it can be used to test the expanding Universe.

Historically, attempts for the test were made with double radio sources, which
hold two radio components and the separation between them was recognized as
rigid rod. However, their observed angular sizes closely follow $\theta\propto
z^{-1}$ (Kapahi, 1987). This contradicts to the expanding cosmology. To
reconcile the tension, it was argued that the separations could evolve with
redshift, because the ambient interstellar and intergalactic gas could be so
dense in the past that the radio components could not penetrate as far as they
can at present time.

Later studies have been focusing on ultracompact radio sources observed by
Very Long Baseline Interferometry (Kellermann, 1993; Gurvits, 1994; Gurvits et
al., 1999; Jackson & Jannetta, 2006). They differ from extended sources in
that their radio components are deeply embedded in the central region of the
host galaxy, so that there is no need to penetrate intergalactic gas. It is
hence claimed that the separation between their radio components could be free
of evolution. However, the radio luminosity was found to systematically
increase with redshift (Kellermann, 1993) due to a selection effect, given the
observed radio sources are flux limited (Gurvits, 1994). The systematics has
been largely ignored and people keep assuming the source size is independent
of redshift (Melia & Yennapureddy, 2018; Li & Lin, 2018). Even so, the data
strongly disagree with the expanding metric at $z<0.5$ (Gurvits et al., 1999).
This casts doubts on the unjustified assumption.

In this paper, we use an iterative approach to investigate the evolution of
radio sources. We start by assuming the luminosity density does not evolve
with redshift and derive the DDR. By comparing the DDR with the prediction, we
infer the evolution of the luminosity density. This way, our results do not
rely on a specific assumption, so that we can robustly test the expected DDR
and study the evolution of radio sources in the meantime.

##  2 Samples and Method

We test the distance duality relation using two independent source samples
that are available in the literature: (1) 613 radio sources at 2.29 GHz
(hereafter JJ2.29) compiled in Jackson & Jannetta (2006) selected from Preston
et al. (1985); (2) 330 radio sources at 5.0 GHz (hereafter GKF5.0) compiled in
Gurvits et al. (1999). In the JJ2.29 sample, the radio sources are not really
resolved, so they appear as a point represented by two $\mu\nu$ points, the
spatial frequencies, along east-west and north-south direction, respectively.
Since more extended sources correspond to lower fringe visibility (see e.g.
Thompson et al., 2017), source size can be characterized by fringe visibility,
which is measured by the ratio of correlated flux density to total flux
density ($\Gamma=S_{c}/S_{t}$). The source structure is modeled as a single
Gaussian profile, so the observable angular size $\theta_{\rm obs}$ is given
by (Preston et al., 1985; Gurvits, 1994)

$\theta_{\rm obs}=\frac{2\sqrt{-\ln\Gamma\ln 2}}{\pi B},$ (1)

where $B$ is the interferometer baseline. In the GKF5.0 sample, the
characteristic angular sizes of the radio sources are defined as the distance
between the brightest core and the most distant component whose peak
brightness is greater than or equal to 2% of the brightness of the core. Some
sources are not resolved by the inteferometer, so only the estimated upper
limits are given (Gurvits et al., 1999).

All sources in the GKF5.0 sample have measured spectral index $\alpha$
($S_{t}\propto\nu^{\alpha}$), and the distribution of their values can be
found in Gurvits et al. (1999). In the JJ2.29 sample, 572 sources out of 613
have measured spectral index as listed in Preston et al. (1985). We plot the
distribution of their values in Figure 1, which has a similar shape as that
for the GKF5.0 sample. For the rest sources (less than 7%) which do not have
measured spectral index, we set their values as the median index ($\alpha_{\rm
med}=-0.1$).

Figure 1: The histogram of spectral index ($S\propto\nu^{\alpha}$) for 572
radio sources at 2.29 GHz compiled in Jackson & Jannetta (2006). The values of
$\alpha$ are picked from Preston et al. (1985).

We recalculate the radio luminosity for both samples from the measured total
flux density $S_{t}$ by

$L_{t}=4\pi\frac{D_{L}^{2}}{(1+z)^{(1+\alpha)}}S_{t}.$ (2)

The factor $(1+z)$ in the denominator is included to account for the finite
band width, and $(1+z)^{\alpha}$ guarantees the derived luminosities for all
the sources are at the same rest wavelength. The luminosity distance $D_{L}$
is numerically calculated (open package “astropy”) according to

$D_{L}(z)=\frac{c}{H_{0}}(1+z)\int^{z}_{0}\frac{{\rm
d}z^{\prime}}{\sqrt{\Omega_{m}(1+z^{\prime})^{3}+\Omega_{\Lambda}}},$ (3)

using the best-fit cosmological parameters from Type Ia supernovae (SNe Ia):
$H_{0}=70.0$ km s-1Mpc-1, $\Omega_{m}=0.277$ and $\Omega_{\Lambda}=0.723$
(Suzuki et al., 2012). One can also use other fitting functions to calculate
luminosity distance, as long as it is consistent with the supernova data.

In Figure 2, we plot the radio luminosity against redshift for both the JJ2.29
(left) and GKF5.0 (right) samples. Since both samples are flux limited
(Gurvits, 1994; Gurvits et al., 1999), they present similar systematics in the
observed radio luminosity. This suggests that the source size cannot be
constant with redshift.

Figure 2: The systematic dependence of radio luminosity on redshift for 613
radio sources at 2.29 GHz (left) from Jackson & Jannetta (2006) and 330
sources at 5.0 GHz (right) from Gurvits et al. (1999). The solid lines are the
best-fit relations in the given parametrization form.

To circumvent possible systematic effect indicated by the observed luminosity-
redshift relation, the widely adopted method is to cut the sample to include
only sources with $z>0.5$ (Gurvits, 1994; Gurvits et al., 1999; Jackson &
Jannetta, 2006; Cao et al., 2015; Li & Lin, 2018) in which range the
systematical dependence is relatively weak. It is then claimed that the
selected radio sources could have constant separations with redshift. In fact,
sources at high-redshift end are also slightly inconsistent with the expanding
Universe (Cao et al., 2017a). As a result, only sources at intermediate
redshift range ($0.46<z<2.80$) in the JJ2.29 sample could possibly have
constant linear separations. This further cuts down the size of useful radio
sources. Eventually, less than 20% of the total sources are used for model
selections and cosmological tests (Cao et al., 2017b; Qi et al., 2017; Zheng
et al., 2017; Xu et al., 2018; Cao et al., 2018; Melia & Yennapureddy, 2018;
Qi et al., 2019; Liu et al., 2021; Tonghua et al., 2023).

As shown in Figure 2, both samples show apparent systematics in radio
luminosity at all redshift range with a smooth transition at $z=0.5$ rather
than a break. Moreover, the huge discrepancy between the observed and expected
$\theta-z$ relations at z $<$ 0.5 has remained unexplored. Although some
possible reasons have been given in the literature (Gurvits, 1994; Jackson,
2004; Jackson & Jannetta, 2006), none of them has been quantified or compared
to the data, so those claims remain unjustified.

The radio luminosity of the compact cores is the product of their volume and
the mean luminosity density, so that more luminous radio sources can possibly
be larger. In a nonexpanding Universe, the radio sources at high and low
redshift are expected to be intrinsically similar. Therefore, we expect the
intrinsic property, luminosity density, is independent of redshift; while
source size is an extensive quantity, so there is no reason to believe it
remains constant given the total luminosity evolves with redshift. In an
expanding Universe, radio sources could evolve with redshift, but it is
unclear how the luminosity density evolves. If one assumes a constant source
size, the energy density of the sources at $z\sim 4.0$ would be $\sim 10^{5}$
larger than those at $z\sim 0$. The extremely high ratio requires an evolution
that is too quick to be true.

Given the above considerations, here we adopt a different assumption: the
luminosity density does not depend on redshift, so the median value for a
group of radio sources is approximately constant. This is a nature expectation
in a nonexpanding Universe. It is also a good starting point for the expanding
cosmological model, which can be tested and adjusted through the resultant
DDR. Since the samples are flux limited, only large sources at high redshift
are observable. Therefore, the source size correlates with the radio
luminosity by $L_{t}\propto d^{3}$. We adopt the same assumption for the two
independent source samples, so that they can serve as a test for each other.

In order to quantify the evolution of the linear size $d$, we parameterize its
dependence on redshift as

$d=d_{0}(1+z)^{a}[\ln(1+z)]^{b},$ (4)

where $d_{0}$, $a$, $b$ are unknown constants. One can choose a different
fitting function with more fitting parameters, as long as it can describe the
observed luminosity-redshift relation. The selection of fitting functions does
not affect our results. To determine their values, we fit the parameterized
luminosity-redshift relation,

$L_{t}h^{2}=L_{0}\Big{[}(1+z)^{a}[\ln(1+z)]^{b}\Big{]}^{3},$ (5)

in log-space using non-linear least squares (python “curve_fit”). The fit
results are shown as solid lines in Figure 2. Both samples can be well
described by the chosen fitting function. We list the best-fit parameters in
Table 1.

Table 1: The best-fit parameters of the parameterized luminosity-redshift relation and the linear size scaling factor for both samples. Sample | $a$ | $b$ | $\log L_{0}$ | $d_{0}$
---|---|---|---|---
|  |  | [W/Hz] | (pc)
JJ2.29 | 0.39 $\pm$ 0.06 | 0.60 $\pm$ 0.02 | 27.1 $\pm$ 0.1 | 21.6 $\pm$ 0.7
GKF5.0 | 0.02 $\pm$ 0.08 | 0.72 $\pm$ 0.04 | 27.4 $\pm$ 0.1 | 63.7 $\pm$ 4.4

Figure 3: The median values of the calibrated angular size versus redshift.
Left: The total 613 data points are nearly equally populated into 34 bins;
Right: 330 data points are distributed into 19 bins (18 sources within each
bin for the first 17 redshift bins and 12 sources within each bin for the rest
two bins at high redshift). The uncertainties on the median angular size and
redshift are estimated according to $\sigma_{\rm
med}=1.2533\frac{\sigma}{\sqrt{N}}$, where $\sigma$ is the scatter around the
mean value and $N$ is the sample size within each bin. The predictions of the
expanding and nonexpanding Universes are presented as the solid and dashed
lines, respectively.

##  3 Angular Size-Redshift Relation

The relation between the observed angular size $\theta_{\rm obs}$ and the
linear separation $d$ is expected to be

$\theta_{\rm obs}=\frac{d_{0}(1+z)^{a}[\ln(1+z)]^{b}}{D_{A}},$ (6)

where the values of $a$ and $b$ are listed in Table 1 and the angular diameter
distance $D_{A}$ is numerically calculated using the same parameters for
calculating $D_{L}$. For better illustrations, we define

$\theta=\theta_{\rm obs}(1+z)^{-a}[\ln(1+z)]^{-b}.$ (7)

The newly defined angular size corresponds to a constant linear separation
$d_{0}$, so that its evolution is solely determined by $D_{A}$.

We bin the calibrated angular sizes $\theta$ and calculate their median values
within each bins. We plot the binned median angular sizes against redshift in
Figure 3. Since the JJ2.29 sample is larger and has more homogeneous
definition about source size, it presents smaller uncertainties than the
GKF5.0 sample. Both samples present a $\theta-z$ relation that monotonically
decrease with redshift. However, there should exist a minimum value of angular
size in an expanding Universe, since the angular diameter distance has a
maximum value at $z\simeq 1.7$. Lacking a minimum angular size contradicts to
the expectation of the standard cosmology.

To quantitatively test the standard cosmology, we fit the following relation,

$\theta=\frac{d_{0}}{D_{A}(z)}.$ (8)

The best-fit relations (dashed lines in Figure 3) show apparent discrepancies
from the observed ones. We also fit the predicted relation of a nonexpanding
Universe,

$\theta=\frac{d_{0}}{(1+z)D_{A}(z)}.$ (9)

It turns out the single-parameter function fully agrees with the observed
$\theta-z$ relations for both samples over the whole available redshift range
(solid lines in Figure 3). Therefore, the data are consistent with a
nonexpanding Universe when assuming a constant luminosity density. To
reconcile this discrepancy with the expanding Universe, one has to decrease
the value of $a$ by one in equation 7. However, this is a 17 $\sigma$
discrepancy for the JJ2.29 sample and 13 $\sigma$ discrepancy for the GKF5.0
sample. We also carry out the analysis using recent Planck results (Planck
Collaboration et al., 2020) and the results remain the same. The difference
between the SNe Ia and Planck cosmologies is too small to change our
conclusion. Hubble constant is just a proportional constant in distance
measurements whose value only affects the derived value of $d_{0}$ rather than
the shape of the $\theta-z$ curve. As such, it is unlikely to compensate the
extra factor (1+z) by varying the cosmological parameters without changing the
initial assumption for the evolution of luminosity density.

The best-fit values of the only fitting parameter $d_{0}$ are given in Table 1
for both samples. The characteristic source size depends on redshift as
described in equation 4. It is noticed that the characteristic source size in
the JJ2.29 sample is systematically smaller than that in the GKF5.0 sample at
same redshift, contrary to the expectation of the Blandford-Konigl model
(Blandford & Königl, 1979), which points out the observed angular size should
be inversely proportional to the observed frequency, $\theta\propto\nu^{-1}$.
As mentioned in Section 2, the angular size of the two source samples are
defined in different ways. The JJ2.29 sample consistently uses the fringe
visibility to define the characteristic source size assuming a Gaussian light
profile; while the GKF5.0 sample defines the size of the resolved source as
the distance between the core and a 2%-component, and estimate an upper limit
of the size for the unresolved source considering the synthesized beam
(Gurvits et al., 1999). The “2%” definition for the GKF5.0 sample makes their
source size systematically larger than that in the JJ2.29 sample.

##  4 Distance Duality Relation

Now that the two independent samples present the same angular size-redshift
relation, we move forward to investigate the distance duality relation. In a
metric-expanding Universe, the distance duality relation is given by

$D_{L}=(1+z)^{2}D_{A},$ (10)

The factor $(1+z)^{2}$ has two origins: (1) redshift and time dilation cause
luminosity distance greater than proper distance by $(1+z)$: After being
redshifted, photon’s energy decreases by a factor of $(1+z)$, so its period
increases by the same factor. As a result, the required detection time dilates
by a factor of $(1+z)$. The equivalent luminosity, when observed, becomes
$L_{\rm obs}=\frac{\delta E_{\rm obs}}{\delta t_{\rm obs}}=\frac{\delta E_{\rm
emit}}{\delta t_{\rm emit}(1+z)^{2}}=L_{\rm emit}/(1+z)^{2}$. The extra factor
of $(1+z)^{2}$ goes into the definition of luminosity distance. (2) the
expanding metric makes angular diameter distance smaller than proper distance
by $(1+z)$: when the photons from an object are detected, the proper distance
has expanded by a factor of $(1+z)$, while its physical size and the angle it
subtends remain invariant. In a nonexpanding Universe, the second origin
disappears, but the first holds. This is because redshift is directly
observed, so it should be accepted as a fact, no matter how we explain it.
Therefore, the expected distance duality relation is given by,

$D_{L}=(1+z)D_{A}.$ (11)

These are strict relations and they hold exactly in each models. It does not
depend on any cosmological effects such that no mechanism can be introduced to
compensate any possible deviations. Therefore, it provides the most robust and
direct test to the cosmic expansion.

We calculate the luminosity distance $D_{L}$ for the radio sources in both
samples based on their redshift using the SNe Ia cosmology mentioned earlier.
Their angular diameter is calculated by $D_{A}=d_{0}/\theta$, where $d_{0}$ is
the linear size scaling factor as shown in Table 1 and $\theta$ is the
calibrated angular size. Therefore, both $D_{A}$ and $D_{L}$ are directly
calculated. There are no fitting parameters.

We plot $D_{L}(z)/D_{A}(z)$ against redshift rather than $D_{A}(z)$ against
$D_{L}(z)$. This has a key advantage: both axes are independent of
cosmological parameters, so that using a different set of parameters does not
affect our results. We combine the two separate samples together, given they
are supposed to follow the same relation. The combined sample is larger and
thereby helps minimize the scatter.

Figure 4: The distance duality relation for the combined sample from Jackson &
Jannetta (2006) and Gurvits et al. (1999). The total 943 radio sources are
distributed into 27 bins. The median values of $D_{L}(z)/D_{A}(z)$ within each
bins are plotted against redshift. The predictions of the nonexpanding and
expanding universes are presented as the solid and dashed lines, respectively.
The data shows a linear distance duality relation, consistent with Euclidean
geometry.

The total 943 radio sources (613 from the JJ2.29 sample and 330 from the
GKF5.0 sample) are nearly equally populated into 27 bins. The median values of
$D_{L}(z)/D_{A}(z)$ within each bins are plotted in Figure 4. Remarkably, the
data are fully consistent with a linear distance duality relation throughout
all the available redshift range. This is the signal of Euclidean geometry. It
implies a nonexpanding Universe.

The above results are derived by assuming a constant luminosity density, which
is a nature expectation in a nonexpanding Universe but may not be true if the
Universe is expanding. In order to reproduce the expected relation in an
expanding Universe, the angular diameter distance of the radio sources must be
smaller by a factor of $(1+z)$. This is equivalent to requiring the linear
separation smaller by $(1+z)$, i.e.

$d=d_{0}(1+z)^{a-1}[\ln(1+z)]^{b}.$ (12)

To reproduce the observed radio luminosity, the mean luminosity density has to
increase with redshfit according to

$\rho_{L}(z)=\rho_{L}(0)(1+z)^{3}.$ (13)

This is the evolution of luminosity density required by the DDR expected in an
expanding Universe.

However, our results present a perfect coincidence. Suppose that the true
evolution of the source size follows eq. 12, but the evolution of the
luminosity density follows a different relation other than eq. 13, e.g.
$\rho_{L}(z)=\rho_{L}(0)(1+z)^{4}$. We would have observed the same angular
sizes but a different evolution of the total luminosity,
$L_{t}h^{2}=L_{0}\big{[}(1+z)^{a}[\ln(1+z)]^{b}\big{]}^{3}(1+z)$, instead of
eq. 5. This would still reproduce the distance duality relation in an
expanding Universe. In a nonexpanding Universe, we would keep assuming the
mean luminosity density is constant with redshift, and it leads to
$d=d_{0}(1+z)^{a+\frac{1}{3}}[\ln(1+z)]^{b}$. This includes an extra factor of
$(1+z)^{\frac{1}{3}}$ with respect to eq. 4. Given $D_{A}=d/\theta_{\rm obs}$,
it would result in a different distance duality relation:
$D_{L}/D_{A}=(1+z)^{\frac{2}{3}}$, inconsistent with a nonexpanding Universe.
Similarly, if the luminosity density follows eq. 13, but the source size
evolves in a different way, the resultant distance duality relation would be
inconsistent with a nonexpanding Universe either. Therefore, the perfect
agreement between the data and the prediction of a nonexpanding Universe (Fig.
4) poses a challenge for the explanation in an expanding Universe: the source
size and luminosity density evolve in the way that precisely mimics a
nonexpanding Universe. This is not only a conspiracy but also a fine-tuning
problem.

##  5 Discussion and Conclusion

In this paper, we tested the distance duality relation with ultracompact radio
sources. There are many other tests for this relation. A general problem in
many tests is that the target relation is often assumed in the methodology.
For example, the test using the baryonic acoustic oscillation (Eisenstein et
al., 2005; Beutler et al., 2011; Martinelli et al., 2020; Azevedo & Avelino,
2021) has to assume the DDR in order to convert the measured angular scales
into distances. The same problem occurs in Hodgson et al. (2020), which used
the distance traveled by photons to estimate the linear size of the emitting
region in an active galactic nuclei, as they have to include an extra factor
of $(1+z)$ in the size measurements (their equation 5 and 6) to account for
the effect of the expanding Universe.

Liao et al. (2016) attempted to test this relation using strong gravitational
lensing, which measures the ratio of the angular diameter distance between the
lens and the source $D^{A}_{ls}$ to the angular diameter distance of the
source $D^{A}_{s}$, i.e. $R_{A}=D^{A}_{ls}/D^{A}_{s}$. However, the formula
they tested (their eq. 7) does not distinguish the expanding and nonexpanding
Universes. In a nonexpanding Universe, angular diameter distance is
essentially the same as comoving distance, so that

$R_{A}=\frac{D^{A}_{ls}}{D^{A}_{s}}=\frac{D^{A}_{s}-D^{A}_{l}}{D^{A}_{s}}=1-\frac{D^{L}_{l}(1+z_{s})}{D^{L}_{s}(1+z_{l})},$
(14)

where we have used the distance duality relation $D^{L}=(1+z)D^{A}$ in a
nonexpanding Universe. When $\eta_{0}=0$, their eq. 7 becomes exactly the same
as the above formula. Their fitting results indeed show $\eta_{0}\simeq 0$
(see their Table 1). Lima et al. (2021) modified their equation 7 by replacing
luminosity distance with angular diameter distance using the DDR expected in
the expanding Universe. As a result, their parametrization scheme,
$\frac{D_{L}}{D_{A}(1+z)^{2}}=1+\eta_{0}z$, excludes the DDR in a nonexpanding
Universe. And yet, their results show inconsistency with the expanding
Universe (see their Figure 2,3,4).

Sunyaev-Zeldovich effect has also been used to test this relation. This
approach requires a careful treatment of the inverse Compton scattering and
robust modellings of the charged particle distributions in clusters of
galaxies. These inevitably introduce many uncertainties. The measured angular
diameter distances are therefore not expected to be accurate. Holanda et al.
(2010) indeed found the distance duality relation derived with the clusters
from De Filippis et al. (2005) is consistent with neither theory. Their second
parametrization scheme in eq. 2, $\eta(z)=1+\eta_{0}\frac{z}{1+z}$, can
describe both the expanding and nonexpanding Universes, where $\eta_{0}=0$ and
$\eta_{0}=-1$ correspond to the expanding and nonexpanding Universe,
respectively. They found $\eta_{0}=-0.66$ when modelling clusters with
spherical geometry. Even if the clusters are modelled with elliptical
geometry, they found $\eta_{0}=-0.43$, which is far from the prediction of
either model. Yang et al. (2013) proposed to include the parameters that
determine the distance modulus of SNe Ia (i.e. absolute B magnitude $M_{B}$,
stretch factor $\alpha$ and color parameter $\beta$) into the fitting. By
opening a larger parameter space, they are eventually able to obtain the value
of $\eta_{0}$ close to zero. However, their best-fit parameters that are used
to determine the luminosity distance are hence inconsistent with those
parameters De Filippis et al. (2005) used to derive the angular diameter
distance. In fact, the new parameters they obtained lead to systematically
larger luminosity distances, which can compensate the discrepancy Holanda et
al. (2010) found.

In our test, the angular diameter distances of ultracompact radio sources are
determined solely from geometry; the luminosity distance are calculated by
interpolating or extrapolating the observed distance-redshift relation from
SNe Ia, which have been extensively accepted as standard candles (Perlmutter
et al., 1999). The properties of SNe Ia are derived from stellar evolutions.
Therefore, our test is entirely independent of cosmological models. The
observed evolution of the angular size and total radio luminosity can be
explained in both expanding and non-expnading Universe: in a nonexpanding
Universe, the luminosity density does not evolve with redshift; in an
expanding Universe, the mean luminosity density evolves as:
$\rho_{L}\propto(1+z)^{3}$. Interestingly, the luminosity density and the
source size conspire to fine tune their evolution to make them
indistinguishable from a nonexpanding Universe.

The problem identified in this paper points two ways forward: (1) the Universe
is not expanding, so we would need to re-explain the observed cosmological
redshift and other related observations; (2) the Universe is expanding, so we
would have to explain why the evolution of ultracompact radio sources
precisely mimics a nonexpanding Universe. Weighing the price, option (2) is
obviously more attractive. However, science is not about bargaining a better
deal, but finding out the truth. The current analysis alone cannot determine
which is the right way to proceed. Similar analyses for other astronomical
objects, e.g. galaxies and/or galaxy clusters, should be able to provide
useful hints.

I thank Hai-Nan Lin for providing the machine-readable form of the radio
sources. This work is made possible through the support from the Alexander von
Humboldt foundation.

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