In this paper, we compute the nef cone and the pseudo-effective cone of $C\times J$ for a smooth projective curve $C$ and its Jacobian variety $J$ such that $C\times J$ has the minimal Picard number.
As a consequence, we also compute the successive minima of a height function for the relative setting $C\times J\to J$, and our result shows that Zhang's theorem of  successive minima does not hold in this case.

\subsection{The cones}

By a \emph{variety} over a field, we mean an integral scheme, separated and of finite type over the field. By a \emph{curve}, we mean a variety of dimension 1.

Let us review the classical notions of nef cones and the \pef cones. We refer to Lazarsfeld \cite{Laz04} for more details. Let $X$ be a smooth projective variety over a field $k$. Let
$\mathrm{NS}(X)=\mathrm{Pic}(X)/\mathrm{Pic}^0(X)$
be its N\'eron-Severi group. It is well-known that $\NS(X)$ is a finitely generated abelian group, whose rank is called the \emph{Picard number} $\rho(X)$ of $X$.

Denote $\mathrm{NS}(X)_\mathbb{R}=\NS(X) \otimes_\mathbb{Z}\mathbb{R}$.
Denote by
$$\Amp(X),\quad \Nef(X), \quad \BBig(X), \quad \Psef(X)$$
the subsets of $\xi\in \NS(X)_\mathbb{R}$ which are ample, nef, big, and pseudo-effective on $X$ respectively.
They are respectively called the \emph{ample cone}, the \emph{nef cone}, the \emph{big cone}, and the \emph{pseudo-effective cone} of $X$.
All of them are convex cones in $\NS(X)_\mathbb{R}$.
Moreover, $\Nef(X)$ is the closure of $\Amp(X)$, and $\Amp(X)$ is the interior of $\Nef(X)$;  $\Psef(X)$ is the closure of $\BBig(X)$, and $\BBig(X)$ is the interior of $\Psef(X)$.

It is known that an element $\xi\in \NS(X)_\mathbb{R}$ is \pef if and only if $\xi + \BBig(X)\subseteq \BBig(X)$. In particular, if a line bundle $L$ on $X$ is effective in that $\Gamma(X,L)\neq 0$, then it is also pseudo-effective.
In practice, it is usually very difficult to determine these cones for a general variety.
In the following, we will provide an explicit result for a very special example.

Let $C$ be a smooth projective curve  of genus $g>1$ over an algebraically closed field $k$. Let $J$ be the Jacobian variety of $C$.
Then the Picard number $\rho(C\times J)\geq 3$ by the classical decomposition
$$
\NS(C\times J) = \NS(C)\oplus \NS(J) \oplus \End(J).
$$
This decomposition is scattered in \cite[\S VI.4]{Lan83}, and we refer to \cite[Lem. 2.2.1]{Zha10} and \cite[\S A-2]{MNP13} for modern treatments.

It is also easy to construct 3 linearly independent elements of $\NS(C\times J)$.
We will introduce them under our notation explicitly, which will be used throughout this paper.
Take a line bundle $\alpha\in \mathrm{Pic}(C)$ such that $(2g-2)\alpha=\omega_{C/k}$  in $\mathrm{Pic}(C)$. Denote the Abel-Jacobian map
\[
i_\alpha:C\longrightarrow J,\quad x\longmapsto x-\alpha.
\]
Let $\theta$ be the image of the morphism
\[
C^{g-1}\longrightarrow J,\quad (x_1,\cdots,x_{g-1})\longmapsto x_1+\cdots+x_{g-1}-(g-1)\alpha.
\]
It is well-known that $\theta$ is an ample divisor on $J$, which gives a principal polarization
\[
\phi:J\stackrel{\thicksim}\longrightarrow J^\vee,\quad  x\longmapsto T^*_{x}\theta-\theta.
\]
Let $P$ be the pull-back of the usual Poincar\'e bundle on $J\times J^\vee$ via
\[
\id\times \phi:J\times J\longrightarrow J\times J^\vee.
\]
Then we have a Poincar\'e line bundle
\[
Q=(i_\alpha\times \id)^*P
\]
 on $C\times J$.

Denote by $p_1:C\times J\to C$ and $p_2:C\times J\to J$ the projections.
Then $\NS(C\times J)$ contains 3 linearly independent elements
$$
 \alpha_1:= p_1^*\alpha, \quad \theta_2:=p_2^*\theta,\quad Q.
$$
Each of them lies in a component in the above decomposition of  $\NS(C\times J)$.

In our main results, we will assume the minimal case $\rho(C\times J)=3$.
We remark that there are plenty of curves $C$ with $\rho(C\times J)=3$.
In fact, if $C$ is a very general smooth projective curve over $\mathbb{C}$ in the sense that it is represented by a complex point of the coarse moduli scheme $\CM_g$ over $\mathbb{C}$ outside a countable union of proper algebraic subsets, then
$\rho(J)=1$ by a classical theorem of Lefschetz
(cf. \cite{Pir88}) and $\End(J)=\mathbb{Z}$ by a classical theorem of Hurwitz
(cf. \cite[\S3]{Cil89} and \cite{Zar00}).
For such complex curves $C$, we have $\rho(C\times J)=3$.
Similar results hold in positive characteristics.

The following is our first theorem, which described the cones of $C\times J$ explicitly.

In this paper, we compute the nef cone and the pseudo-effective cone of $C\times J$ for a smooth projective curve $C$ and its Jacobian variety $J$ such that $C\times J$ has the minimal Picard number.
As a consequence, we also compute the successive minima of a height function for the relative setting $C\times J\to J$, and our result shows that Zhang's theorem of successive minima does not hold in this case.

Denote $\mathrm{NS}(X)_\mathbb{R}=\NS(X) \otimes_\mathbb{Z}\mathbb{R}$.
Denote by
$$\Amp(X),\quad \Nef(X), \quad \BBig(X), \quad \Psef(X)$$
the subsets of $\xi\in \NS(X)_\mathbb{R}$ which are ample, nef, big, and pseudo-effective on $X$ respectively.
They are respectively called the \emph{ample cone}, the \emph{nef cone}, the \emph{big cone}, and the \emph{pseudo-effective cone} of $X$.
All of them are convex cones in $\NS(X)_\mathbb{R}$.
Moreover, $\Nef(X)$ is the closure of $\Amp(X)$, and $\Amp(X)$ is the interior of $\Nef(X)$; $\Psef(X)$ is the closure of $\BBig(X)$, and $\BBig(X)$ is the interior of $\Psef(X)$.

Let $C$ be a smooth projective curve of genus $g>1$ over an algebraically closed field $k$. Let $J$ be the Jacobian variety of $C$.
Then the Picard number $\rho(C\times J)\geq 3$ by the classical decomposition
$$
\NS(C\times J) = \NS(C)\oplus \NS(J) \oplus \End(J).
$$
This decomposition is scattered in \cite[\S VI.4]{Lan83}, and we refer to \cite[Lem. 2.2.1]{Zha10} and \cite[\S A-2]{MNP13} for modern treatments.

Denote by $p_1:C\times J\to C$ and $p_2:C\times J\to J$ the projections.
Then $\NS(C\times J)$ contains 3 linearly independent elements
$$
 \alpha_1:= p_1^*\alpha, \quad \theta_2:=p_2^*\theta,\quad Q.
$$
Each of them lies in a component in the above decomposition of $\NS(C\times J)$.

In our main results, we will assume the minimal case $\rho(C\times J)=3$.
We remark that there are plenty of curves $C$ with $\rho(C\times J)=3$.
In fact, if $C$ is a very general smooth projective curve over $\mathbb{C}$ in the sense that it is represented by a complex point of the coarse moduli scheme $\CM_g$ over $\mathbb{C}$ outside a countable union of proper algebraic subsets, then
$\rho(J)=1$ by a classical theorem of Lefschetz
(cf. \cite{Pir88}) and $\End(J)=\mathbb{Z}$ by a classical theorem of Hurwitz
(cf. \cite[\S3]{Cil89} and \cite{Zar00}).
For such complex curves $C$, we have $\rho(C\times J)=3$.
Similar results hold in positive characteristics.

The following is our first theorem, which described the cones of $C\times J$ explicitly.

\begin{theorem}[Theorem \ref{first main thm}]\label{cone thm}
Assume that $\rho(C\times J)=3$. Then
	\[
	\Amp(C\times J)=\BBig(C\times J)=\{a\cdot \alpha_1+b\cdot \theta_2+c\cdot Q:a>0,\ b> 0, \ ab> gc^2\}
	\]
and
	\[
	\Nef(C\times J)=\Psef(C\times J)=\{a\cdot \alpha_1+b\cdot \theta_2+c\cdot Q:a\geq0,\ b\geq 0, \ ab\geq gc^2\}.
	\]
\end{theorem}

The following is our first theorem, which described the cones of $C\times J$ explicitly.

The main idea of our proof of the theorem is to construct all nef line bundles on $C\times J$ by geometric operations. In fact, for $m,n\in \ZZ$, consider the morphism
$$
f_{m,n}: C\times J\longrightarrow J,\quad (x,y)\longmapsto m(x-\alpha)+ny.
$$
Then the pull-back
\[
f_{m,n}^*\theta=gm^2\cdot \alpha_1+n^2\cdot \theta_2+mn\cdot Q.
\]
is nef on $C\times J$.
This essentially gives all the line bundles on the boundary of the nef cone.

\begin{theorem}[Theorem \ref{second main}]
\label{second main thm}
Assume that $\rho(C\times J)=3$. Then
\[
e_1(h_{\CL}^\theta)=e_2(h_{\CL}^\theta)=(g-\frac{1}{g})\cdot (g-1)!,
\]
and
\[
h_{\CL}^\theta(C_K)=(g-1)\cdot (g-1)!.
\]
Moreover, there is an infinite sequence $(x_n)_n$ in $C_K(\overline K)$ with $\deg(x_n)\to \infty$ and with
$$
h^\theta_\CL(x_n) = (g-\frac{1}{g})\cdot (g-1)!.
$$
\end{theorem}

\begin{theorem}[Theorem \ref{first main thm}]\label{cone thm}
Assume that $\rho(C\times J)=3$. Then
 \[
 \Amp(C\times J)=\BBig(C\times J)=\{a\cdot \alpha_1+b\cdot \theta_2+c\cdot Q:a>0,\ b> 0, \ ab> gc^2\}
 \]
and
 \[
 \Nef(C\times J)=\Psef(C\times J)=\{a\cdot \alpha_1+b\cdot \theta_2+c\cdot Q:a\geq0,\ b\geq 0, \ ab\geq gc^2\}.
 \]
\end{theorem}

\begin{proof}
Denote
\[
\Sigma= \{a\cdot \alpha_1+b\cdot \theta_2+c\cdot Q:a>0,\ b> 0, \ ab> gc^2\}.
\]
It suffices to prove
\[
\Psef(C\times J) \ \subseteq \
\Sigma
\ \subseteq \
\Nef(C\times J).
\]
In fact, this forces equalities by $\Nef(C\times J) \ \subseteq
\Psef(C\times J).$
The interiors give the expression for $\Amp(C\times J)$ and $\BBig(C\times J)$.

Now we prove $\Psef(C\times J) \subseteq \Sigma$.
By taking the closure, it suffices to prove $\BBig(C\times J) \subseteq \Sigma$.
Let
$$L=a\cdot \alpha_1+b\cdot \theta_2+c\cdot Q$$
be an element of $\BBig(C\times J)$ with $a,b,c\in \RR$.
We first claim that $a,b>0$.
In fact, by Lemma \ref{computing Q},
\[
L|_{x\times J}=b\cdot \theta+c \cdot \phi(x-\alpha), \quad x\in C(k)
\]
and
\[
L|_{C\times y}=a\cdot \alpha+c \cdot y, \quad y\in J(k).
\]
By the Kodaira lemma (cf. \cite[Cor. 2.2.7]{Laz04}), we can write $L=A+E$ for an ample class $A$ and an effective class $E$ on $J$.
It follows that $L|_{x\times J}$ (resp. $L|_{C\times y}$) is big as long as $x\times J$
(resp. ${C\times y}$) is not contained in the support of $E$.
These imply $a,b>0$.

\begin{theorem}[Theorem \ref{second main thm}]
\label{second main}
Assume that $\rho(C\times J)=3$. Then
\[
e_1(h_{\CL}^\theta)=e_2(h_{\CL}^\theta)=(g-\frac{1}{g})\cdot (g-1)!,
\]
and
\[
h_{\CL}^\theta(C_K)=(g-1)\cdot(g-1)!.
\]
Moreover, there is an infinite sequence $(x_n)_n$ in $C_K(\overline K)$ with $\deg(x_n)\to \infty$ and with
$$
h^\theta_\CL(x_n) = (g-\frac{1}{g})\cdot (g-1)!.
$$
\end{theorem}

For any point $x\in C_K(\ol K)$, its Zariski closure $\bar x$ is an effective divisor on $C\times J$. Suppose that in $\NS(C\times J)_\RR$, we have
\[
\bar x\equiv a\cdot \alpha_1+b\cdot\theta_2+c\cdot Q, \quad a,b,c\in \QQ.
\]
We have
$$
\deg(x)= \deg(\bar x|_{C\times y})=a>0.
$$
By Theorem \ref{cone thm} for the pseudo-effective cone, we have $b\geq 0$, and $ab\geq gc^2$.
Write
$s=b/a$ and $t=c/a$. Then
$$
\bar x\equiv a( \alpha_1+s\cdot\theta_2+t\cdot Q), \quad s\geq gt^2.
$$
It follows that
\begin{align*}
h^\theta_\CL(x)
&=\frac{1}{\deg(x)}\bar x\cdot\CL\cdot \theta_2^{g-1}\\
&=( \alpha_1+s\cdot\theta_2+t\cdot Q)\cdot\CL\cdot \theta_2^{g-1}\\
&= (1+gs-2t)\cdot g! \\
&\geq (1+g^2t^2-2t)\cdot g!\\
&\geq (1-\frac{1}{g^2})\cdot g!\\
&= (g-\frac{1}{g})\cdot (g-1)!.
\end{align*}
As a consequence,
\[
e_1(h_{\CL}^\theta)\geq e_2(h_{\CL}^\theta)\geq (g-\frac{1}{g})\cdot (g-1)!.
\]
Note that the last statement of the theorem implies
\[
e_1(h_{\CL}^\theta)\leq (g-\frac{1}{g})\cdot (g-1)!,
\]
which forces equalities.

\begin{theorem}[Theorem \ref{first main thm}]\label{cone thm}
Assume that $\rho(C\times J)=3$. Then
	\[
	\Amp(C\times J)=\BBig(C\times J)=\{a\cdot \alpha_1+b\cdot \theta_2+c\cdot Q:a>0,\ b> 0, \ ab> gc^2\}
	\]
and
	\[
	\Nef(C\times J)=\Psef(C\times J)=\{a\cdot \alpha_1+b\cdot \theta_2+c\cdot Q:a\geq0,\ b\geq 0, \ ab\geq gc^2\}.
	\]
\end{theorem}

\begin{theorem}[Theorem \ref{cone thm}]
 \label{first main thm}
Assume that $\rho(C\times J)=3$. Then
	\[
	\Amp(C\times J)=\BBig(C\times J)=\{a\cdot \alpha_1+b\cdot \theta_2+c\cdot Q:a>0,\ b> 0, \ ab> gc^2\}
	\]
and
	\[
	\Nef(C\times J)=\Psef(C\times J)=\{a\cdot \alpha_1+b\cdot \theta_2+c\cdot Q:a\geq0,\ b\geq 0, \ ab\geq gc^2\}.
	\]
\end{theorem}

The main idea of our proof of the theorem is to construct all nef line bundles on $C\times J$ by geometric operations. In fact, for $m,n\in \mathbb{Z}$, consider the morphism
$$
f_{m,n}: C\times J\longrightarrow J,\quad (x,y)\longmapsto m(x-\alpha)+ny.
$$
Then the pull-back
\[
f_{m,n}^*\theta=gm^2\cdot \alpha_1+n^2\cdot \theta_2+mn\cdot Q.
\]
is nef on $C\times J$.
This essentially gives all the line bundles on the boundary of the nef cone.

\subsection{Successive minima}

To introduce our second result, let us first review a geometric version of Zhang's theorem of successive minima. For simplicity, we restrict to curves over function fields.

Let $B$ be a projective variety  of dimension $d>0$ over a field $k$, and let $K=k(B)$ be the function field of $B$. Let $X$ be a projective curve over $K$.
Let $\pi:\CX\to B$ be an integral model of $X$ over $B$; namely, $\CX$ is a projective variety over $k$, and $\pi:\CX\to B$ is $k$-morphism whose generic fiber is isomorphic to $X\to \Spec K$.
Let $\CL$ be a $\pi$-nef line bundle on $\CX$, i.e. $\CL$ has a non-negative degree on every projective curve in fibers of $\pi$.
Let $\CM$ be an ample line bundle on $B$.
This defines a height function
\[
h^\CM_\CL:C(\ol K)\longrightarrow\mathbb{R},\quad x\longmapsto \frac{1}{\deg(x)}\bar x\cdot \CL\cdot(\pi^*\CM)^{d-1}.
\]
Here $\bar x$ is the Zariski closure of $x$ in $\CX$.

The \emph{essential minimum} of $h_{\CL}^\CM$ is defined as
\[
e_1(h_{\CL}^\CM)=\liminf_{x\in C(\overline K)}h_{\CL}^\CM(x).
\]
The \emph{absolute minimum} of $h_{\CL}^\CM$ is defined as
\[
e_2(h_{\CL}^\CM)=\inf_{x\in C_K(\ol K)}h_{\CL}^\CM(x).
\]

Assume that the generic fiber $\CL_K=\CL|_X$ has a positive degree. Then the height of $X$ is defined by
$$
h^\CM_\CL(X)=\frac{1}{2\deg(\CL_K)} \CL^2\cdot(\pi^*\CM)^{d-1}.
$$

If $d=1$, then $K$ is the function field of one variable, and $\CM$ does not show up in the above definitions, so we drop the dependence on $\CM$ in $h^\CM_\CL$ and $e_i(h_{\CL}^\CM)$. The situation is parallel to the case of number fields treated by Zhang \cite{Zha92}. Then a geometric analogue of Zhang's theorem of successive minima in \cite[Thm. 6.3]{Zha92} asserts that
$$
e_1(h_{\CL}) \geq h_\CL(X) \geq \frac{1}{2}(e_1(h_{\CL})+e_2(h_{\CL})).
$$
This holds if $d=1$, $\CL$ is $\pi$-nef, and $\deg(\CL_K)>0$, and the proof is similar to the loc. cit..

If $d>1$, it is natural to speculate
$$
e_1(h_{\CL}^\CM) \geq h^\CM_\CL(X) \geq \frac{1}{2}(e_1(h_{\CL}^\CM)+e_2(h_{\CL}^\CM)),
$$
assuming $\CL$ is $\pi$-nef, $\deg(\CL_K)>0$, and $\CM$ is ample on $B$.
In fact, the first inequality is claimed in \cite[Lem. 4.1]{Gub07}, and the second inequality is claimed in \cite[Prop. 4.3]{Gub07}.
In the following, we prove that the second inequality is wrong by a counterexample.

To introduce the counterexample,
return to the situation of $C$ and $J$.
Namely, let $C$ be a smooth projective curve  of genus $g>1$ over an algebraically closed field $k$, and let $J$ be the Jacobian variety of $C$.

To compare utilize the above terminology of successive minima, we take $\pi:\CX\to B$ to be $p_2:C\times J\to J$.
Then $X=C_K$ for $K=k(J).$
Take
$$\CM=\theta \in \mathrm{Pic}(J)$$
and
\[
\CL=g\cdot \alpha_1+ \theta_2+Q\in\mathrm{Pic}(C\times J).
\]
Here for simplicity, we write $\theta$ for the line bundle $\CO(\theta)$ in the Picard group.
Then we have the following computational results, which disprove the second inequality speculated above.

\begin{theorem}[Theorem \ref{second main}]
\label{second main thm}
Assume that $\rho(C\times J)=3$. Then
\[
e_1(h_{\CL}^\theta)=e_2(h_{\CL}^\theta)=(g-\frac{1}{g})\cdot (g-1)!,
\]
and
\[
h_{\CL}^\theta(C_K)=(g-1)\cdot (g-1)!.
\]
Moreover, there is an infinite sequence $(x_n)_n$ in $C_K(\overline K)$ with $\deg(x_n)\to \infty$ and with
$$
h^\theta_\CL(x_n) = (g-\frac{1}{g})\cdot (g-1)!.
$$
\end{theorem}

To prove Theorem \ref{second main thm}, the key is that in the height formula
\[
h^\theta_\CL(x)= \frac{1}{\deg(x)}\bar x\cdot \CL\cdot(\pi^*\theta)^{d-1},
\]
the term $\bar x$ is an effective divisor on $C\times J$.
Then we can use Theorem \ref{first main thm} to express $\bar x$ in terms $\alpha_1, \theta_2, Q$, and estimate the height by explicit calculations.

In the end, we make a few remarks on the truth of the inequalities
$$
e_1(h_{\CL}^\CM) \geq h^\CM_\CL(X) \geq \frac{1}{2}(e_1(h_{\CL}^\CM)+e_2(h_{\CL}^\CM)).
$$
Here we assume that  $\CL$ is $\pi$-nef, $\deg(\CL_K)>0$, and $\CM$ is ample on $B$.
As mentioned above, both inequalities hold for $\dim B=1$.
The first inequality always holds for general $\dim B$ by reducing it to $\dim B=1$
following \cite[Prop. 5.11]{Gub08}.
Our counterexample disproves the second inequality for $\dim B=2$.

Now we consider the arithmetic setting, where the projective varieties over $k$ are replaced projective arithmetic varieties over $\mathbb{Z}$ and the line bundles are replaced by hermitian line bundles.
If $\dim B=1$, this is Zhang's original theorem in \cite{Zha92}.
If $\dim B>1$, the height $h_{\CL}^\CM$ is the Moriwaki height originally introduced by Moriwaki \cite{Mor00}, and we refer to \cite[Cor. 5.2]{Mor00} for a weaker version of the inequalities.
With some efforts, we should prove the first inequality for all $\dim B$ and disprove the second inequality for $\dim B>1$.

Finally, we refer to Zhang \cite[Thm. 1.10]{Zha95} and Yuan--Zhang \cite[\S5.3]{YZ21} for adelic versions of the inequalities.

\subsubsection*{Acknowledgments}
The authors are grateful to Walter Gubler and Junyi Xie for communications related to this paper.
The authors would like to thank the support of the China--Russia Mathematics Center. The second author is supported by grants NO. 12250004 and NO. 12321001
from the National Science Foundation of China, and by the Xplorer Prize from the New Cornerstone Science Foundation.

The post-theorem introduction explains that the proof of Theorem~\ref{first main thm} proceeds by “construct[ing] all nef line bundles on $C\times J$ by geometric operations.” Concretely, for $m,n\in\mathbb{Z}$ one considers the morphisms
\[
 f_{m,n}: C\times J\to J,\quad (x,y)\mapsto m(x-\alpha)+ny,
\]
so that
\[
 f_{m,n}^*\theta = gm^2\cdot \alpha_1+n^2\cdot \theta_2+mn\cdot Q
\]
“is nef on $C\times J$.” The text then states that “This essentially gives all the line bundles on the boundary of the nef cone,” yielding the boundary description needed for the cone statement in Theorem~\ref{first main thm}.

The post-theorem introduction explains that, to prove the main theorem, the proof proceeds by “construct[ing] all nef line bundles on $C\times J$ by geometric operations.” Concretely, for $m,n\in\mathbb{Z}$ one considers the morphisms
\[
 f_{m,n}: C\times J\to J,\quad (x,y)\mapsto m(x-\alpha)+ny,
\]
so that
\[
 f_{m,n}^*\theta = gm^2\cdot \alpha_1+n^2\cdot \theta_2+mn\cdot Q
\]
“is nef on $C\times J$.” The text then states that “This essentially gives all the line bundles on the boundary of the nef cone,” yielding the boundary description needed for the cone statement in the main theorem.

In establishing the main theorem, assume that $\rho(C\times J)=3$. Then
	\[
	\Amp(C\times J)=\BBig(C\times J)=\{a\cdot \alpha_1+b\cdot \theta_2+c\cdot Q:a>0,\ b> 0, \ ab> gc^2\}
	\]
and
	\[
	\Nef(C\times J)=\Psef(C\times J)=\{a\cdot \alpha_1+b\cdot \theta_2+c\cdot Q:a\geq0,\ b\geq 0, \ ab\geq gc^2\}.
	\]

Classification or Bijection
Biconditional or Equivalence

Question: Let $C$ be a smooth projective curve of genus $g>1$ over an algebraically closed field, and let $J$ be its Jacobian. Write $p_1:C\times J\to C$ and $p_2:C\times J\to J$ for the projections, let $\alpha_1=p_1^*\alpha$ for a divisor class $\alpha$ of degree $1$ on $C$, let $\theta_2=p_2^*\theta$ where $\theta$ is the principal polarization on $J$, and let $Q\in \mathrm{NS}(C\times J)_\mathbb{R}$ be the third class such that, when $\rho(C\times J)=3$, the classes $\alpha_1,\theta_2,Q$ span $\mathrm{NS}(C\times J)_\mathbb{R}$. Under the assumption $\rho(C\times J)=3$, which explicit description gives the ample cone and big cone, and also the nef cone and pseudo-effective cone, of $C\times J$?

$$\Amp(C\times J)=\BBig(C\times J)=\{a\alpha_1+b\theta_2+cQ:\ a>0,\ b>0,\ ab>gc^2\},$$
$$\Nef(C\times J)=\Psef(C\times J)=\{a\alpha_1+b\theta_2+cQ:\ a\ge 0,\ b\ge 0,\ ab\ge gc^2\}.$$

Choices:
B.
$$\Amp(C\times J)=\BBig(C\times J)=\{a\alpha_1+b\theta_2+cQ:\ a>0,\ b>0,\ ab\ge gc^2\},$$
$$\Nef(C\times J)=\Psef(C\times J)=\{a\alpha_1+b\theta_2+cQ:\ a\ge 0,\ b\ge 0,\ ab\ge gc^2\}.$$
C.
$$\Amp(C\times J)\subseteq\BBig(C\times J)=\{a\alpha_1+b\theta_2+cQ:\ a>0,\ b>0,\ ab>gc^2\},$$
$$\Nef(C\times J)=\Psef(C\times J)=\{a\alpha_1+b\theta_2+cQ:\ a\ge 0,\ b\ge 0,\ ab\ge gc^2\}.$$
D.
$$\Amp(C\times J)=\BBig(C\times J)=\{a\alpha_1+b\theta_2+cQ:\ a>0,\ b>0,\ ab>gc^2\},$$
$$\Nef(C\times J)=\Psef(C\times J)=\{a\alpha_1+b\theta_2+cQ:\ a\ge 0,\ b\ge 0,\ ab\ge g^2c^2\}.$$
E.
$$\Amp(C\times J)=\BBig(C\times J)=\{a\alpha_1+b\theta_2+cQ:\ a>0,\ b>0,\ ab>gc^2\},$$
$$\Nef(C\times J)=\Psef(C\times J)=\{a\alpha_1+b\theta_2+cQ:\ a\ge 0,\ b\ge 0,\ \text{and } a,b,c\in\mathbb{Q},\ ab\ge gc^2\}.$$

open-vs-closed boundary condition for ample/big cone

dropped the equality \(\Amp(C\times J)=\BBig(C\times J)\) to a one-sided inclusion

quadratic boundary coefficient from \(f_{m,n}^*\theta=gm^2\alpha_1+n^2\theta_2+mnQ\)

real cone coefficients replaced by rational coefficients


\label{thm-characterization}
Let $(M, \varphi)$ be a diffuse separable $\mathrm{W}^*$-probability space. The following assertions are equivalent:
\begin{enumerate}[{\rm (i)}]
\item $(M, \varphi)$ is selfless.

\item The first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (N, \psi)$ is existentially closed for some nontrivial $\mathrm{W}^*$-probability space $(N, \psi)$.

\item The first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (\operatorname{ L}(\mathbb{Z}), \tau_\mathbb{Z})$ is existentially closed, where $\tau_\mathbb{Z}$ is the canonical trace on $\operatorname{ L}(\mathbb{Z})$.

\item $\operatorname{B}(M, \varphi) = \mathbb{C} 1$.
\end{enumerate}

\begin{letterthm} \label{thm-characterization}
Let $(M, \varphi)$ be a diffuse separable $\mathrm{W}^*$-probability space. The following assertions are equivalent:
\begin{enumerate}[{\rm (i)}]
\item $(M, \varphi)$ is selfless.

\item The first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (N, \psi)$ is existentially closed for some nontrivial $\mathrm{W}^*$-probability space $(N, \psi)$.

\item The first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (\rL(\Z), \tau_\Z)$ is existentially closed, where $\tau_\Z$ is the canonical trace on $\rL(\Z)$.

\item $\rB(M, \varphi) = \C 1$.
\end{enumerate}
\end{letterthm}

In \cite{Ro23}, Robert introduced a new class of C$^*$-probability spaces, which he called \emph{selfless}, characterized by the existence of a copy of themselves in their ultrapower that is freely independent from the diagonal copy (thus being ``free from themselves"). This property quickly attracted the attention of numerous researchers as it implies many important regularity properties and is satisfied by a large class of examples (see \cite{AGKEP24, HKER25, RTV25, Vi25, Oz25}).

In this short note, we introduce a parallel notion of selfless W$^*$-probability space and we relate this notion to Connes'\! bicentralizer problem.

A W$^*$-{\em probability space} is a pair $(M, \varphi)$ that consists of a von Neumann algebra $M$ endowed with a faithful normal state $\varphi \in M_\ast$. For W$^*$-probability spaces $(M, \varphi)$ and $(N, \psi)$, we say that $(M, \varphi) \subset (N, \psi)$ is an {\em inclusion} of W$^*$-probability spaces if $M \subset N$ and if there exists a faithful normal conditional expectation $\operatorname{ E} : N \to M$ such that $\varphi\circ \operatorname{ E} = \psi$. In that case, $\operatorname{ E} : N \to M$ is the unique faithful normal conditional expectation such that $\varphi \circ \operatorname{ E} = \psi$.

Following \cite{GH21}, we say that an inclusion of W$^*$-probability spaces $(M, \varphi) \subset (N, \psi)$ is {\em existentially closed} if there exists a nonprincipal ultrafilter $\mathcal U$ on some directed set $I$ such that $(M, \varphi) \subset (N, \psi) \subset (M, \varphi)^{\mathcal U}$, where $(M, \varphi) \subset (M, \varphi)^{\mathcal U}$ is the diagonal inclusion. Note that if  $N$ is separable (i.e.\! $N$ has separable predual), then $\mathcal U$ can be chosen to be a nonprincipal ultrafilter on $\mathbb{N}$.

Adapting \cite[Definition 2.1]{Ro23} to the von Neumann algebraic realm, we say that a W$^*$-probability space $(M, \varphi)$ is {\em selfless} if the first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (M, \varphi)$ is existentially closed.

Popa's seminal work \cite{Po95} shows that $(M, \tau)$ is selfless for any separable type ${\rm II_1}$ factor $M$ endowed with its canonical trace $\tau$. Houdayer--Isono \cite{HI14} extended Popa's result by showing that $(M, \varphi)$ is selfless for any separable factor $M$ endowed with a faithful normal state $\varphi \in M_\ast$ for which $(M_\varphi)' \cap M = \C1$.

In this note, we show that a diffuse separable W$^*$-probability space $(M, \varphi)$ is selfless if and only if it has a trivial \emph{bicentralizer}. Recall from \cite{Co80, Ha85}, that the {\em bicentralizer} $\operatorname{B}(M, \varphi)$ of a W$^*$-probability space $(M, \varphi)$ is the set of all elements $x \in M$ that satisfy the following condition:
\begin{verse}
\noindent
For every $\varepsilon > 0$, there exists $\delta > 0$ such that for every $u \in \mathscr U(M)$, if $\|u \varphi - \varphi u\| < \delta$, then $\|u x - x u\|_\varphi < \varepsilon$.
\end{verse}

We then obtain the following characterization.

A W$^*$-{\em probability space} is a pair $(M, \varphi)$ that consists of a von Neumann algebra $M$ endowed with a faithful normal state $\varphi \in M_\ast$. For W$^*$-probability spaces $(M, \varphi)$ and $(N, \psi)$, we say that $(M, \varphi) \subset (N, \psi)$ is an {\em inclusion} of W$^*$-probability spaces if $M \subset N$ and if there exists a faithful normal conditional expectation $\operatorname{ E} : N \to M$ such that $\varphi\circ \operatorname{ E} = \psi$. In that case, $\operatorname{ E} : N \to M$ is the unique faithful normal conditional expectation such that $\varphi \circ \operatorname{ E} = \psi$.

Following \cite{GH21}, we say that an inclusion of W$^*$-probability spaces $(M, \varphi) \subset (N, \psi)$ is {\em existentially closed} if there exists a nonprincipal ultrafilter $\mathcal U$ on some directed set $I$ such that $(M, \varphi) \subset (N, \psi) \subset (M, \varphi)^{\mathcal U}$, where $(M, \varphi) \subset (M, \varphi)^{\mathcal U}$ is the diagonal inclusion. Note that if $N$ is separable (i.e.\! $N$ has separable predual), then $\mathcal U$ can be chosen to be a nonprincipal ultrafilter on $\mathbb{N}$.

Adapting \cite[Definition 2.1]{Ro23} to the von Neumann algebraic realm, we say that a W$^*$-probability space $(M, \varphi)$ is {\em selfless} if the first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (M, \varphi)$ is existentially closed.

Popa's seminal work \cite{Po95} shows that $(M, \tau)$ is selfless for any separable type ${\rm II_1}$ factor $M$ endowed with its canonical trace $\tau$. Houdayer--Isono \cite{HI14} extended Popa's result by showing that $(M, \varphi)$ is selfless for any separable factor $M$ endowed with a faithful normal state $\varphi \in M_\ast$ for which $(M_\varphi)' \cap M = \C1$.

In this note, we show that a diffuse separable W$^*$-probability space $(M, \varphi)$ is selfless if and only if it has a trivial \emph{bicentralizer}. Recall from \cite{Co80, Ha85}, that the {\em bicentralizer} $\operatorname{B}(M, \varphi)$ of a W$^*$-probability space $(M, \varphi)$ is the set of all elements $x \in M$ that satisfy the following condition:
\begin{verse}
\noindent
For every $\varepsilon > 0$, there exists $\delta > 0$ such that for every $u \in \mathscr U(M)$, if $\|u \varphi - \varphi u\| < \delta$, then $\|u x - x u\|_\varphi < \varepsilon$.
\end{verse}

We then obtain the following characterization.

We then obtain the following characterization.

\item The first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (N, \psi)$ is existentially closed for some nontrivial $\mathrm{W}^*$-probability space $(N, \psi)$.

\item The first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (\rL(\Z), \tau_\Z)$ is existentially closed, where $\tau_\Z$ is the canonical trace on $\rL(\Z)$.

We point out that the first three conditions above are analogous to the ones appearing in \cite[Theorem 2.6]{Ro23} for C$^*$-probability spaces.

Observe that if $\rB(M, \varphi) = \C1$, then $M$ must be a factor. Moreover, according to \cite{Ok21}, one and exactly one of the following assertions hold:
\begin{itemize}
\item $(M, \varphi)$ is a tracial factor of type ${\rm I}_n$ for $n \in \N^*$ or of type ${\rm II_1}$.
\item There exists $\lambda \in (0, 1)$ such that $(M, \varphi)$ is a type ${\rm III_\lambda}$ factor endowed with its $\frac{2\pi}{|\log(\lambda)|}$-periodic faithful normal state. In that case, we have $(M_\varphi)' \cap M = \C 1$.
\item $M$ is a type ${\rm III_1}$ factor. In that case, using \cite[Corollary 1.5]{Ha85}, we further have $\rB(M, \psi) = \C1$ for \emph{every} faithful normal state $\psi \in M_\ast$.
\end{itemize}

\begin{lettercor}\label{cor}
Let $M$ be a separable type ${\rm III_1}$ factor satisfying Connes'\! bicentralizer conjecture. Then for {\em every} faithful normal state $\varphi \in M_\ast$, the $\mathrm{W}^*$-probability space $(M, \varphi)$ is selfless.
\end{lettercor}

$(\rm ii) \Rightarrow (\rm iii)$ Let $(N, \psi)$ be a nontrivial W$^*$-probability space such that the first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (N, \psi)$ is existentially closed. We may assume that $N$ is separable. Choose a nonprincipal ultrafilter $\mathcal U$ on $\N$ such that we have $(M, \varphi) \subset (M, \varphi) \ast (N, \psi) \subset (M, \varphi)^{\mathcal U}$, where $(M, \varphi) \subset (M, \varphi)^{\mathcal U}$ is the diagonal inclusion. There are two cases to consider.

Secondly, assume that $N_\psi \neq \C 1$. Upon replacing $(N, \psi)$ by $(N_\psi, \psi)$, we may assume that $(N, \psi)$ is tracial. Reasoning as is the first case, we obtain
\begin{align*}
(M, \varphi) &\subset (M, \varphi) \ast (N, \psi)^{\ast 2} \\
&= \left( (M, \varphi) \ast (N, \psi) \right ) \ast (N, \psi) \\
&\subset (M, \varphi)^{\mathcal U} \ast (N, \psi) \\
&\subset (M, \varphi)^{\mathcal U} \ast (N, \psi)^{\mathcal U} \\
& \subset (M^{\mathcal U}, \varphi^{\mathcal U})^{\mathcal U} = (M, \varphi)^{\mathcal W}.
\end{align*}
By \cite[Lemma 2.5]{Ro23}, there exists $n \in \N$ large enough so that the iterated free product $(N, \psi)^{\ast n}$ is diffuse and so $(\rL(\Z), \tau_{\Z}) \subset (N, \psi)^{\ast n}$. Upon iterating $n$ times the ultraproduct construction and replacing $\mathcal U$ by the appropriate ultrafilter $\mathcal W = \mathcal U^{\otimes n}$, we obtain
$$(M, \varphi) \subset (M, \varphi) \ast (\rL(\Z), \tau_\Z) \subset (M, \varphi) \ast (N, \psi)^{\ast n} \subset (M, \varphi)^{\mathcal W},$$ where $(M, \varphi) \subset (M, \varphi)^{\mathcal W}$ is the diagonal inclusion. Therefore, the first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (\rL(\Z), \tau_\Z)$ is existentially closed.

We point out that the first three conditions above are analogous to the ones appearing in \cite[Theorem 2.6]{Ro23} for C$^*$-probability spaces.

The bicentralizer $\operatorname{B}(M, \varphi) \subset M$ is a von Neumann subalgebra such that $\operatorname{B}(M, \varphi) \subset (M_\varphi)' \cap M$ (see \cite[Proposition 1.3]{Ha85}). Thus Theorem \ref{thm-characterization} strengthens Houdayer--Isono's result \cite[Theorem A]{HI14}. In fact, the proof of Theorem \ref{thm-characterization} relies on \cite[Theorem A]{HI14} combined with a diagonal argument.

Observe that if $\operatorname{B}(M, \varphi) = \C1$, then $M$ must be a factor. Moreover, according to \cite{Ok21}, one and exactly one of the following assertions hold:
\begin{itemize}
\item $(M, \varphi)$ is a tracial factor of type ${\rm I}_n$ for $n \in \mathbb{N}^*$ or of type ${\rm II_1}$.
\item There exists $\lambda \in (0, 1)$ such that $(M, \varphi)$ is a type ${\rm III_\lambda}$ factor endowed with its $\frac{2\pi}{|\log(\lambda)|}$-periodic faithful normal state. In that case, we have $(M_\varphi)' \cap M = \mathbb{C} 1$.
\item $M$ is a type ${\rm III_1}$ factor. In that case, using \cite[Corollary 1.5]{Ha85}, we further have $\operatorname{B}(M, \psi) = \C1$ for \emph{every} faithful normal state $\psi \in M_\ast$.
\end{itemize}

In particular, Theorem \ref{thm-characterization} implies that no W$^*$-probability space of type ${\rm II_\infty}$ or type ${\rm III_0}$ can be selfless (this also follows by combining \cite[Theorem 3.5]{GH21} and \cite[Theorem 4.1]{Ue11}).

On the other hand, a famous conjecture of Connes, known as Connes'\! bicentralizer problem, claims that $\operatorname{B}(M,\varphi)=\mathbb{C} 1$ for \emph{every} type ${\rm III_1}$ factor $M$ and every faithful normal state $\varphi \in M_*$.  This conjecture has been verified for several families of type ${\rm III_1}$ factors such as amenable factors \cite{Ha85}, factors with a Cartan subalgebra, free products \cite{HU15}, semisolid factors \cite{HI15}, $q$-deformed Araki--Woods factors \cite{HI20, Bi24} and tensor products of type ${\rm III_1}$ factors \cite{Ma25}.

\begin{lettercor}\label{cor}
Let $M$ be a separable type ${\rm III_1}$ factor satisfying Connes'\! bicentralizer conjecture. Then for {\em every} faithful normal state $\varphi \in M_\ast$, the $\mathrm{W}^*$-probability space $(M, \varphi)$ is selfless.
\end{lettercor}

The post-theorem introduction explicitly says: ``the proof of Theorem \ref{thm-characterization} relies on \cite[Theorem A]{HI14} combined with a diagonal argument.'' No further steps or outline are provided there.

The post-theorem introduction explicitly says: ``the proof of Theorem \label{thm-characterization} relies on \cite[Theorem A]{HI14} combined with a diagonal argument.'' No further steps or outline are provided there.

Biconditional or Equivalence
Classification or Bijection

Question:
Let $(M,\varphi)$ be a diffuse separable $\mathrm{W}^*$-probability space. Call $(M,\varphi)$ selfless if the first factor inclusion
\[
(M,\varphi)\subset (M,\varphi)\ast (M,\varphi)
\]
is existentially closed, meaning that there exists a nonprincipal ultrafilter $\mathcal U$ such that
\[
(M,\varphi)\subset (M,\varphi)\ast (M,\varphi)\subset (M,\varphi)^{\mathcal U}
\]
with the last inclusion the diagonal one. The bicentralizer $\operatorname{B}(M,\varphi)$ is the set of all $x\in M$ such that for every $\varepsilon>0$ there exists $\delta>0$ with
\[
\|u\varphi-\varphi u\|<\delta \implies \|ux-xu\|_{\varphi}<\varepsilon
\]
for every unitary $u\in\mathscr U(M)$. Which statement is equivalent to $(M,\varphi)$ being selfless?

The bicentralizer is trivial: $\operatorname{B}(M,\varphi)=\mathbb C1$.

Choices:
B. There exists a nontrivial \mathrm{W}^*\text{-probability space }(N,\psi)\text{ such that for every nonprincipal ultrafilter }\mathcal U,\text{ one has }(M,\varphi)\subset (M,\varphi)\ast (N,\psi)\subset (M,\varphi)^{\mathcal U}\text{ with the last inclusion diagonal.}
C. There exists a nontrivial \mathrm{W}^*\text{-probability space }(N,\psi)\text{ such that the first factor inclusion }(M,\varphi)\subset (M,\varphi)\ast (N,\psi)\text{ is existentially closed.
D. The first factor inclusion \[(M,\varphi)\subset (M,\varphi)\ast (\operatorname{L}(\mathbb Z),\tau_{\mathbb Z})\] is existentially closed for every faithful normal state on \operatorname{L}(\mathbb Z).
E. The relative commutant of the centralizer is trivial: \((M_{\varphi})'\cap M=\mathbb C1\).

specific free-product witness (M,\varphi) with itself / bicentralizer conclusion dropped to mere existence of some nontrivial free factor

canonical trace on \operatorname{L}(\mathbb Z) replaced by arbitrary faithful normal state

bicentralizer condition conflated with trivial relative commutant of the centralizer


\label{thm:intro-keller}
    Let $X$ be a quasi-compact separated scheme over a base field $\mathbb{K}$. Then there exist canonical isomorphisms
    \begin{align*}
        & HH_\bullet\big(\Perf(X)\big) \to HH_\bullet(X),\\
        & HC_\bullet\big(\Perf(X)\big) \to HC_\bullet(X).
    \end{align*}

\begin{thm}\label{thm:intro-keller}
    Let $X$ be a quasi-compact separated scheme over a base field $\bbK$. Then there exist canonical isomorphisms
    \begin{align*}
        & HH_\bullet\big(\Perf(X)\big) \to HH_\bullet(X),\\
        & HC_\bullet\big(\Perf(X)\big) \to HC_\bullet(X).
    \end{align*}
\end{thm}

Let $X$ be a quasi-compact separated scheme over a base field $\mathbb{K}$. An important set of invariants associated with $X$ is its Hochschild homology $HH_\bullet(X)$ and cyclic homology $HC_\bullet(X)$. They are defined as the hyper-cohomology groups of the sheafification of the Hochschild complexes and cyclic chain complexes respectively.

On the other hand, let $\Perf(X)$ denote the dg category of perfect complexes over $X$. We emphasize that this is a dg-enhancement of the full subcategory of the bounded derived category $D^b(X)$ consisting of perfect complexes over $X$. For any small $\mathbb{K}$-linear dg category, we may also define purely algebraically its own Hochschild type invariants. In the case of $\Perf(X)$, we are naturally lead to the question if $HH_\bullet\big(\Perf(X)\big)$ (respectively $HC_\bullet\big(\Perf(X)\big)$) would be naturally isomorphic to the geometrically defined invariants $HH_\bullet(X)$ (respectively $HC_\bullet(X)$). In~\cite{keller1998cyclic}, Keller obtained the following result.

\begin{abstract}
Let $X$ be a quasi-compact separated scheme over a base field. Keller proved a theorem stating that the cyclic homology of $X$ is canonically isomorphic to the cyclic homology of the dg category $\Perf(X)$ consisting of perfect complexes over $X$. This theorem shows the categorical nature of the cyclic homology. In this note, we generalize Keller's theorem to allow $X$ be defined over a base commutative ring.
\end{abstract}

On the other hand, let $\Perf(X)$ denote the dg category of perfect complexes over $X$. We emphasize that this is a dg-enhancement of the full subcategory of the bounded derived category $D^b(X)$ consisting of perfect complexes over $X$. For any small $\bbK$-linear dg category, we may also define purely algebraically its own Hochschild type invariants. In the case of $\Perf(X)$, we are naturally lead to the question if $HH_\bullet\big(\Perf(X)\big)$ (respectively $HC_\bullet\big(\Perf(X)\big)$) would be naturally isomorphic to the geometrically defined invariants $HH_\bullet(X)$ (respectively $HC_\bullet(X)$). In~\cite{keller1998cyclic}, Keller obtained the following result.

In the study of logrithmic Hochschild homology~\cite{Olsson,HABLICSEK2026127}, one is often lead to consider Hochschild homology relative to a base space~\cite{OLSSON2003747}. This serves as at least one of the motivations to obtain a generalization of Keller's theorem above in the relative setting. More precisely, we prove the following

\begin{thm}
 Let $\mathfrak{X}$ be a quasi-compact separated scheme over a base commutative ring $R$. Then there are canonical isomorphisms
 \begin{align*}
 & HH_\bullet\big(\Perf(\mathfrak{X})/R\big) \to HH_\bullet(\mathfrak{X}/R),\\
 & HC_\bullet\big(\Perf(\mathfrak{X})/R\big) \to HC_\bullet(\mathfrak{X}/R).
 \end{align*}
\end{thm}

\subsection{Conventions and Notations.} We work over a base field $\mathbb{K}$. For a dg category $\CC$ over $\mathbb{K}$, denote by $\underline{\CC}$ its homotopy category. This is a category with the same underlying objects as $\CC$ and with
\[ {\sf Hom}_{\underline{\CC}}(X,Y):= H^0\big(\hom^\bullet_\CC(X,Y)\big).\]
For a scheme $X$, in Keller's original treatment~\cite{keller1998cyclic}, he uses the notion of localization pairs, i.e. a pair of dg categories $\big(\per(X),{\sf acyc}(X)\big)$ formed by the category of perfect complexes over $X$ and its full subcategory of acyclic perfect complexes. By a result of Drinfeld~\cite{drinfeld2004dg} on the construction of dg quotients of dg categories, we may first take the dg quotient $\per(X)/{\sf acyc}(X)$ as our definition of $\Perf(X)$. Due to this reason, in this paper we shall work exclusively with dg categories instead of localization pairs. In other words, in our setting, we have
\[ \underline{\Perf(X)} \cong \underline{\per(X)}/\underline{{\sf acyc}(X)}.\]
Thus no further localization is needed when working with $\Perf(X)$.

Let $X$ be a quasi-compact separated scheme over a base field $\bbK$. Keller considers a pair of dg categories $\big(\per(X),{\sf acyc}(X)\big)$ formed by the category of perfect complexes over $X$ and its full subcategory of acyclic perfect complexes. Then he defines the associated mixed complex as
\[ M\big(\per(X),{\sf acyc}(X)\big):={\sf cone}\big( M({\sf acyc}(X))\to M(\per(X))\big),\]
using the canonical inclusion functor. Throughout the paper, we shall denote the pair above by $\Perf(X):= \big(\per(X),{\sf acyc}(X)\big)$. By a result of Drinfeld~\cite{drinfeld2004dg} on the construction of dg quotients of dg categories, this choice of notation should cause no confusion. In other words, we may first take the dg quotient $\per(X)/{\sf acyc}(X)$ and then just take its associated mixed complex.

\begin{thm}\label{thm:main}
 Let $X$ be a quasi-compact separated scheme over a base field $\bbK$. Then trace map $\tau: M\big(\Perf(X)\big)\to R\Gamma(X,M\big(\cO)^\sharp)$ defined in Diagram~\eqref{diagram:tau} is invertible in $\DD Mix$.
\end{thm}

\begin{thm}\label{thm:main-relative}
 Let $\mathfrak{X}$ be a quasi-compact separated scheme over a base commutative ring $R$. Then trace map $\tau: M\big(\Perf(\mathfrak{X})\big)\to R\Gamma(\mathfrak{X},M\big(\cO)^\sharp)$ defined in Diagram~\eqref{diagram:tau} is invertible in $\DD Mix(R)$. Note that the mixed complex functor $M$ should be understood in the sense of Equation~\eqref{eq:mixed-ring}.
\end{thm}

In the study of logrithmic Hochschild homology~\cite{Olsson,HABLICSEK2026127}, one is often lead to consider Hochschild homology relative to a base space~\cite{OLSSON2003747}. This serves as at least one of the motivations to obtain a generalization of Keller's theorem above in the relative setting. More precisely, we prove the following

\begin{thm}
    Let $\mathfrak{X}$ be a quasi-compact separated scheme over a base commutative ring $R$. Then there are canonical isomorphisms
    \begin{align*}
        & HH_\bullet\big(\Perf(\mathfrak{X})/R\big) \to HH_\bullet(\mathfrak{X}/R),\\
        & HC_\bullet\big(\Perf(\mathfrak{X})/R\big) \to HC_\bullet(\mathfrak{X}/R).
    \end{align*}
\end{thm}

Note that in defining the Hochschild invariants above, when the structure sheaf $\cO_\mathfrak{X}$ is not flat over $R$, we need to replace it by a flat resolution. This homology theory is sometimes called Shukla-Hochschild homology~\cite{Shuk}. Similarly, such a replacement is also needed for the dg category $\Perf(\mathfrak{X})$. The proof of the relative case follows the same line as that of Theorem~\ref{thm:intro-keller}. Indeed, in Section~\ref{sec:review} we briefly sketch Keller's original proof. In Section~\ref{sec:relative} we deduce our main result by taking care of flatness using semi-free resolutions of dg categories.

\subsection{Conventions and Notations.} We work over a base field $\mathbb{K}$. For a dg category $\mathcal{C}$ over $\mathbb{K}$, denote by $\underline{\mathcal{C}}$ its homotopy category. This is a category with the same underlying objects as $\mathcal{C}$ and with
\[ {\sf Hom}_{\underline{\mathcal{C}}}(X,Y):= H^0\big(\hom^\bullet_\mathcal{C}(X,Y)\big).\]
For a scheme $X$, in Keller's original treatment~\cite{keller1998cyclic}, he uses the notion of localization pairs, i.e. a pair of dg categories $\big(\per(X),{\sf acyc}(X)\big)$ formed by the category of perfect complexes over $X$ and its full subcategory of acyclic perfect complexes. By a result of Drinfeld~\cite{drinfeld2004dg} on the construction of dg quotients of dg categories, we may first take the dg quotient $\per(X)/{\sf acyc}(X)$ as our definition of $\Perf(X)$. Due to this reason, in this paper we shall work exclusively with dg categories instead of localization pairs. In other words, in our setting, we have
\[ \underline{\Perf(X)} \cong \underline{\per(X)}/\underline{{\sf acyc}(X)}.\]
Thus no further localization is needed when working with $\Perf(X)$.

The post-theorem introduction does not sketch a proof of Theorem~\ref{thm:intro-keller} itself, but indicates the strategy for the relative generalization: it says that “The proof of the relative case follows the same line as that of Theorem~\ref{thm:intro-keller}.” It highlights the extra issue that “when the structure sheaf $\cO_\mathfrak{X}$ is not flat over $R$, we need to replace it by a flat resolution” (Shukla-Hochschild homology), and that “such a replacement is also needed for the dg category $\Perf(\mathfrak{X})$.” The plan of the argument is then: “in Section~\ref{sec:review} we briefly sketch Keller's original proof,” and “in Section~\ref{sec:relative} we deduce our main result by taking care of flatness using semi-free resolutions of dg categories.”

The post-theorem introduction does not sketch a proof of the main theorem itself, but indicates the strategy for the relative generalization: it says that “The proof of the relative case follows the same line as that of the main theorem.” It highlights the extra issue that “when the structure sheaf $\cO_\mathfrak{X}$ is not flat over $R$, we need to replace it by a flat resolution” (Shukla-Hochschild homology), and that “such a replacement is also needed for the dg category $\Perf(\mathfrak{X})$.” The plan of the argument is then: “next we briefly sketch Keller's original proof,” and “later we deduce our main result by taking care of flatness using semi-free resolutions of dg categories.”

Question: Let $X$ be a quasi-compact separated scheme over a base field $\mathbb{K}$, and let $\Perf(X)$ denote the dg category of perfect complexes on $X$. Which invariants are canonically isomorphic to the Hochschild homology $HH_\bullet(\Perf(X))$ and the cyclic homology $HC_\bullet(\Perf(X))$?

They are canonically isomorphic to the geometrically defined Hochschild and cyclic homology of $X$, respectively; that is, there are canonical isomorphisms $$HH_\bullet\big(\Perf(X)\big) \to HH_\bullet(X), \qquad HC_\bullet\big(\Perf(X)\big) \to HC_\bullet(X).$$

Choices:
B. They are canonically isomorphic to the geometrically defined Hochschild and cyclic homology of $X$ relative to the base field; that is, there are canonical isomorphisms $$HH_\bullet\big(\Perf(X)/\mathbb{K}\big) \to HH_\bullet(X/\mathbb{K}), \qquad HC_\bullet\big(\Perf(X)/\mathbb{K}\big) \to HC_\bullet(X/\mathbb{K}).$$
C. At least the Hochschild homology agrees with the geometrically defined Hochschild homology of $X$; namely, there is a canonical isomorphism $$HH_\bullet\big(\Perf(X)\big) \to HH_\bullet(X).$$
D. They are canonically isomorphic after passing to a flat or semi-free resolution of the structure sheaf and of the dg category; in particular, one has canonical isomorphisms only up to such resolutions, rather than directly for $\Perf(X)$ and $X$ themselves.
E. They are canonically isomorphic to the geometrically defined Hochschild and cyclic homology of $X$ for affine quasi-compact separated schemes, and more generally after imposing a flatness hypothesis on $\mathcal{O}_X$ over $\mathbb{K}$; namely, under those assumptions one has $$HH_\bullet\big(\Perf(X)\big) \to HH_\bullet(X), \qquad HC_\bullet\big(\Perf(X)\big) \to HC_\bullet(X).$$


Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\mathbb R[x_1,\dots,x_n],$ contains more simple modules than $\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\mathbb C$-algebra $A_{\mathbb R}$ such that $\simp(A_{\mathbb R})\cong\mathbb R^n.$ Thus the points in $\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\mathbb R}$-modules for which $\aspec(A_{\mathbb R})$ is a fine moduli, see the book \cite{S23} or the preprint \cite{S241}. For any real manifold $M$ we define an associative variety $(\mathcal M,\mathcal O^A_L)$ over $\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\mathcal M,$ and such that the charts $U$ in $\mathcal M$ corresponds to $\mathcal O^A_L(U)=A_{\mathbb R}.$

In short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$

Because of the discovery of localization in associative rings, \cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.
Let $k$ be a field and consider the polynomial algebra in $n\geq 1$ variables $$A=k[x_1,\dots,x_n]=k[\underline x].$$ For a point $P=(p_1,\dots,p_n)\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\dots,x_n-p_n).$$

\begin{abstract}
In the preprint \cite{S252} we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\oplus_{i=1}^rM_i$ a direct sum of $r\geq 1$ simple right $A$-modules. For a homomorphism of associative rings $A\rightarrow B$ we define the contraction of a simple $B$-module to $A.$
Then we define the set of aprime right $A$-modules $\aspec A$ to be the set of simple $A$-modules together with contractions of such. When $A$ is commutative, $\aspec A=\spec A,$ and we define a topology on $\aspec A$ such that when $A$ is commutative, this is the Zariski topology. In the preprint \cite{S251}, we proved that when we have a topology and a localizing subcategory, there exists a sheaf of associative rings $\mathcal O_X$ on $\aspec A,$ agreeing with the usual sheaf of rings on $\spec A.$ In this text, we write out this construction, and we see that we can restrict the sheaf and topology to any subset $V\subseteq\aspec A.$ In particular, this
proves that we can use complex varieties in real algebraic geometry, by restricting in accordance with $\mathbb R\subseteq\mathbb C.$ Thus the theory of schemes over algebraically closed fields and its associative generalization can be applied to real (algebraic) geometry.
\end{abstract}

In short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$

\begin{proof} From \cite{S23} we know that $$\ext^1_A(M_1,M_2)\simeq\der_k(A,\hmm_k(M_1,M_2))/\inner$$ where $\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\phi(m)=\phi(am), \phi a(m)=\phi(m)a.$

When $P\neq Q$ we can consider $M_{\underline 0}, M_P$ with $p_1\neq 0.$
For $$\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P)$$ we have $$\begin{aligned}\delta(x_ix_1)&=\delta(x_1x_i)\\
&\Updownarrow\\
\delta(x_i)x_1+x_i\delta(x_1)&=\delta(x_1)x_i+x_1\delta(x_i)\\
&\Updownarrow\\
\delta(x_i)p_1&=\delta(x_1)p_i\\
&\Updownarrow\\
\delta(x_i)&=\delta(x_i)\frac{p_i}{p_1}.
\end{aligned}$$
This is also true for the inner derivations, proving that $\dim_k\inner=1$ so that $\ext^1_A(M_P,M_Q)=0$ when $P\neq Q.$
\end{proof}

\begin{proof} We have a homomorphism $f:A\rightarrow A/\mathfrak m$ such for every $s\in A\setminus\mathfrak m,$ $f(s)$ is a unit. By the universal property of localization, there exists a unique homomorphism $\phi:A_\mathfrak m\rightarrow A/\mathfrak m$ such that $\phi(\frac{a}{s})=\iota(a)\iota(s)^{-1}.$ This homomorphism is clearly surjective, and its kernel is $\mathfrak m A_\mathfrak m$ giving the wanted isomorphism.
\end{proof}

\begin{proof} We send the prime ideal $\mathfrak p\subset A$ to $A_\mathfrak p/\mathfrak p A_\mathfrak p$ which is $A$-prime by definition. On the other hand, let
$M$ be an aprime $A$-module, defined by $\iota_M: A\rightarrow B$ such that $M$ is a simple $B$-module. Let $\mathfrak m\subset B$ be the maximal ideal defining $M$ as a simple $B$-module. Then $\iota_M^{-1}(\mathfrak m)$ is a prime ideal in $A.$ That these two operations are inverses to each other follows from the fact that for a maximal ideal in a ring $B$ we have $B/\mathfrak m\simeq B_{\mathfrak m}/\mathfrak m B_\mathfrak m$ as proven i Lemma \ref{anotherloclemma}.
\end{proof}

\begin{definition} Define the subset of $k$-points in $\mathbb X$ by $$\tilde{\mathbb X}(k)=\{x\in X\subseteq\mathbb X|x\text{ is simple}\}\subseteq\mathbb X.$$ Then the induced associative subscheme $\mathbb X(k)$ is called the associative subscheme of $k$-points.
\end{definition}

\begin{proof} Because $k\subseteq\Bbbk$ is a sub-algebra, it follows that if $\phi\otimes\id$ is an isomorphism, then $\dim_k V_1=\dim_k V_2,$ and that choosing corresponding bases, $$0\neq\det(\phi\otimes\id)=\det\phi.$$
\end{proof}

\begin{proof} From \cite{S23} we know that $$\ext^1_A(M_1,M_2)\simeq\der_k(A,\hmm_k(M_1,M_2))/\inner$$ where $\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\phi(m)=\phi(am), \phi a(m)=\phi(m)a.$

When $P=Q$ we can assume $M_1=M_2=k[\underline x]/(\underline x).$ Then any inner derivation is on the form $\ad_\phi$ for which $\ad_\phi(x_i)=[\phi,x_i]=0.$ Thus the inner derivations is of dimension zero and $\ext^1_A(M_1,M_2)\simeq\der_k(A,k)$ where
$\operatorname d_i(x_i)=1,\ i=1,\dots,n$ gives a basis for the derivations.

When $P\neq Q$ we can consider $M_{\underline 0}, M_P$ with $p_1\neq 0.$
For $$\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P)$$ we have $$\begin{aligned}\delta(x_ix_1)&=\delta(x_1x_i)\\
&\Updownarrow\\
\delta(x_i)x_1+x_i\delta(x_1)&=\delta(x_1)x_i+x_1\delta(x_i)\\
&\Updownarrow\\
\delta(x_i)p_1&=\delta(x_1)p_i\\
&\Updownarrow\\
\delta(x_i)&=\delta(x_i)\frac{p_i}{p_1}.
\end{aligned}$$
This is also true for the inner derivations, proving that $\dim_k\inner=1$ so that $\ext^1_A(M_P,M_Q)=0$ when $P\neq Q.$
\end{proof}

It follows from the lemma that not all finite dimensional simple modules over a noncommutative $k$-algebra can be classified by a finitely generated commutative algebra. See \cite{S23} for a lot of examples.

From \cite{S23}: \(\ext^1_A(M_1,M_2)\simeq \der_k(A,\hmm_k(M_1,M_2))/\inner\), with \(\hmm_k(M_1,M_2)\) an \(A\)-\(A\) bimodule via \(a\phi(m)=\phi(am)\), \(\phi a(m)=\phi(m)a\).

- Case \(P=Q\): assume \(M_1=M_2=k[\underline x]/(\underline x)\). Any inner derivation is \(\ad_\phi\) and \(\ad_\phi(x_i)=[\phi,x_i]=0\), so \(\dim_k\inner=0\). Hence \(\ext^1_A(M_1,M_2)\simeq \der_k(A,k)\), and the derivations \(\operatorname d_i\) with \(\operatorname d_i(x_i)=1\) for \(i=1,\dots,n\) give a basis, so the dimension is \(n\).

- Case \(P\neq Q\): consider \(M_{\underline 0}, M_P\) with \(p_1\neq 0\). For \(\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P))\), comparing \(\delta(x_ix_1)=\delta(x_1x_i)\) yields
\[\delta(x_i)x_1+x_i\delta(x_1)=\delta(x_1)x_i+x_1\delta(x_i)\Rightarrow \delta(x_i)p_1=\delta(x_1)p_i\Rightarrow \delta(x_i)=\delta(x_1)\frac{p_i}{p_1}.\]
This relation also holds for inner derivations, and it is concluded that \(\dim_k\inner=1\), so \(\ext^1_A(M_P,M_Q)=0\) when \(P\neq Q\).

For two points $P,Q\in k^n$ we have that $$\dim_k\ext^1_A(M_P,M_Q)=\begin{cases}n,\ P=Q\\0,\ P\neq Q.\end{cases}$$,

Question:
Let $k$ be a field, let $A=k[x_1,\dots,x_n]$ with $n\ge 1$, and for each point $P=(p_1,\dots,p_n)\in k^n$ define the simple $A$-module
$$M_P:=A/(x_1-p_1,\dots,x_n-p_n).$$
Which statement correctly describes $\dim_k\operatorname{Ext}^1_A(M_P,M_Q)$ for two points $P,Q\in k^n$?

$$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\0,& P\ne Q.\end{cases}$$

Choices:
B. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\1,& P\ne Q.\end{cases}$$
C. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=0\quad\text{for all }P\ne Q.$$
D. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n-1,& P=Q,\\0,& P\ne Q.\end{cases}$$
E. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\0,& P-Q\in (k^	imes)^n,\end{cases}$$


\label{th_homo_image} The relation   $\{\langle A,B \rangle  \colon A \leftarrow\mathrel{\mkern-14mu}\leftarrow B\}$  on the Borel space of countable   groups      is  a  complete   analytic quasi-order. Thus, the relation of bi-epimorphism on the same Borel space  is  a complete      analytic equivalence relation.

The set of groups with domain $\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\cite{gao}), one can attempt to determine the (Borel) complexity of equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\cite{friedman_and_stanley} proved that the isomorphism relation on countable groups is Borel bireducible with the isomorphism relation between countable graphs on $\omega$, using a construction of Mekler~\cite{mekler}.
 In fact they show that also isomorphism relation on the class of $2$-nilpotent group of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently,
 the fourth author of the present paper and S.\ Shelah~\cite{1205} proved that the same applies to countable torsion-free abelian groups. The isomorphism relation between finitely generated groups is a universal essentially countable Borel equivalence relation~\cite{veli}. In another direction, Calderoni and Thomas~\cite{calderoni} show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability: being an epimorphic image.

\smallskip

Throughout, when we say a structure is countable we mean that it has $\omega$ as its domain. Classes of structures we are interested in will always form a standard Borel space. For countable groups $A,B$, we write $A \leftarrow\mathrel{\mkern-14mu}\leftarrow B$ if there is a surjective homomorphism from $B$ onto $A$; equivalently, $A$ is isomorphic to a quotient of $B$. This relation is easily shown to be analytic. Our main result is that it has the maximum possible complexity. Camerlo et al. \cite{camerlo2} showed such a result for several classes of structures. Notably, the case of countable groups was left open.

\begin{abstract}
 We prove that the epimorphism relation is a complete analytic quasi-order on the space of countable groups.
 In the process we obtain the result of indepent interest showing that the epimorphism relation on pointed reflexive graph is complete.
\end{abstract}

The set of groups with domain $\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\cite{gao}), one can attempt to determine the (Borel) complexity of equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\cite{friedman_and_stanley} proved that the isomorphism relation on countable groups is Borel bireducible with the isomorphism relation between countable graphs on $\omega$, using a construction of Mekler~\cite{mekler}.
 In fact they show that also isomorphism relation on the class of $2$-nilpotent group of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently,
 the fourth author of the present paper and S.\ Shelah~\cite{1205} proved that the same applies to countable torsion-free abelian groups. The isomorphism relation between finitely generated groups is a universal essentially countable Borel equivalence relation~\cite{veli}. In another direction, Calderoni and Thomas~\cite{calderoni} show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability: being an epimorphic image.

\smallskip

Our reduction proceeds in two steps. In the first step, we use a result of Louveau and Rosendal \cite{LR} to show that the epimorphism relation on countable pointed reflexive graphs is analytic complete; here a pointed reflexive graph is one that has a distinguished vertex that is adajacent to each vertex. In the second step, we Borel reduce this quasi-order to the epimorphism relation on countable groups, using
 a new group theoretic construction based on certain countably generated Coxeter groups.

\begin{question} Is the epimorphism relation a complete-analytic quasi-order on the Borel space of countable {\em abelian} groups?
\end{question}

\begin{remark} \label{LRrem}
Louveau and Rosendal \cite[Theorem 3.5]{LR} proved that the homomorphism relation between countable graphs is a complete analytic quasi-order. They give a Borel reduction $T \to G_T$ of their complete quasi-order $\leq_{max} $ on normal trees (see \cite[Definition 2.3 and above]{LR}) to it. One observes from the proof that for each normal tree $T$, the constructed graph $G_T$ has not isolated vertex. As they remark at the end of their proof, if $S \le_{max} T$ then $G_S$ is in fact isomorphic to an induced subgraph of $G_T$, and so in particular $G_S\preceqc_1 G_T$. Thus, their proof shows that \emph{the relation $\preceqc_1$ on countable graphs with no isolated vertices is a complete analytic quasi-order.}
\end{remark}

\begin{proposition}\label{reduction}
 There is a Borel map $F$ from the class of countable graphs without isolated vertices to the class of countable pointed reflexive graphs, such that for any countable connected graphs $\Gamma$ and $\Delta$, we have
 $$\Gamma \preceqc_1 \Delta \iff F(\Gamma) \leftarrowdbl F(\Delta). $$
 Thus, the relation $\leftarrowdbl$ between countable pointed reflexive graphs is a complete analytic quasi-order using Remark~\ref{LRrem}.
\end{proposition}

\begin{fact}[{\cite[3.1.7]{krammer}}]\label{krammer_fact} Let $(G, S)$ be a Coxeter system and let $I, J \subseteq S$. Then $G_I = \langle I \rangle_G$ and $G_J = \langle J \rangle_G$ are conjugate in $G$ if and only if
$I$ and $J$ are in the same connected component of $\mathcal{K}_S$ (cf. \ref{def_K}).
\end{fact}

Our reduction proceeds in   two steps.  In the first step, we use a result of Louveau and Rosendal \cite{LR} to show that the epimorphism relation  on countable    pointed reflexive graphs  is analytic complete; here a pointed reflexive graph is one that has a distinguished vertex that is adajacent to each vertex.  In the second step, we Borel reduce this quasi-order to the epimorphism relation  on countable groups, using
	a new group theoretic construction based on certain countably generated Coxeter groups.

We leave open the following question.

	\begin{question}   Is      the   epimorphism relation  a complete-analytic quasi-order on  the Borel space of countable {\em abelian} groups?
\end{question}

Our reduction proceeds in two steps. In the first step, we use a result of Louveau and Rosendal \cite{LR} to show that the epimorphism relation on countable pointed reflexive graphs is analytic complete (where a pointed reflexive graph has a distinguished vertex adjacent to each vertex). In the second step, we Borel reduce this quasi-order to the epimorphism relation on countable groups, using a new group theoretic construction based on certain countably generated Coxeter groups.

Question: Let countable groups be groups with underlying set \(\omega\), viewed as a standard Borel space. For countable groups \(A,B\), write \(A \leftarrow\!\!\leftarrow B\) if there is a surjective homomorphism from \(B\) onto \(A\) (equivalently, \(A\) is isomorphic to a quotient of \(B\)). Which statement correctly describes the descriptive-set-theoretic complexity of this epimorphism relation and of the induced bi-epimorphism relation?

The relation \(\{\langle A,B\rangle : A \leftarrow\!\!\leftarrow B\}\) on the Borel space of countable groups is a complete analytic quasi-order. Consequently, the bi-epimorphism relation on the same space—defined by \(A \leftarrow\!\!\leftarrow B\) and \(B \leftarrow\!\!\leftarrow A\)—is a complete analytic equivalence relation.

Choices:
B. The relation \(\{\langle A,B\rangle : A \leftarrow\!\!\leftarrow B\}\) on the Borel space of countable groups is an analytic quasi-order, but it is not complete analytic; in fact its restriction to countable groups is Borel bireducible with graph isomorphism. Consequently, the bi-epimorphism relation on the same space is analytic but not a complete analytic equivalence relation.
C. The relation \(\{\langle A,B\rangle : A \leftarrow\!\!\leftarrow B\}\) on the Borel space of countable groups is an analytic quasi-order. Consequently, the bi-epimorphism relation on the same space—defined by \(A \leftarrow\!\!\leftarrow B\) and \(B \leftarrow\!\!\leftarrow A\)—is an analytic equivalence relation.
D. The relation \(\{\langle A,B\rangle : A \leftarrow\!\!\leftarrow B\}\) on the Borel space of countable groups is a complete analytic quasi-order only after restricting to finitely generated groups; on the full Borel space of countable groups it is merely analytic. Consequently, the bi-epimorphism relation on all countable groups is not complete analytic.
E. The relation \(\{\langle A,B\rangle : A \leftarrow\!\!\leftarrow B\}\) on the Borel space of countable groups is a complete analytic quasi-order, and moreover there is a Borel reduction witnessing completeness already from the epimorphism relation on arbitrary countable graphs directly to groups. Consequently, the bi-epimorphism relation on the same space is a complete analytic equivalence relation.


\begin{thm}\label{XiaFock}
		We have
		\begin{equation*}
			C^{*} (\mathcal{WL})=\mathcal{T}^{(1)}.
		\end{equation*}
	\end{thm}

For some \(\alpha>0\), \(p>0\) and \(dV\) the standard volume measure on \(\mathbb{C}^n\), let \(L^{p}_{\alpha}(\mathbb{C}^n,dV)\) be the Lebesgue space of measurable functions \(f\) on \(\mathbb{C}^n\) such that
	\begin{equation}\label{Fnorm1}
		\left\lVert f \right\rVert^{p}_{\alpha} := \left(\frac{p\alpha}{2\pi}\right)^{n} \int_{\mathbb{C}^n} |f(z)e^{-\frac{\alpha}{2}|z|^2}|^{p} dV(z) < \infty~.
	\end{equation}
	Similarly, for \(\alpha>0\) and \(p=\infty\), we denoted by \(L^{\infty}_{\alpha}\) the space of Lebesgue measurable function \(f\) on \(\mathbb{C}^n\) such that
	\begin{equation*}
		\left\lVert f \right\rVert_{\infty,\alpha} := \text{esssup}\{|f(z)|e^{-\frac{\alpha|z|^2}{2}} : ~z \in \mathbb{C}^n\}< \infty ~.
	\end{equation*}
	The classical Fock space \(F^{p}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \( L^{p}_{\alpha}(\mathbb{C}^n, dV)\). Similarly, the Fock space \(F^{\infty}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \(L^{\infty}_{\alpha}\).

	Let \(d\mu\) be the Gaussian measure on \(\mathbb{C}^n\), \(n\ge 1\). In terms of the standard volume measure \(dV\) on \(\mathbb{C}^n\), it is given by
	\begin{equation*}
		d\mu(z) = \pi^{-n} e^{-|z|^2} dV(z)~.
	\end{equation*}

	 The Fock space \(H^{2}(\mathbb{C}^n,d\mu)\) is defined to be the subspace of the (Hilbert-) Lebesgue space \(L^{2}(\mathbb{C}^n,d\mu)\) consisting of entire functions. Notice that \(H^{2}(\mathbb{C}^n,d\mu) = F^{2}_{1} \). The symbol \(K_z\) denotes the reproducing kernel and the symbol \(k_z\) denotes the normalized reproducing kernel for \(H^{2}(\mathbb{C}^n,d\mu)\). That is,
	\begin{equation*}
		K_z(\zeta) = e^{\langle \zeta, z \rangle}, \quad k_{z}(\zeta) = e^{\langle \zeta, z\rangle}e^{-\frac{|z|^2}{2}}~, \quad z,\zeta \in \mathbb{C}^n.
	\end{equation*}

	In \cite{xia2015localization}, J. Xia showed in the case of the Bergman space on the unit ball of \(\mathbb{C}^n\) that the norm closure of \(\{T_{f}: f \in L^{\infty}(B,dv) \}\) coincides with the \(C^{*}\)-algebra of weakly localized operators. Also, he stated in \cite[Section 4 ]{xia2015localization}  that  the analogue of \cite[Theorem 1.5]{xia2015localization} on the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\) was true. In this paper, we define the notion of weakly localized operators, state Xia's theorem for the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\) and provide details of its proof. Further, we present a consequence of this theorem on the compactness of operators on \(H^{2}(\mathbb{C}^n,d\mu)\).  We begin with the following definitions and we state the main theorem, the proof of which will retain our attention in the following sections.

	\begin{defn}
		For \(f \in L^{\infty}(\mathbb{C}^n,dV) \), the \textbf{Toeplitz operator} \(T_{f}\) is defined by the formula
		\begin{equation*}
			T_{f}h = P(fh)~,\quad h \in H^{2}(\mathbb{C}^n,d\mu)~,
		\end{equation*}
		where \(P \colon L^{2}(\mathbb{C}^n,d\mu) \rightarrow H^{2}(\mathbb{C}^n,d\mu)\) is the orthogonal projection.
	\end{defn}

	The \textbf{standard lattice} in \(\mathbb{C}^n\) is denoted by
	\begin{equation*}
		\mathbb{Z}^{2n} = \{(m_{1}+il_{1}, \dots, m_{n}+il_{n}): m_{1}, l_{1}, \dots, m_{n},l_{n} \in \mathbb{Z}\}~.
	\end{equation*}
	We fix an orthonormal set \(\{e_{u}: u \in \mathbb{Z}^{2n}\}\) in \(H^{2}(\mathbb{C}^n,d\mu)\). We let \(S\) denote the \textbf{fundamental unit cube in }\(\mathbb{C}^{n}\). That is,
	\begin{equation*}
		S = \{(x_{1}+iy_{1}, \dots, x_{n}+iy_{n}) : x_{1},y_{1}, \dots , x_{n},y_{n} \in [0,1)\}.
	\end{equation*}
	With \(\mathbb{Z}^{2n}\) and \(S\), we have
	\begin{equation*}
		\cup_{u \in \mathbb{Z}^{2n}} \{S+u\} = \mathbb{C}^{n} = \cup_{u \in \mathbb{Z}^{2n}} \{u-S\} ~,
	\end{equation*}
	which is a tiling of the space, meaning that there is no overlap between \(S+u\) and \(S+v\) for \(u \neq v\) in \(\mathbb{Z}^{2n}\) (resp. between \(u-S\) and \(v-S\) for \(u \neq v \in \mathbb{Z}^{2n}\) ).

	\begin{defn}
		Let \(\mathcal{T}^{(1)}\) denote the norm closure of \(\{T_{f}: f\in L^{\infty}(\mathbb{C}^n,dV)\}\) in \(\mathcal{B}(H^2(\mathbb{C}^n,d\mu))\) with respect to the operator norm. That is
		\begin{equation*}
			\mathcal{T}^{(1)} = \{B : \lim_{k \to \infty } \norm{B-T_{b_{k}}}=0 , b_{k} \in L^{\infty}(\mathbb{C}^{n},dV)\}.
		\end{equation*}
	\end{defn}

	\begin{defn}
		We denote by \(\mathcal{S}\) the linear span of the normalized reproducing kernels \(k_z\). A linear operator \(B : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) is said to be \textbf{admissible} on \( H^{2}(\mathbb{C}^n,d\mu) \) if there exists a linear operator \( B^{*} : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) such that the duality relation
		\begin{equation}\label{R1}
			\langle Bk_{z}, k_{w} \rangle = \langle k_{z}, B^{*}k_w \rangle
		\end{equation}
		holds for all \(z,w \in \mathbb{C}^n\).
	\end{defn}
	The inner product here is with respect to \(d\mu\).

	We define below sufficiently localized operators following J. Xia and D. Zheng (XZ) in \cite{xia2013localization}.
	\begin{defn}
		A bounded linear operator \(B\) on \(H^{2}(\mathbb{C}^n,d\mu)\) is said to be XZ-\textbf{sufficiently localized} if there exist constants \(2n<\beta<\infty\) and \(0<C<\infty\) such that
		\begin{equation}\label{SL1}
			|\langle Bk_{z},k_{w}\rangle| \le \frac{C}{(1+|z-w|)^{\beta}}
		\end{equation}
		for all \(z,w\in \mathbb{C}^n\).
	\end{defn}

		\begin{defn}
		An admissible operator \(B\) on \(H^{2}(\mathbb{C}^n,d\mu)\) is said to be \textbf{weakly localized} if it satisfies the following four conditions
		\begin{equation*}
			\sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle Bk_{z}, k_{w} \rangle| dV(w) <\infty, \quad \sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle B^{*} k_{z}, k_{w} \rangle| dV(w) <\infty
		\end{equation*}

		and
		\begin{equation*}
			\lim_{r\to\infty}\sup_{z\in \mathbb{C}^n}\int_{|z-w|\ge r} |\langle Bk_{z}, k_{w} \rangle| dV(w) =0, \quad \lim_{r\to\infty}\sup_{z\in \mathbb{C}^n}\int_{|z-w|\ge r} |\langle B^{*} k_{z}, k_{w} \rangle| dV(w) =0~.
		\end{equation*}
	\end{defn}

	\begin{exa}\label{Ex1}
		If \(f\) is a \textbf{bounded measurable function} on \(\mathbb{C}^n\), then there is a positive constant \(0<C=C(f)<\infty\) such that
		\begin{equation*}
			|\langle T_{f}k_{z}, k_{w}\rangle| \le C e^{-(1/8) |z-w|^2}
		\end{equation*}
		for all \(z,w \in \mathbb{C}^n\). That is, \(T_{f} \in \text{XZ-}\mathcal{SL}\). Also, \(T_{f} \in \mathcal{L}_{p}\) for \(2<p<\frac{8}{3}\).
	\end{exa}

	\begin{proof}
		For each \(z, w, \xi \in \mathbb{C}^n\), we have
		\begin{equation*}
			|k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = |e^{\langle \xi , z \rangle - \frac{|z|^2}{2} } e^{\langle \xi, w \rangle - \frac{|w|^2}{2}} | e^{-|\xi|^2} = e^{-\frac{1}{2}\left(|z-\xi|^2 + |w-\xi|^2\right)} ~.
		\end{equation*}
		By the triangular inequality we have
		\begin{equation*}
			|z-w|^2 \le |z-\xi + \xi -w|^2 \le (|z-\xi| + |w-\xi|)^2 \le 2 (|z-\xi|^2 + |w-\xi|^2)~.
		\end{equation*}
		It follows that
		\begin{eqnarray}\label{kzw}
			|k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)}e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} \le e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} e^{-\frac{1}{8}|z-w|^2}~.
		\end{eqnarray}
		Let \(f\) be a bounded measurable function on \(\mathbb{C}^n\), then for all \(g,h\in H^{2}(\mathbb{C}^n,d\mu)\), it holds
		\begin{equation}\label{Toepfgh}
			\langle T_{f}g, h \rangle = \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
		\end{equation}
		In fact,
		\begin{eqnarray*}
			\langle T_{f}g, h \rangle &=&  \int_{\mathbb{C}^{n}} T_{f}g(w)\overline{h(w)} d\mu(w) \\
			&=& \int_{\mathbb{C}^{n}}\int_{\mathbb{C}^{n}} K(w,\xi) f(\xi)g(\xi) d\mu(\xi) \overline{h(w)} d\mu(w) \\
			&=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \int_{\mathbb{C}^{n}} K_{\xi}(w) \overline{h(w)} d\mu(w) d\mu(\xi) \\
			&=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \langle K_{\xi} , h \rangle d\mu(\xi) = \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \overline{\langle  h, K_{\xi}  \rangle} d\mu(\xi) \\
			&=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
		\end{eqnarray*}
		Therefore,
		\begin{eqnarray*}
			|\langle T_{f}k_{z}, k_{w} \rangle| &=& \left|\int_{\mathbb{C}^n}
			f(\xi)k_{z}(\xi) \overline{k_{w}(\xi)} d\mu(\xi)\right|  \\
			&\le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} \int_{\mathbb{C}^n} |k_{z}(\xi) k_{w}(\xi)| e^{-|\xi|^2} dV(\xi) \\
			& \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n}  e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)}  dV(\xi) \\
			& \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n}  e^{-\frac{1}{8}|z-w|^2} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|z-\xi|^2} dV(\xi) \right)^{\frac{1}{2}} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|w-\xi|^2}dV(\xi) \right)^{\frac{1}{2}} \\
			&=& \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n}  e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{2}|\zeta|^2} dV(\zeta) \\
			&=& (\sqrt{2})^n \left\lVert f \right\rVert_{\infty} e^{-\frac{1}{8}|z-w|^2}~.
		\end{eqnarray*}
		This completes the proof.
	\end{proof}

	 \begin{pro}
	 	Any XZ-sufficiently localized operator is weakly localized.
	 \end{pro}
	\begin{proof}
	Let \(B\) be a XZ-sufficiently localized operator, then  for all \(z\in \mathbb{C}^n\), we have
	\begin{eqnarray*}
		\int_{\mathbb{C}^n} |\langle Bk_z, k_w \rangle| dV(w) &\le& \frac{C}{\pi^n} \int_{\mathbb{C}^n} \frac{1}{(1+|\zeta|)^{\beta}} dV(\zeta) \\
		&=& \frac{Cc_n}{\pi^n} \int_{0}^{\infty} \frac{r^{2n-1}}{(1+r)^{\beta}} dr \\
		&=& C_n \left[ \int_{0}^{1} \frac{r^{2n-1}}{(1+r)^{\beta}} dr  + \int_{1}^{\infty } \frac{r^{2n-1}}{(1+r)^{\beta}} dr\right] \\
		&\le & C_n \left[ \int_{0}^{1} \frac{1}{(1+r)^{\beta}} dr +  \int_{1}^{\infty } \frac{r^{2n-1}}{r^{\beta}} dr\right] \\
		&=& C_n \left\{\frac{1}{\beta-1}\left(1-\frac{1}{2^{\beta}-1}\right) + \frac{1}{\beta-2n}\right\} \\
		&:=& C(n,\beta)~,
	\end{eqnarray*}
	which does not depend on \(z\). Therefore,
	\begin{equation*}
		\sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle Bk_z, k_w \rangle| dV(w) \le C(n,\beta)  <\infty~.
	\end{equation*}
	And in the same way, we also have
	\begin{equation*}
		\sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle B^*k_z, k_w \rangle| dV(w) \le C(n,\beta) <\infty~.
	\end{equation*}
	We now show that \(B\) satisfies the third condition of weakly localized operator.

	Let \(z\in \mathbb{C}^n\), using the change of variables \(\zeta=z-w\) and the spherical coordinates, it holds
	\begin{eqnarray*}
		\int_{|z-w|\ge r} |\langle Bk_z, k_w \rangle dV(w) &\le& C \int_{|z-w|\ge r} \frac{1}{(1+|z-w|)^{\beta}} dV(w) \\
		&=& C \int_{|\zeta|\ge r} \frac{1}{(1+|\zeta|)^{\beta}} dV(\zeta) \\
		&=& C c_n \int_{\rho\ge r} \frac{\rho^{2n-1}}{(1+\rho)^{\beta}} d \rho \\
		&\le& C c_n \int_{\rho\ge r} \frac{\rho^{2n-1}}{\rho^{\beta}} d \rho =  C c_n \int_{\rho\ge r} \frac{1}{\rho^{\beta-2n+1}} d \rho \\
		&=& \frac{Cc_n}{\beta-2n}\frac{1}{r^{\beta-2n}}~.
	\end{eqnarray*}
	It follows that,
	\begin{equation*}
		\lim_{r\to\infty} \sup_{z\in \mathbb{C}^n} \int_{|z-w|\ge r} |\langle Bk_z, k_w \rangle dV(w) \le \frac{Cc_n}{\beta-2n} \lim_{r\to\infty} \frac{1}{r^{\beta-2n}} =0~.
	\end{equation*}
	Using the duality relation (\ref{R1}) of an admissible operator and similarly like with \(B\), we also have:
	\begin{equation*}
		\lim_{r\to\infty} \sup_{z\in \mathbb{C}^n} \int_{|z-w|\ge r} |\langle B^*k_z, k_w \rangle dV(w) \le \frac{Cc_n}{\beta-2n} \lim_{r\to\infty} \frac{1}{r^{\beta-2n}} =0~.
	\end{equation*}
\end{proof}

	We denote by \(\mathcal{WL}\) the collection of weakly localized operators on \(H^{2}(\mathbb{C}^n,d\mu)\). We recall the following definition from \cite{zhu2007}.

		\begin{defn}
		A \textbf{Banach algebra} is a complex algebra together with a complete norm satisfying the condition \(\left\lVert xy \right\rVert \le \left\lVert x \right\rVert\left\lVert y \right\rVert\).
		A \(C^{*}\)\textbf{-algebra} is a Banach algebra \(\mathcal{A}\) with an involution \(x \mapsto x^{*}\) on it satisfying the following conditions:
		\begin{enumerate}[label=$\blacktriangleright$]
			\item \(x^{**}=x\) for all vectors \(x \in \mathcal{A} \).
			\item \((ax + by)^{*} = \bar{a}x^{*} + \bar{b}y^{*} \) for all vectors \(x,y \in \mathcal{A}\) and \(a,b \in \mathbb{C}\).
			\item \((xy)^{*}=y^{*}x^{*}\) for all vectors \(x,y \in \mathcal{A}\).
			\item \(\norm{xx^{*}} = \left\lVert x \right\rVert^{2}\) for all vector \(x \in \mathcal{A}\).
		\end{enumerate}
	\end{defn}

	\begin{defn}
		A Banach algebra \(\mathcal{A}\) is called a \textbf{\(\star\)-algebra} if for every \(A\in \mathcal{A}\), we have \(A^{*} \in \mathcal{A} \).
	\end{defn}

	\begin{defn}
		We denote by \(C^{*}(\mathcal{WL})\) the \(C^{*}\)-algebra generated by weakly localized operators on \(H^{2}(\mathbb{C}^n, d\mu)\). Also, \(C^{*}(\mathcal{WL})\) is actually the norm closure of \(\mathcal{WL}\) since \(\mathcal{WL} \) is a \(\star\)-algebra.
	\end{defn}

	We will prove the following main result.

For some \(\alpha>0\), \(p>0\) and \(dV\) the standard volume measure on \(\mathbb{C}^n\), let \(L^{p}_{\alpha}(\mathbb{C}^n,dV)\) be the Lebesgue space of measurable functions \(f\) on \(\mathbb{C}^n\) such that
 \begin{equation}\label{Fnorm1}
 \left\lVert f \right\rVert^{p}_{\alpha} := \left(\frac{p\alpha}{2\pi}\right)^{n} \int_{\mathbb{C}^n} |f(z)e^{-\frac{\alpha}{2}|z|^2}|^{p} dV(z) < \infty~.
 \end{equation}
 Similarly, for \(\alpha>0\) and \(p=\infty\), we denoted by \(L^{\infty}_{\alpha}\) the space of Lebesgue measurable function \(f\) on \(\mathbb{C}^n\) such that
 \begin{equation*}
 \left\lVert f \right\rVert_{\infty,\alpha} := \text{esssup}\{|f(z)|e^{-\frac{\alpha|z|^2}{2}} : ~z \in \mathbb{C}^n\}< \infty ~.
 \end{equation*}
 The classical Fock space \(F^{p}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \( L^{p}_{\alpha}(\mathbb{C}^n, dV)\). Similarly, the Fock space \(F^{\infty}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \(L^{\infty}_{\alpha}\).

\begin{defn}
 We denote by \(\mathcal{S}\) the linear span of the normalized reproducing kernels \(k_z\). A linear operator \(B : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) is said to be \textbf{admissible} on \( H^{2}(\mathbb{C}^n,d\mu) \) if there exists a linear operator \( B^{*} : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) such that the duality relation
 \begin{equation}\label{R1}
 \langle Bk_{z}, k_{w} \rangle = \langle k_{z}, B^{*}k_w \rangle
 \end{equation}
 holds for all \(z,w \in \mathbb{C}^n\).
 \end{defn}
 The inner product here is with respect to \(d\mu\).

\begin{exa}\label{Ex1}
 If \(f\) is a \textbf{bounded measurable function} on \(\mathbb{C}^n\), then there is a positive constant \(0<C=C(f)<\infty\) such that
 \begin{equation*}
 |\langle T_{f}k_{z}, k_{w}\rangle| \le C e^{-(1/8) |z-w|^2}
 \end{equation*}
 for all \(z,w \in \mathbb{C}^n\). That is, \(T_{f} \in \text{XZ-}\mathcal{SL}\). Also, \(T_{f} \in \mathcal{L}_{p}\) for \(2<p<\frac{8}{3}\).
 \end{exa}

\begin{proof}
 For each \(z, w, \xi \in \mathbb{C}^n\), we have
 \begin{equation*}
 |k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = |e^{\langle \xi , z \rangle - \frac{|z|^2}{2} } e^{\langle \xi, w \rangle - \frac{|w|^2}{2}} | e^{-|\xi|^2} = e^{-\frac{1}{2}\left(|z-\xi|^2 + |w-\xi|^2\right)} ~.
 \end{equation*}
 By the triangular inequality we have
 \begin{equation*}
 |z-w|^2 \le |z-\xi + \xi -w|^2 \le (|z-\xi| + |w-\xi|)^2 \le 2 (|z-\xi|^2 + |w-\xi|^2)~.
 \end{equation*}
 It follows that
 \begin{eqnarray}\label{kzw}
 |k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)}e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} \le e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} e^{-\frac{1}{8}|z-w|^2}~.
 \end{eqnarray}
 Let \(f\) be a bounded measurable function on \(\mathbb{C}^n\), then for all \(g,h\in H^{2}(\mathbb{C}^n,d\mu)\), it holds
 \begin{equation}\label{Toepfgh}
 \langle T_{f}g, h \rangle = \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
 \end{equation}
 In fact,
 \begin{eqnarray*}
 \langle T_{f}g, h \rangle &=& \int_{\mathbb{C}^{n}} T_{f}g(w)\overline{h(w)} d\mu(w) \\
 &=& \int_{\mathbb{C}^{n}}\int_{\mathbb{C}^{n}} K(w,\xi) f(\xi)g(\xi) d\mu(\xi) \overline{h(w)} d\mu(w) \\
 &=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \int_{\mathbb{C}^{n}} K_{\xi}(w) \overline{h(w)} d\mu(w) d\mu(\xi) \\
 &=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \langle K_{\xi} , h \rangle d\mu(\xi) = \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \overline{\langle h, K_{\xi} \rangle} d\mu(\xi) \\
 &=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
 \end{eqnarray*}
 Therefore,
 \begin{eqnarray*}
 |\langle T_{f}k_{z}, k_{w} \rangle| &=& \left|\int_{\mathbb{C}^n}
 f(\xi)k_{z}(\xi) \overline{k_{w}(\xi)} d\mu(\xi)\right| \\
 &\le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} \int_{\mathbb{C}^n} |k_{z}(\xi) k_{w}(\xi)| e^{-|\xi|^2} dV(\xi) \\
 & \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} dV(\xi) \\
 & \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} e^{-\frac{1}{8}|z-w|^2} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|z-\xi|^2} dV(\xi) \right)^{\frac{1}{2}} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|w-\xi|^2}dV(\xi) \right)^{\frac{1}{2}} \\
 &=& \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{2}|\zeta|^2} dV(\zeta) \\
 &=& (\sqrt{2})^n \left\lVert f \right\rVert_{\infty} e^{-\frac{1}{8}|z-w|^2}~.
 \end{eqnarray*}
 This completes the proof.
 \end{proof}

\begin{defn}
 We denote by \(C^{*}(\mathcal{WL})\) the \(C^{*}\)-algebra generated by weakly localized operators on \(H^{2}(\mathbb{C}^n, d\mu)\). Also, \(C^{*}(\mathcal{WL})\) is actually the norm closure of \(\mathcal{WL}\) since \(\mathcal{WL} \) is a \(\star\)-algebra.
 \end{defn}

We will prove the following main result.

\begin{defn}
 We denote by \(C^{*}(\mathcal{WL})\) the \(C^{*}\)-algebra generated by weakly localized operators on \(H^{2}(\C^n, d\mu)\). Also, \(C^{*}(\mathcal{WL})\) is actually the norm closure of \(\mathcal{WL}\) since \(\mathcal{WL} \) is a \(\star\)-algebra.
 \end{defn}

The organization of this paper is as follows. In Section 2, we will give propositions in the case of the Fock space \(H^{2}(\C^n, d\mu)\) which are the analogue of those given by Xia in \cite{xia2015localization} in the Bergman space case of the unit ball. Later, using these propositions in Section 3, we establish the proof of Theorem \ref{XiaFock} and present a consequence.

\begin{defn}\label{normstar}
 For an entire function \(h\) in \(\C^{n}\), we write
 \begin{equation*}
 \norm{h}_{*} = \left(\int_{\C^{n}} |h(\zeta)|^{2} e^{-\frac{1}{2}|\zeta|^2} dV(\zeta)\right)^{\frac{1}{2}}~.
 \end{equation*}
 We denote by \(\mathcal{H}_{*}\) the collection of entire functions \(h\) on \(\C^n\) satisfying \(\norm{h}_{*}<\infty\).
 \end{defn}
 \begin{rem}
 The norm \(\norm{\cdot}_{*}\) is equivalent to the norm on the Fock space \(F^{2}_{1/2}\) given in relation (\ref{Fnorm1}) which is an Hilbert space. More precisely, \(\norm{\cdot}_{*} = (2\pi)^{n/2} \norm{\cdot}_{F^{2}_{1/2}}\). This ensures the continuity of \(\norm{\cdot}_{*}\).
 \end{rem}
 In what follows, we will use the operator \(U_{z}\) defined by
 \begin{equation*}
 U_{z}f(w) = f(z-w) k_{z}(w)~, \quad f \in H^{2}(\C^n, d\mu)~.
 \end{equation*}

\begin{pro}\label{proYz}
 Suppose \(\{c_{u}: u \in \Z^{2n}\}\) are complex numbers satisfying the condition \mbox{\(\sup_{u\in \Z^{2n}}|c_{u}| < \infty\)}. Then for each \(z\in \C^{n}\), the operator
 \begin{equation}\label{Yz}
 Y_{z} = \sum_{u \in \Z^{2n}} c_{u} k_{u-z} \otimes k_{u-z}
 \end{equation}
 is bounded on \(H^{2}(\C^n, d\mu)\). Moreover, the map \(z \mapsto Y_{z}\) from \(\C^{n}\) to \(\mathcal{B}(H^{2}(\C^{n},d\mu))\) is continuous with respect to the operator norm.
 \end{pro}
 \begin{proof}
 Let \(\{c_{u} : u \in \Z^{2n}\}\) be complex numbers such that \mbox{\(\sup_{u\in \Z^{2n}}|c_{u}| < \infty\)}. For \(z\in \C^{n}\), we define
 \begin{equation*}
 A_{z} = \sum_{u \in \Z^{2n}} c_{u} (U_{u}k_{z}) \otimes e_{u} \quad \text{ and } \quad B_{z} = \sum_{u \in \Z^{2n}}(U_{u}k_{z})\otimes e_{u}~.
 \end{equation*}
 Using the relation (\ref{Uu2}), we have \(Y_{z} = A_{z}B^{*}_{z}\). Set \(h_{u} = c_{u}k_{z}\), then using Definition \ref{normstar}, we have
 \begin{equation*}
 \norm{h_{u}}_{*} = |c_{u}|\norm{k_{z}}_{*} = (2\pi)^{n/2} e^{\frac{|z|^2}{2}} |c_{u}|~.
 \end{equation*}
 Therefore, \(\sup_{u\in \Z^{2n}} \norm{h_{u}}_{*} < \infty\). Applying Lemma \ref{LemUh} to \(h_{u} = c_{u}k_{z}\), we see that \(\norm{A_{z}} \le C \sup_{u\in \Z^{2n}}|c_{u}|\). Thus each \(A_{z}\) is bounded. Since \(B_z\) is just a special case of \(A_{z}\) (with \(c_{u}=1\) for all \(u \in \Z^{2n}\)), then it is also bounded and hence \(Y_{z} = A_{z}B^{*}_{z}\) is bounded.

\begin{lem}\label{Lim=0}
 Let \(B \in \mathcal{WL} \), then for every \(z,w \in \C^n \), we have
 \begin{equation*}
 \lim_{R\to \infty} \sup_{u\in \Z^{2n}} \sum_{\underset{|u-v|>R}{v\in \Z^{2n} }} |\langle B k_{u-z}, k_{v-w} \rangle| =0 \quad \text{ and } \quad \lim_{R\to \infty} \sup_{u\in \Z^{2n}} \sum_{\underset{|u-v|>R}{v\in \Z^{2n} }} |\langle k_{u-z}, Bk_{v-w}\rangle| = 0~.
 \end{equation*}
 \end{lem}
 \begin{proof}
 By \cite[Lemma 2.32]{zhu2012}, for any entire function \(f\) on \(\C^n\), we have
 \begin{equation*}
 \left|f(z)e^{-\frac{\alpha}{2}|z|^2}\right|^p \le C \int_{B(z,\delta)} |f(w)e^{-\frac{\alpha}{2}|w|^2}|^p dV(w) \quad \text{ for } z \in \C^n.
 \end{equation*}
 Hence for \(\alpha=p=1\) and \(\delta\) small such that the balls \(\{B(v-w,\delta): v \in \Z^{2n} \}\) are mutually disjoint, we have
 \begin{equation*}
 |\langle B k_{u-z}, k_{v-w} \rangle| = |Bk_{u-z} (v-w) |e^{-\frac{|v-w|^2}{2}} \le C \int_{B(v-w,\delta)} |Bk_{u-z}(\zeta) | e^{-\frac{|\zeta|^2}{2}} dV(\zeta).
 \end{equation*}
 Indeed, for \(\delta < \frac{1}{2} \), the balls \(\{B(v-w,\delta) : v \in \Z^{2n} \}\) are mutually disjoint. Otherwise, there would exist \(v,v' \in \Z^{2n} \) such that \(v \neq v' \), and a point \(\xi\) such that \(\xi \in B(v-w,\delta) \cap B(v'-w,\delta) \). In other words:
 \begin{equation*}
 |v-w-\xi| < \delta \quad \text{ and } \quad |v'-w - \xi| < \delta~.
 \end{equation*}
 This implies that
 \begin{equation*}
 |v-v'|=|(v-w-\xi)-(v'-w-\xi)|\le |v-w-\xi| + |v'-w-\xi| < \delta + \delta= 2 \delta < 1~.
 \end{equation*}
 That is \(|v-v'|<1\). This contradicts the well-known fact that \(|v-v'|\ge 1\). This result actually implies that there exists \(N\in \N\) such that each \(\zeta\in \C^n \) belongs to at most \(N\) balls in \(\{B(v-w,\delta): v\in \Z^{2n}\}\). That is \(\sum_{v \in \Z^{2n}} \chi_{B(v-w,\delta)}(\zeta) \le N \) for each \(\zeta \in \C^n \).

\begin{proof}[\textbf{Proof of Proposition \ref{EwBEzD} }]
 From (\ref{EwBEz}), we have
 \begin{equation*}
 E_{w} BE_{z} = \frac{1}{\pi^{2n}} \sum_{u,v \in \Z^{2n}} \langle Bk_{u-z}, k_{v-w} \rangle k_{v-w} \otimes k_{u-z}~.
 \end{equation*}
 Thus for any \(R>0\), we can write \(E_{w}BE_{z} = V_{R} + W_{R}\), where
 \begin{equation*}
 V_{R} = \frac{1}{\pi^{2n}} \sum_{\underset{|u-v|\le R}{u,v \in \Z^{2n}}} \langle Bk_{u-z}, k_{v-w} \rangle k_{v-w} \otimes k_{u-z} \quad \text{ and }
 \end{equation*}
 \begin{equation*}
 \quad W_{R} = \frac{1}{\pi^{2n}} \sum_{\underset{|u-v|>R}{u,v \in \Z^{2n}}} \langle Bk_{u-z}, k_{v-w} \rangle k_{v-w} \otimes k_{u-z}~.
 \end{equation*}
 To complete the proof, it suffices to prove that:
 \begin{enumerate}[label=(\alph*)]
 \item \label{itm:first1} \(\lim_{R\to \infty} \norm{W_{R}}=0\).
 \item \label{itm:second1} \(V_{R} \in \) span(\(\mathcal{D}_{0}\)) for every \(R>0\).
 \end{enumerate}

\begin{pro}\label{DoT1}
 We have \(\mathcal{D}_{0} \subset \mathcal{T}^{(1)}\).
 \end{pro}
 To establish the proof of this proposition, we will need the next three propositions.
 \begin{pro}\label{YzT1}
 Suppose that \(\{c_{u}: u \in \Z^{2n}\}\) is a bounded set of complex coefficients. Then for each \(z \in \C^{n}\), the operator \(Y_{z}\) defined in (\ref{Yz}) belongs to \(\mathcal{T}^{(1)}\).
 \end{pro}
 \begin{proof}
 \begin{enumerate}[label=(\alph*)]
 \item Let us first show that \(Y_{0} \in \mathcal{T}^{(1)}\). We have \(|u-v|\ge 1\) for all \(u \neq v \in \Z^{2n}\). Hence \(B(u,\frac{1}{2}) \cap B(v,\frac{1}{2}) = \emptyset \) for \(u\neq v\). For each \(0 < \varepsilon < \frac{1}{2}\), define the operator
 \begin{equation*}
 A_{\varepsilon} = \frac{1}{|B(0,\varepsilon)|} \int_{B(0,\varepsilon)} Y_{z} dV(z) .
 \end{equation*}
 From Proposition \ref{proYz}, we have the norm continuity of the map \(z\mapsto Y_{z}\) and it implies that
 \begin{equation*}
 \lim_{\varepsilon \to 0} \norm{Y_{0}-A_{\varepsilon}} = 0.
 \end{equation*}
 This comes from the fact that
 \begin{eqnarray*}
 \norm{Y_{0}-A_{\varepsilon}} &=& \norm{\frac{1}{|B(0,\varepsilon)|} \int_{B(0,\varepsilon)} (Y_{0}-Y_{z}) ~dV(z)} \le \frac{1}{|B(0,\varepsilon)|} \int_{B(0,\varepsilon)} \norm{Y_{0}-Y_{z}} ~dV(z)
 \end{eqnarray*}
 and \(\lim_{z\to 0}\norm{Y_{z}-Y_{0}}=0\) .

The organization of this paper is as follows. In Section 2, we will give propositions in the case of the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\) which are the analogue of those given by Xia in \cite{xia2015localization} in the Bergman space case of the unit ball. Later, using these propositions in Section 3, we establish the proof of Theorem \ref{XiaFock} and present a consequence.

To establish the proof of Theorem~\ref{XiaFock}, the paper first "give[s] propositions in the case of the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\)" in Section 2, as analogues of results of Xia \cite{xia2015localization} for the Bergman space of the unit ball. Then, "using these propositions in Section 3," it "establish[es] the proof of Theorem~\ref{XiaFock}."

To establish the proof of the main theorem, the paper first "give[s] propositions in the case of the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\)" in Section 2, as analogues of results of Xia \cite{xia2015localization} for the Bergman space of the unit ball. Then, "using these propositions in Section 3," it "establish[es] the proof of the main theorem."

Question: In the Fock space \(H^{2}(\mathbb{C}^{n},d\mu)\), let \(C^{*}(\mathcal{WL})\) denote the \(C^{*}\)-algebra generated by all weakly localized operators on \(H^{2}(\mathbb{C}^{n},d\mu)\). Which algebra is exactly equal to \(C^{*}(\mathcal{WL})\)?

Choices:
B. the norm closure of the weakly localized operators themselves, i.e. \(\overline{\mathcal{WL}}^{\|\cdot\|}\).
C. a subalgebra of \(\mathcal{T}^{(1)}\).
D. the algebra of all admissible operators on \(H^{2}(\mathbb{C}^{n},d\mu)\).
E. \(\mathcal{T}^{(p)}\) for every \(p>0\).

equality_to_\mathcal{T}^{(1)} dropped to mere inclusion

specific index 1 replaced by uniform all-p statement