Let $X,\, Y$ be closed oriented surfaces and $f \,:\, Y\,\longrightarrow\, X$ a degree $d$ branched
covering. Given a flat vector
bundle $(V,\, \nabla)$ of rank $r$ on $Y$, by taking the direct image (pushforward) we get a flat bundle
$(f_*V,\, f_*\nabla)$ of rank $dr$ on $X\setminus B$ by Grauert's Theorem, where $B$ is the branch locus of $f$.

Denote $G_m\,=\,\mathrm{GL}(m,\mathbb{C})$. The isomorphism class of the bundle $(V, \,\nabla)$ is determined by the
$G_r$--conjugation orbit of its holonomy $\rho\,:\,\pi_1(Y)\,\longrightarrow\, G_r$. Denote this orbit
(conjugacy class) by $[\rho]$. Consequently, the moduli space of such bundles is identified with the (affine)
Geometric Invariant Theory (GIT) quotient $\mathcal{M}_r(Y)\,:=\,\mathrm{Hom}(\pi_1(Y),\,G_r)/\!\!/G_r$.
We note that the {\it affine} GIT quotient is the affine variety whose coordinate ring is the ring of
$\mathrm{Inn}(G_r)$-invariant polynomials on $\mathrm{Hom}(\pi_1(Y),\,G_r)$. The above GIT quotient
$\mathrm{Hom}(\pi_1(Y),\,G_r)/\!\!/G_r$ is also known as a {\it character variety} or more precisely
the $G_r$--character variety of $\pi_1(Y)$. We likewise consider the character variety
$\mathcal{M}_{dr}(X-B)\,:=\,\mathrm{Hom}(\pi_1(X-B),\,G_{dr})/\!\!/G_{dr}$.

By \cite{Go, La, BJ}, we know $\mathcal{M}_r(Y)$ is (holomorphic) symplectic and $\mathcal{M}_{dr}(X-B)$ is
(holomorphic) Poisson on their smooth loci. By sending the holonomy of $(V,\, \nabla)$ to the holonomy of
$(f_*V,\, f_*\nabla)$, we get a natural map $\mathsf{Frob} \,:\, \mathcal{M}_r(Y)
\,\longrightarrow\, \mathcal{M}_{dr}(X-B)$ which we call the {\it Frobenius reciprocity map}.

\begin{abstract}
We show that a Frobenius reciprocity map on character varieties of surfaces is a Poisson embedding.
\end{abstract}

By \cite{Go, La, BJ}, we know $\mathcal{M}_r(Y)$ is (holomorphic) symplectic and $\mathcal{M}_{dr}(X-B)$ is
(holomorphic) Poisson on their smooth loci. By sending the holonomy of $(V,\, \nabla)$ to the holonomy of
$(f_*V,\, f_*\nabla)$, we get a natural map $\mathsf{Frob} \,:\, \mathcal{M}_r(Y)
\,\longrightarrow\, \mathcal{M}_{dr}(X-B)$ which we call the {\it Frobenius reciprocity map}.

Let $Y_0\,=\,Y-f^{-1}(B)$ and $\mathcal{M}_r(Y_0)\,:=\,\mathrm{Hom}(\pi_1(Y_0),\,G_r)/\!\!/G_r$. Again,
$\mathcal{M}_r(Y_0)$ is (holomorphic) Poisson on its smooth locus. Since $Y_0\,\subset\, Y$ there is a
natural map $\iota\,:\,\pi_1(Y_0)\,\longrightarrow\, \pi_1(Y)$. The map $\iota$ is surjective (and hence has a
right inverse) because any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ since $\dim f^{-1}(B)\,=\,0$.

Let $\mathsf{Res}\,:\,\mathcal{M}_r(Y)\,\longrightarrow\, \mathcal{M}_r(Y_0)$ be given by $\mathsf{Res}([h])
\,=\,[h\circ \iota]$, where $h\,:\,\pi_1(Y)\,\longrightarrow\, G_r$. If
$\mathsf{Res}([h_1])\,=\,\mathsf{Res}([h_1])$, then
$[h_1\circ\iota]\,=\,[h_2\circ\iota]$ and so $[h_1]=[h_2]$ since $\iota$ has a right inverse. Therefore,
$\mathsf{Res}$ is an embedding. The inclusion map $Y_0\,\subset\, Y$ preserves transversality of based loops
and thus by \cite[Theorem A]{BHJL} the ``restriction" map $\mathsf{Res}$ is Poisson.

Since $f\,:\,Y_0\,\longrightarrow\, X-B$ is an unramified covering map, $\pi_1(Y_0)$ is isomorphic to a
subgroup of $\pi_1(X-B)$. Therefore, there
is a map $\mathsf{Ind}\,:\,\mathcal{M}_r(Y_0)\,\longrightarrow\, \mathcal{M}_{dr}(X-B)$ given by
$\mathsf{Ind}([h])\,=\,[\mathsf{ind}(h)]$, where $\mathsf{ind}(h)\,:\,\pi_1(X-B)\,\longrightarrow\, G_{dr}$
is the representation induced by $h\,:\,\pi_1(Y_0)\,\longrightarrow\, G_r$. Since $f\,:\,Y_0\,\longrightarrow\,
X-B$ is a $d$--sheeted cover, the index $[\pi_1(X-B):\,\pi_1(Y_0)]\,=\,d$ is finite and
$\pi_1(X-B)/\pi_1(Y_0)\,=\, \{\gamma_1\pi_1(Y_0),\, \cdots,\,\gamma_d\pi_1(Y_0)\}$. And
so $\mathsf{ind}(h)$ is defined
by the natural action of $\pi_1(X-B)$ on $\oplus_{i=1}^d\gamma_i\C^r\,\cong\,\C^{dr}$ by $\gamma\cdot
\sum_{i=1}^d\gamma_iv_i\,:=\,\sum_{i=1}^d\gamma_{j_i}\pi(\delta_i)v_i$, where
$\gamma\gamma_i\,=\,\gamma_{j_i}\delta_i$ with $\delta_i\,\in\,\pi_1(Y_0)$. Thus, $\mathsf{ind}(h)$ takes
values in $\mathsf{Sym}_d\times G_r$ inside $G_{dr}$ where $\mathsf{Sym}_d$ is the symmetric group on $d$ symbols
(realized by permutation matrices).

By construction of the direct image sheaf, we have $\mathsf{Frob}\,=\,\mathsf{Ind}\circ \mathsf{Res}$ (see
Section \ref{sec:directimage} for details). Consequently, $\mathsf{Frob}$ is algebraic. By the Mackey
Formula \cite{Mackey} applied to this context, we see that $\mathsf{Ind}$ is injective. Therefore,
$\mathsf{Frob}$ is an embedding. In Section \ref{sec3}, we show that $\mathsf{Ind}$ is Poisson. Having
both $\mathsf{Res}$ and $\mathsf{Ind}$ injective and Poisson, we have outlined the proof of Theorem
\ref{thm:main}.

Let $\text{Hom}(\Delta,\, \text{GL}(r,{\mathbb C}))^{\rm ir}\, \subset\, \text{Hom}(\Delta,\, \text{GL}
(r,{\mathbb C}))$ be the space of irreducible representations of $\Delta$ in $\text{GL}(r,{\mathbb C})$.
The conjugation action of $\text{GL}(r,{\mathbb C})$ on itself produces an action of $\text{GL}
(r,{\mathbb C})$ on $\text{Hom}(\Delta,\, \text{GL}(r,{\mathbb C}))^{\rm ir}$. The corresponding quotient
\begin{equation}\label{e12}
{\mathcal C}_r(Y_0)\ :=\ \text{Hom}(\Delta,\, \text{GL}(r,{\mathbb C}))^{\rm ir}/\text{GL}(r,{\mathbb C})
\end{equation}
is a complex manifold equipped with a holomorphic Poisson structure \cite{La}, \cite{BJ}, \cite{Go}.
Similarly, $\text{Hom}(\Gamma,\, \text{GL}(rd,{\mathbb C}))^{\rm ir}\, \subset\, \text{Hom}(\Gamma,\, \text{GL}
(rd,{\mathbb C}))$ is the space of irreducible representations of $\Gamma$ in $\text{GL}(rd,{\mathbb C})$.
As before,
\begin{equation}\label{e13}
{\mathcal C}_{rd}(X_0)\ :=\ \text{Hom}(\Gamma,\, \text{GL}(rd,{\mathbb C}))^{\rm ir}/\text{GL}(rd,{\mathbb C})
\end{equation}
is the quotient complex manifold equipped with a holomorphic Poisson structure. Sending
any $(V,\, \nabla)\, \in\, {\mathcal C}_r(Y_0)$ to its direct image $(\phi_* V,\, \phi_*\nabla)\, \in\,
{\mathcal C}_{rd}(X_0)$ (see \eqref{e12} and \eqref{e13}) we obtain a map
\begin{equation}\label{e14}
\Psi\ :\ {\mathcal C}_r(Y_0)\ \longrightarrow\ {\mathcal C}_{rd}(X_0).
\end{equation}

\begin{proof}
The holomorphic tangent (respectively, cotangent) bundle of ${\mathcal C}_r(Y_0)$ will be
denoted by $T{\mathcal C}_r(Y_0)$ (respectively, $T^*{\mathcal C}_r(Y_0)$). The Poisson structure
on ${\mathcal C}_r(Y_0)$ is given by a holomorphic homomorphism
\begin{equation}\label{e15}
P\ :\ T^*{\mathcal C}_r(Y_0) \ \longrightarrow\ T{\mathcal C}_r(Y_0)
\end{equation}
which is skew-symmetric and the Poisson bracket defined by $P$ satisfies the Jacobi identity. Similarly,
\begin{equation}\label{e16}
Q\ :\ T^*{\mathcal C}_{rd}(X_0) \ \longrightarrow\ T{\mathcal C}_{rd}(X_0)
\end{equation}
is the Poisson structure on ${\mathcal C}_{rd}(X_0)$. The map $\Psi$ in \eqref{e14} is said to be
compatible with the Poisson structures on ${\mathcal C}_r(Y_0)$ and ${\mathcal C}_{rd}(X_0)$ if the
following diagram of homomorphisms of vector bundles over ${\mathcal C}_r(Y_0)$ is commutative:
\begin{equation}\label{e17}
\begin{matrix}
T^*{\mathcal C}_r(Y_0) & \xrightarrow{\,\,\, P\,\,\,} & T{\mathcal C}_r(Y_0)\\
\, \,\,\,\,\, \Big\uparrow (d\Psi)^* && \,\,\,\, \Big\downarrow d\Psi\\
\Psi^*T^*{\mathcal C}_{rd}(X_0) & \xrightarrow{\,\,\, \Psi^*Q\,\,\,} & \Psi^* T{\mathcal C}_{rd}(X_0)
\end{matrix}
\end{equation}
where $P$ (respectively, $Q$) is constructed in \eqref{e15} (respectively, \eqref{e16})
and $d\Psi$ is the differential of the map $\Psi$ in \eqref{e14} while $(d\Psi)^*$ is its dual.

\begin{theorem}\label{thm:main}
 $\mathsf{Frob}$ is a Poisson embedding.
\end{theorem}

Let $Y_0\,=\,Y-f^{-1}(B)$ and $\mathcal{M}_r(Y_0)\,:=\,\mathrm{Hom}(\pi_1(Y_0),\,G_r)/\!\!/G_r$. Again,
$\mathcal{M}_r(Y_0)$ is (holomorphic) Poisson on its smooth locus. Since $Y_0\,\subset\, Y$ there is a
natural map $\iota\,:\,\pi_1(Y_0)\,\longrightarrow\, \pi_1(Y)$. The map $\iota$ is surjective (and hence has a
right inverse) because any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ since $\dim f^{-1}(B)\,=\,0$.

Let $\mathsf{Res}\,:\,\mathcal{M}_r(Y)\,\longrightarrow\, \mathcal{M}_r(Y_0)$ be given by $\mathsf{Res}([h])
\,=\,[h\circ \iota]$, where $h\,:\,\pi_1(Y)\,\longrightarrow\, G_r$. If
$\mathsf{Res}([h_1])\,=\,\mathsf{Res}([h_1])$, then
$[h_1\circ\iota]\,=\,[h_2\circ\iota]$ and so $[h_1]=[h_2]$ since $\iota$ has a right inverse. Therefore,
$\mathsf{Res}$ is an embedding. The inclusion map $Y_0\,\subset\, Y$ preserves transversality of based loops
and thus by \cite[Theorem A]{BHJL} the ``restriction" map $\mathsf{Res}$ is Poisson.

Since $f\,:\,Y_0\,\longrightarrow\, X-B$ is an unramified covering map, $\pi_1(Y_0)$ is isomorphic to a
subgroup of $\pi_1(X-B)$. Therefore, there
is a map $\mathsf{Ind}\,:\,\mathcal{M}_r(Y_0)\,\longrightarrow\, \mathcal{M}_{dr}(X-B)$ given by
$\mathsf{Ind}([h])\,=\,[\mathsf{ind}(h)]$, where $\mathsf{ind}(h)\,:\,\pi_1(X-B)\,\longrightarrow\, G_{dr}$
is the representation induced by $h\,:\,\pi_1(Y_0)\,\longrightarrow\, G_r$. Since $f\,:\,Y_0\,\longrightarrow\,
X-B$ is a $d$--sheeted cover, the index $[\pi_1(X-B):\,\pi_1(Y_0)]\,=\,d$ is finite and
$\pi_1(X-B)/\pi_1(Y_0)\,=\, \{\gamma_1\pi_1(Y_0),\, \cdots,\,\gamma_d\pi_1(Y_0)\}$. And
so $\mathsf{ind}(h)$ is defined
by the natural action of $\pi_1(X-B)$ on $\oplus_{i=1}^d\gamma_i\mathbb{C}^r\,\cong\,\mathbb{C}^{dr}$ by $\gamma\cdot
\sum_{i=1}^d\gamma_iv_i\,:=\,\sum_{i=1}^d\gamma_{j_i}\pi(\delta_i)v_i$, where
$\gamma\gamma_i\,=\,\gamma_{j_i}\delta_i$ with $\delta_i\,\in\,\pi_1(Y_0)$. Thus, $\mathsf{ind}(h)$ takes
values in $\mathsf{Sym}_d\times G_r$ inside $G_{dr}$ where $\mathsf{Sym}_d$ is the symmetric group on $d$ symbols
(realized by permutation matrices).

By construction of the direct image sheaf, we have $\mathsf{Frob}\,=\,\mathsf{Ind}\circ \mathsf{Res}$ (see
Section \ref{sec:directimage} for details). Consequently, $\mathsf{Frob}$ is algebraic. By the Mackey
Formula \cite{Mackey} applied to this context, we see that $\mathsf{Ind}$ is injective. Therefore,
$\mathsf{Frob}$ is an embedding. In Section \ref{sec3}, we show that $\mathsf{Ind}$ is Poisson. Having
both $\mathsf{Res}$ and $\mathsf{Ind}$ injective and Poisson, we have outlined the proof of Theorem
\ref{thm:main}.

Work with $Y_0:=Y-f^{-1}(B)$ and the Poisson (on the smooth locus) character variety $\mathcal{M}_r(Y_0)=\mathrm{Hom}(\pi_1(Y_0),G_r)/\!\!/G_r$. Since $Y_0\subset Y$ there is a natural surjection $\iota:\pi_1(Y_0)\to\pi_1(Y)$ (any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ because $\dim f^{-1}(B)=0$), hence $\iota$ has a right inverse.

Define the “restriction” map $\mathsf{Res}:\mathcal{M}_r(Y)\to\mathcal{M}_r(Y_0)$ by $\mathsf{Res}([h])=[h\circ\iota]$. If $\mathsf{Res}([h_1])=\mathsf{Res}([h_2])$ then $[h_1\circ\iota]=[h_2\circ\iota]$, and since $\iota$ has a right inverse this implies $[h_1]=[h_2]$; thus $\mathsf{Res}$ is an embedding. The inclusion $Y_0\subset Y$ “preserves transversality of based loops” and hence by \cite[Theorem A]{BHJL} $\mathsf{Res}$ is Poisson.

Because $f:Y_0\to X-B$ is an unramified $d$--sheeted cover, $\pi_1(Y_0)$ is a subgroup of finite index $d$ in $\pi_1(X-B)$. Define an induction map $\mathsf{Ind}:\mathcal{M}_r(Y_0)\to\mathcal{M}_{dr}(X-B)$ by $\mathsf{Ind}([h])=[\mathsf{ind}(h)]$, where $\mathsf{ind}(h):\pi_1(X-B)\to G_{dr}$ is the induced representation via the action of $\pi_1(X-B)$ on $\oplus_{i=1}^d\gamma_i\mathbb{C}^r\cong\mathbb{C}^{dr}$; thus $\mathsf{ind}(h)$ lands in $\mathsf{Sym}_d\times G_r\subset G_{dr}$. By construction of the direct image sheaf, $\mathsf{Frob}=\mathsf{Ind}\circ\mathsf{Res}$.

Consequently $\mathsf{Frob}$ is algebraic. By the Mackey Formula \cite{Mackey}, $\mathsf{Ind}$ is injective, hence $\mathsf{Frob}$ is an embedding. In Section \ref{sec3} it is shown that $\mathsf{Ind}$ is Poisson. With both $\mathsf{Res}$ and $\mathsf{Ind}$ injective and Poisson, this “outlines the proof of Theorem~\ref{thm:main},” i.e. that $\mathsf{Frob}$ is a Poisson embedding.

Work with $Y_0:=Y-f^{-1}(B)$ and the Poisson (on the smooth locus) character variety $\mathcal{M}_r(Y_0)=\mathrm{Hom}(\pi_1(Y_0),G_r)/\!\!/G_r$. Since $Y_0\subset Y$ there is a natural surjection $\iota:\pi_1(Y_0)\to\pi_1(Y)$ (any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ because $\dim f^{-1}(B)=0$), hence $\iota$ has a right inverse.

Define the “restriction” map $\mathsf{Res}:\mathcal{M}_r(Y)\to\mathcal{M}_r(Y_0)$ by $\mathsf{Res}([h])=[h\circ\iota]$. If $\mathsf{Res}([h_1])=\mathsf{Res}([h_2])$ then $[h_1\circ\iota]=[h_2\circ\iota]$, and since $\iota$ has a right inverse this implies $[h_1]=[h_2]$; thus $\mathsf{Res}$ is an embedding. The inclusion $Y_0\subset Y$ “preserves transversality of based loops” and hence by \cite[Theorem A]{BHJL} $\mathsf{Res}$ is Poisson.

Because $f:Y_0\to X-B$ is an unramified $d$--sheeted cover, $\pi_1(Y_0)$ is a subgroup of finite index $d$ in $\pi_1(X-B)$. Define an induction map $\mathsf{Ind}:\mathcal{M}_r(Y_0)\to\mathcal{M}_{dr}(X-B)$ by $\mathsf{Ind}([h])=[\mathsf{ind}(h)]$, where $\mathsf{ind}(h):\pi_1(X-B)\to G_{dr}$ is the induced representation via the action of $\pi_1(X-B)$ on $\oplus_{i=1}^d\gamma_i\mathbb{C}^r\cong\mathbb{C}^{dr}$; thus $\mathsf{ind}(h)$ lands in $\mathsf{Sym}_d\times G_r\subset G_{dr}$. By construction of the direct image sheaf, $\mathsf{Frob}=\mathsf{Ind}\circ\mathsf{Res}$.

Consequently $\mathsf{Frob}$ is algebraic. By the Mackey Formula \cite{Mackey}, $\mathsf{Ind}$ is injective, hence $\mathsf{Frob}$ is an embedding. In Section \ref{sec3} it is shown that $\mathsf{Ind}$ is Poisson. With both $\mathsf{Res}$ and $\mathsf{Ind}$ injective and Poisson, this outlines the proof of the main theorem, i.e. that $\mathsf{Frob}$ is a Poisson embedding.

\label{thm:main}
 $\mathsf{Frob}$ is a Poisson embedding.,

Question:
Let $X$ and $Y$ be closed oriented surfaces, and let $f:Y\to X$ be a degree $d$ branched covering with branch locus $B\subset X$. For $m\ge 1$, write $G_m=\mathrm{GL}(m,\mathbb C)$ and define the character varieties
\[\mathcal M_r(Y):=\mathrm{Hom}(\pi_1(Y),G_r)/\!/G_r,\qquad \mathcal M_{dr}(X\setminus B):=\mathrm{Hom}(\pi_1(X\setminus B),G_{dr})/\!/G_{dr}.\]
If $(V,\nabla)$ is a rank-$r$ flat vector bundle on $Y$, its holonomy class gives a point of $\mathcal M_r(Y)$, and the direct image flat bundle $(f_*V,f_*\nabla)$ on $X\setminus B$ gives a point of $\mathcal M_{dr}(X\setminus B)$. Let
\[\mathsf{Frob}:\mathcal M_r(Y)\longrightarrow \mathcal M_{dr}(X\setminus B)\]
be the resulting map on character varieties, called the Frobenius reciprocity map. Which statement holds for this map, with respect to the natural holomorphic Poisson structures on these character varieties?

The map $\mathsf{Frob}$ is a Poisson embedding; that is, it is an embedding of $\mathcal M_r(Y)$ into $\mathcal M_{dr}(X\setminus B)$ and it is compatible with the natural holomorphic Poisson structures.

Choices:
B. The map $\mathsf{Frob}$ is a Poisson immersion on the smooth locus of $\mathcal M_r(Y)$, but it need not be injective as a map into $\mathcal M_{dr}(X\setminus B)$.
C. The map $\mathsf{Frob}$ is Poisson; that is, it is compatible with the natural holomorphic Poisson structures on $\mathcal M_r(Y)$ and $\mathcal M_{dr}(X\setminus B)$.
D. The map $\mathsf{Frob}$ is an embedding of $\mathcal M_r(Y)$ into $\mathcal M_{dr}(X\setminus B)$, but in general it is only algebraic and does not preserve the natural holomorphic Poisson structures.
E. The map $\mathsf{Frob}$ is a holomorphic symplectic embedding onto its image; equivalently, its image is a symplectic submanifold of $\mathcal M_{dr}(X\setminus B)$ and $\mathsf{Frob}$ preserves the symplectic forms.

Poisson target is not symplectic in general

The stem does not explicitly state that the map is a Poisson embedding. It defines the construction and asks for the correct structural property, without giving away the conclusion.

This is close to a theorem-recall item: it essentially asks which formal statement about the Frobenius reciprocity map is true. The multiple choices introduce stronger, weaker, and distorted variants, so it is not pure tautology, but it remains a mild reformulation of a likely theorem statement.

Some reasoning is needed to distinguish among 'Poisson', 'Poisson embedding', 'immersion', and 'symplectic embedding', especially since one distractor is a weaker true statement. Still, the item mainly tests recognition of the exact theorem rather than substantial derivation.

The distractors are strong: one is weaker but true in part, others reflect common confusions between injective/immersive, algebraic/Poisson, and Poisson/symplectic. They are distinct and mathematically meaningful.

A solid MCQ with no answer leakage and high-quality distractors, but it leans toward theorem recognition rather than deep generative reasoning.


This section introduces the mathematical formulation of the inverse problem and specifies the functional framework adopted throughout the paper. We also describe the model equation, its physical motivation, and the regularity assumptions required to ensure well-posedness.

Let $\Omega\subset \mathbb{R}^N$ be a bounded domain ($N\geqslant 1$) with boundary $\partial \Omega$ of class $C^2$ and $T>0$. Here and in the sequel, the function spaces refer to spaces of complex-valued functions unless otherwise specified.

We consider $a,b$, $\vec{r}$ and $p$ with the following assumptions:
\begin{itemize}
    \item[{\bf (H1)}] $a,b\in W^{1,\infty}(\Omega;\mathbb{R})$ with
    \begin{align*}
        a(x)\geqslant a_\star >0\quad \text{ almost everywhere in }\Omega,
    \end{align*}
    for some $a_\star$.
    \item[{\bf (H2)}] $\vec{r}=\vec{r}(x)$, $p=p(x)$ and $y_0=y_0(x)$ are complex-valued functions such that
    \begin{align*}
        \vec{r}\in [W^{1,\infty}(\Omega)]^N,\,
        q\in L^\infty(\Omega)\text{ and }y_0\in L^2(\Omega).
    \end{align*}
\end{itemize}

Now, consider the following Ginzburg-Landau equation with Dirichlet boundary conditions:
\begin{align}
    \label{eq:intro:01}
    \begin{cases}
        \partial_t y -\text{div}((a(x)+ib(x))\nabla y) + \vec{r}(x)\cdot \nabla y +p(x)y=f(x,t)&\text{ in }\Omega\times (0,T),\\
        y=0&\text{ on }\partial \Omega\times (0,T),\\
        y(\cdot,0)=y_0&\text{ in }\Omega.
    \end{cases}
\end{align}

Under the assumptions {\bf (H1)}, {\bf (H2)} and $f\in L^2(0,T;L^2(\Omega))$, there exists a unique weak solution of \eqref{eq:intro:01} with the following regularity:
\begin{align*}
    y\in C^0([0,T];L^2(\Omega)) \cap L^2(0,T;H_0^1(\Omega)).
\end{align*}

For completeness and self-containment,  we include in Appendix \ref{section:appendix:existence:uniqueness:results} general statements, together with their proofs, concerning the existence of weak and strong solutions of the equation \eqref{eq:intro:01}.

In this article, we will consider the following {\bf Inverse Source Problem (ISP):} Identify the unknown spatial-temporal source term $f$ in the space $L^2(0,T;L^2(\Omega))$ in \eqref{eq:intro:01} from the following final time measured output:
\begin{align*}
    u_T(x):=y(x,T),\quad x\in \Omega.
\end{align*}

Here, $y(x,T)\equiv y(x,t)\big|_{t=T}$ is a trace appropriately defined of the weak solution $y(x,t)$ of \eqref{eq:intro:01}. In addition, $u_T(x)$ represents the measured output containing a random noise.

Inverse source problems of this type are not only mathematically rich but also arise naturally in applied sciences. In many experimental settings, direct access to distributed sources is often impossible, and only partial or final-time measurements are available. For instance, in nonlinear optics one may measure the profile of a pulse at the end of a cavity, while the internal gain or loss mechanisms driving its formation remain hidden. Similarly, in chemical or biological pattern formation, one often observes the state of the system after some transient dynamics but seeks to infer the localized heterogeneities or forcing terms that generated it. This gap between accessible measurements and hidden dynamics motivates the study of reconstruction methods capable of identifying unknown excitations from indirect data.

The choice of the Ginzburg-Landau equation as the underlying model is particularly significant. It serves as a universal amplitude equation near the onset of instability in diverse physical systems, ranging from fluid dynamics and plasma physics to optics and chemical oscillations \cite{Aranson2002,levermore1996complex,chen1998numerical,bartuccelli1990possibility,mielke2002ginzburg}. Consequently, advances in the understanding of inverse problems for this equation are not limited to a single discipline but carry implications across multiple fields. Identifying unknown sources in such a model not only enriches the theory of dissipative-dispersive PDEs but also provides a mathematical framework for interpreting experiments in which only end-point information is available.

At the mathematical level, the inverse source problem we consider is severely ill-posed. Regulation strategies and variational formulations thus become essential. Our work adopts this perspective, combining the theory of weak solutions, adjoint-based gradient formulas, and Tikhonov regularization to rigorously justify reconstruction algorithms and test their performance through numerical simulations.

\subsection{Literature}

We now review some representative contributions on the complex Ginzburg-Landau equation and its variants, emphasizing their role as canonical amplitude equations and discussing prior work related to inverse or identification problems.

The complex Ginzburg-Landau (GL) equation is the canonical amplitude equation near a Hopf or Turing-Hopf bifurcation. In its cubic form, it reads
\begin{align*}
    \partial_t A=(\mu+i\omega)A + (1+c_1i)\Delta A -(1+c_3 i)|A|^2 A,
\end{align*}
while the cubic-quintic extension adds a saturating nonlinearity:
\begin{align*}
    \partial_t A=(\mu +\omega i)A +(1+c_1i)\Delta A -(1+c_3 i)|A|^2 A -(\nu +c_5i)|A|^4A,
\end{align*}
with real parameters $\mu,\nu,c_j$ ($j=1,3,5$) that encode growth, dispersion/diffusion, and nonlinear gain/loss. The cubic model is the universal normal form near onset and captures phase diffusion, plane-wave selection and Benjamin-Feir/Newell instabilities. On the other hand, the quintic term become relevant farther from threshold to model saturation and complex dissipative structures (e.g. localized pulses). See the reviews \cite{Cross1993} and \cite{Aranson2002}, which also discuss higher-order variants and parameter scalings.

The cubic and cubic-quintic complex GL equations serve as canonical models of pattern formation and nonlinear wave dynamics in nonequilibrium systems. These models describe the slow complex envelope $A(x,t)$ of oscillatory modes near stability. In hydrodynamics and Rayleigh-B\'enard convection, the cubic GL equation explains the emergence of roll patterns, travelling waves, and their stability domains \cite{Newell1969} and \cite{Cross1993}. In nonlinear optics, it governs mode-locked lasers, pulse propagation in fiber cavities, and the formation of dissipative solutions. In this case, the cubic-quintic equation extension accounts for gain saturation and quintic nonlinear effects that are essential for bounded amplitude states \cite{SotoCrespo1997}, \cite{Akhmediev2005}. Similarly, in chemical oscillations and biological excitable media, it models spiral waves, defect turbulence, and spatiotemporal chaos, highlighting its universality across disciplines \cite{Aranson2002}.

From a mathematical viewpoint, the GL equations combine parabolic smoothing (via the Laplacian operator) with dispersive phase dynamics (via imaginary coefficients), which generates a delicate interplay between diffusion, dispersion, and nonlinearity \cite{Akhmediev2005,Cross1993, levermore1996complex, mielke2002ginzburg}. This hybrid character produces a wide spectrum of phenomena: existence of plane-wave solutions, modulational (Benjamin-Feir/Newell) instabilities, turbulence, and complex coherent structures \cite{Akhmediev2005, Cross1993, Newell1969, SotoCrespo1997}. The cubic equation alone already yields chaotic attractors, defect-mediated turbulence, and spiral wave breakup \cite{Aranson2002,bartuccelli1990possibility, SotoCrespo1997}, while the quintic term stabilizes or destabilizes localized patterns depending on parameter regimes \cite{Akhmediev2005,SotoCrespo1997}. The GL equations have been used to study attractors, bifurcations, and long-time dynamics of nonlinear PDEs, serving as a testbed for methods in dissipative systems, control, and inverse problems \cite{chen1998numerical,mielke2002ginzburg,santos2019insensitizing,borzi2005analysis,junge2000synchronization}.
Recovering sources and parameters in GL-type models is crucial in practice: in optics, it enables diagnosing distributed gain/loss or saturable absorbers in cavity models from end-of-pulse measurements; in chemical/biological pattern formation, it helps infer spatially localized forcing or heterogeneities driving wave patterns; in fluid and plasma contexts, it allows one to back-out effective forcing or feedback from limited snapshots near transitions to turbulence. Because the GL framework is a universal amplitude model, successful inversion translates across disciplines, providing interpretable maps of where and how energy is injected or dissipated. From a computational perspective, adjoint-based Tikhonov schemes scale to high-dimensional discretizations and accommodate realistic noise models \cite{Cross1993}, \cite{Aranson2002}, \cite{MR2516528}.

Within this broad literature, inverse problems for GL-type models remain relatively less explored. Classical studies on inverse source problems for parabolic or Schr\"odinger-type equations \cite{hasanouglu2021introduction,hasanov2007simultaneous,GarciaOssesTapia+2013+755+779,chorfi2025identification} provide methodological foundations, yet the adaptation to GL equations, which combine diffusion, dispersion, and nonlinearity, poses unique analytical and numerical challenges.

We also note that other GL-type models, such as those with dynamic boundary conditions, have recently attracted attention in the context of controllability \cite{carreno2025local}. While these studies highlight the richness of the GL framework under more complex boundary interactions, the analysis of inverse problems in such settings remains largely open. Our contribution is thus complementary, as we address the identification of space-time dependent source terms for the standard GL equation, providing a first step toward broader inverse problem formulations in extended GL models.

It is worth emphasizing that the linear GL equation can also be expressed as a system of two coupled real-valued PDE equations, corresponding to the real and imaginary parts of the complex amplitude. In this formulation, the coupling terms are of second order and involve cross-derivative operators, which profoundly alter the analytical structure of the system. As a consequence, classical techniques from the theory of control and inverse problems for scalar parabolic equations (based on Carleman estimates and observability inequalities \cite{fursikov1996carleman}) cannot be directly applied. In particular, the presence of complex coefficients and dispersive-type interactions requires adapted Carleman weights, refined energy estimates, and the development of new stability results tailored to mixed dissipative-dispersive operators.

The present work is intended as a first step toward a comprehensive theory of inverse problems for GL-type equations, focusing here on the standard setting with Dirichlet boundary conditions.

\subsection{The input-output operator}
To formalize the inverse problem, we introduce the input-output operator mapping the source term to the final-state of the system. This operator provides a compact representation of the forward model and allows for a variational reformulation of the inverse problem.

Let $\mathcal{U}\subset L^2(0,T;L^2(\Omega))$ be a non-empty set, which is supposed to be bounded, closed and convex. In addition, we define the operator $\Psi: \mathcal{U}\to L^2(\Omega)$ in the following way:
\begin{align*}
    \Psi(f)=y(\cdot,T)\quad f\in \mathcal{U},
\end{align*}
where $y=y(x,t)$ is the solution of \eqref{eq:intro:01} associated to the source term $f$.

We point out that, the {\bf (ISP)} can be formulated in terms of $\Psi$ as the following functional equation:
\begin{align}
    \label{eq:Input-output:op}
    \Psi(f)=u_T(x),\quad x\in \Omega,\quad  f\in \mathcal{U}.
\end{align}

We emphasize that, due to measurement errors in the output $u_T$, exact equality in the equation \eqref{eq:Input-output:op} cannot be satisfied in general. Moreover, we have the following result

Let $\Omega\subset \mathbb{R}^N$ be a bounded domain ($N\geqslant 1$) with boundary $\partial \Omega$ of class $C^2$ and $T>0$. Here and in the sequel, the function spaces refer to spaces of complex-valued functions unless otherwise specified.

Now, consider the following Ginzburg-Landau equation with Dirichlet boundary conditions:
\begin{align}
 \label{eq:intro:01}
 \begin{cases}
 \partial_t y -\text{div}((a(x)+ib(x))\nabla y) + \vec{r}(x)\cdot \nabla y +p(x)y=f(x,t)&\text{ in }\Omega\times (0,T),\\
 y=0&\text{ on }\partial \Omega\times (0,T),\\
 y(\cdot,0)=y_0&\text{ in }\Omega.
 \end{cases}
\end{align}

\subsection{The input-output operator}
To formalize the inverse problem, we introduce the input-output operator mapping the source term to the final-state of the system. This operator provides a compact representation of the forward model and allows for a variational reformulation of the inverse problem.

Let $\mathcal{U}\subset L^2(0,T;L^2(\Omega))$ be a non-empty set, which is supposed to be bounded, closed and convex. In addition, we define the operator $\Psi: \mathcal{U}\to L^2(\Omega)$ in the following way:
\begin{align*}
 \Psi(f)=y(\cdot,T)\quad f\in \mathcal{U},
\end{align*}
where $y=y(x,t)$ is the solution of \eqref{eq:intro:01} associated to the source term $f$.

We point out that, the {\bf (ISP)} can be formulated in terms of $\Psi$ as the following functional equation:
\begin{align}
 \label{eq:Input-output:op}
 \Psi(f)=u_T(x),\quad x\in \Omega,\quad f\in \mathcal{U}.
\end{align}

We emphasize that, due to measurement errors in the output $u_T$, exact equality in the equation \eqref{eq:Input-output:op} cannot be satisfied in general. Moreover, we have the following result

\begin{align}
    \label{eq:Input-output:op}
    \Psi(f)=u_T(x),\quad x\in \Omega,\quad  f\in \mathcal{U}.
\end{align}

\begin{align}
    \label{eq:intro:01}
    \begin{cases}
        \pt y -\text{div}((a(x)+ib(x))\nabla y) + \vec{r}(x)\cdot \nabla y +p(x)y=f(x,t)&\text{ in }\Omega\times (0,T),\\
        y=0&\text{ on }\partial \Omega\times (0,T),\\
        y(\cdot,0)=y_0&\text{ in }\Omega.
    \end{cases}
\end{align}

Let $\mathcal{U}\subset L^2(0,T;L^2(\Omega))$ be a non-empty set, which is supposed to be bounded, closed and convex. In addition, we define the operator $\Psi: \mathcal{U}\to L^2(\Omega)$ in the following way:
\begin{align*}
 \Psi(f)=y(\cdot,T)\quad f\in \mathcal{U},
\end{align*}
where $y=y(x,t)$ is the solution of \eqref{eq:intro:01} associated to the source term $f$.

We point out that, the {\bf (ISP)} can be formulated in terms of $\Psi$ as the following functional equation:
\begin{align}
 \label{eq:Input-output:op}
 \Psi(f)=u_T(x),\quad x\in \Omega,\quad f\in \mathcal{U}.
\end{align}

\begin{proof}
 Let $(f_k)_{k\in \mathbb{N}}$ be a bounded sequence in $L^2(\Omega\times (0,T))$. Then, by Proposition \ref{proposition:existence:strong:solutions}, the sequence of associated weak solutions $(y(\cdot,,\cdot,f_k))_{k\in \mathbb{N}}$ is bounded in $C^0([0,T];H_0^1(\Omega))$. In particular, the sequence $(Y_k)_{k\in \mathbb{N}}$ given by $Y_k:=y(\cdot,T,f_k)$ is bounded in $H_0^1(\Omega)\cap H^2(\Omega)$. Thus, using the Sobolev-Gagliardo-Nirenberg compact embedding $H_0^1(\Omega) \hookrightarrow L^2(\Omega)$, there exists a subsequence of $ (Y_k)_{k\in \mathbb{N}}$ which converges strongly in $L^2(\Omega)$. This implies that the input-output operator $\Psi$ is compact and the proof of Proposition \ref{Proposition:Phi:compact} is finished.
\end{proof}

In view of the Proposition \ref{Proposition:Phi:compact}, it is evident that the inverse problem {\bf (ISP)} is ill-posed in the sense of Hadamard. For this reason, one needs to introduce the functional $\mathcal{J}:\mathcal{U}\to \mathbb{R}$ by
\begin{align}
 \label{def:functional:J}
 \mathcal{J}(f):=\frac{1}{2}\int_\Omega |\Psi(f)-u_T|^2\,dx,
\end{align}
and reformulate {\bf (ISP)} in terms of the quasi-solution method, i.e., we minimize the following extremal problem
\begin{align*}
 \mathcal{J}(f^\star)=\inf_{f\in \mathcal{U}} \mathcal{J}(f).
\end{align*}

Since the operator $\Psi$ is compact, small perturbations in the data may cause large variations in the reconstructed source. Therefore, regularization becomes essential to obtain stable approximations. In this context, it is customary to consider a regularized Tikhonov version of the functional $J$ in \eqref{def:functional:J}. For $\epsilon>0$, we introduce the regularized functional
\begin{align}\label{regfunct}
 \mathcal{J}_\epsilon (f):=\frac{1}{2}\int_\Omega |\Psi(f)-u_T|^2\,dx + \frac{\epsilon}{2}\int_0^T\int_\Omega |f|^2\,dx\,dt ,\quad f\in \mathcal{U}.
\end{align}

\begin{proposition}
 \label{proposition:Frechet:formula}
 Consider the assumptions {\bf (H1)} and {\bf (H2)}. Then, the functional $\mathcal{J}:\mathcal{U}\to \mathbb{R}$ is Fr\'echet differentiable and its gradient at each $f\in \mathcal{U}$ is given by
 \begin{align}
 \label{eq:frechet:gradient:formula}
 \mathcal{J}'(f)=\phi,\quad f\in \mathcal{U},
 \end{align}
 where $\phi$ is the unique weak solution of the following adjoint system
 \begin{align}
 \label{problem:Frechet:dif}
 \begin{cases}
 -\pt \phi -\text{div}((a(x)-ib(x))\nabla \phi) - \overline{\vec{r}(x)}\cdot \nabla \phi +(\overline{p(x)}- \overline{\text{div}(\vec{r})(x))}\phi =0&\text{ in }\Omega\times (0,T),\\
 \phi=0&\text{ on }\partial\Omega \times (0,T),\\
 \phi(\cdot,T)=y(\cdot,T;f)-u_T&\text{ in }\Omega.
 \end{cases}
 \end{align}
\end{proposition}

Using the complex identity
\begin{align*}
 \frac{1}{2}(|x-z|^2-|y-z|^2)=\Re [(x-y)\overline{(y-z)}] +\frac{1}{2} |x-y|^2,\quad \forall x,y,z\in \mathbb{C},
\end{align*}
we have
\begin{align}
 \nonumber
 \delta \mathcal{J}(f)=&\Re \int_\Omega (y(\cdot,T;f+\delta f)-y(\cdot,T;f)) \overline{y(\cdot,T;f)-u_T}\,dx \\
 \nonumber
 &+\frac{1}{2}\int_\Omega |y(\cdot,T;f+\delta f)-y(\cdot,T;f)|^2\,dx \\
 \label{eq:Frechet:for:01}
 =&-\Re \int_\Omega \delta y(\cdot,T;f)\overline{\phi (\cdot,T;f)}\,dx +\frac{1}{2}\int_\Omega |\delta y(\cdot,T;f)|^2\,dx
\end{align}
where $\phi(\cdot,\cdot,f)$ is the weak solution of \eqref{problem:Frechet:dif} (and thanks to Proposition \ref{proposition:existence:weak:solutions}, $\phi(\cdot,T,f)=y(\cdot,T;f)-u_T$ in $L^2(\Omega)$) and $\delta y$ is the solution of the following \textit{sensitivity problem}
\begin{align}
 \label{eq:sensitivity:problem}
 \begin{cases}
 \pt \delta y -\text{div}((a(x)+b(x)i)\nabla \delta y)+\vec{r}(x)\cdot \nabla \delta y + p(x)\delta y=\delta f(x,t)&\text{ in }\Omega\times (0,T),\\
 \delta y=0&\text{ on }\partial \Omega\times (0,T),\\
 \delta y(\cdot,0)=0&\text{ in }\Omega.
 \end{cases}
\end{align}

\begin{proof}
 Let us fix $f,\delta f\in \mathcal{U}$ such that $f+\delta f\in \mathcal{U}$. Then, the function $\delta \phi=\delta \phi (\cdot,\cdot,f)$ is the solution of
 \begin{align}
 \label{eq:Lipschitz:01}
 \begin{cases}
 -\pt \delta \phi -\text{div}((a(x)-ib(x))\nabla \phi) - \overline{\vec{r}(x)}\cdot \nabla \phi +(\overline{p(x)}-\overline{\text{div}(\vec{r}(x))}\phi =0&\text{ in }\Omega\times (0,T),\\
 \delta \phi =0&\text{ on }\partial \Omega \times (0,T),\\
 \delta \phi(\cdot,T)=\delta y(\cdot,T;f)&\text{ in }\Omega,
 \end{cases}
 \end{align}
 where $\delta y$ is the solution of \eqref{problem:Frechet:dif} associated to $\delta f$. We remark that, by Proposition \ref{proposition:existence:strong:solutions}, $\delta \phi$ is a strong solution of \eqref{eq:Lipschitz:01}. Thus, by Proposition \ref{proposition:Frechet:formula} we have, for some constant $C>0$:
 \begin{align*}
 |\mathcal{J}'(f+\delta f)-\mathcal{J}'(f)|= \|\delta \phi\|_{L^2(\Omega\times (0,T))}\leq C\|\delta y(\cdot,T;f)\|_{L^2(\Omega)}\leq C\|\delta f\|_{L^2(0,T;L^2(\Omega))},
 \end{align*}
 where we have used the continuity of the strong solutions applied to $\delta y$ respect to the data. This proves the assertion of the Lemma \ref{Lemma:Lipschitz:gradient}.
\end{proof}

\begin{proposition}
    \label{Proposition:Phi:compact}
    The input-output operator $\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega)$ is compact.
\end{proposition}

\begin{align}
    \label{def:functional:J}
    \mathcal{J}(f):=\frac{1}{2}\int_\Omega |\Psi(f)-u_T|^2\,dx,
\end{align}

\begin{align}
    \label{eq:intro:01}
    \begin{cases}
        \pt y -\text{div}((a(x)+ib(x))\nabla y) + \vec{r}(x)\cdot \nabla y +p(x)y=f(x,t)&\text{ in }\Omega\times (0,T),\\
        y=0&\text{ on }\partial \Omega\times (0,T),\\
        y(\cdot,0)=y_0&\text{ in }\Omega.
    \end{cases}
\end{align}

\begin{proposition}
    \label{proposition:existence:strong:solutions}
    Let $\Omega\subset \mathbb{R}^N$ be a bounded domain with $C^2$ boundary and let $T>0$. Assume that the coefficients $\alpha$ and $\beta$ satisfy
    \begin{align}
        \label{assumption:prop:existence:02:alpha:beta}
        \alpha,\beta\in W^{1,\infty}(\Omega;\mathbb{R}),\quad \alpha(x)\geq \alpha_0 >0\text{ almost everywhere in }\Omega.
    \end{align}
    Moreover, consider \eqref{prop:existence:statement:02}, $v_0\in H_0^1(\Omega)$ and $h\in L^2(0,T;L^2(\Omega))$. Then, there exists a strong solution $v$ of \eqref{problem:existence:uniqueness} satisfying
    \begin{align*}
        v\in H^1(0,T;L^2(\Omega)) \cap C^0([0,T];H_0^1(\Omega)) \cap L^2(0,T; H^2(\Omega) \cap H_0^1(\Omega)).
    \end{align*}

    Moreover, there exists a constant $C>0$ depending only on $\alpha_0$, $\|\alpha\|_{W^{1,\infty}}$, $\||\vec{\sigma}|\|_{L^\infty}$, $\|p\|_\infty$, $\Omega$ and $T$ such that
    \begin{align*}
        \begin{split}
        &\|v\|_{L^2(0,T;H^2(\Omega))} + \|v\|_{C^0([0,T];H_0^1(\Omega))} + \|\pt v\|_{L^2(0,T;L^2(\Omega))} \\
        \leq &C\left(\|h\|_{L^2(0,T;L^2(\Omega))} + \|v_0\|_{H_0^1(\Omega)} \right).
        \end{split}
    \end{align*}
\end{proposition}

\begin{proof}
    Let $(f_k)_{k\in \mathbb{N}}$ be a bounded sequence in $L^2(\Omega\times (0,T))$. Then, by Proposition \ref{proposition:existence:strong:solutions}, the sequence of associated weak solutions $(y(\cdot,,\cdot,f_k))_{k\in \mathbb{N}}$ is bounded in $C^0([0,T];H_0^1(\Omega))$. In particular, the sequence $(Y_k)_{k\in \mathbb{N}}$ given by $Y_k:=y(\cdot,T,f_k)$ is bounded in $H_0^1(\Omega)\cap H^2(\Omega)$. Thus, using the Sobolev-Gagliardo-Nirenberg compact embedding $H_0^1(\Omega) \hookrightarrow L^2(\Omega)$, there exists a subsequence of $ (Y_k)_{k\in \mathbb{N}}$ which converges strongly in $L^2(\Omega)$. This implies that the input-output operator $\Psi$ is compact and the proof of Proposition  \ref{Proposition:Phi:compact} is finished.
\end{proof}

In view of the Proposition \ref{Proposition:Phi:compact}, it is evident that the inverse problem {\bf (ISP)} is ill-posed in the sense of Hadamard. For this reason, one needs to introduce the functional $\mathcal{J}:\mathcal{U}\to \mathbb{R}$ by
\begin{align}
    \label{def:functional:J}
    \mathcal{J}(f):=\frac{1}{2}\int_\Omega |\Psi(f)-u_T|^2\,dx,
\end{align}
and reformulate {\bf (ISP)} in terms of the quasi-solution method, i.e., we minimize the following extremal problem
\begin{align*}
    \mathcal{J}(f^\star)=\inf_{f\in \mathcal{U}} \mathcal{J}(f).
\end{align*}

Since the operator $\Psi$ is compact, small perturbations in the data may cause large variations in the reconstructed source. Therefore, regularization becomes essential to obtain stable approximations. In this context, it is customary to consider a regularized Tikhonov version of the functional $J$ in \eqref{def:functional:J}. For $\epsilon>0$, we introduce the regularized functional
\begin{align}\label{regfunct}
    \mathcal{J}_\epsilon (f):=\frac{1}{2}\int_\Omega |\Psi(f)-u_T|^2\,dx + \frac{\epsilon}{2}\int_0^T\int_\Omega |f|^2\,dx\,dt ,\quad f\in \mathcal{U}.
\end{align}

\subsection{Outline}
The rest of the paper is as follows. In Section \ref{section:Frechet:formula}, we focus on the properties of $J$, i.e., a detailed characterization of its Fr\'echet derivative via a suitable adjoint system. In Section \ref{section:Existence:ISP}, we give sufficient conditions for the existence and uniqueness of quasi-solutions to {\bf (ISP)}. In Section \ref{section:Numerical:experiments}, we validate our theoretical results by some numerical experiments to reconstruct source terms for 1-D and 2-D case. Finally, in Section \ref{section:Summary:perspectives} we give additional comments concerning the theoretical and numerical results obtained in this article.

Take a bounded sequence $(f_k)\subset L^2(\Omega\times(0,T))$. By Proposition \ref{proposition:existence:strong:solutions}, the associated weak solutions $y(\cdot,\cdot,f_k)$ are bounded in $C^0([0,T];H_0^1(\Omega))$, hence the terminal states $Y_k:=y(\cdot,T,f_k)$ form a bounded sequence in $H_0^1(\Omega)\cap H^2(\Omega)$. Using the compact embedding $H_0^1(\Omega)\hookrightarrow L^2(\Omega)$, extract a subsequence of $(Y_k)$ converging strongly in $L^2(\Omega)$. Therefore $\Psi(f_k)=Y_k$ has a strongly convergent subsequence in $L^2(\Omega)$ for every bounded $(f_k)$, which implies $\Psi$ is compact (Proposition~\ref{Proposition:Phi:compact}).

Take a bounded sequence $(f_k)\subset L^2(\Omega\times(0,T))$. We first use the following result.

\begin{proposition}
    \label{proposition:existence:strong:solutions}
    Let $\Omega\subset \mathbb{R}^N$ be a bounded domain with $C^2$ boundary and let $T>0$. Assume that the coefficients $\alpha$ and $\beta$ satisfy
    \begin{align}
        \label{assumption:prop:existence:02:alpha:beta}
        \alpha,\beta\in W^{1,\infty}(\Omega;\mathbb{R}),\quad \alpha(x)\geq \alpha_0 >0\text{ almost everywhere in }\Omega.
    \end{align}
    Moreover, consider \eqref{prop:existence:statement:02}, $v_0\in H_0^1(\Omega)$ and $h\in L^2(0,T;L^2(\Omega))$. Then, there exists a strong solution $v$ of \eqref{problem:existence:uniqueness} satisfying
    \begin{align*}
        v\in H^1(0,T;L^2(\Omega)) \cap C^0([0,T];H_0^1(\Omega)) \cap L^2(0,T; H^2(\Omega) \cap H_0^1(\Omega)).
    \end{align*}

    Moreover, there exists a constant $C>0$ depending only on $\alpha_0$, $\|\alpha\|_{W^{1,\infty}}$, $\||\vec{\sigma}|\|_{L^\infty}$, $\|p\|_\infty$, $\Omega$ and $T$ such that
    \begin{align*}
        \begin{split}
        &\|v\|_{L^2(0,T;H^2(\Omega))} + \|v\|_{C^0([0,T];H_0^1(\Omega))} + \|\pt v\|_{L^2(0,T;L^2(\Omega))} \\
        \leq &C\left(\|h\|_{L^2(0,T;L^2(\Omega))} + \|v_0\|_{H_0^1(\Omega)} \right).
        \end{split}
    \end{align*}
\end{proposition}

By this proposition, the associated weak solutions $y(\cdot,\cdot,f_k)$ are bounded in $C^0([0,T];H_0^1(\Omega))$, hence the terminal states $Y_k:=y(\cdot,T,f_k)$ form a bounded sequence in $H_0^1(\Omega)\cap H^2(\Omega)$. Using the compact embedding $H_0^1(\Omega)\hookrightarrow L^2(\Omega)$, extract a subsequence of $(Y_k)$ converging strongly in $L^2(\Omega)$. Therefore $\Psi(f_k)=Y_k$ has a strongly convergent subsequence in $L^2(\Omega)$ for every bounded $(f_k)$, which implies compactness. This completes the proof of the main theorem.

\label{Proposition:Phi:compact}
    The input-output operator $\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega)$ is compact.,

Question:
Let \(\Omega\subset \mathbb{R}^N\) be a bounded domain with \(C^2\) boundary, let \(T>0\), and fix the coefficients and initial datum in the Dirichlet Ginzburg--Landau problem
\[
\begin{cases}
\partial_t y-\operatorname{div}((a(x)+ib(x))\nabla y)+\vec r(x)\cdot \nabla y+p(x)y=f(x,t) & \text{in } \Omega\times(0,T),\\
y=0 & \text{on } \partial\Omega\times(0,T),\\
y(\cdot,0)=y_0 & \text{in } \Omega.
\end{cases}
\]
For each source term \(f\in L^2(0,T;L^2(\Omega))\), let \(y\) denote the corresponding solution and define the input-output operator
\[
\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega),\qquad \Psi(f)=y(\cdot,T).
\]
Which statement holds for this operator?

The operator \(\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega)\) is compact; equivalently, it sends every bounded subset of \(L^2(0,T;L^2(\Omega))\) into a relatively compact subset of \(L^2(\Omega)\).

Choices:
B. The operator \(\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega)\) is bounded but not compact; in fact, for every bounded sequence \((f_k)\subset L^2(0,T;L^2(\Omega))\), the sequence \((\Psi(f_k))\) is only guaranteed to be bounded in \(L^2(\Omega)\), not to admit a strongly convergent subsequence there.
C. The operator \(\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega)\) is continuous (in particular, bounded as a linear map).
D. The operator \(\Psi:L^2(0,T;L^2(\Omega))\to H_0^1(\Omega)\) is compact; equivalently, it sends every bounded subset of \(L^2(0,T;L^2(\Omega))\) into a relatively compact subset of \(H_0^1(\Omega)\).
E. The operator \(\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega)\) is compact only when the source set is restricted to a bounded, closed, convex subset \(\mathcal U\subset L^2(0,T;L^2(\Omega))\); on the whole space \(L^2(0,T;L^2(\Omega))\) one can assert continuity but not compactness.

upgrade from boundedness of terminal states to relative compactness via compact embedding

compactness is only obtained after embedding into L^2, not in H_0^1

domain of the operator: compactness proved on the whole bounded-sequence criterion in L^2, not only on a preset bounded closed convex subset

The stem only defines the PDE and the terminal-state map; it does not explicitly state or strongly hint that compactness is the key conclusion.

This is not a bare restatement of a theorem in the stem. The solver must distinguish among continuity, compactness in L^2, and a stronger false compactness claim in H_0^1.

Identifying A requires some reasoning about parabolic regularization and compact embeddings, so it is not immediate. However, the presence of C, which is also true as a weaker statement, reduces the need to generate the uniquely strongest conclusion in a clean way.

B, D, and E are plausible and reflect common misconceptions about boundedness vs compactness and target-space regularity. But C is a weakly true statement rather than a genuine distractor, which weakens the single-best-answer design.

A conceptually solid question with no answer leakage and good mathematical discrimination, but its quality is notably reduced by having a weaker true option (C), making the distractor set only partially effective.


\begin{theorem}\label{thm:MAIN}
For any stratum $\Omega\mathcal M_{g}(\mu)$ of holomorphic differentials, the projectivized stratum $\mathbb P\Omega\mathcal M_{g}(\mu)$ contains no positive-dimensional complete subvariety.
\end{theorem}

Let $\mu$ be a partition of $2g-2$. Denote by $\Omega\mathcal M_{g}(\mu)$ the stratum of Abelian differentials (i.e., differential one-forms) on smooth, connected, genus-$g$ complex curves whose zero and pole orders are prescribed by~$\mu$, and let $\mathbb P\Omega\mathcal{M}_{g}(\mu) = \Omega\mathcal{M}_{g}(\mu)/\mathbb C^{*}$ be the corresponding projectivized stratum.

It is a natural and meaningful question to study how the strata of differentials appear from the perspective of affine geometry—for instance, whether a given stratum can contain a positive-dimensional complete algebraic subvariety.

For a holomorphic signature~$\mu$, that is, when all entries of~$\mu$ are nonnegative, Gendron showed that the unprojectivized stratum $\Omega\mathcal M_{g}(\mu)$ of holomorphic differentials contains no positive-dimensional complete subvariety, by applying the maximum modulus principle to shortest saddle connections; see~\cite{G20}. In this case, an alternative proof was later given by the first-named author, using the positivity of certain divisor classes on the moduli space of curves; see~\cite{C23}.

When $\mu$ is a signature of (strictly) meromorphic differentials, again by exploiting the positivity of divisor classes, the first-named author proved that both the projectivized and unprojectivized strata, $\mathbb P\Omega\mathcal M_{g}(\mu)$ and $\Omega\mathcal M_{g}(\mu)$, of strictly meromorphic differentials contain no positive-dimensional complete subvarieties; see~\cite{C19, C24}. Therefore, the remaining question is whether the projectivized strata of holomorphic differentials can contain a positive-dimensional complete subvariety.

This remaining problem is not only the most challenging case, but also significant from the viewpoint of the geometry of canonical divisors. For instance, for $\mu = (2g-2)$, the minimal stratum $\mathbb P\Omega\mathcal M_{g}(2g-2)$ parameterizes subcanonical points $z$, where $(2g-2)z$ is a canonical divisor. Harris asked whether there exist complete families of such subcanonical points (see~\cite[Proof of Corollary~5]{harris84}), a question that has remained open for four decades.

In this paper, we resolve this problem as follows.

In this paper, we resolve this problem as follows.

\begin{lemma}\label{lem:LocalHarmonic}
Let $M$ be a positive-dimensional subvariety of the projectivized stratum $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ of holomorphic differentials.
For any $(X,\omega) \in M$ and any cylinder $\mathcal{C}$ in $(X,\omega)$, exactly one of the following holds:
\begin{enumerate}
\item There exists a neighborhood $U$ of $(X,\omega)$ in $M$ such that the conformal modulus of $\mathcal{C}$ remains constant on $U$ (\textbf{rigid cylinder});
\item The conformal modulus of $\mathcal{C}$ is a non-constant positive pluriharmonic function in a neighborhood of $(X,\omega)$ in $M$ (\textbf{flexible cylinder}).
\end{enumerate}
\end{lemma}

\begin{proposition}\label{prop:RigidCylinder}
Let $M$ be a positive-dimensional subvariety of the projectivized stratum $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ of holomorphic differentials.
Then for every $(X,\omega) \in M$, any cylinder $\mathcal{C}$ in $(X,\omega)$ has constant conformal modulus on a neighborhood of $(X,\omega)$ in $M$; equivalently, every cylinder in $(X,\omega)$ is rigid.
\end{proposition}

\begin{lemma}\label{lem:DichotomyPeriods}
Let $M$ be a positive-dimensional subvariety of the projectivized stratum $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ of holomorphic differentials.
For any $(X,\omega) \in M$, and for any pair of cylinders $\mathcal{C}_{1},\mathcal{C}_{2}$ in $(X,\omega)$ with waist curves $\alpha_{1}$ and $\alpha_{2}$, exactly one of the following holds:
\begin{enumerate}
\item There exists a neighborhood $U$ of $(X,\omega)$ in $M$ such that the ratio $ \frac{\int_{\alpha_{1}}\omega}{\int_{\alpha_{2}}\omega}$ is constant on $U$ (\textbf{rigid pair of cylinders});
\item The function $ \log\left|\frac{\int_{\alpha_{1}}\omega}{\int_{\alpha_{2}}\omega}\right|$ is a non-constant pluriharmonic function in a neighborhood of $(X,\omega)$ in $M$ (\textbf{flexible pair of cylinders}).
\end{enumerate}
\end{lemma}

\begin{proposition}\label{prop:RigidPair}
Let $M$ be a positive-dimensional complete subvariety contained in the projectivized stratum $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ of holomorphic differentials. Then, for any $(X,\omega) \in M$, every pair of cylinders in $(X,\omega)$ is rigid.
\end{proposition}

\begin{proof}[Proof of Theorem~\ref{thm:MAIN}]
We first consider the case of the projectivized stratum of flat tori $\mathbb{P}\Omega\mathcal{M}_{1}(\emptyset)$, which is one-dimensional. Any holomorphic curve contained in $\mathbb{P}\Omega\mathcal{M}_{1}(\emptyset)$ must coincide with it, and hence is not compact (since flat tori degenerate to a nodal sphere). In the remainder of the proof, we will therefore assume that $\mu$ is nonempty, so that the results of Section~\ref{sec:Cylinders} apply.
\par
We assume, for the sake of contradiction, that there exists a positive-dimensional complete subvariety $M$ in $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$. In the definition of the normalized length, we use the set $MaxCyl(X,\omega)$ of the cylinders of largest width among those having the largest conformal modulus. According to Propositions~\ref{prop:RigidCylinder} and~\ref{prop:RigidPair}, these cylinders remain the ones of largest width among those with the largest conformal modulus as $(X,\omega)$ is deformed in $M$. Deformation along a closed path in $M$ can, however, permute the elements of $MaxCyl(X,\omega)$.
\par
A saddle connection $\alpha$ is called \textbf{rigid} if its normalized length $l(\alpha)$ remains constant in a neighborhood of $(X,\omega)$ in $M$, and \textbf{flexible} otherwise. Observe that for any rigid (resp. flexible) saddle connection $\alpha$ of $(X,\omega)$, there exists a neighborhood of $(X,\omega)$ in $M$ where $\alpha$ persists as a saddle connection and remains rigid (resp. flexible).
This follows from the fact that the ratio $\frac{\int_{\alpha} \omega}{\int_{\beta} \omega}$ is holomorphic in the local coordinates of $M$.
\par
By hypothesis, any translation surface in $M$ can be deformed nontrivially within $M$, so every $(X,\omega)$ in $M$ contains at least one flexible saddle connection. We define the function
$$L\colon M \to \mathbb R^{\geq 0}$$
by assigning to $(X,\omega)$ the normalized length $l(\alpha)$, where $\alpha$ is one of the shortest flexible saddle connections in $(X,\omega)$.
Since a unit-area translation surface contains only finitely many saddle connections of length at most $K$ for any $K>0$, the function $L$ is well defined.
We first show that $L$ is a continuous function on $M$.
\par
We consider a sequence of surfaces $(X_{n},\omega_{n})_{n \in \mathbb{N}}$ in $M$ converging to a surface $(X_{\infty},\omega_{\infty})$ in $M$.
In each $(X_{n},\omega_{n})$, let $\alpha_{n}$ be a flexible saddle connection such that $L(X_{n},\omega_{n}) = l(\alpha_{n})$.
Denote by $\beta_{n}$ the waist curve of a cylinder in $MaxCyl(X_{n},\omega_{n})$.
Up to passing to a subsequence of $(X_{n},\omega_{n})_{n \in \mathbb{N}}$, we may assume that
$$l(\alpha_{n})_{n \in \mathbb{N}}=\frac{|\int_{\alpha_{n}} \omega_n|}{|\int_{\beta_{n}} \omega_n|}$$
converges to some limit $\lambda$.
Let $\alpha_{\infty}$ be a flexible saddle connection such that $L(X_{\infty},\omega_{\infty}) = l(\alpha_{\infty})$.
Since $\alpha_{\infty}$ persists as a flexible saddle connection in a sufficiently small neighborhood of $(X_{\infty},\omega_{\infty})$ in $M$, we deduce that $\lambda \leq l(\alpha_{\infty})$.
It remains to prove that $\lambda \geq l(\alpha_{\infty})$.
\par
Up to passing to a subsequence again, we may assume that the ratios
$$\frac{\int_{\alpha_{n}} \omega_n}{\int_{\beta_n} \omega_n}$$
converge to some limit value $\nu$ satisfying $0 \leq |\nu| = \lambda$. Lemma~\ref{lem:NormalizedLength} already establishes that the normalized length of any saddle connection is bounded below by some positive constant. We therefore deduce that $|\nu|>0$.
\par
Since this convergence takes place in an arbitrarily small contractible neighborhood of $(X_{\infty},\omega_{\infty})$, we can mark the zeros of $\omega_{\infty}$ and, up to taking a further subsequence, require that all the saddle connections $(\alpha_{n})_{n \in \mathbb{N}}$ join the same oriented pair of zeros. The cylinders of $MaxCyl(X_{n},\omega_{n})$ can be marked within this neighborhood, so we may also assume that all the waist curves $\beta_{n}$ belong to the same cylinder. We denote by $\beta$ a waist curve of this cylinder in $(X_{\infty},\omega_{\infty})$.
\par
A subsequence of saddle connections $(\alpha_{\phi(n)})_{n \in \mathbb{N}}$ accumulates on an arc $\gamma$ satisfying $$\frac{\int_{\gamma} \omega_{\infty}}{\int_{\beta} \omega_{\infty}}=\nu,$$
which is formed by one or several oriented saddle connections of the same oriented slope. For $n$ sufficiently large, the relative homology classes $[\alpha_{\phi(n)}]$ and $[\gamma]$ coincide, so at least one saddle connection forming $\gamma$ must be flexible, which we denote by $\delta$. Then this saddle connection $\delta$ satisfies
$l(\delta) \leq |\nu| = \lambda$,
and hence $\lambda \geq l(\alpha_{\infty})$.
Consequently, we obtain $L(X_{n},\omega_{n}) \rightarrow L(X_{\infty},\omega_{\infty})$ as $n \to \infty$, thereby establishing the continuity of $L$.
\par
Since $L$ is continuous on $M$, which is compact, it attains a global minimum realized by some flexible saddle connection $\alpha$ on a translation surface $(X,\omega)$ in $M$. Because the ratio $\frac{\int_{\alpha} \omega}{\int_{\beta} \omega}$ is holomorphic in the local coordinates of $M$, its modulus can decrease again along certain local deformation paths, leading to a contradiction. Therefore, no flexible saddle connection can exist on any translation surface in $M$, which implies that no nontrivial deformation exists for any translation surface in $M$. In conclusion, we deduce that there is no positive-dimensional complete subvariety in $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$.
\end{proof}

Besides Abelian differentials, one can also study the strata of $k$-differentials with prescribed zero and pole orders, where a $k$-differential is a section of the $k$-th power of the canonical bundle. Note that a $k$-differential with pole orders at most $k-1$ (i.e., when the corresponding $\frac{1}{k}$-translation surface has finite area) can be lifted via the canonical cyclic covering construction to the $k$-th power of a holomorphic one-form; see~\cite[Section 2]{BCGGM-k}. Theorem~\ref{thm:MAIN} therefore implies the following corollary, which settles the remaining case concerning complete subvarieties in the strata of $k$-differentials; see~\cite[Remark]{C23}.

\begin{corollary}\label{cor:k-diff}
    For any stratum $\Omega^k\mathcal M_{g}(\mu)$ of $k$-differentials whose pole orders are bounded by $k-1$, the projectivized stratum $\mathbb P\Omega^k\mathcal M_{g}(\mu)$ contains no positive-dimensional complete subvarieties.
\end{corollary}

To prove Theorem~\ref{thm:MAIN}, we refine Gendron’s argument from~\cite{G20}. By applying the maximum principle to suitably chosen plurisubharmonic functions, we show that, in any complete subvariety contained in a projectivized stratum, the following quantities are rigid:
\begin{itemize}
\item the conformal moduli of cylinders (see Proposition~\ref{prop:RigidCylinder});
\item the periods of closed geodesics, up to a global scaling (see Proposition~\ref{prop:RigidPair});
\item suitably normalized lengths of saddle connections (see Section~\ref{sec:Main}).
\end{itemize}

Finally, we remark that since $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ is a quasi-projective variety, its complete algebraic subvarieties and complex analytic subvarieties   coincide by the GAGA principle. Moreover, if a complete subvariety $M$ is singular, we may work with a resolution of singularities of $M$ and pull back the family of differentials accordingly. Alternatively, one can intersect a higher-dimensional complete subvariety with ample hypersurfaces until obtaining a complete algebraic curve. Therefore, Theorem~\ref{thm:MAIN} is equivalent to showing that $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ contains no complete algebraic curves, in which case we may work with the normalization of such a curve, which is smooth. Therefore, in these senses, even when $M$ is singular, we can still speak of and make use of holomorphic local coordinates on $M$. We shall do so without further comment.

To prove Theorem~\ref{thm:MAIN}, the authors “refine Gendron’s argument from~\cite{G20}.” The strategy is to “apply[] the maximum principle to suitably chosen plurisubharmonic functions” and deduce that on any complete subvariety contained in a projectivized stratum, several geometric quantities are rigid: (1) “the conformal moduli of cylinders” (Proposition~\ref{prop:RigidCylinder}); (2) “the periods of closed geodesics, up to a global scaling” (Proposition~\ref{prop:RigidPair}); and (3) “suitably normalized lengths of saddle connections” (Section~\ref{sec:Main}). They also reduce to the curve case: since $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ is quasi-projective, complete algebraic and complex analytic subvarieties coincide by GAGA; if needed one passes to a resolution/normalization, and by intersecting with ample hypersurfaces, Theorem~\ref{thm:MAIN} is “equivalent to showing that $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ contains no complete algebraic curves.”

\label{thm:MAIN}
For any stratum $\Omega\mathcal M_{g}(\mu)$ of holomorphic differentials, the projectivized stratum $\mathbb P\Omega\mathcal M_{g}(\mu)$ contains no positive-dimensional complete subvariety.,

Question:
Let $g$ be a genus and let $\mu$ be any holomorphic signature with total order $2g-2$ (so $\Omega\mathcal M_g(\mu)$ is the stratum of holomorphic Abelian differentials $\omega$ on smooth, connected, genus-$g$ complex curves $X$ whose zero orders are prescribed by $\mu$). Let
$$\mathbb P\Omega\mathcal M_g(\mu)=\Omega\mathcal M_g(\mu)/\mathbb C^*$$
be the projectivized stratum, obtained by identifying differentials that differ by nonzero scalar multiplication. Which statement holds for every such projectivized stratum?

The projectivized stratum $\mathbb P\Omega\mathcal M_g(\mu)$ contains no positive-dimensional complete subvariety.

Choices:
B. The projectivized stratum $\mathbb P\Omega\mathcal M_g(\mu)$ contains no complete subvariety of codimension one.
C. The projectivized stratum $\mathbb P\Omega\mathcal M_g(\mu)$ is not itself a positive-dimensional complete variety.
D. The projectivized stratum $\mathbb P\Omega\mathcal M_g(\mu)$ contains no positive-dimensional complete subvariety whenever $g\ge 2$.
E. The unprojectivized stratum $\Omega\mathcal M_g(\mu)$ contains no positive-dimensional complete subvariety.

absence_of_all_positive_dimensional_complete_subvarieties weakened to non-completeness of the whole stratum

The stem gives definitions and asks for a universal property, but it does not state or strongly hint at the correct conclusion. There is no explicit answer leakage.

The correct option is essentially a theorem-style statement about projectivized strata, so the item is close to direct recall. However, the presence of nearby alternatives prevents it from being a pure restatement.

Some reasoning is needed to distinguish the strongest true statement from weaker or altered variants, especially versus the weaker true option and the projectivized/unprojectivized confusion. Still, this is more theorem recognition than deep generative reasoning.

The distractors are well designed: one is a weaker true statement, others reflect common confusions about codimension, genus restrictions, and projectivized versus unprojectivized strata. They are plausible and mathematically distinct.

A solid MCQ with no answer leakage and strong distractors, but it mainly tests precise theorem recall rather than substantial generative mathematical reasoning.


\begin{thm}\label{W-WeylProjInv}
Weakly-Weyl curvature is a projective invariant quantity in Finsler spaces.
\end{thm}

The birth of a new projective invariant in any space marks a significant advancement in the field of geometry. Hermann Weyls pioneering work in the 1920s laid the foundation for the concept of Weyl curvature\cite{HermanWeyl}, a projective invariant tailored for Riemannian spaces, significantly influencing the advancement of Finsler geometry, a discipline with roots in the early 20th century \cite{PaulFinsler}.

In the 1920s, J. Douglas extended Weyl curvature to Finsler metrics, introducing the concept of Weyl and Douglas metrics. Weyl Finsler spaces are a special class of Finsler spaces that are characterized by the vanishing of the projective Weyl curvature.

The introduction of Weyl Finsler spaces has opened up new avenues for research and exploration in the field, making them a valuable tool for understanding geometric structures and their applications in various scientific disciplines.

The investigation of Weyl curvature in Finsler geometry persisted, leading Z. Szab\'{o} to confirm in the 1970s that Weyl metrics are precisely Finsler metrics with scalar flag curvature \cite{Weyl=Scalar}.
According to \cite{Sh2}, the Weyl curvature and Douglas curvature are considered among the fundamental projective invariants in Finsler geometry. This means that If a Finsler metric meets the criteria for Weyl curvature or Douglas curvature disappearing, then any Finsler metric that is projectively equivalent to it will also meet these criteria. Other notable projective invariants include the generalized Douglas-Weyl metrics and the class of Finsler metrics satisfying a certain equation involving the Weyl and Douglas metrics \cite{GDW}, \cite{SadeDouglas}. \\

The generalization of important geometric quantities and their applications is a remarkable phenomenon. In the field of projective invariant quantities, a notable example of this generalization is the class of $GDW$-metrics, which encompasses the important class of projective invariant Finsler metrics, including Weyl metrics. This raises a natural and important question: are there any other generalizations of Weyl structures that have a significant relationship with other projective invariant Finsler metrics? In other words, is there a broader framework that encompasses Weyl metrics or other projective invariant Finsler metrics, and if so, what are its implications for our understanding of geometric structures and their applications?

In this paper, we introduce a new perspective by developing projective invariant Weyl metrics. We define the attributes of both weakly-Weyl and generalized weakly-Weyl Finsler metrics, presenting a comprehensive approach to analyzing their relations with various types of other projective invariants. In particular, we establish the following theorems.

In the 1920s, J. Douglas extended Weyl curvature to Finsler metrics, introducing the concept of Weyl and Douglas metrics. Weyl Finsler spaces are a special class of Finsler spaces that are characterized by the vanishing of the projective Weyl curvature.

The generalization of important geometric quantities and their applications is a remarkable phenomenon. In the field of projective invariant quantities, a notable example of this generalization is the class of $GDW$-metrics, which encompasses the important class of projective invariant Finsler metrics, including Weyl metrics. This raises a natural and important question: are there any other generalizations of Weyl structures that have a significant relationship with other projective invariant Finsler metrics? In other words, is there a broader framework that encompasses Weyl metrics or other projective invariant Finsler metrics, and if so, what are its implications for our understanding of geometric structures and their applications?

\section{Weakly-Weyl Finsler metrics}
In this section, we introduce a new projective invariant quantity in Finsler geometry, known as weakly-Weyl Finsler metrics. This quantity is a generalization of the well-known Weyl curvature, which is characterized by Finsler metrics of scalar flag curvature. Weakly-Weyl Finsler metrics are defined as Finsler metrics for which the curvature $\tilde{W}$, referred to as the weakly-Weyl curvature, vanishes. The weakly-Weyl curvature is defined as follows.
\begin{Def}\label{DefW-Weyl}
For a Finsler space $(M,F)$ with the Weyl curvature $W=\{W_y\}_{y \in T_xM\setminus{0}}$, the weakly-Weyl curvature is defined as,
\be\label{W-Weyl}
\tilde{W}_y: T_xM \times T_xM \times T_xM \longrightarrow T_xM\\
\ee
\[
\tilde{W}_y (u, v, w)= \tilde{W}_j{^i}_{kl}u^j v^k w^l \frac{\partial} {\partial x^i},
\]
where $\tilde{W}_j{^i}_{kl}=W_j{^i}_{pl.k}y^p$ and $W_j{^i}_{pl}=\frac{1}{3}(W{^i}_{p.l}-W{^i}_{l.p})_{.j}$.
\end{Def}
Based on the definition provided above, we can conclude the following proposition
\begin{prop}
A Finsler metric has a vanishing weakly-Weyl curvature vanishes if and only if it is of $W$-quadratic.
\end{prop}
\begin{proof}
Assume that the Finsler metric \( F \) has a vanishing weakly-Weyl curvature. This implies that \( W_j{^i}_{pl.k} y^p = 0 \). Differentiating this expression with respect to \( y^m \), we obtain
\[
0= (W_j{^i}_{pl.k}y^p)_{.m}= W_j{^i}_{ml.k} + W_j{^i}_{pl.k.m}y^p= W_j{^i}_{ml.k} +( W_j{^i}_{pl.m}y^p)_{.k}- W_j{^i}_{kl.m}.
\]
Since we have \( W_j{^i}_{pl.m} y^p = 0 \), the equation simplifies to
\[
W_j{^i}_{ml.k} - W_j{^i}_{kl.m}=(W{^i}_{ml.k} - W{^i}_{kl.m})_{.j}= \frac{1}{3} (W{^i}_{m.k} - W{^i}_{k.m})_{.l.j} =0.
\]
This indicates the existence of a tensor \( \Omega_l{^i}_{km}= \Omega_l{^i}_{km}(x) \), with the property $\Omega_l{^i}_{km}=-\Omega_l{^i}_{mk}$, such that
\[
\frac{1}{3}(W{^i}_{m.k} - W{^i}_{k.m})_{.l}=\Omega_l{^i}_{km}(x).
\]
By contracting the above equation with \( y^l \) and \( y^m \), we derive
\[
W{^i}_{k} = -\Omega_l{^i}_{km}(x) y^l y^m,
\]
which completes the proof of the proposition.
\end{proof}
Then we define
\begin{Def}\label{DefW-WeylMetric}
A Finsler metric $F$ is called a weakly-Weyl Finsler metric if its weakly-Weyl curvature satisfies the following equation,
\[
\tilde{W}_j{^i}_{kl}=\omega_{jkl} y^i,
\]
where \( \omega_{jkl}=\omega_{jkl}(x,y) \) represents the tensor coefficients, and it holds that
\be\label{DefomegaW-Weyl}
\omega_{jkl|0} + \mu F\omega_{jkl}=0,
\ee
for a scalar function \( \mu \) defined on the tangent bundle \( TM \).
\end{Def}
Given that Weyl curvature is a projective invariant quantity in Finsler spaces, and recognizing the concept of weakly-Weyl curvature, it is simple to establish the subsequent Theorem.
\subsection{\textbf{Proof of Theorem \ref{W-WeylProjInv}}}
\begin{proof}
Consider the Finsler metrics $F$ and $\bar{F}$, which are projectively related through their geodesic coefficients, denoted by $G^{i}$ and $\bar{G}^{i}$, respectively. We have
\be\label{ProjG}
\bar{G}^{i}=G^{i}+P y^{i},
\ee
with projective factor $P$. After differentiating with regards to $y^{j}$, we will have
\begin{equation}\label{ProjGij}
\bar{G}^{i} \cdot{ }_{j}=G^{i}{ }_{. j}+P \delta^{i}{ }_{j}+P_{. j} y^{i},
\end{equation}
The Weyl curvature of both metrics, \( F \) and \( \bar{F} \), is identical. According to Definition \ref{DefW-WeylMetric}, it can be expressed as
\begin{equation}\label{Weylomega}
\bar{W}_j{^i}_{ml.k}y^m = W_j{^i}_{ml.k}y^m = \omega_{jkl} y^i,
\end{equation}
with the condition that \( \omega_{jkl|0} =- \mu F \omega_{jkl} \) for some scalar function \( \mu \) defined on the tangent bundle \( TM \). We will now demonstrate that
\[
\omega_{jkl||0} + \bar{\mu} \bar{F} \omega_{jkl}=0,
\]
for a scalar function \( \bar{\mu} \) on \( TM \). Here, the symbol \( || \) denotes the horizontal derivative with respect to the metric \( \bar{F} \). Based on equations \eqref{ProjG} and \eqref{ProjGij}, we obtain
\[
\omega_{jkl||0} = \omega_{jkl;0} - 2 (G^{r}+P y^{r}) \omega_{jkl.r} - (G^{r}_{. j}+P \delta^{r}_{j}+P_{. j} y^{r}) \omega_{rkl} - (G^{r}_{. k}+P \delta^{r}_{k}+P_{. k} y^{r}) \omega_{jrl}
\]
\[
- (G^{r}_{. l}+P \delta^{r}_{l}+P_{. l} y^{r}) \omega_{jkr}.
\]
According to equation \eqref{Weylomega}, we find that \( \omega_{jkl.r} y^r = -\omega_{jkl} \) and \( \omega_{jkl} y^l = \omega_{jkl} y^k = \omega_{jkl} y^j = 0 \). Thus, from the above equation, we get
\[
\omega_{jkl||0} = \omega_{jkl|0}-P\omega_{jkl}.
\]
According to assumption, $F$ is a weakly-Weyl metric, then there is a scalar function $\mu$ on $TM$ such that $\omega_{jkl|0} + \mu F \omega_{jkl}=0$. Then we conclude
\[
\omega_{jkl||0} = - (\mu F + P) \omega_{jkl} = - \bar{\mu} \bar{F} \omega_{jkl}.
\]
This completes the proof.
\end{proof}
According to the Definition \ref{DefW-WeylMetric}, it is straightforward to deduce the following proposition.
\begin{prop}
Weyl Finsler metrics and $W$-quadratic Finsler metrics are weakly-Weyl metrics.
\end{prop}

By the equation \eqref{WeylDotheta} in Theorem \ref{WeylDouglasTheta}, we have
\[
\tilde{W}_j{^i}_{kl\mid 0}+ \mu F \tilde{W}_j{^i}_{kl} - \lambda_r \tilde{W}_j{^r}_{kl} y^i
\]
\[
=D_j{^i}_{kl|0|0}+ \mu F D_{j}{^i}_{kl\mid 0}- \lambda_r D_j{^r}_{kl|0} y^i-\frac{1}{n+1} \big(\theta_{jkl|0} +(\mu F- \lambda_0) \theta_{jkl}\big) y^i
\]
The above equation suggests that there may exist some generalized weakly-Weyl Finsler metrics that are not $GDW$-metrics. It also shows that there may exist a $GDW$-metric $F$ with $D_j{^i}_{kl|0}=d_{jkl}y^i$, that is not a generalized weakly-Weyl Finsler metric. Based on the equation \eqref{WeylDotheta}, for this metric we have
\[
\tilde{W}_j{^i}_{kl|0}+ \mu F \tilde{W}_j{^i}_{kl}- \lambda_r \tilde{W}_j{^r}_{kl}y^i= \big(\tilde{d}_{jkl\mid 0}+(\mu F- \lambda_0)\tilde{d}_{jkl}\big)y^i,
\]
where $\tilde{d}_{jkl}=d_{jkl}-\frac{1}{n+1}\theta_{jkl}$ and $\lambda_0=\lambda_r y^r$. The formula for $\theta_{jkl}$ is presented in the Theorem \ref{WeylDouglasTheta}.\\
The class of generalized weakly-Weyl Finsler metrics is a projective invariant class that has non-empty intersections with the class of $GDW$-metrics, such as the class of weakly-Weyl metrics. However, generalized weakly-Weyl metrics are not necessarily a subset of or contained within the class of $GDW$-metrics.
\section*{Declarations}
\subsection*{\textbf{Competing Interests}}
\quad The authors declare that they have no competing interests related to the work submitted for publication.
\subsection*{\textbf{Funding}}
\quad The authors acknowledge that there were no financial supports received for this study.
\subsection*{\textbf{Ethical Approval}}
\quad This research did not involve human participants or animals, and therefore ethical approval was not required.
\subsection*{\textbf{Informed Consent}}
\quad As this study did not involve human participants, informed consent was not applicable.

Additionally, we address the class of generalized weakly-Weyl Finsler metrics. Theorem \ref{W-WeylProjInv} can be understood as a direct consequence of Theorem \ref{GW-WeylProjInv} below. Specifically, since the class of weakly-Weyl Finsler metrics is a subclass of the generalized weakly-Weyl Finsler metrics, the projective invariance of the weakly-Weyl curvature follows naturally from the closure property of the broader class under projective changes. This logical dependency highlights that the invariance result presented in Theorem \ref{W-WeylProjInv} is subsumed by the more general statement on the stability of the entire generalized family in Theorem \ref{GW-WeylProjInv}.

\begin{thm}\label{GW-WeylProjInv}
The class of generalized weakly-Weyl Finsler metrics is closed under projective changes.
\end{thm}
The method of establishing generalized weakly-Weyl metrics is somewhat similar to the technique utilized in the study on \cite{SadeDouglas}, which introduced relatively isotropic $\tilde{D}$ metrics.
Our framework provides a unified approach to the study of projective invariant Finsler metrics, including Weyl metrics and other notable examples. This research leads to a different and more straightforward proof of Sakaguchi's Theorem, along with its generalization. We first prove the following Theorem.
\begin{thm} \label{WeylDouglasTheta}\label{PreSakaguchi}
For every Finsler metric $F$, with Weyl curvature $W=\{W_y\}_{y \in T_xM\setminus{0}}$ and Douglas curvature $D=\{D_y\}_{y \in T_xM\setminus{0}}$, it follows that
\begin{equation}\label{WeylDotheta}
W_j{^i}_{ml.k}y^m=D_j{^i}_{kl|0} -\frac{1}{n+1} \theta_{jkl} y^i,
\end{equation}
where $\theta_{jkl}=2E_{jk|l}-\frac{1}{3}(R{^s}_{l.s}-(n+2)R_{.l})_{.j.k}$.
\end{thm}
The subsequent theorem is a straightforward consequence of the previous one.
\begin{thm}\label{GSakaguchi}(\textbf{Generalized Sakaguchi Theorem})\\
Every weakly-Weyl Finsler metric is a $GDW$-metric.
\end{thm}
Additionally, we explore several examples within this new class of projective invariant Finsler metrics, focusing particularly on the class of weakly-Weyl spherically symmetric Finsler metrics.
\begin{thm}\label{W-WeylSphThm}
Every spherically symmetric Finsler metrics in $\mathbb{R}^n$ is weakly-Weyl if and only if it is of $W$-quadratic.
\end{thm}
The key question we explore in this paper is whether there exist other generalizations of Weyl structures that have a significant relationship with other projective invariant Finsler metrics. Our novel framework allows us to address this question by introducing a unified perspective on these important geometric structures.
\\
The symbols ${}_{.}$ and ${}_{|}$ in this article denote the vertical and horizontal derivatives with respect to the Berwald connection within the Finsler metric $F$. \\
Moreover, the subscript ${}_0$ represents the operation of contraction by $y^m$ as shown by the subscript ${}{m}$, while the symbol ${}_{;m}$ signifies differentiation in relation to $x^m$.\\
Additionally, the subscript ${}_{s}$ and ${}_{r}$ signify the derivative with respect to $s$ and $r$, respectively.

Theorem~\ref{W-WeylProjInv} is presented as a corollary of Theorem~\ref{GW-WeylProjInv}: since “the class of weakly-Weyl Finsler metrics is a subclass of the generalized weakly-Weyl Finsler metrics,” the “projective invariance of the weakly-Weyl curvature follows naturally from the closure property of the broader class under projective changes.” Equivalently, the invariance result in Theorem~\ref{W-WeylProjInv} is “subsumed by the more general statement on the stability of the entire generalized family in Theorem~\ref{GW-WeylProjInv}.”

To prove the main theorem, we present it as a corollary of the following theorem.

\begin{thm}\label{GW-WeylProjInv}
The class of generalized weakly-Weyl Finsler metrics is closed under projective changes.
\end{thm}

Since “the class of weakly-Weyl Finsler metrics is a subclass of the generalized weakly-Weyl Finsler metrics,” the “projective invariance of the weakly-Weyl curvature follows naturally from the closure property of the broader class under projective changes.” Equivalently, the invariance result in the main theorem is “subsumed by the more general statement on the stability of the entire generalized family in the theorem above.”

\label{W-WeylProjInv}
Weakly-Weyl curvature is a projective invariant quantity in Finsler spaces.,

Question:
For a Finsler metric \((M,F)\), let \(W\) denote its Weyl curvature, and define the weakly-Weyl curvature by
\[
\tilde{W}_y(u,v,w)=\tilde{W}_j{}^i{}_{kl}u^jv^kw^l\frac{\partial}{\partial x^i},
\]
where
\[
\tilde{W}_j{}^i{}_{kl}=W_j{}^i{}_{pl.k}y^p,
\qquad
W_j{}^i{}_{pl}=\frac{1}{3}(W^i_{p.l}-W^i_{l.p})_{.j}.
\]
Suppose \(F\) and \(\bar F\) are projectively equivalent Finsler metrics on the same manifold, meaning their sprays have the same geodesics as point sets (equivalently, their spray coefficients satisfy \(\bar G^i=G^i+P(x,y)y^i\) for some scalar function \(P\)). Which statement holds for every such pair?

Their weakly-Weyl curvatures coincide; equivalently, the weakly-Weyl curvature \(\tilde W\) is a projective invariant quantity in Finsler geometry.

Choices:
B. Their weakly-Weyl curvatures coincide provided the projective factor \(P\) depends only on \(x\); in general, \(\tilde W\) need not be preserved under an arbitrary projective change \(\bar G^i=G^i+P(x,y)y^i\).
C. If \(F\) has vanishing weakly-Weyl curvature, then every Finsler metric \(\bar F\) projectively equivalent to \(F\) also has vanishing weakly-Weyl curvature.
D. Projective equivalence preserves only the vanishing of the weakly-Weyl curvature: one has \(\tilde W=0\iff \bar{\tilde W}=0\), but the tensors \(\tilde W\) and \(\bar{\tilde W}\) need not coincide in general.
E. For any projectively equivalent pair \(F\) and \(\bar F\), there exists a scalar function \(\mu\) on \(TM\) such that their weakly-Weyl curvatures satisfy \(\bar{\tilde W}_j{}^i{}_{kl}=e^{\mu}\tilde W_j{}^i{}_{kl}\); hence the vanishing of \(\tilde W\) is projectively invariant, though \(\tilde W\) itself is only relatively invariant.

The stem does not state or strongly hint that weakly-Weyl curvature is projectively invariant; it only gives definitions and asks which conclusion is valid under projective equivalence.

The item is close to a direct theorem recall question: the correct option is essentially the standard invariance statement. Still, the presence of weaker and distorted variants means it is not a pure verbatim restatement.

Some reasoning is needed to distinguish full tensor equality from weaker consequences like preservation of vanishing or relative invariance. However, for a student who knows the theorem, the answer is fairly immediate.

Several distractors are mathematically plausible and reflect common confusions, especially between absolute invariance and weaker zero-locus preservation. But choice C is also true if A is true, so the item is not cleanly single-correct as written.

A reasonably strong theorem-recall MCQ with plausible distractors, but weakened by ambiguity: one distractor states a weaker true consequence of the correct answer, so the question does not cleanly enforce a unique choice.


\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 a countable group mean a group with domain \(\omega\), viewed in the standard Borel space of all countable groups. For countable groups \(A\) and \(B\), write \(A \leftarrow\!\!\leftarrow B\) if there exists a surjective homomorphism from \(B\) onto \(A\) (equivalently, \(A\) is isomorphic to a quotient of \(B\)). Let bi-epimorphism mean the symmetric relation \(A \leftarrow\!\!\leftarrow B\) and \(B \leftarrow\!\!\leftarrow A\). Which statement holds for these relations on the Borel space of countable groups? Here, “complete analytic quasi-order” means an analytic quasi-order to which every analytic quasi-order on a standard Borel space Borel reduces, and “complete analytic equivalence relation” is defined analogously among analytic equivalence relations.

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

Choices:
B. The relation \(\{\langle A,B\rangle : A \leftarrow\!\!\leftarrow B\}\) on countable groups is an analytic quasi-order, but it is not complete analytic. Consequently, the bi-epimorphism relation on the same Borel space is analytic, though not a complete analytic equivalence relation.
C. The relation \(\{\langle A,B\rangle : A \leftarrow\!\!\leftarrow B\}\) on countable groups is an analytic quasi-order. Consequently, the bi-epimorphism relation on the same Borel space is an analytic equivalence relation.
D. The relation \(\{\langle A,B\rangle : A \leftarrow\!\!\leftarrow B\}\) on countable groups is a complete analytic equivalence relation. Consequently, the bi-epimorphism relation on the same Borel space is also a complete analytic equivalence relation.
E. The relation \(\{\langle A,B\rangle : A \leftarrow\!\!\leftarrow B\}\) on countable groups is a complete analytic quasi-order, and moreover there is a direct Borel reduction from every analytic quasi-order to epimorphism on countable groups without passing through pointed reflexive graphs.

analytic-completeness obtained via the two-step Borel reduction

dropped completeness in both the quasi-order and equivalence-relation conclusions

necessity of the intermediate pointed reflexive graph reduction step

The stem defines the epimorphism and bi-epimorphism relations and asks for the correct descriptive-set-theoretic classification, but it does not explicitly or implicitly reveal that the relation is complete analytic.

The question is not a direct restatement of a theorem from the stem itself; the student must choose among materially different claims about analyticity, completeness, and quasi-order versus equivalence-relation status.

Some reasoning is needed to compare the strength and type of the options, but the item mainly tests recall of a known theorem rather than derivation. Generative pressure is also weakened because one distractor states a weaker claim that would still be true if the correct choice is true.

Several distractors are plausible and target real confusions (analytic vs complete analytic, quasi-order vs equivalence relation, overclaiming a direct reduction). However, choice C is a weaker true statement implied by A, so it is not a clean distractor and introduces ambiguity.

Good on avoiding leakage and tautology, but only moderate as a reasoning item. Its main flaw is ambiguity: option C appears true as a weaker consequence of A, so the MCQ does not cleanly enforce a unique correct answer.