\begin{theorem}
\label{T2}
Assume \eqref{(2)}-\eqref{(6)} hold. Then,
\begin{itemize}
\item[(i)] If $0<q<1-\theta$, there exists a distributional solution $u\in W_{0}^{1,(r_{j})}(\Omega)$ to problem \eqref{(1)}, such that
\begin{align}
\label{(17)}
r_{j}=\frac{N(\overline{p}-\theta)}{\overline{p}(N-\theta)}p_{j},\quad \forall \; j\in J.
\end{align}
\item[(ii)] If $1-\theta<q\leq1$, there exists a distributional solution $u\in W_{0}^{1,(\eta_{j})}(\Omega)$ to problem \eqref{(1)}, such that
\begin{align}
\label{(017)}
r_{j}<\frac{N(\overline{p}-1+q)}{\overline{p}(N+q-1)}p_{j},\quad \forall \; j\in J.
\end{align}
\item[(iii)] If $q>1$, there exists a distributional solution $u\in W_{0}^{1,(p_{j})}(\Omega)$ to problem \eqref{(1)}.
\end{itemize}
\end{theorem}

Anisotropic elliptic equations describe physical and biological phenomena where
diffusion or conductivity varies with direction, such as flow in heterogeneous
porous media \cite{R18}, epidemic propagation across irregular landscapes
\cite{R19}, and thermistor models exhibiting directional heat–electric
responses \cite{HHH1,HHH2,HHH3}. Singular elliptic equations also arise in many
applications, featuring blow-up behaviour, boundary singularities
\cite{CD,ZD}. These problems often require
techniques beyond the classical isotropic theory \cite{V}, especially when
combined with nonstandard growth conditions \cite{R1,R2}.

Let \(\Omega \subset \mathbb{R}^{N}\) be a bounded open set, where \(N \geq 2\), and let
\((p_{j}) = (p_{1}, \ldots, p_{N}) \in \mathbb{R}^{N}\) satisfy
\[
1 \leq p^{-} = \min_{j \in J}\{p_{j}\} \leq p^{+} = \max_{j \in J}\{p_{j}\} < \infty .
\]

The anisotropic Sobolev spaces \(W^{1,(p_{j})}(\Omega)\) and \(W_{0}^{1,(p_{j})}(\Omega)\) are defined respectively by
\[
\begin{aligned}
W^{1,(p_{j})}(\Omega)
&= \big\{ v \in W^{1,1}(\Omega) :\; D_{j}v \in L^{p_{j}}(\Omega)\ \text{for all } j \in J \big\},\\[2mm]
W_{0}^{1,(p_{j})}(\Omega)
&= \big\{ v \in W_{0}^{1,1}(\Omega) :\; D_{j}v \in L^{p_{j}}(\Omega)\ \text{for all } j \in J \big\}.
\end{aligned}
\]

Equivalently, \(W_{0}^{1,(p_{j})}(\Omega)\) is the closure of \(C_{0}^{\infty}(\Omega)\) with respect to the norm
\[
\|v\|_{W_{0}^{1,(p_{j})}(\Omega)}
= \sum_{j \in J} \|D_{j}v\|_{L^{p_{j}}(\Omega)}.
\]

Endowed with this norm, \(W_{0}^{1,(p_{j})}(\Omega)\) is a separable and reflexive Banach space.
Its dual space is denoted by \(W^{-1,(p'_{j})}(\Omega)\), where for each \(j \in J\), \(p_{j}'\) is the conjugate exponent of \(p_{j}\), that is,
\[
\frac{1}{p_{j}} + \frac{1}{p_{j}'} = 1.
\]

This article's main goal is to investigate the following problem.
\begin{equation}
\label{(1)}
\left\{\begin{array}{ll}\displaystyle-\sum\limits_{j\in J} D_{j}\left(\left[a(x)+u^{q}\right]\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)+b(x)\sum\limits_{j\in J}\frac{\vert D_{j}u\vert^{p_{j}}}{ u^{\theta}}=f& \hbox{in}\;\Omega, \\
u>0& \hbox{in}\;\Omega, \\
 u  =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation}
where $\Omega$ is a bounded open subset of $\mathbb{R}^{N}$ ($N\geq 3$), $f$, $q$, $\theta$, and $p_{j}$ are satisfies  the following conditions
\begin{align}
&\label{(2)}
q>0,\quad 0<\theta<1,\\
&\label{(3)}
f\geq0,\;\; f\not\equiv0,\quad f\in L^{1}(\Omega),\\
&\label{(4)}
2\leq p_{1}\leq p_{2}\leq ... \leq p_{N-1}\leq p_{N}\quad \mbox{and}\quad 2\leq \overline{p}:=N\left(\sum\limits_{j\in J}\frac{1}{p_{j}}\right)^{-1}<N.
\end{align}
For some positive numbers $\alpha,\beta,\mu$, and $\nu$, we assume in this study that $a,b$ are measurable functions satisfying the following conditions.
\begin{align}
&\label{(5)}
 0<\alpha\leq a(x)\leq \beta,\\
&\label{(6)}
0<\mu\leq b(x)\leq \nu.
\end{align}
The isotropic case of problem \eqref{(1)}, where $p_j$ is constant for all $j \in J$, has been extensively studied by several authors, with notable contributions in \cite{olivia, boca92, boca97, dalill}. Also, a problem like \eqref{(1)} without the singular term was studied in more general context in \cite{FR1,FA1}.

A further motivation for this work comes from the isotropic analogue of problem \eqref{(1)}, studied in \cite{R0}. There, the authors considered
\begin{equation}
\label{P3}
\left\{\begin{array}{ll}\displaystyle-\mbox{div}\left(\left[a(x)+u^{q}\right]\nabla u\right)+b(x)\frac{\vert \nabla u\vert^{2}}{ u^{\theta}}=f& \hbox{in}\;\Omega, \\
u>0& \hbox{in}\;\Omega, \\
 u  =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation}
with $q>0$, $0<\theta<1$, and $f\in L^{1}(\Omega)$. They proved the existence of nonnegative solutions and derived precise summability results depending on the balance between the exponent of the  absorption term $q$ and the singular term's exponent $\theta$.
The present work investigates the same type of phenomenon in an \textit{anisotropic setting}, where the diffusion operator involves directional exponents $(p_{j})$ and the harmonic mean $\overline{p}$ replaces the Laplacian structure.

The problem \eqref{P3} in the case of $q = 0$ has been addressed by several authors, including \cite{A1, A2, A3, A4, G1, G2}.

 In the context of anisotropic elliptic problems like \eqref{(1)} with no absorption term (i.e., when $q=0$, $a\equiv 1$, and $b\equiv 0)$, Di Castro in \cite{R2} addressed the following problem:
\begin{equation*}
\left\{\begin{array}{ll}\displaystyle-\sum\limits_{j\in J} D_{j}\left(\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)=f& \hbox{in}\;\Omega, \\
 u  =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation*}
where $p_j$ increases with respect to $j$ such that $2 - \frac{1}{N} \leq \overline{p} < N$, and $f \in L^1(\Omega)$. The author demonstrated that this problem admits a  solution $u \in W_0^{1, (s_j)}(\Omega)$, where
\begin{align}
\label{HH}
1\leq s_{j}<\frac{N(\overline{p}-1)}{\overline{p}(N-1)}p_{j},\quad\forall\; j\in J.
\end{align}
and for all  $\Phi \in C_0^1(\Omega),$
\begin{equation}\label{HH1}
\sum\limits_{j\in J}\int_{\Omega}\vert D_{j}u\vert^{p_{j}-2}D_{j}u D_{j}\Phi\,dx =\int_{\Omega}f\Phi\,dx .
\end{equation}

Further, this problem has been studied under different assumptions on the data in \cite{Stamp, BO1, BO2, OR1, BO3}, particularly when $p_j$ is constant for all $j \in J$.

 In the case where $q = 0$ and with lower order terms with sign conditions, Di Castro in \cite{R1} explored the following  general anisotropic problem
\begin{equation}
\label{P2}
\left\{\begin{array}{ll}\displaystyle-\sum\limits_{j\in J} D_{j}\left(\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)+\sum\limits_{j\in J}g_{j}(x,u,\nabla u)=f& \hbox{in}\;\Omega, \\
 u  =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation}
where $f \in L^1(\Omega)$ and $g_j: \Omega \times \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}$ is a Carath\'eodory function satisfying the following conditions
\begin{itemize}
\item[(H1)] $g_{j}(x,s,\xi)\mbox{sgn}(s)\geq0$ for a.e. $x\in \Omega$, for all $s\in \mathbb{R}$ and $\xi\in \mathbb{R}^{N}$;
\item[(H2)] $\vert g_{j}(x,s,\xi)\vert\leq h(\vert s\vert)\vert\xi_{j}\vert^{p_{j}}$, for a.e. $x\in \Omega$, for all $s\in \mathbb{R}$ and $\xi\in \mathbb{R}^{N}$, where $h:\mathbb{R}\rightarrow \mathbb{R}^{+}$ is a continuous, nondecreasing function such that $h(t)>\gamma>0$ for $\vert t\vert$ sufficiently large;
\item[(H3)] There exists $\nu>0$ such that $\vert g_{j}(x,s,\xi)\vert\geq \nu \vert \xi_{j}\vert^{p_{j}}$ for a.e. $x\in \Omega$, for all $s\in \mathbb{R}$ and $\xi\in \mathbb{R}^{N}$.
\end{itemize}
The author investigated the existence and regularity of solutions to this problem, yielding the following results
\begin{itemize}
\item[$\bullet$] If conditions (H1) and (H2) hold, and if $p_j$ satisfies the inequality
$$\frac{\overline{p}(N-1)}{N(\overline{p}-1)}<p_{j}<\frac{\overline{p}(N-1)}{N-\overline{p}},\quad \forall\; j\in J,$$
then there exists at least one  solution $u \in W_0^{1, (s_j)}(\Omega)$, where $s_j$ is defined in \eqref{HH} and \eqref{HH1} holds.
\item[$\bullet$] If conditions (H1), (H2), and (H3) hold, then a weak solution $u$ exists in $W_0^{1, (p_j)}(\Omega)$.
\end{itemize}

On the other hand, when \(b = 0\) and \(q \neq 0\), we have recently addressed, see \cite{FH}, a corresponding problem involving a singular right-hand side of the type \(\displaystyle\frac{f}{u^{l}}\), with \(l > 0\).

We end this  brief history of the problem by recalling the paper \cite{bacii}, in their work, the authors establish the existence of weak solutions for a broad class of anisotropic Dirichlet problems of the type
\[
\mathcal{A}u + \Phi(x,u,\nabla u) = \Psi(u,\nabla u) + \mathcal{L}u + f \quad \text{in } \Omega,
\]
where \(\Omega \subset \mathbb{R}^N\) \((N \ge 2)\) is a bounded domain and \(f \in L^1(\Omega)\) is arbitrary.
The anisotropic principal operator is
\[
\mathcal{A}u = -\sum_{j=1}^N \partial_j\left( \vert\partial_j u\vert^{p_j - 2}\partial_j u \right),
\]
and the lower-order term \(\Phi\) is given by
\[
\Phi(x,u,\nabla u) = \left( 1 + \sum_{j=1}^N \mathfrak{a}_j \vert\partial_j u\vert^{p_j} \right)|u|^{m-2}u,
\]
with parameters \(m,p_j>1\), \(\mathfrak{a}_j \ge 0\), and \(\sum_{k=1}^N \frac{1}{p_k} > 1\).
A key novelty of the study is the inclusion of a possibly singular gradient-dependent nonlinearity
\[
\Psi(u,\nabla u) = \sum_{j=1}^N |u|^{\theta_j - 2}u , \vert\partial_j u\vert^{q_j},
\]
where \(\theta_j>0\) and \(0 \le q_j < p_j\).
The lower-order operator $\mathcal{L}$ belongs to the general class $\mathcal{BC}$ introduced in \cite{baci}.
The class $\mathcal{BC}$ consists of \emph{bounded} operators
\[
\mathcal{L} : W_0^{1,(p_j)}(\Omega) \rightarrow W^{-1,(p_j')}(\Omega)
\]
that satisfy the following structural conditions:

\begin{itemize}
    \item[(P$_1$)]
    The operator $\mathcal{A} - \mathcal{L}$ is coercive on $W_0^{1,(p_j)}(\Omega)$, namely,
    \[
    \frac{\langle \mathcal{A}u - \mathcal{L}u,\, u \rangle}{\|u\|_{W_0^{1,(p_j)}(\Omega)}} \to \infty
    \qquad \text{as } \|u\|_{W_0^{1,(p_j)}(\Omega)} \to \infty .
    \]

    \item[(P$_2$)]
    If $u_l \rightharpoonup u$ and $v_l \rightharpoonup v$ weakly in $W_0^{1,(p_j)}(\Omega)$, then
    \[
    \lim_{l \to \infty} \langle \mathcal{L}u_l,\, v_l \rangle
    = \langle \mathcal{L}u,\, v \rangle .
    \]
\end{itemize}

These assumptions allow the authors to include a broad family of lower-order terms while preserving the coercivity and compactness properties needed for their existence theory.

The authors obtain existence results under two different regimes:
\begin{enumerate}
\item when \(\theta_j > 1\) for every \(j = 1,\dots, N\);

\item when \(\theta_j \le 1\) for at least one index \(j\).
\end{enumerate}
   In the latter case, by assuming \(f \ge 0\) almost everywhere in \(\Omega\), they further prove the existence of nonnegative weak solutions.

 In our study, we face three main difficulties. The first comes from the anisotropic structure of the operator, which requires estimates adapted to different growth conditions in each direction. The second difficulty is the presence of the singular lower-order term depending on the gradient $\frac{\vert D_{j} u\vert^{p_{j}}}{\vert u\vert^{\theta}}$, which complicates the analysis, especially near the points where the unknown $u$ may approach zero. The third difficulty is related to the additional term
 $$
\displaystyle-\sum\limits_{j\in J} D_{j}\left(u^{q}\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)$$
whose nonlinear form interacts with the previous terms and makes the compactness arguments more delicate.
\begin{definition}
\label{Definition3.1}
Let the condition in \eqref{(3)} hold. A function $u\in W_{0}^{1,1}(\Omega)$ is a distributional solution to problem \eqref{(1)} if
\begin{align}
\label{(13)}
& u>0\quad\quad \mbox{ a.e. in}\; \Omega,\\
\label{(14)}
& \sum\limits_{j\in J}\left[a(x)+u^{q}\right]\vert D_{j}u\vert^{p_{j}-2}D_{j}u\in (L^{\sigma_{j}}(\Omega))^N,\quad \mbox{for all}\; \sigma_{j}<p_{j}',\\
\label{(15)}
& \sum\limits_{j\in J} b(x)\frac{\vert D_{j}u\vert^{p_{j}}}{u^{\theta}}\in L^{1}(\Omega),
\end{align}
and for all  $\Phi \in W_0^{1,d_j}(\Omega),\; d_j>p_j$
\begin{align}
\label{(16)}
&\displaystyle\sum\limits_{j\in J}\int_{\Omega}\left[a(x)+u^{q}\right]\vert D_{j}u\vert^{p_{j}-2}D_{j}u D_{j}\Phi +\sum\limits_{j\in J}\int_{\Omega} b(x)\frac{\vert D_{j}u\vert^{p_{j}}}{u^{\theta}}\Phi =\int_{\Omega}f\Phi\,dx .
\end{align}
\end{definition}
\par Our main results are as follows

A further motivation for this work comes from the isotropic analogue of problem \eqref{(1)}, studied in \cite{R0}. There, the authors considered
\begin{equation}
\label{P3}
\left\{\begin{array}{ll}\displaystyle-\mbox{div}\left(\left[a(x)+u^{q}\right]\nabla u\right)+b(x)\frac{\vert \nabla u\vert^{2}}{ u^{\theta}}=f& \hbox{in}\;\Omega, \\
u>0& \hbox{in}\;\Omega, \\
 u =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation}
with $q>0$, $0<\theta<1$, and $f\in L^{1}(\Omega)$. They proved the existence of nonnegative solutions and derived precise summability results depending on the balance between the exponent of the absorption term $q$ and the singular term's exponent $\theta$.
The present work investigates the same type of phenomenon in an \textit{anisotropic setting}, where the diffusion operator involves directional exponents $(p_{j})$ and the harmonic mean $\overline{p}$ replaces the Laplacian structure.

In the context of anisotropic elliptic problems like \eqref{(1)} with no absorption term (i.e., when $q=0$, $a\equiv 1$, and $b\equiv 0)$, Di Castro in \cite{R2} addressed the following problem:
\begin{equation*}
\left\{\begin{array}{ll}\displaystyle-\sum\limits_{j\in J} D_{j}\left(\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)=f& \hbox{in}\;\Omega, \\
 u =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation*}
where $p_j$ increases with respect to $j$ such that $2 - \frac{1}{N} \leq \overline{p} < N$, and $f \in L^1(\Omega)$. The author demonstrated that this problem admits a solution $u \in W_0^{1, (s_j)}(\Omega)$, where
\begin{align}
\label{HH}
1\leq s_{j}<\frac{N(\overline{p}-1)}{\overline{p}(N-1)}p_{j},\quad\forall\; j\in J.
\end{align}
and for all $\Phi \in C_0^1(\Omega),$
\begin{equation}\label{HH1}
\sum\limits_{j\in J}\int_{\Omega}\vert D_{j}u\vert^{p_{j}-2}D_{j}u D_{j}\Phi\,dx =\int_{\Omega}f\Phi\,dx .
\end{equation}

In the case where $q = 0$ and with lower order terms with sign conditions, Di Castro in \cite{R1} explored the following general anisotropic problem
\begin{equation}
\label{P2}
\left\{\begin{array}{ll}\displaystyle-\sum\limits_{j\in J} D_{j}\left(\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)+\sum\limits_{j\in J}g_{j}(x,u,\nabla u)=f& \hbox{in}\;\Omega, \\
 u =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation}
where $f \in L^1(\Omega)$ and $g_j: \Omega \times \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}$ is a Carath\'eodory function satisfying the following conditions
\begin{itemize}
\item[(H1)] $g_{j}(x,s,\xi)\mbox{sgn}(s)\geq0$ for a.e. $x\in \Omega$, for all $s\in \mathbb{R}$ and $\xi\in \mathbb{R}^{N}$;
\item[(H2)] $\vert g_{j}(x,s,\xi)\vert\leq h(\vert s\vert)\vert\xi_{j}\vert^{p_{j}}$, for a.e. $x\in \Omega$, for all $s\in \mathbb{R}$ and $\xi\in \mathbb{R}^{N}$, where $h:\mathbb{R}\rightarrow \mathbb{R}^{+}$ is a continuous, nondecreasing function such that $h(t)>\gamma>0$ for $\vert t\vert$ sufficiently large;
\item[(H3)] There exists $\nu>0$ such that $\vert g_{j}(x,s,\xi)\vert\geq \nu \vert \xi_{j}\vert^{p_{j}}$ for a.e. $x\in \Omega$, for all $s\in \mathbb{R}$ and $\xi\in \mathbb{R}^{N}$.
\end{itemize}
The author investigated the existence and regularity of solutions to this problem, yielding the following results
\begin{itemize}
\item[$\bullet$] If conditions (H1) and (H2) hold, and if $p_j$ satisfies the inequality
$$\frac{\overline{p}(N-1)}{N(\overline{p}-1)}<p_{j}<\frac{\overline{p}(N-1)}{N-\overline{p}},\quad \forall\; j\in J,$$
then there exists at least one solution $u \in W_0^{1, (s_j)}(\Omega)$, where $s_j$ is defined in \eqref{HH} and \eqref{HH1} holds.
\item[$\bullet$] If conditions (H1), (H2), and (H3) hold, then a weak solution $u$ exists in $W_0^{1, (p_j)}(\Omega)$.
\end{itemize}

\item when \(\theta_j \le 1\) for at least one index \(j\).
\end{enumerate}
 In the latter case, by assuming \(f \ge 0\) almost everywhere in \(\Omega\), they further prove the existence of nonnegative weak solutions.

In our study, we face three main difficulties. The first comes from the anisotropic structure of the operator, which requires estimates adapted to different growth conditions in each direction. The second difficulty is the presence of the singular lower-order term depending on the gradient $\frac{\vert D_{j} u\vert^{p_{j}}}{\vert u\vert^{\theta}}$, which complicates the analysis, especially near the points where the unknown $u$ may approach zero. The third difficulty is related to the additional term
 $$
\displaystyle-\sum\limits_{j\in J} D_{j}\left(u^{q}\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)$$
whose nonlinear form interacts with the previous terms and makes the compactness arguments more delicate.
\begin{definition}
\label{Definition3.1}
Let the condition in \eqref{(3)} hold. A function $u\in W_{0}^{1,1}(\Omega)$ is a distributional solution to problem \eqref{(1)} if
\begin{align}
\label{(13)}
& u>0\quad\quad \mbox{ a.e. in}\; \Omega,\\
\label{(14)}
& \sum\limits_{j\in J}\left[a(x)+u^{q}\right]\vert D_{j}u\vert^{p_{j}-2}D_{j}u\in (L^{\sigma_{j}}(\Omega))^N,\quad \mbox{for all}\; \sigma_{j}<p_{j}',\\
\label{(15)}
& \sum\limits_{j\in J} b(x)\frac{\vert D_{j}u\vert^{p_{j}}}{u^{\theta}}\in L^{1}(\Omega),
\end{align}
and for all $\Phi \in W_0^{1,d_j}(\Omega),\; d_j>p_j$
\begin{align}
\label{(16)}
&\displaystyle\sum\limits_{j\in J}\int_{\Omega}\left[a(x)+u^{q}\right]\vert D_{j}u\vert^{p_{j}-2}D_{j}u D_{j}\Phi +\sum\limits_{j\in J}\int_{\Omega} b(x)\frac{\vert D_{j}u\vert^{p_{j}}}{u^{\theta}}\Phi =\int_{\Omega}f\Phi\,dx .
\end{align}
\end{definition}
\par Our main results are as follows

\item when \(\theta_j \le 1\) for at least one index \(j\).
\end{enumerate}
 In the latter case, by assuming \(f \ge 0\) almost everywhere in \(\Omega\), they further prove the existence of nonnegative weak solutions.

We first note that hypothesis \eqref{(4)} already ensures that
\[
1 < r_{j} \le p_{j} \qquad \text{for every } j \in J.
\]
Moreover, by combining assumptions \eqref{(2)} and \eqref{(4)}, we obtain the stronger condition
\[
r_{j} > p_{j} - 1, \qquad \forall\, j \in J.
\]

The following lemma will be of central importance in the remainder of the proof, and it will be especially needed in the passage to the limit.
\begin{lem}\label{lmq}
Assume that the assumptions \eqref{(2)}-\eqref{(6)} hold true.
Then, we have
\begin{equation}\label{ugest}
\int_{\Omega} u_{n}^{q\sigma_j}\left\vert D_{j} u_{n}\right\vert^{\sigma_j(p_{j}-1)}\;dx \leq C\quad\quad \forall \sigma_j<p_j^\prime.
\end{equation}
\end{lem}
\begin{proof}
 Taking $ \varphi(u_n)=1-\frac{1}{(1-u_n)^\lambda},$ with $\lambda>1,$ as test function in the weak formulation \eqref{(21)}, we get
\begin{equation*}
(\lambda-1)\sum_{j\in J}\int_{\Omega} \frac{[a(x)+u_n^q]\vert D_j u_n\vert^{p_{j}}}{(1+u_n)^\lambda}+\sum_{j\in J}\int_{\Omega}b(x)\frac{u_{n}\vert D_{j}u_{n}\vert^{p_{j}}}{\left( u_{n}+\frac{1}{n}\right)^{\theta+1}}\varphi =\int_\Omega f_n \varphi(u_n).
\end{equation*}
We recall that there exists a positive constant $\mathcal{C}$ such that
$$(a(x)+u_n^q) \geq \mathcal{C}(1+u_n)^q,\quad \quad \forall q>1.$$
Dropping the positive term and using the fact that $\vert \varphi(u_n)\vert\leq 1,$ we obtain
\begin{equation}\label{est1}
\sum_{j\in J}\int_{\Omega} \frac{\vert D_j u_n\vert^{p_j}}{(1+u_n)^{\lambda-q}}\leq\frac{1}{\mathcal{C}(\lambda-1)}\int_\Omega \vert f_n\vert .
\end{equation}
For $1 < \sigma_{j} < p'_{j} = \frac{p_{j}}{p_{j} - 1}$, an application of H\"older's inequality together with estimate \eqref{est1} yields, for every $j \in J$,
\begin{align*}
\int_{\Omega} u_{n}^{q \sigma_{j}(p_{j}-1)}\left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)} \leq C\left( \int_{\Omega} (1+u_{n})^{\frac{\sigma_{j}(p_{j}-1)(\lambda-q+p_{j} q)}{p_{j}-\sigma_{j}(p_{j}-1)}}\right)^{\frac{p_{j}-\sigma_{j}(p_{j}-1)}{p_{j}}}.
\end{align*}
We now introduce the parameter $\sigma_{j} = \theta\frac{p_{j}}{p_{j}-1}$, with $\theta \in (0,1)$. Under this choice, one readily verifies that
$$
\frac{p_{j}-\sigma_{j}(p_{j}-1)}{p_{j}} = 1 - \theta,\quad \forall\; j\in J,$$
and
$$ \frac{\sigma_{j}(p_{j}-1)(\lambda-q+p_{j} q)}{p_{j}-\sigma_{j}(p_{j}-1)} = \frac{\theta}{1-\theta}[\lambda+q(p_{j}-1)],\quad \forall\; j\in J.
$$
Substituting into the previous estimate leads to
\begin{equation}
\label{E11}
\int_{\Omega} u_{n}^{q \sigma_{j}(p_{j}-1)}\left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)} \leq C\left( \int_{\Omega}(1+ u_{n})^{\frac{\theta}{1-\theta}[\lambda+q(p_{j}-1)]}\right)^{1-\theta},
\end{equation}
for every $j\in J$. Hence, we deduce the following inequality:
\begin{align*}
&\prod_{i\in J}\left(\int_{\Omega} u_{n}^{q \sigma_{j}(p_{j}-1)}\left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)}\;dx\right)^{\frac{1}{\theta p_{j}}}\\
&\quad \leq C\prod_{i\in J}\left( \int_{\Omega} (1+u_{n})^{\frac{\theta}{1-\theta}[\lambda+q(p_{j}-1)]}\;dx\right)^{\frac{1-\theta}{\theta p_{j}}}.
\end{align*}
By invoking the anisotropic inequality \eqref{(9)} with $\delta_{j} = \sigma_{j}(p_{j}-1) \leq p_{j}$ (due to $\theta < 1$), $t_{j} = q \geq 0$, and $\overline{\delta} = \theta \overline{p}$, we obtain
\begin{equation}\label{E12}
\left(\int_{\Omega}u_{n}^{r}\right)^{\frac{N}{\theta\overline{p}}-1} \leq C\prod_{i\in J}\left( \int_{\Omega} (1+u_{n})^{\frac{\theta}{1-\theta}[\lambda+q(p_{j}-1)]}\right)^{\frac{1-\theta}{\theta p_{j}}}.
\end{equation}
In order for the preceding inequality to hold uniformly for all $j \in J$, we must enforce that
$$
r = \frac{\theta}{1-\theta}[\lambda+q(p_{j}-1)], \quad \forall\, j\in J.
$$
This leads to the following system of equations:
\begin{align}
& r=\frac{1+q}{b_{j}(N-1)-1+\frac{1}{\theta p_{j}}}, \quad \forall j\in J, \label{S1} \\
& r=\frac{\theta}{1-\theta}[\lambda+q(p_{j}-1)], \quad \forall j\in J, \label{S2} \\
&\sum_{i\in J}b_{j}=1, \quad b_{j} \geq 0, \quad \forall j\in J. \label{S3}
\end{align}
By combining equations \eqref{S1} and \eqref{S3}, one deduces
$$
r = \frac{N(1+q)\theta \overline{p}}{N-\theta\overline{p}}.
$$
In parallel, equation \eqref{S2} yields
$$
\theta = \frac{N(\overline{p}-\lambda+q)}{\overline{p}[N(1+q)-q\overline{p}-\lambda+q]}.
$$
Since $q \geq 0$ and $\lambda > 1$, it follows that $0 < \theta < 1$. Consequently, by combining these expressions, we arrive at
$$
r = \frac{N(\overline{p}-\lambda+q)}{N-\overline{p}}.
$$
Using this value of $r$ in inequality \eqref{E12}, we find
$$
\left(\int_{\Omega}u_{n}^{r}\right)^{\frac{N}{\theta\overline{p}}-1} \leq C\left(\int_{\Omega}u_{n}^{r}\right)^{\frac{(1-\theta)N}{\theta \overline{p}}}.
$$
Since the assumption $\overline{p} < N$ ensures that the left-hand exponent exceeds the right-hand one, we conclude that $\{u_{n}\}$ is bounded in $L^{r}(\Omega)$. This bound, together with estimate \eqref{E11}, implies that
$$
\int_{\Omega} u_{n}^{q \sigma_{j}(p_{j}-1)}\left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)} \leq C, \quad \forall\, j\in J.
$$
Given that $p_{j} > 2$ and $\sigma_{j} > 1$ for all $j\in\mathcal{E}$, we apply Hölder’s inequality to obtain
\begin{align*}
\int_{\Omega} u_{n}^{q \sigma_{j}} \left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)}
&= \int_{\left\{u_{n}<1\right\}} u_{n}^{q \sigma_{j}} \left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)} \\
&\qquad+ \int_{\left\{u_{n} \geq 1\right\}} u_{n}^{q \sigma_{j}} \left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)}\\
&\leq \int_{\Omega} \left\vert D_{j} \mathcal{T}_{1}(u_{n})\right\vert^{\sigma_{j}(p_{j}-1)}
+ \int_{\Omega} u_{n}^{q \sigma_{j}(p_{j}-1)} \left\vert D_{j} u_{n}\right\vert^{\sigma_{j}(p_{j}-1)} \\
&\leq C \left(\int_{\Omega} \left\vert D_{j} \mathcal{T}_{1}(u_{n})\right\vert^{p_{j}} \right)^{\frac{\sigma_{j}(p_{j}-1)}{p_{j}}} + C \\
&\leq C.
\end{align*}
Thus, it follows that the sequence $\left\lbrace u_{n}^{q}\left\vert D_{j} u_{n}\right\vert^{p_{j}-1}\right\rbrace$ is uniformly bounded in $L^{\sigma_{j}}(\Omega)$ for all $1 < \sigma_{j} < p'_{j}$ and for each $j \in J$.
\end{proof}
Let us specify some useful notation we will use from now on. For any \(n\in \mathbb{N}^{*}\) we define the following auxiliary functions:
\begin{equation*}
\mathcal{H}_{n}(l)=\int_{0}^{l}\frac{dt}{\left(t+\frac{1}{n}\right)^{\theta}},
\qquad
\mathcal{H}_{\infty}(l)=\int_{0}^{l}\frac{dt}{t^{\theta}},
\end{equation*}
and, let $\gamma=\frac{\nu}{\alpha}>0,$ we denote
\begin{equation*}
\Psi_{n}(l)=e^{-\gamma \mathcal{H}_{n}(l)},\qquad \Psi_{\infty}(l)=e^{-\gamma \mathcal{H}_{\infty}(l)}.
\end{equation*}
Note that, the function \(\mathcal{H}_{\infty}\) is well defined because \(\theta<1\). Moreover, it's clear that
$$\lim_{n\rightarrow \infty}\mathcal{H}_{n}(l)=\mathcal{H}_{\infty}(l)\text{ and }\lim_{n\rightarrow \infty}\Psi_{n}(l)=\Psi_{\infty}(l).$$
Observe that, for any \(\phi, u_{n}\in W_{0}^{1,(p_{j})}(\Omega)\cap L^{\infty}(\Omega)\) with \(\phi\ge0\) , we have \(\Psi_{n}(u_{n})\phi\in W_{0}^{1,(p_{j})}(\Omega)\cap L^{\infty}(\Omega)\).

We now give a brief description of the proof of our main theorem. The first step consists in removing the degeneracy of the principal operator and the singularity of the lower-order gradient term. For this purpose, we consider a sequence of approximate problems associated with \eqref{(1)}, where both difficulties are regularized. In the second step, we establish suitable a priori estimates for the approximate solutions. The main challenge here is to control the term
$$-\sum_{j \in J} D_{j}\left(u^{q}\vert D_{j}u\vert^{p_{j}-2}D_{j}u\right)$$
in an appropriate anisotropic Lebesgue space. Finally, by using Fatou’s lemma together with a standard compactness argument, we pass to the limit in the approximate problems and obtain a weak solution to the original equation.
\begin{remark}
We collect here several observations related to the assumptions of Theorem~\ref{T2}.

We first note that hypothesis \eqref{(4)} already ensures that
\[
1 < r_{j} \le p_{j} \qquad \text{for every } j \in J.
\]
Moreover, by combining assumptions \eqref{(2)} and \eqref{(4)}, we obtain the stronger condition
\[
r_{j} > p_{j} - 1, \qquad \forall\, j \in J.
\]

We also observe that the condition \(1 - \theta < q\) appearing in Theorem~\ref{T2}(ii) is equivalent to
\[
\frac{N(\overline{p}-\theta)}{\overline{p}(N-\theta)} p_{j}
\;<\;
\frac{N(\overline{p}-1+q)}{\overline{p}(N+q-1)} p_{j},
\qquad \forall\, j \in J.
\]
This relation will be useful when comparing the integrability exponents arising in the proof.

Furthermore, the hypothesis \(q \le 1\) in Theorem~\ref{T2}(ii) automatically implies that
\[
1 < \eta_{j} < p_{j},
\]
which plays an important role in establishing the required estimates.

It is also worth mentioning that, in the isotropic case \(p_{j} = 2\) for all \(j \in J\), Theorem~\ref{T2} reduces to known regularity results for classical elliptic equations. In particular, our conclusions coincide with those of Theorem~1.1 in \cite{R0}.

Finally, by applying the Sobolev embedding theorem, we can describe the integrability of the solutions.
For Theorem~\ref{T2}(i), the solution \(u\) also satisfies
\[
u \in L^{\overline{r}^{*}}(\Omega),
\qquad
\overline{r}^{*} = \frac{N(\overline{p}-\theta)}{N-\overline{p}}.
\]
For Theorem~\ref{T2}(ii), the solution enjoys the additional property
\[
u \in L^{s}(\Omega),
\qquad \text{for every }\; s < \frac{N(\overline{p}-1+q)}{\,N-\overline{p}\,}.
\]
\end{remark}
For ease of reading, Section 2 begins with a brief recall of the main analytical tools of anisotropic Sobolev spaces and the definition of the approximate problem. We then state the essential a priori estimates Lemmas as well as some convergences results, which form the main step on the proof, and finally we pass to the limit.

To prove Theorem~\ref{T2}, the authors give the following outline. First, they "remove the degeneracy of the principal operator and the singularity of the lower-order gradient term" by introducing "a sequence of approximate problems associated with \eqref{(1)}, where both difficulties are regularized." Second, they "establish suitable a priori estimates for the approximate solutions"; the main difficulty is "to control the term
$$-\sum_{j \in J} D_{j}\left(u^{q}\vert D_{j}u\vert^{p_{j}-2}D_{j}u\right)$$
in an appropriate anisotropic Lebesgue space." Finally, "by using Fatou’s lemma together with a standard compactness argument," they "pass to the limit in the approximate problems and obtain a weak solution to the original equation." They further indicate that Section~2 recalls tools on anisotropic Sobolev spaces and defines the approximate problem, then states the key a priori estimates and convergence results, and "finally we pass to the limit."

To prove the main theorem, the authors give the following outline. First, they “remove the degeneracy of the principal operator and the singularity of the lower-order gradient term” by introducing “a sequence of approximate problems associated with
\begin{equation}
\label{(1)}
\left\{\begin{array}{ll}\displaystyle-\sum\limits_{j\in J} D_{j}\left(\left[a(x)+u^{q}\right]\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)+b(x)\sum\limits_{j\in J}\frac{\vert D_{j}u\vert^{p_{j}}}{ u^{\theta}}=f& \hbox{in}\;\Omega, \\
u>0& \hbox{in}\;\Omega, \\
 u  =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation}
, where both difficulties are regularized.” Second, they “establish suitable a priori estimates for the approximate solutions”; the main difficulty is “to control the term
$$-\sum_{j \in J} D_{j}\left(u^{q}\vert D_{j}u\vert^{p_{j}-2}D_{j}u\right)$$
in an appropriate anisotropic Lebesgue space.” Finally, “by using Fatou’s lemma together with a standard compactness argument,” they “pass to the limit in the approximate problems and obtain a weak solution to the original equation.” They further indicate that the next part recalls tools on anisotropic Sobolev spaces and defines the approximate problem, then states the key a priori estimates and convergence results, and “finally we pass to the limit.”

\label{T2}
Assume
\begin{align}
&\label{(2)}
q>0,\quad 0<\theta<1,\\
&\label{(3)}
f\geq0,\;\; f\not\equiv0,\quad f\in L^{1}(\Omega),\\
&\label{(4)}
2\leq p_{1}\leq p_{2}\leq ... \leq p_{N-1}\leq p_{N}\quad \mbox{and}\quad 2\leq \overline{p}:=N\left(\sum\limits_{j\in J}\frac{1}{p_{j}}\right)^{-1}<N.
\end{align}
--
\begin{align}
&\label{(5)}
 0<\alpha\leq a(x)\leq \beta,\\
&\label{(6)}
0<\mu\leq b(x)\leq \nu.
\end{align}
hold. Then,
\begin{itemize}
\item[(i)] If $0<q<1-\theta$, there exists a distributional solution $u\in W_{0}^{1,(r_{j})}(\Omega)$ to problem
\begin{equation}
\label{(1)}
\left\{\begin{array}{ll}\displaystyle-\sum\limits_{j\in J} D_{j}\left(\left[a(x)+u^{q}\right]\vert D_{j}u\vert^{p_{j}-2} D_{j}u\right)+b(x)\sum\limits_{j\in J}\frac{\vert D_{j}u\vert^{p_{j}}}{ u^{\theta}}=f& \hbox{in}\;\Omega, \\
u>0& \hbox{in}\;\Omega, \\
 u  =0 & \hbox{on}\; \partial\Omega, \end{array}
 \right.
\end{equation}
such that
\begin{align}
\label{(17)}
r_{j}=\frac{N(\overline{p}-\theta)}{\overline{p}(N-\theta)}p_{j},\quad \forall \; j\in J.
\end{align}
\item[(ii)] If $1-\theta<q\leq1$, there exists a distributional solution $u\in W_{0}^{1,(\eta_{j})}(\Omega)$ to the above problem such that
\begin{align}
\label{(017)}
r_{j}<\frac{N(\overline{p}-1+q)}{\overline{p}(N+q-1)}p_{j},\quad \forall \; j\in J.
\end{align}
\item[(iii)] If $q>1$, there exists a distributional solution $u\in W_{0}^{1,(p_{j})}(\Omega)$ to the above problem.
\end{itemize}

Question:
Let \(J=\{1,\dots,N\}\). Consider the anisotropic Dirichlet problem
\[
\begin{cases}
-\displaystyle\sum_{j\in J} D_j\!
\left(\left[a(x)+u^{q}\right]\,|D_j u|^{p_j-2}D_j u\right)
+ b(x)\displaystyle\sum_{j\in J}\frac{|D_j u|^{p_j}}{u^{\theta}}=f & \text{in } \Omega,\\[2mm]
u>0 & \text{in } \Omega,\\
u=0 & \text{on } \partial\Omega.
\end{cases}
\]
Assume
\[
q>0,\qquad 0<\theta<1,
\]
\[
f\ge 0,\quad f\not\equiv 0,\quad f\in L^1(\Omega),
\]
\[
2\le p_1\le p_2\le \cdots \le p_N,
\qquad
2\le \overline p:=N\left(\sum_{j\in J}\frac1{p_j}\right)^{-1}<N,
\]
and the coefficients satisfy, a.e. in \(\Omega\),
\[
0<\alpha\le a(x)\le \beta,
\qquad
0<\mu\le b(x)\le \nu.
\]
Here \(W_0^{1,(s_j)}(\Omega)\) denotes the anisotropic Sobolev space of zero-trace functions such that \(D_j u\in L^{s_j}(\Omega)\) for each \(j\), and a distributional solution means a function with zero trace, positive a.e. in \(\Omega\), satisfying the equation in the sense of distributions. Which conclusion about existence and regularity is valid under these hypotheses?

The problem admits a distributional solution with the following case-by-case regularity: (i) if \(0<q<1-\theta\), then there exists \(u\in W_0^{1,(r_j)}(\Omega)\) with
\[
r_j=\frac{N(\overline p-\theta)}{\overline p\,(N-\theta)}\,p_j,\qquad \forall j\in J;
\]
(ii) if \(1-\theta<q\le 1\), then there exists a distributional solution \(u\in W_0^{1,(\eta_j)}(\Omega)\) with directional exponents satisfying
\[
\eta_j<\frac{N(\overline p-1+q)}{\overline p\,(N+q-1)}\,p_j,\qquad \forall j\in J;
\]
(iii) if \(q>1\), then there exists a distributional solution \(u\in W_0^{1,(p_j)}(\Omega)\).

Choices:
B.
The problem admits a distributional solution with the following case-by-case regularity: (i) if \(0<q\le 1-\theta\), then there exists \(u\in W_0^{1,(r_j)}(\Omega)\) with
\[
r_j=\frac{N(\overline p-\theta)}{\overline p\,(N-\theta)}\,p_j,\qquad \forall j\in J;
\]
(ii) if \(1-\theta<q\le 1\), then there exists a distributional solution \(u\in W_0^{1,(\eta_j)}(\Omega)\) with directional exponents satisfying
\[
\eta_j\le \frac{N(\overline p-1+q)}{\overline p\,(N+q-1)}\,p_j,\qquad \forall j\in J;
\]
(iii) if \(q>1\), then there exists a distributional solution \(u\in W_0^{1,(p_j)}(\Omega)\).
C.
The problem admits a distributional solution with the following case-by-case regularity: (i) if \(0<q<1-\theta\), then there exists \(u\in W_0^{1,(r_j)}(\Omega)\) with
\[
r_j=\frac{N(\overline p-\theta)}{\overline p\,(N-\theta)}\,p_j,\qquad \forall j\in J;
\]
(ii) if \(1-\theta<q\le 1\), then there exists a distributional solution \(u\in W_0^{1,(\eta_j)}(\Omega)\) for some directional exponents \((\eta_j)_{j\in J}\);
(iii) if \(q>1\), then there exists a distributional solution \(u\in W_0^{1,(p_j)}(\Omega)\).
D.
The problem admits a distributional solution with the following case-by-case regularity: (i) if \(0<q<1-\theta\), then there exists \(u\in W_0^{1,(r_j)}(\Omega)\) with
\[
r_j=\frac{N(\overline p-\theta)}{\overline p\,(N-\theta)}\,p_j,\qquad \forall j\in J;
\]
(ii) if \(1-\theta<q\le 1\), then there exists a distributional solution \(u\in W_0^{1,(\eta_j)}(\Omega)\) with directional exponents satisfying
\[
\eta_j<\frac{N(\overline p-1+q)}{\overline p\,(N+q-1)}\,p_j,\qquad \forall j\in J;
\]
(iii) if \(q>1\), then there exists a distributional solution \(u\in W_0^{1,(p_j)}(\Omega)\), and moreover for every \(q>0\) one may choose the solution in the full energy space \(W_0^{1,(p_j)}(\Omega)\).
E.
The problem admits a distributional solution with the following case-by-case regularity: (i) if \(0<q<1-\theta\), then there exists \(u\in W_0^{1,(r_j)}(\Omega)\) with
\[
r_j=\frac{N(\overline p-\theta)}{\overline p\,(N-\theta)}\,p_j,\qquad \forall j\in J;
\]
(ii) if \(1-\theta\le q\le 1\), then there exists a distributional solution \(u\in W_0^{1,(\eta_j)}(\Omega)\) with directional exponents satisfying
\[
\eta_j<\frac{N(\overline p-1+q)}{\overline p\,(N+q-1)}\,p_j,\qquad \forall j\in J;
\]
(iii) if \(q>1\), then there exists a distributional solution \(u\in W_0^{1,(p_j)}(\Omega)\).

critical threshold and strict upper bound in the intermediate regime

explicit quantitative upper bound on the exponents in case (ii)

loss of full anisotropic energy regularity for small/intermediate q caused by the delicate estimate on the nonlinear term


\label{Hao_Kodaira_codim_1}\cite[][Thm. 1.3]{Hao_Nowhere_Vanishing_Holomorphic_One-Forms_on_Varieties_of_Kodaira_Codimension_One}
    Let $X$ be a smooth projective variety of Kodaira dimension $\kappa(X)=\dim X-1$. If $X$ has a nowhere vanishing holomorphic 1-form, then for any minimal model $X^{min}$ of $X$ there exists a finite quasi-\'etale covering $X'\rightarrow X^{min}$ such that any $\mathbb{Q}$-factorialization $X''$ of $X'$ is a product $B\times E$, where $B$ is a minimal model of general type and $E$ is an elliptic curve.

\begin{theorem}\label{Hao_Kodaira_codim_1}\cite[][Thm. 1.3]{Hao_Nowhere_Vanishing_Holomorphic_One-Forms_on_Varieties_of_Kodaira_Codimension_One}
    Let $X$ be a smooth projective variety of Kodaira dimension $\kappa(X)=\dim X-1$. If $X$ has a nowhere vanishing holomorphic 1-form, then for any minimal model $X^{min}$ of $X$ there exists a finite quasi-\'etale covering $X'\rightarrow X^{min}$ such that any $\Q$-factorialization $X''$ of $X'$ is a product $B\times E$, where $B$ is a minimal model of general type and $E$ is an elliptic curve.
\end{theorem}

\begin{theorem}\label{precise_main_result}
    Let $X$ be a compact K\"ahler manifold of Kodaira dimension $\kappa(X)=\dim X-1$ that admits a holomorphic 1-form without zeros. Then there is a smooth projective variety $Y$ of general type and of dimension $\dim Y=\dim X-1$, an elliptic curve $E=\C/\Lambda$, a cohomology class $\eta\in H^1(Y,\OO_Y/\Lambda)$ and a finite \'etale cover $X'\rightarrow X$ such that $X'$ is bimeromorphic to $(Y\times E)^\eta$.
\end{theorem}

\begin{remark}\label{example_no_splitting}
    \Cref{fundamental_properties_special_torus_bundles} allows us to construct a compact K\"ahler manifold $X$ of Kodaira dimension $\kappa(X)=\dim X-1$ that admits a holomorphic 1-form without zeros such that no finite \'etale cover of $X$ splits into a product where one factor is an elliptic curve. This shows that \Cref{Hao_Kodaira_codim_1} fails in the K\"ahler case. Indeed, let $C$ be a smooth projective curve of genus $g\geq 2$, and let $E=\C/\Lambda$ be an elliptic curve. Consider the exact sequence
    \begin{equation*}
        H^1(C,\Lambda)\longrightarrow H^1(C,\OO_C)\stackrel{\mathrm{exp}}{\longrightarrow}H^1(C,\OO_C/\Lambda)\stackrel{c}{\longrightarrow}H^2(C,\Lambda),
    \end{equation*}
    and choose a class $\alpha\in H^1(C,\OO_C)$ that is not in the image of
    \begin{equation*}
        H^1(C,\Lambda)\otimes\Q\longrightarrow H^1(C,\OO_C).
    \end{equation*}
    Since $H^1(C,\OO_C)$ is a non-trivial complex vector space, and $H^1(C,\Lambda)$ is a finitely generated abelian group, such an $\alpha$ exists. Define $\eta:=\mathrm{exp}(\alpha)$. Note that the choice of $\alpha$ implies that $\eta$ is non-torsion. The exactness of the above sequence implies that the Chern class $c(\eta)$ vanishes. Thus, by \Cref{fundamental_properties_special_torus_bundles} \cref{torus_bundle_kaehler} and \cref{1-forms_on_torus_bundles}, the space $X:=(C\times E)^\eta$ is a compact K\"ahler manifold that admits a holomorphic 1-form without zeros. Since $C$ is of genus $g\geq 2$, $\kappa(X)=\dim C=\dim X-1$. We claim that no finite \'etale cover of $X$ splits into a product where one factor is an elliptic curve. To show this, suppose that a finite \'etale cover $X'\rightarrow X$ splits into a product $X'\cong D\times F$, where $F$ is an elliptic curve. Since $g(C)\geq 2$, the composition
    \begin{equation*}
        \{x\}\times F\longhookrightarrow D\times F\longrightarrow X\longrightarrow C
    \end{equation*}
    must be constant for every $x\in D$, and hence there exists a morphism $D\rightarrow C$ such that the composition $D\times F\rightarrow X\rightarrow C$ factors through $D\times F\rightarrow D\rightarrow C$. Since $D\times F\rightarrow X\rightarrow C$ is surjective, the morphism $D\rightarrow C$ cannot be constant. Therefore, the image $S\subset X$ of $D\times\{0\}\rightarrow X$ defines a multisection of $X\rightarrow C$. Thus, by \Cref{fundamental_properties_special_torus_bundles} \cref{torus_bundle_multisection_a}, the cohomology class $\eta$ is torsion, which is a contradiction.
\end{remark}

Let $X$ be a smooth projective variety over $\mathbb{C}$ with a nowhere vanishing holomorphic 1-form $\omega\in H^0(X,\Omega^1_X)$. In \cite{Popa_Schnell_Kodaira_dimension_zeros_1-forms}, Popa and Schnell showed that the Kodaira dimension of $X$ is bounded by $\kappa(X)\leq\dim X-1$. To prove this result, Popa-Schnell applied the theory of mixed Hodge modules. A fortiori, Hao gave the following geometric explanation of this behavior, for $\kappa(X)=\dim X-1$.

\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\OO}{\mathcal{O}}
\newcommand{\pic}{\mathrm{Pic}}
\newcommand{\im}{\mathrm{im}}
\newcommand{\Hom}{\mathrm{Hom}}
\newcommand{\Ext}{\mathrm{Ext}}
\newcommand{\id}{\mathrm{id}}

\title{Holomorphic one-forms without zeros on K\"ahler manifolds of Kodaira codimension one}
\author{Simon Pietig}
\subjclass{32Q15, 32J18, 32Q57, 32H04}
\address{Leibniz Universit\"at Hannover, Institut f\"ur Algebraische Geometrie, Welfengarten 1, 30167 Hannover}
\address{\href{mailto:pietig@math.uni-hannover.de}{pietig@math.uni-hannover.de}}
\address{\href{https://orcid.org/0009-0001-1800-6799}{Orcid iD}}

@book{Eisenbud_Harris_3264,
 author = {Eisenbud, David and Harris, Joe},
 title = {3264 and all that: Intersection theory in algebraic geometry},
 year = {2016},
 publisher = {Cambridge Univ. Press}
}

@article{Hao_Nowhere_Vanishing_Holomorphic_One-Forms_on_Varieties_of_Kodaira_Codimension_One,
 author = {Hao, Feng},
 title = {Nowhere Vanishing Holomorphic One-Forms on Varieties of Kodaira Codimension One},
 journal = {Int. Math. Res. Not. IMRN},
 volume = {2024},
 number = {6},
 pages = {4501--4515},
 year = {2023}
}

@article{Popa_Schnell_Kodaira_dimension_zeros_1-forms,
 author = {Popa, Mihnea and Schnell, Christian},
 title = {Kodaira dimension and zeros of holomorphic one-forms},
 journal = {Ann. of Math.},
 volume = {179},
 number = {3},
 pages = {1109--1120},
 year = {2017}
}

@online{Hao_Schreieder_erratum,
 author = {Hao, Feng and Schreieder, Stefan},
 title = {Corrections to "Holomorphic one-forms" without zeros on threefolds},
 url = {https://www.iag.uni-hannover.de/fileadmin/iag/homepages/schreieder/publications/Correction_for_nowhere_vanishing_1-forms_on_3-folds_2.pdf},
 year = {2025}
}

@misc{church_gmm,
 title={Nowhere vanishing 1-forms on varieties admitting a good minimal model},
 author={Church, Benjamin},
 year={2024},
 eprint={2410.22753},
 archivePrefix={arXiv},
 primaryClass={math.AG},
 url={https://arxiv.org/abs/2410.22753},
}

@misc{Hao_Wang_Zhang_gmm,
 title={Good minimal models with nowhere vanishing holomorphic $1$-forms},
 author={Hao, Feng and Wang, Zichang and Zhang, Lei},
 year={2024},
 eprint={2412.12582},
 archivePrefix={arXiv},
 primaryClass={math.AG},
 url={https://arxiv.org/abs/2412.12582},
}

Recent work of Church \cite{church_gmm} and of Hao, Wang and Zhang \cite{Hao_Wang_Zhang_gmm} generalized Hao's result to a birational classification of smooth projective varieties with $g$ everywhere linearly independent holomorphic 1-forms assuming the abundance conjecture without any restrictions of the Kodaira dimension of $X$.\par
The main result of our paper is the following generalization of \Cref{Hao_Kodaira_codim_1} to the K\"ahler case. For a more precise version, see \Cref{precise_main_result} down below.
\begin{theorem}\label{main_result}
    Let $X$ be a compact K\"ahler manifold of Kodaira dimension $\kappa(X)=\dim X-1$. If $X$ has a nowhere vanishing holomorphic 1-form, then there exist a finite \'etale cover $X'\rightarrow X$ such that $X'$ is bimeromorphic to an elliptic fiber bundle $Z\rightarrow B$ with trivial monodromy over a smooth projective base $B$ of general type.
\end{theorem}
\begin{remark}
    In contrast to the projective situation, one cannot expect that the finite \'etale covering space $X'$ of $X$ in \Cref{main_result} is bimeromorphic to a product $Z\cong B\times E$, see \Cref{example_no_splitting} below.
\end{remark}
\subsection{Outline of the argument}
In the projective case, Hao argues as follows: Let $X$ be a smooth projective variety over $\mathbb{C}$ of Kodaira codimension 1 that admits a holomorphic 1-form $\omega\in H^0(X,\Omega^1_X)$ without zeros. Let $X^{min}$ be a good minimal model of $X$, and consider the Iitaka fibration $f\colon X^{min}\rightarrow Y$. By \cite[][Thm. 2.1]{Popa_Schnell_Kodaira_dimension_zeros_1-forms}, the holomorphic 1-form $\omega$ cannot come from $Y$. In particular, the general fiber of $f$ is not contracted by the Albanese morphism. Therefore, it is mapped to a translate of a fixed elliptic curve $E\subset Alb(X)$. As $Alb(X)$ is projective, we can dualize the inclusion $E\subset Alb(X)$ to get a morphism
\begin{equation*}
    \varphi\colon X^{min}\longrightarrow Alb(X)\longrightarrow E.
\end{equation*}
The variety $B$ in \Cref{Hao_Kodaira_codim_1}, as well as the splitting $X''\cong B\times E$ of a $\mathbb{Q}$-factorialization $X''\rightarrow X'$ of a finite quasi-\'etale cover $X'\rightarrow X^{min}$ are then constructed from a general fiber of $\varphi\colon X^{min}\rightarrow E$.\par
Now, let $X$ be a compact connected K\"ahler manifold of Kodaira codimension 1 that admits a holomorphic 1-form $\omega\in H^0(X,\Omega^1_X)$ without zeros. Note that we cannot apply the result by Popa-Schnell \cite[][Thm. 2.1]{Popa_Schnell_Kodaira_dimension_zeros_1-forms}. Moreover, $Alb(X)$ is not projective, and therefore, the morphism $\varphi$ does not exists in general.\par
To overcome these obstacles, we invoke Lin's construction of an algebraic approximation via tautological models and families \cite{Lin_algebraic_approximation_codim_1} to obtain the following diagram
\[\begin{tikzcd}
	{\mathcal{X}} && {\mathcal{T}/G} \\
	& V && {Y/G.}
	\arrow["\sim", dashed, from=1-1, to=1-3]
	\arrow["\pi"', from=1-1, to=2-2]
	\arrow["{\Pi/G}", from=1-3, to=2-2]
	\arrow["p/G"', from=1-3, to=2-4]
\end{tikzcd}\]
Here, $\pi\colon\mathcal{X}\rightarrow V$ is a deformation of $X\cong\mathcal{X}_0:=\pi^{-1}(0)$ over a complex vector space $V$, and $\mathcal{T}$ is a complex analytic variety on which a finite group $G$ acts. The morphism
\begin{equation*}
    \Pi\colon\mathcal{T}\longrightarrow Y\times V\longrightarrow V
\end{equation*}
is an explicitly constructed $G$-equivariant family of elliptic fibrations $p_v\colon\Pi^{-1}(v)\rightarrow Y$ over a fixed smooth projective base $Y$, called the tautological family. The central elliptic fibration
\begin{equation*}
    p_0/G\colon\Pi^{-1}(0)/G\longrightarrow Y/G
\end{equation*}
is bimeromorphic to the Iitaka fibration of $X$. Moreover, the deformation $\pi\colon\mathcal{X}\rightarrow V$ has the property that the set of points $v\in V$ such that $\mathcal{X}_v:=\pi^{-1}(v)$ is a projective manifold, is dense in the euclidean topology. The two families $\pi\colon\mathcal{X}\rightarrow V$ and $\Pi/G\colon\mathcal{T}/G\rightarrow V$ are bimeromorphic over $V$. The elliptic fibrations $p_v\colon\Pi^{-1}(v)\rightarrow Y$ are locally isomorphic over $Y$, i.e. there is an open cover $\{U_i\}_{i\in I}$ such that for every $U_i$ the isomorphism class of the elliptic fibrations
\begin{equation*}
    p_v\colon\Pi^{-1}(v)\vert_{U_i}\longrightarrow U_i
\end{equation*}
does not depend on $v\in V$.\par
We then show that a small deformation of $X$ also admits a holomorphic 1-form without zeros. In particular, there is a point $v_0\in V$ such that $\mathcal{X}_{v_0}$ is a projective manifold that admits a holomorphic 1-form without zeros. As the Iitaka fibration of $\mathcal{X}_{v_0}$ is birational to $p_{v_0}/G\colon\Pi^{-1}(v_0)/G\rightarrow Y/G$, and the elliptic fibrations $p_v\colon\Pi^{-1}(v)\rightarrow Y$ are locally isomorphic over $Y$, we conclude that the general fiber of the Iitaka fibration of any $\mathcal{X}_v$ is isomorphic to a fixed elliptic curve. This overcomes the first obstacle.\par
To finish the argument, we show that the elliptic fibration $p_0\colon\Pi^{-1}(0)\rightarrow Y$ is locally trivial, i.e. locally isomorphic to a product. It remains to carefully describe the $G$-action on this fibration. Taking a carefully chosen subquotient, yields the desired elliptic fiber bundle $Z\rightarrow B$.
\subsection{Conventions and notation}
By a complex analytic variety, we mean a Hausdorff, second-countable, irreducible, and reduced complex space. A complex analytic variety is called K\"ahler if it admits a K\"ahler metric in the sense of \cite[][II. 1]{Varouchas_Kaehler_spaces_proper_open_morphisms}. A compact complex analytic K\"ahler variety $X$ is called minimal if it is $\mathbb{Q}$-factorial, has terminal singularities, and the canonical divisor $K_X$ is nef. It is furthermore called a good minimal model, if a positive multiple of $K_X$ is globally generated
\subsection*{Acknowledgments}
I would like to thank Stefan Schreieder for suggesting this problem to me, for very helpful discussions, and for comments on this manuscript. The research was conducted in the framework of the DFG-funded research training group RTG 2965: From Geometry to Numbers.

In the projective case, Hao’s argument is summarized as follows: take a good minimal model $X^{\min}$ and its Iitaka fibration $f\colon X^{\min}\to Y$. By \cite[][Thm. 2.1]{Popa_Schnell_Kodaira_dimension_zeros_1-forms}, the nowhere-vanishing $1$-form $\omega$ “cannot come from $Y$”. Hence “the general fiber of $f$ is not contracted by the Albanese morphism”, so it maps to “a translate of a fixed elliptic curve $E\subset \operatorname{Alb}(X)$”. Since $\operatorname{Alb}(X)$ is projective, one “dualize[s] the inclusion $E\subset \operatorname{Alb}(X)$ to get a morphism” $\varphi\colon X^{\min}\to \operatorname{Alb}(X)\to E$, and then “$B$ … as well as the splitting $X''\cong B\times E$ … are then constructed from a general fiber of $\varphi\colon X^{\min}\to E$.

For the K\"ahler generalization (Theorem~\ref{main_result}), two obstacles are noted: one cannot apply Popa–Schnell, and $\operatorname{Alb}(X)$ is not projective so “the morphism $\varphi$ does not exists in general.” To overcome this, the authors “invoke Lin’s construction of an algebraic approximation via tautological models and families” to produce a deformation $\pi\colon \mathcal{X}\to V$ of $X$ and a $G$-equivariant tautological family of elliptic fibrations $\Pi\colon \mathcal{T}\to Y\times V\to V$, with $\Pi^{-1}(0)/G\to Y/G$ bimeromorphic to the Iitaka fibration of $X$, and with $\mathcal{X}\to V$ having a dense set of projective fibers; moreover $\mathcal{X}$ and $\mathcal{T}/G$ are “bimeromorphic over $V$” and the elliptic fibrations $p_v\colon \Pi^{-1}(v)\to Y$ are “locally isomorphic over $Y$” (independent of $v$ on an open cover).

They then “show that a small deformation of $X$ also admits a holomorphic 1-form without zeros”, so there exists $v_0\in V$ with $\mathcal{X}_{v_0}$ projective and carrying such a form. Since the Iitaka fibration of $\mathcal{X}_{v_0}$ is birational to $p_{v_0}/G$ and the $p_v$ are locally isomorphic, they “conclude that the general fiber of the Iitaka fibration of any $\mathcal{X}_v$ is isomorphic to a fixed elliptic curve,” addressing the first obstacle. “To finish the argument,” they “show that the elliptic fibration $p_0\colon \Pi^{-1}(0)\to Y$ is locally trivial” (locally a product), then “carefully describe the $G$-action,” and “taking a carefully chosen subquotient, yields the desired elliptic fiber bundle $Z\to B$” with trivial monodromy.

In the projective case, Hao’s argument is summarized as follows: take a good minimal model $X^{\min}$ and its Iitaka fibration $f\colon X^{\min}\to Y$. By Popa–Schnell, “Kodaira dimension and zeros of holomorphic 1-forms” (Thm. 2.1), the nowhere-vanishing $1$-form $\omega$ “cannot come from $Y$”. Hence “the general fiber of $f$ is not contracted by the Albanese morphism”, so it maps to “a translate of a fixed elliptic curve $E\subset \operatorname{Alb}(X)$”. Since $\operatorname{Alb}(X)$ is projective, one “dualize[s] the inclusion $E\subset \operatorname{Alb}(X)$ to get a morphism” $\varphi\colon X^{\min}\to \operatorname{Alb}(X)\to E$, and then “$B$ … as well as the splitting $X''\cong B\times E$ … are then constructed from a general fiber of $\varphi\colon X^{\min}\to E$.

For the K\"ahler generalization (Theorem~\ref{main_result}), two obstacles are noted: one cannot apply Popa–Schnell, and $\operatorname{Alb}(X)$ is not projective so “the morphism $\varphi$ does not exists in general.” To overcome this, the authors “invoke Lin’s construction of an algebraic approximation via tautological models and families” to produce a deformation $\pi\colon \mathcal{X}\to V$ of $X$ and a $G$-equivariant tautological family of elliptic fibrations $\Pi\colon \mathcal{T}\to Y\times V\to V$, with $\Pi^{-1}(0)/G\to Y/G$ bimeromorphic to the Iitaka fibration of $X$, and with $\mathcal{X}\to V$ having a dense set of projective fibers; moreover $\mathcal{X}$ and $\mathcal{T}/G$ are “bimeromorphic over $V$” and the elliptic fibrations $p_v\colon \Pi^{-1}(v)\to Y$ are “locally isomorphic over $Y$” (independent of $v$ on an open cover).

They then “show that a small deformation of $X$ also admits a holomorphic 1-form without zeros”, so there exists $v_0\in V$ with $\mathcal{X}_{v_0}$ projective and carrying such a form. Since the Iitaka fibration of $\mathcal{X}_{v_0}$ is birational to $p_{v_0}/G$ and the $p_v$ are locally isomorphic, they “conclude that the general fiber of the Iitaka fibration of any $\mathcal{X}_v$ is isomorphic to a fixed elliptic curve,” addressing the first obstacle. “To finish the argument,” they “show that the elliptic fibration $p_0\colon \Pi^{-1}(0)\to Y$ is locally trivial” (locally a product), then “carefully describe the $G$-action,” and “taking a carefully chosen subquotient, yields the desired elliptic fiber bundle $Z\to B$” with trivial monodromy.

\label{Hao_Kodaira_codim_1}\\cite[][Thm. 1.3]{Hao_Nowhere_Vanishing_Holomorphic_One-Forms_on_Varieties_of_Kodaira_Codimension_One}
    Let $X$ be a smooth projective variety of Kodaira dimension $\kappa(X)=\dim X-1$. If $X$ has a nowhere vanishing holomorphic 1-form, then for any minimal model $X^{min}$ of $X$ there exists a finite quasi-\'etale covering $X'\rightarrow X^{min}$ such that any $\mathbb{Q}$-factorialization $X''$ of $X'$ is a product $B\times E$, where $B$ is a minimal model of general type and $E$ is an elliptic curve.,

Question: Let $X$ be a smooth projective variety with Kodaira dimension $\kappa(X)=\dim X-1$, and assume that $X$ admits a nowhere vanishing holomorphic $1$-form. A finite quasi-\'etale covering means a finite surjective morphism that is \'etale in codimension $1$, and a $\mathbb{Q}$-factorialization of a normal variety means a small birational morphism from a $\mathbb{Q}$-factorial variety. Under these assumptions, which existence statement holds for every minimal model $X^{\min}$ of $X$?

For any minimal model $X^{\min}$ of $X$, there exists a finite quasi-\'etale covering $X'\to X^{\min}$ such that every $\mathbb{Q}$-factorialization $X''$ of $X'$ is isomorphic to a product $B\times E$, where $B$ is a minimal model of general type and $E$ is an elliptic curve.

Choices:
B. For any minimal model $X^{\min}$ of $X$, there exists a finite quasi-\'etale covering $X'\to X^{\min}$ such that $X'$ itself is isomorphic to a product $B\times E$, where $B$ is a minimal model of general type and $E$ is an elliptic curve.
C. For any minimal model $X^{\min}$ of $X$, there exists a finite quasi-\'etale covering $X'\to X^{\min}$ and a $\mathbb{Q}$-factorialization $X''$ of $X'$ such that $X''$ is isomorphic to a product $B\times E$, where $B$ is a minimal model of general type and $E$ is an elliptic curve.
D. There exists a finite quasi-\'etale covering $X'\to X$ such that for every minimal model $X^{\min}$ of $X$, every $\mathbb{Q}$-factorialization $X''$ of $X'$ is isomorphic to a product $B\times E$, where $B$ is a minimal model of general type and $E$ is an elliptic curve.
E. For any minimal model $X^{\min}$ of $X$, there exists a finite quasi-\'etale covering $X'\to X^{\min}$ such that every $\mathbb{Q}$-factorialization $X''$ of $X'$ is isomorphic to a product $B\times E$, where $B$ is a smooth projective variety of general type and $E$ is an elliptic curve.


\begin{lem}
\label{prop:The-crucial-involution}Let $\Gamma_{4}\subset\mathbb{P}^{4}$
be a rational normal quartic curve and let $T\subset\mathrm{Aut}(\mathbb{P}^{4},\Gamma_{4})\cong\mathrm{Aut}(\Gamma_{4})\cong\mathrm{PGL}_{2}$
be a maximal torus. Then the following hold:

1) For every smooth $T$-stable quadric $Q$ containing $\Gamma_{4}$,
the linear system of quadric sections of $Q$ containing $\Gamma_{4}$
determines a birational involution $\iota_{Q}:Q\dashrightarrow Q$
contracting the unique cubic section $S_{\Gamma_{4}}$ of $Q$ singular
along $\Gamma_{4}$ onto $\Gamma_{4}$ and restricting to an automorphism
of $Q\setminus S_{\Gamma_{4}}$ which semi-commutes with the induced
$T$-action on it.

2) Let $r,r'\in\Gamma_{4}$ be the two $T$-fixed points. Then the
proper transform $(\iota_{Q}^{-1})_{*}H_{r'}$ of the tangent hyperplane
section $H_{r'}$ to $Q$ at $r'$ is the unique integral quadric
section of $Q$ containing $\Gamma_{4}$ and singular along the tangent
line $L_{r}$ to $\Gamma_{4}$ at $r$. The scheme $H_{r'}\cap(\iota_{Q}^{-1})_{*}H_{r'}$
is the union of the line $L_{r',s}$ passing through $r'$ and the
unique $T$-fixed point $s\in L_{r}\setminus\{r\}$ and of a rational
normal cubic curve $C_{2r',s}$ with tangent line $L_{r'}$ at $r'$
and passing through $s$. Moreover, $\iota_{Q}$ maps $C_{2r',s}$
isomorphically onto itself.
\end{lem}

\begin{thm}
\label{thm:KPS} A smooth prime Fano threefold of degree $22$ belonging
to ${\mathscr{M}}_{22}^{\circ}$ is isomorphic to one of the following
threefolds:

$\bullet$ A threefold $X_{22}^{{\rm MU}}\in{\mathscr{M}}_{22}^{\circ}$
with ${\rm Aut}^{0}(X_{22}^{{\rm MU}})={\rm Aut}(X_{22}^{{\rm MU}})\cong{\rm PGL}_{2}$
(the Mukai-Umemura threefold),

$\bullet$ A threefold $X_{22}^{a}\in{\mathscr{M}}_{22}^{\circ}$
with ${\rm Aut}^{0}(X_{22}^{a})\cong\mathbb{G}_{a}$. Moreover, ${\rm Aut}(X_{22}^{a})$
is isomorphic to a subgroup $\mathbb{G}_{a}\rtimes\mu_{4}$ of a Borel
subgroup $\mathbb{G}_{a}\rtimes\mathbb{G}_{m}$ of $\mathrm{PGL}_{2}$.

$\bullet$ A member of a one-parameter family of pairwise non isomorphic
threefold $X_{22}^{m}(v)\in{\mathscr{M}}_{22}^{\circ}$ with ${\rm Aut}^{0}(X_{22}^{m}(v))\cong\mathbb{G}_{m}$
parametrized by $v\in{\mathbb{P}}^{1}\backslash\{0,1,\infty,-4\}$.
Moreover, for every $v\in{\mathbb{P}}^{1}\backslash\{0,1,\infty,-4\}$,
${\rm Aut}(X_{22}^{m}(v))$ is isomorphic to the normalizer $\mathbb{G}_{m}\rtimes\mu_{2}$
of a maximal torus $\mathbb{G}_{m}$ of $\mathrm{PGL}_{2}$ .
\end{thm}

Smooth prime Fano threefolds $X$ of degree $(-K_{X})^{3}=22$ over
an algebraically closed field $k$ of characteristic zero form a family
${\mathscr{M}}_{22}$ of dimension 6 whose members are rational \cite{Mu04}.
Our main interest lies in the subfamily ${\mathscr{M}}_{22}^{\circ}$
consisting of threefolds with infinite automorphism groups, whose
classification was established by Kuznetsov, Prokhorov and Shramov
in \cite{KP18,KPS18} in a form which can be summarized as follows
(see Theorem \ref{prop:Second-Isomorphism-Classification}):

\begin{thm}
\label{prop:Second-Isomorphism-Classification} Up to isomorphism,
the following hold:

1) There exists a unique threefold $X_{22}^{MU}$ such that $\mathrm{Aut}^{0}(X_{22}^{MU})$
contains a Borel subgroup isomorphic to $\mathbb{G}_{a}\rtimes\mathbb{G}_{m}$.
Moreover, every line on $X_{22}^{MU}$ is of special type and $\mathrm{Aut}(X_{22}^{MU})\cong\mathrm{PGL}_{2}$.

2) There exists a unique threefold $X_{22}^{a}$ such that $\mathrm{Aut}^{0}(X_{22}^{a})$
contains a Borel subgroup isomorphic to $\mathbb{G}_{a}$. Moreover,
$X_{22}^{a}$ contains a unique line of special type and $\mathrm{Aut}(X_{22}^{a})\cong\mathbb{G}_{a}\rtimes\mu_{4}$,
where the action of $\mu_{4}$ on $\mathbb{G}_{a}$ is given by the
natural injective homomorphism $\mu_{4}\to\mathrm{Aut}(\mathbb{G}_{a})\cong\mathbb{G}_{m}$.

3) There exist pairwise non-isomorphic threefolds $X_{22}^{m}(v)$,
$v\in\mathbb{P}^{1}\setminus\{0,1,\infty,-4\}$, such that $\mathrm{Aut}^{0}(X_{22}^{m}(v))$
contains a Borel subgroup isomorphic to $\mathbb{G}_{m}$. Each threefold
$X_{22}^{m}(v)$ contains exactly two lines of special type and $\mathrm{Aut}(X_{22}^{m}(v))\cong\mathbb{G}_{m}\rtimes\mu_{2}$,
where the group $\mu_{2}$ is generated by an involution $\theta$
which exchanges the two lines of special type in $X_{22}^{m}(v)$.
\end{thm}

\section*{Introduction}

$\bullet$ A threefold $X_{22}^{a}\in{\mathscr{M}}_{22}^{\circ}$
with ${\rm Aut}^{0}(X_{22}^{a})\cong\mathbb{G}_{a}$. Moreover, ${\rm Aut}(X_{22}^{a})$
is isomorphic to a subgroup $\mathbb{G}_{a}\rtimes\mu_{4}$ of a Borel
subgroup $\mathbb{G}_{a}\rtimes\mathbb{G}_{m}$ of $\mathrm{PGL}_{2}$.

The possibilities for $X$ were initially found by Prokhorov in \cite{Pr90}
and the articles \cite{KPS18} and \cite{KP18} completed later on
the structures of the automorphism groups ${\rm Aut}(X_{22}^{a})$
and ${\rm Aut}(X_{22}^{m}(v))$. Our purpose is to give a self-contained
alternative proof of Theorem \ref{thm:KPS} and a complementary view
on certain of the constructions and results in these articles.

3) There exist pairwise non-isomorphic threefolds $X_{22}^{m}(v)$,
$v\in\mathbb{P}^{1}\setminus\{0,1,\infty,-4\}$, such that $\mathrm{Aut}^{0}(X_{22}^{m}(v))$
contains a Borel subgroup isomorphic to $\mathbb{G}_{m}$. Each threefold
$X_{22}^{m}(v)$ contains exactly two lines of special type and $\mathrm{Aut}(X_{22}^{m}(v))\cong\mathbb{G}_{m}\rtimes\mu_{2}$,
where the group $\mu_{2}$ is generated by an involution $\theta$
which exchanges the two lines of special type in $X_{22}^{m}(v)$.
\end{thm}

$\quad$ - If $\Gamma=\Gamma_{p}$ then $\mathrm{Aut}(X,Z)=G_{p}$
and $\mathrm{Aut}(X,Z')\subset\mathrm{Aut}(X)$ is the conjugate of
$\mathrm{Aut}(X,Z)$ by the involution $\theta$. Since the latter
normalizes $T$, $\mathrm{Aut}(X,Z)$ and $\theta$ generate a subgroup
of $\mathrm{Aut}(X)$ isomorphic to $\mathrm{PGL}_{2}$, from which
it follows, by Lemma \ref{prop:line}, that $\mathrm{Aut}^{0}(X)\cong\mathrm{PGL}_{2}$.
Finally, since an element of $\mathrm{Aut}(X)$ which stabilizes every
line in $X$ is contained in the intersection of all Borel subgroups
of $\mathrm{Aut}^{0}(X)$, it must be trivial, which implies that
the canonically induced action of $\mathrm{Aut}(X)$ on $(\mathcal{H}_{X})_{\mathrm{red}}\cong\mathbb{P}^{1}$
is faithful. Thus, $\mathrm{Aut}(X)$ identifies with a subgroup of
the automorphism group of $(\mathcal{H}_{X})_{\mathrm{red}}\cong\mathbb{P}^{1}$,
hence is connected, equal to $\mathrm{PGL}_{2}$.
\end{proof}
In the proof of Theorem \ref{prop:Second-Isomorphism-Classification}
above, we have used the following auxiliary result on the birational
geometry of smooth quadric threefolds with an effective action of
a $1$-dimensional torus.
\begin{lem}
\label{prop:The-crucial-involution}Let $\Gamma_{4}\subset\mathbb{P}^{4}$
be a rational normal quartic curve and let $T\subset\mathrm{Aut}(\mathbb{P}^{4},\Gamma_{4})\cong\mathrm{Aut}(\Gamma_{4})\cong\mathrm{PGL}_{2}$
be a maximal torus. Then the following hold:

To prove assertion 2), we can assume that $r=[0:0:0:0:1]$ and $r'=[1:0:0:0:0]$
so that $H_{r'}=\{w_{4}=0\}\cap Q$, $L_{r'}=\{w_{2}=w_{3}=w_{4}=0\}$
and $L_{r}=\{w_{0}=w_{1}=w_{2}=0\}$. Then $(\iota_{Q}^{-1})_{*}H_{r'}=\{w_{0}w_{2}-w_{1}^{2}=0\}\cap Q$
is an anti-canonically embedded del Pezzo surface of degree $4$ singular
along $L_{r}$ and $H_{r'}\cap(\iota_{Q}^{-1})_{*}H_{r'}=\{w_{4}=w_{0}w_{2}-w_{1}^{2}=0\}\cap Q$
is the union of the line $L_{r',s}=\{w_{1}=w_{2}=w_{4}=0\}$ where
$s=[0:0:0:1:0]\in L_{r}\setminus\{r\}$ and of the rational normal
cubic curve $C_{2r',s}$ with tangent line $L_{r'}$ at $r'$ passing
through $s$ given by the image of the morphism
\[
\alpha:\mathbb{P}_{[u_{0}:u_{1}]}^{1}\to\mathbb{P}^{4},\,[u_{0}:u_{1}]\mapsto[u_{0}^{3}:u_{0}^{2}u_{1}:u_{0}u_{1}^{2}:(1-c^{2})u_{1}^{3}:0].
\]
Now it is routine to verify from (\ref{eq:The-involution}) that $\iota_{Q}\circ\alpha=\alpha\circ\iota_{c}$,
where $\iota_{c}$ is the involution $[u_{0}:u_{1}]\mapsto[u_{1}:\frac{c}{1-c^{2}}u_{0}]$
of $\mathbb{P}^{1}$.
\end{proof}
\begin{rem}
\label{rem:Link-remark}In the proof of Theorem \ref{prop:Second-Isomorphism-Classification},
we constructed for a threefold $X=X_{22}^{m}(v)$, $v\in\mathbb{P}^{1}\setminus\{0,1,\infty\}$,
where $X_{22}^{m}(-4)=X_{22}^{MU}$, and a pair $(Z,Z')$ of $\mathbb{G}_{m}$-stable
lines of special type in $X$, a biregular involution $\theta$ of
$X$ semi-commuting with the $\mathbb{G}_{m}$-action and exchanging
the lines $Z$ and $Z'$. Let us briefly indicate a complementary
interpretation of the construction of this involution $\theta$. With
the notation of the proof of Theorem \ref{prop:Second-Isomorphism-Classification},
the composition of the double projection $\psi_{Z}:X\dashrightarrow V_{5}$
from $Z$ with the projection $\pi_{(\psi_{Z})_{*}Z'}:V_{5}\dashrightarrow Q$
from the line $(\psi_{Z})_{*}Z'$ is a $\mathbb{G}_{m}$-equivariant
birational map which contracts the unique hyperplane section $S_{Z}$
of $X$ of multiplicity $3$ along $Z$ onto a rational normal quartic
curve $\Gamma^{Q}$, contracts the unique hyperplane section $R_{\{Z,Z'\}}$
of $X$ singular along $Z\cup Z'$ onto a rational normal cubic curve
$C$ and maps the the unique hyperplane section $S_{Z'}$ of $X$
singular along $Z'$ onto a cubic section $S_{Z'}^{Q}$ of $Q$ singular
along $\Gamma^{Q}$. Letting $\alpha_{\Gamma^{Q}}:\hat{Q}\to Q$ be
the blow-up of $\Gamma^{Q}$, where $\hat{Q}\subset\mathbb{P}^{16}$
is a Fano threefold of Picard rank $2$ and degree $28$ of type 2.21
in \cite{MoMu81}, we obtain an induced $\mathbb{G}_{m}$-equivariant
birational map
\[
\hat{\gamma}=\alpha_{(\pi_{(\psi_{Z})_{*}Z'}\circ\psi_{Z})_{*}S_{Z}}^{-1}\circ\pi_{(\psi_{Z})_{*}Z'}\circ\psi_{Z}:X\dashrightarrow\hat{Q}
\]
which fits into the following commutative diagram of $\mathbb{G}_{m}$-equivariant
birational maps
\[
\xymatrix{ & & R'_{\{Z,Z'\}}\subseteq Y\ar@{.>}[r]^{\hat{\chi}}\ar@{->}[lld]_{\hat{\sigma}} & Y^{+}\supseteq R^{+}{}_{\{Z,Z'\}}\ar@{->}[rrd]^{\hat{\tau}}\\
Z\cup Z'\subset X\ar@{-->}[rrrrr]^{\hat{\gamma}} & & & & & \hat{Q}\supset\hat{C}\\
}
\]
where $\hat{\sigma}:Y\to X$ is the blow-up of $Z\cup Z'$, $\hat{\chi}:Y\dashrightarrow Y^{+}$
is a small $\mathbb{Q}$-factorial modification and $\hat{\tau}:Y^{+}\to\hat{Q}$
is the contraction of the proper transform $R^{+}{}_{\{Z,Z'\}}$ of
$R_{\{Z,Z'\}}$ onto a smooth curve $\hat{C}\subset\hat{S}$ of bi-degree
$(3,3)$. One can further check from the construction that $\hat{\gamma}$
is given by the linear system of quadric sections of $X$ having multiplicity
$3$ along $Z$ and $Z'$. The birational involution $\iota_{Q}:Q\dashrightarrow Q$
in the proof of Theorem \ref{prop:Second-Isomorphism-Classification}
lifts to a biregular involution $\iota_{\hat{Q}}$ of $\hat{Q}$ which
normalizes the lifted $\mathbb{G}_{m}$-action and makes $\hat{Q}$
into a Fano threefold of $(\mathbb{G}_{m}\rtimes\mu_{2})$-Picard
rank $1$, and by the construction of the associated biregular involution
$\theta$ of $X$, the above diagram is then a $(\mathbb{G}_{m}\rtimes\mu_{2})$-equivariant
Sarkisov link. We refer the reader to the forthcoming article \cite{DFK-next}
for a for a more in-depth and detailed study of these birational links
between smooth prime Fano threefolds of degree 22 and blow-ups of
smooth quadric threefolds along rational normal quartic curves.
\end{rem}

The possibilities for $X$ were initially found by Prokhorov in \cite{Pr90}
and the articles \cite{KPS18} and \cite{KP18} completed later on
the structures of the automorphism groups ${\rm Aut}(X_{22}^{a})$
and ${\rm Aut}(X_{22}^{m}(v))$. Our purpose is to give a self-contained
alternative proof of Theorem \ref{thm:KPS} and a complementary view
on certain of the constructions and results in these articles.

\medskip

Our approach to the classification of isomorphism types of prime Fano
threefolds of degree $22$ with infinite automorphism follows the
same initial path as in \cite{KP18,KPS18,Pr90}. It builds essentially
on Iskovskikh's \emph{double projection from lines} \cite{Isk89}
which are birational maps relating the geometry of $X$ to that of
the smooth quintic del Pezzo threefold $V_{5}$ in $\mathbb{P}^{6}$.
The principle stems from the observation that such a double projection
$\psi_{Z}:X\dashrightarrow V_{5}$ from a line $Z$ in $X$ induces
a correspondence between subgroups of the stabilizer $\mathrm{Aut}(X,Z)$
of $Z$ which stabilize the base locus of $\psi_{Z}$ and subgroups
of the stabilizer $\mathrm{Aut}(V_{5},\Gamma)$ of a rational normal
quintic curve $\Gamma\subset V_{5}$ contained in the base locus of
$\psi_{Z}^{-1}$ which stabilize the base locus of $\psi_{Z}^{-1}$
. This leads to determine equivalence classes of rational normal quintic
curves in $V_{5}$ with infinite stabilizers under the action of $\mathrm{Aut}(V_{5})$.
In \cite{KPS18}, this is done by exhibiting representatives of these
classes in the form of orbit closures of certain polynomials in Mukai-Umemura's
description \cite{MuUm83} of $V_{5}$ as an orbit closure of the
action of $\mathrm{PGL}_{2}$ on the projective space of homogeneous
polynomials of degree $6$ in two variables. We propose a complementary
approach building on the study of the equivariant geometry of inverse
images of rational normal quintic curves in $V_{5}$ in the universal
family on the Hilbert scheme of lines on $V_{5}$ constructed by Furushima-Nakayama
\cite{FuNa89}.

To determine the exact automorphism group for each isomorphism type,
the main difficulty essentially reduces to establish the existence
of a special involution of $X$ generating the displayed subgroup
$\mu_{2}$ in the case $X=X_{22}^{m}(v)$. In \cite{KPS18}, the existence
is taken for granted by an external result \cite[Proposition 5.1]{DKK}
which depends itself on Mukai's theory of varieties of sums of powers.
Later, \cite{KP18} provided a different argument for the existence
which depends in particular on the fact established in \cite{KPS18}
that the Hilbert schemes of conics in $X$ is isomorphic to $\mathbb{P}^{2}$.
In contrast, we produce such a special involution in a geometric way,
as a conjugate by an appropriate $\mathbb{G}_{m}$-equivariant birational
map of a known involution on certain quadric threefolds with $\mathbb{G}_{m}$-actions
normalizing the $\mathbb{G}_{m}$-action, see the proof of Theorem
\ref{prop:Second-Isomorphism-Classification} and Lemma \ref{prop:The-crucial-involution}
as well as Remark \ref{rem:Link-remark} for these constructions.

\medskip

We henceforth work over a fixed algebraically closed base field $k$
of characteristic zero. The scheme of the article is the following.
In Section \ref{sec:V5-Stuff}, we first review basic properties of
the construction of the Hilbert scheme of lines on $V_{5}$ and then
proceed to the classification of rational normal quintic curves $\Gamma\subset V_{5}$
with infinite stabilizers up to the action of the automorphism group
of $V_{5}$. In Section \ref{sec:X_22-stuff}, we begin with a review
of classical properties of the double projection from a line in a
smooth prime Fano threefold $X$ of degree $22$ and of the correspondence
it induces between certain families of lines in $X$ and $V_{5}$
and then re-derive Theorem \ref{thm:KPS} from these results.\\

\textit{Acknowledgements.} We would like to express our sincere gratitude
to Sasha Kuznetsov for his thorough readings of successive drafts
of this article and his numerous constructive comments and suggestions,
which contributed significantly to the clarity and mathematical accuracy
of the results.

The present research was initiated during visits of the first and
the second authors at Saitama University and continued during a one-month
visit of the second and third authors at the University of Poitiers
during fall 2024 funded by the University of Poitiers . The authors
are grateful to these institutions for their generous support and
the excellent working conditions offered. The second author was supported
by JSPS KAKENHI Grant Number 22K03269, Royal Society International
Collaboration Award ICA\textbackslash 1\textbackslash 23109 and
Asian Young Scientist Fellowship. The third author was partially funded
by JSPS KAKENHI Grant Number 23K03047.

The post-theorem introduction outlines an alternative proof of Theorem~\ref{thm:KPS} based on Iskovskikh's \emph{double projection from lines}. Concretely, one studies a double projection \(\psi_Z\colon X\dashrightarrow V_5\) from a line \(Z\subset X\), relating \(X\) to the smooth quintic del Pezzo threefold \(V_5\subset\mathbb P^6\). The key principle is that \(\psi_Z\) induces a correspondence between subgroups of the stabilizer \(\mathrm{Aut}(X,Z)\) that stabilize the base locus of \(\psi_Z\) and subgroups of \(\mathrm{Aut}(V_5,\Gamma)\) (where \(\Gamma\subset V_5\) is a rational normal quintic curve contained in the base locus of \(\psi_Z^{-1}\)) that stabilize the base locus of \(\psi_Z^{-1}\). This reduces the classification to determining equivalence classes of rational normal quintic curves \(\Gamma\subset V_5\) with infinite stabilizers under \(\mathrm{Aut}(V_5)\).

Instead of the method of \cite{KPS18} using Mukai--Umemura's description of \(V_5\) as an orbit closure for the \(\mathrm{PGL}_2\)-action on degree-6 binary forms, the authors propose a \"complementary approach\" via \"the equivariant geometry of inverse images of rational normal quintic curves in \(V_5\) in the universal family on the Hilbert scheme of lines on \(V_5\)\" (Furushima--Nakayama).

For determining the \emph{exact} automorphism groups in each isomorphism type, the introduction singles out one main difficulty: in the case \(X=X_{22}^m(v)\), one must establish the existence of a \"special involution\" generating the subgroup \(\mu_2\). The authors' approach is to \"produce such a special involution in a geometric way\", namely as \"a conjugate by an appropriate \(\mathbb G_m\)-equivariant birational map of a known involution on certain quadric threefolds with \(\mathbb G_m\)-actions normalizing the \(\mathbb G_m\)-action\".

Finally, the proof strategy is organized by sections: first classify rational normal quintic curves \(\Gamma\subset V_5\) with infinite stabilizers (Section~\ref{sec:V5-Stuff}); then review the double projection from a line in \(X\) and the induced correspondence between families of lines in \(X\) and \(V_5\), and \"re-derive Theorem~\ref{thm:KPS} from these results\" (Section~\ref{sec:X_22-stuff}).

The post-theorem introduction outlines an alternative proof of the main theorem based on Iskovskikh's \emph{double projection from lines}. Concretely, one studies a double projection \(\psi_Z\colon X\dashrightarrow V_5\) from a line \(Z\subset X\), relating \(X\) to the smooth quintic del Pezzo threefold \(V_5\subset\mathbb P^6\). The key principle is that \(\psi_Z\) induces a correspondence between subgroups of the stabilizer \(\mathrm{Aut}(X,Z)\) that stabilize the base locus of \(\psi_Z\) and subgroups of \(\mathrm{Aut}(V_5,\Gamma)\) (where \(\Gamma\subset V_5\) is a rational normal quintic curve contained in the base locus of \(\psi_Z^{-1}\)) that stabilize the base locus of \(\psi_Z^{-1}\). This reduces the classification to determining equivalence classes of rational normal quintic curves \(\Gamma\subset V_5\) with infinite stabilizers under \(\mathrm{Aut}(V_5)\).

Instead of the method of \cite{KPS18} using Mukai--Umemura's description of \(V_5\) as an orbit closure for the \(\mathrm{PGL}_2\)-action on degree-6 binary forms, the authors propose a "complementary approach" via "the equivariant geometry of inverse images of rational normal quintic curves in \(V_5\) in the universal family on the Hilbert scheme of lines on \(V_5\)" (Furushima--Nakayama).

For determining the \emph{exact} automorphism groups in each isomorphism type, the introduction singles out one main difficulty: in the case \(X=X_{22}^m(v)\), one must establish the existence of a "special involution" generating the subgroup \(\mu_2\). The authors' approach is to "produce such a special involution in a geometric way", namely as "a conjugate by an appropriate \(\mathbb G_m\)-equivariant birational map of a known involution on certain quadric threefolds with \(\mathbb G_m\)-actions normalizing the \(\mathbb G_m\)-action".

Finally, the proof strategy is organized by sections: first classify rational normal quintic curves \(\Gamma\subset V_5\) with infinite stabilizers; then review the double projection from a line in \(X\) and the induced correspondence between families of lines in \(X\) and \(V_5\), and re-derive the main theorem from these results.

\label{thm:KPS} A smooth prime Fano threefold of degree $22$ belonging
to ${\mathscr{M}}_{22}^{\circ}$ is isomorphic to one of the following
threefolds:

$\bullet$ A threefold $X_{22}^{{\rm MU}}\in{\mathscr{M}}_{22}^{\circ}$
with ${\rm Aut}^{0}(X_{22}^{{\rm MU}})={\rm Aut}(X_{22}^{{\rm MU}})\cong{\rm PGL}_{2}$
(the Mukai-Umemura threefold),

$\bullet$ A threefold $X_{22}^{a}\in{\mathscr{M}}_{22}^{\circ}$
with ${\rm Aut}^{0}(X_{22}^{a})\cong\mathbb{G}_{a}$. Moreover, ${\rm Aut}(X_{22}^{a})$
is isomorphic to a subgroup $\mathbb{G}_{a}\rtimes\mu_{4}$ of a Borel
subgroup $\mathbb{G}_{a}\rtimes\mathbb{G}_{m}$ of $\mathrm{PGL}_{2}$.

$\bullet$ A member of a one-parameter family of pairwise non isomorphic
threefold $X_{22}^{m}(v)\in{\mathscr{M}}_{22}^{\circ}$ with ${\rm Aut}^{0}(X_{22}^{m}(v))\cong\mathbb{G}_{m}$
parametrized by $v\in{\mathbb{P}}^{1}\backslash\{0,1,\infty,-4\}$.
Moreover, for every $v\in{\mathbb{P}}^{1}\backslash\{0,1,\infty,-4\}$,
${\rm Aut}(X_{22}^{m}(v))$ is isomorphic to the normalizer $\mathbb{G}_{m}\rtimes\mu_{2}$
of a maximal torus $\mathbb{G}_{m}$ of $\mathrm{PGL}_{2}$ .,

Question: Let $k$ be an algebraically closed field of characteristic $0$, and let ${\mathscr M}_{22}^{\circ}$ denote the class of smooth prime Fano threefolds $X$ of degree $(-K_X)^3=22$ whose automorphism group is infinite. Which statement holds for every such threefold $X$?

Every $X\in {\mathscr M}_{22}^{\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\mathrm{MU}}$, with $\operatorname{Aut}^0(X_{22}^{\mathrm{MU}})=\operatorname{Aut}(X_{22}^{\mathrm{MU}})\cong \operatorname{PGL}_2$; or (ii) a threefold $X_{22}^a$ with $\operatorname{Aut}^0(X_{22}^a)\cong \mathbb G_a$, and $\operatorname{Aut}(X_{22}^a)$ is isomorphic to a subgroup $\mathbb G_a\rtimes \mu_4$ of a Borel subgroup $\mathbb G_a\rtimes \mathbb G_m$ of $\operatorname{PGL}_2$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\in \mathbb P^1\setminus\{0,1,\infty,-4\}$, with $\operatorname{Aut}^0(X_{22}^m(v))\cong \mathbb G_m$, and for every such $v$, $\operatorname{Aut}(X_{22}^m(v))\cong \mathbb G_m\rtimes \mu_2$, the normalizer of a maximal torus $\mathbb G_m$ in $\operatorname{PGL}_2$.

Choices:
B. Every $X\in {\mathscr M}_{22}^{\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\mathrm{MU}}$, with $\operatorname{Aut}^0(X_{22}^{\mathrm{MU}})=\operatorname{Aut}(X_{22}^{\mathrm{MU}})\cong \operatorname{PGL}_2$; or (ii) a threefold $X_{22}^a$ with $\operatorname{Aut}^0(X_{22}^a)\cong \mathbb G_a$, and $\operatorname{Aut}(X_{22}^a)\cong \mathbb G_a\rtimes \mu_4$ as a Borel subgroup of $\operatorname{PGL}_2$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\in \mathbb P^1\setminus\{0,1,\infty\}$, with $\operatorname{Aut}^0(X_{22}^m(v))\cong \mathbb G_m$, and for every such $v$, $\operatorname{Aut}(X_{22}^m(v))\cong \mathbb G_m\rtimes \mu_2$, the normalizer of a maximal torus $\mathbb G_m$ in $\operatorname{PGL}_2$.
C. Every $X\in {\mathscr M}_{22}^{\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\mathrm{MU}}$; or (ii) a threefold $X_{22}^a$ with $\operatorname{Aut}^0(X_{22}^a)\cong \mathbb G_a$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\in \mathbb P^1\setminus\{0,1,\infty,-4\}$, with $\operatorname{Aut}^0(X_{22}^m(v))\cong \mathbb G_m$.
D. Every $X\in {\mathscr M}_{22}^{\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\mathrm{MU}}$, with $\operatorname{Aut}^0(X_{22}^{\mathrm{MU}})=\operatorname{Aut}(X_{22}^{\mathrm{MU}})\cong \operatorname{PGL}_2$; or (ii) a threefold $X_{22}^a$ with $\operatorname{Aut}^0(X_{22}^a)\cong \mathbb G_a$, and $\operatorname{Aut}(X_{22}^a)$ is isomorphic to a subgroup $\mathbb G_a\rtimes \mu_4$ of a Borel subgroup $\mathbb G_a\rtimes \mathbb G_m$ of $\operatorname{PGL}_2$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\in \mathbb P^1\setminus\{0,1,\infty,-4\}$, with $\operatorname{Aut}^0(X_{22}^m(v))\cong \mathbb G_m$, and for every such $v$, $\operatorname{Aut}(X_{22}^m(v))\cong \mathbb G_m$, a maximal torus in $\operatorname{PGL}_2$.
E. Every $X\in {\mathscr M}_{22}^{\circ}$ is isomorphic to one of the following: (i) the Mukai--Umemura threefold $X_{22}^{\mathrm{MU}}$, with $\operatorname{Aut}^0(X_{22}^{\mathrm{MU}})=\operatorname{Aut}(X_{22}^{\mathrm{MU}})\cong \operatorname{PGL}_2$; or (ii) a threefold $X_{22}^a$ with $\operatorname{Aut}^0(X_{22}^a)\cong \mathbb G_a$, and $\operatorname{Aut}(X_{22}^a)$ is isomorphic to a subgroup $\mathbb G_a\rtimes \mu_4$ of a Borel subgroup $\mathbb G_a\rtimes \mathbb G_m$ of $\operatorname{PGL}_2$; or (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\in \mathbb P^1\setminus\{0,1,\infty,-4\}$, with $\operatorname{Aut}^0(X_{22}^m(v))\cong \mathbb G_m$, and there exists a choice of involution $\theta_v$ such that after replacing $X_{22}^m(v)$ by an isomorphic model one has $\operatorname{Aut}(X_{22}^m(v))\cong \mathbb G_m\rtimes \mu_2$.


Question:
Let $X$ be a smooth prime Fano threefold over an algebraically closed field of characteristic $0$ such that $(-K_X)^3=22$ and $
obreak\operatorname{Aut}(X)$ is infinite. Up to isomorphism, which objects occur as such threefolds?

Exactly the following three types occur: (i) the Mukai--Umemura threefold $X_{22}^{\mathrm{MU}}$, with $\operatorname{Aut}^0(X_{22}^{\mathrm{MU}})=\operatorname{Aut}(X_{22}^{\mathrm{MU}})\cong \mathrm{PGL}_2$; (ii) a threefold $X_{22}^a$ with $\operatorname{Aut}^0(X_{22}^a)\cong \mathbb{G}_a$, and $\operatorname{Aut}(X_{22}^a)$ is isomorphic to a subgroup $\mathbb{G}_a\rtimes \mu_4$ of a Borel subgroup $\mathbb{G}_a\rtimes \mathbb{G}_m$ of $\mathrm{PGL}_2$; (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\in \mathbb{P}^1\setminus\{0,1,\infty,-4\}$, with $\operatorname{Aut}^0(X_{22}^m(v))\cong \mathbb{G}_m$ and $\operatorname{Aut}(X_{22}^m(v))\cong \mathbb{G}_m\rtimes \mu_2$, the normalizer of a maximal torus $\mathbb{G}_m$ in $\mathrm{PGL}_2$.

Choices:
B. Exactly the following three types occur: (i) the Mukai--Umemura threefold $X_{22}^{\mathrm{MU}}$, with $\operatorname{Aut}^0(X_{22}^{\mathrm{MU}})=\operatorname{Aut}(X_{22}^{\mathrm{MU}})\cong \mathrm{PGL}_2$; (ii) a threefold $X_{22}^a$ with $\operatorname{Aut}^0(X_{22}^a)\cong \mathbb{G}_a$, and $\operatorname{Aut}(X_{22}^a)\cong \mathbb{G}_a\rtimes \mu_4$ as a Borel subgroup of $\mathrm{PGL}_2$; (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\in \mathbb{P}^1\setminus\{0,1,\infty\}$, with $\operatorname{Aut}^0(X_{22}^m(v))\cong \mathbb{G}_m$ and $\operatorname{Aut}(X_{22}^m(v))\cong \mathbb{G}_m\rtimes \mu_2$.
C. The following occur: the Mukai--Umemura threefold $X_{22}^{\mathrm{MU}}$; a threefold $X_{22}^a$ with $\operatorname{Aut}^0(X_{22}^a)\cong \mathbb{G}_a$; and a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\in \mathbb{P}^1\setminus\{0,1,\infty,-4\}$, with $\operatorname{Aut}^0(X_{22}^m(v))\cong \mathbb{G}_m$.
D. Exactly the following three types occur: (i) the Mukai--Umemura threefold $X_{22}^{\mathrm{MU}}$, with $\operatorname{Aut}^0(X_{22}^{\mathrm{MU}})=\operatorname{Aut}(X_{22}^{\mathrm{MU}})\cong \mathrm{PGL}_2$; (ii) a threefold $X_{22}^a$ with $\operatorname{Aut}^0(X_{22}^a)\cong \mathbb{G}_a$, and $\operatorname{Aut}(X_{22}^a)$ is isomorphic to a subgroup $\mathbb{G}_a\rtimes \mu_4$ of a Borel subgroup $\mathbb{G}_a\rtimes \mathbb{G}_m$ of $\mathrm{PGL}_2$; (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\in \mathbb{P}^1\setminus\{0,1,\infty,-4\}$, with $\operatorname{Aut}^0(X_{22}^m(v))\cong \mathbb{G}_m$ and $\operatorname{Aut}(X_{22}^m(v))\cong \mathbb{G}_m$.
E. Exactly the following three types occur: (i) the Mukai--Umemura threefold $X_{22}^{\mathrm{MU}}$, with $\operatorname{Aut}^0(X_{22}^{\mathrm{MU}})=\operatorname{Aut}(X_{22}^{\mathrm{MU}})\cong \mathrm{PGL}_2$; (ii) a threefold $X_{22}^a$ with $\operatorname{Aut}^0(X_{22}^a)\cong \mathbb{G}_a$, and $\operatorname{Aut}(X_{22}^a)$ is isomorphic to a subgroup $\mathbb{G}_a\rtimes \mu_4$ of a Borel subgroup $\mathbb{G}_a\rtimes \mathbb{G}_m$ of $\mathrm{PGL}_2$; (iii) a member of a one-parameter family of pairwise non-isomorphic threefolds $X_{22}^m(v)$, parametrized by $v\in \mathbb{P}^1\setminus\{0,1,\infty,-4\}$, with $\operatorname{Aut}^0(X_{22}^m(v))\cong \mathbb{G}_m$ and $\operatorname{Aut}(X_{22}^m(v))\cong \mathbb{G}_m\rtimes \mu_4$, the normalizer of a maximal torus $\mathbb{G}_m$ in $\mathrm{PGL}_2$.