\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 over $\mathbb{C}$ 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 one, and a $\mathbb{Q}$-factorialization of a normal variety $X'$ means a small birational morphism $X''\to X'$ with $X''$ $\mathbb{Q}$-factorial. Under these assumptions, which statement about minimal models of $X$ is valid?

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 minimal model $X^{\min}$ of $X$ such that for every finite quasi-\'etale covering $X'\to X^{\min}$, 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 \`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 with $\kappa(B)=\dim B-1$ and $E$ is an elliptic curve.

The stem does not explicitly state the target conclusion; it only gives assumptions and definitions needed to parse the options. There is no direct textual leakage of choice A.

This is not a mere restatement of a theorem in the stem. The options differ by subtle quantifier structure and strength ('every' vs. 'some', quasi-étale vs. étale, cover vs. factorialization), so the item asks the student to distinguish competing conclusions.

The item requires some reasoning about logical strength and theorem precision, especially around quantifiers and the role of Q-factorialization. However, it is closer to exact theorem recognition than to substantial generative mathematical reasoning, and the ambiguity introduced by another true option weakens the pressure.

The distractors are not fully well-formed because choice C is a weaker statement implied by A, so it is also valid. That makes the single-correct-answer format defective, even though B, D, and E are plausibly designed around common quantifier and regularity mistakes.

Conceptually sophisticated and non-tautological, with no answer leakage, but flawed as an MCQ because at least one distractor is also true, undermining uniqueness and assessment quality.


\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 $\operatorname{Aut}(X_{22}^m(v))\cong \mathbb G_m\rtimes \mu_2$.

The stem states only the hypotheses and asks for the classification; it does not reveal the listed cases or the automorphism-group structure. There is no explicit or trivial answer leakage.

This is essentially a direct multiple-choice restatement of a classification theorem: under the stated hypotheses, select the exact classified objects. It primarily asks for recall of the theorem rather than application to a new situation.

There is some pressure to distinguish the fully correct statement from subtle variants (e.g. the excluded parameter value -4, the discrete automorphism factor, and the weaker-true option), but the task is still mostly exact recall/discrimination rather than genuine generative reasoning.

The distractors are plausible and mathematically targeted: one is weaker but true, while others alter specific structural details in ways that reflect realistic confusion about parameter ranges and automorphism groups.

A technically well-constructed recall/discrimination MCQ with strong distractors and no answer leakage, but it is largely tautological as a direct restatement of a classification theorem and only modestly tests generative reasoning.


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 $X$?

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 $\operatorname{Aut}(X_{22}^m(v))\cong \mathbb G_m\rtimes \mu_2$.

The stem does not explicitly or implicitly reveal the keyed answer. It only specifies the class of Fano threefolds and asks which statement holds for all of them.

The item is not a direct restatement of the stem, but it mainly asks for recall of a known classification theorem in nearly theorem-statement form. It tests recognition of the exact classification rather than a genuinely competing mathematical conclusion derived from the setup.

There is some reasoning pressure because the options differ in subtle but meaningful ways (excluded parameter values, full automorphism groups, normalizer vs torus). However, the task is mostly theorem recall/verification, and the presence of a weaker true option reduces the need for strong generative reasoning.

Several distractors are mathematically plausible and target realistic errors, such as omitting the excluded value -4 or confusing the full automorphism group with its identity component. But choice C is a weaker true statement, so the distractor set is not cleanly single-correct, which significantly weakens item quality.

A mathematically sophisticated item with subtle near-miss options, but it is weakened by theorem-recall framing and, more seriously, by including a weaker true alternative, making the single-best-answer format ambiguous.


\begin{theorem}[\textbf{Dynamical stability of $n$-solitons}]\label{thm1.1}
Given $ n\in \mathbb{N} $, $ n\geq 1 $, a collection of wave speeds $\textbf{c} = (c_{1},\dots,c_{n})$ with $ 0 < c_{1} < \dots < c_{n} $ and a collection of space transitions $\textbf{x} = (x_{1},\dots,x_{n})\in \mathbb{R}^{n}$, let $ U^{(n)}(\cdot,\cdot;\textbf{c},\textbf{x}) $ be the corresponding $n$-soliton solutions of (\ref{ILW}). Then for any $ \epsilon > 0 $, there exists $ \delta > 0 $ such that for any $ u_{0}\in H^{\frac{n}{2}}(\mathbb{R}) $, the following stability property holds, if

\begin{align}
	&\bigl\lVert u_{0} - U^{(n)}(0,\cdot;\textbf{c},\textbf{x}) \bigr\rVert_{H^{\frac{n}{2}}} < \delta,
\end{align}
then for any $t \in \mathbb{R}$, the corresponding solution of (\ref{ILW}) verifies
\begin{align}
\inf_{\tau \in \mathbb{R},\  \textbf{y} \in \mathbb{R}^{n}} \bigl\lVert u(t) - U^{(n)}(\tau,\cdot;\textbf{c},\textbf{y}) \bigr\rVert_{H^{\frac{n}{2}}} < \epsilon.
\end{align}
\end{theorem}

\label{sec.1}
In this work, we consider the stability of $n$-soliton solutions for the intermediate long wave (ILW) equation\cite{J,KAS,KKD}
\begin{align}
	u_t+\frac{1}{\delta}u_x+2uu_x+T^\delta u_{xx}=0,\quad  (t,x)\in \mathbb{R} \times \mathbb{R},\label{ILW}
\end{align}
where $u=u(t,x)\in\mathbb{R}$ is a real-valued function, $T^\delta $ is the inverse of Tilbert  transform $T$ with a large scale parameter $\delta>0$. These two operators are defined as
\begin{align}
	(T^\delta f)(x)&=\frac{1}{2\delta}\text{P.V.}\int_{\mathbb{R}}\coth \frac{\pi (y-x)}{2\delta}f(y)dy,\\\nonumber
		(Tf)(x)&=\frac{1}{2\delta}\text{P.V.}\int_{\mathbb{R}}\mathrm{cosech} \frac{\pi (y-x)}{2\delta}f(y)dy,
\end{align}
and P.V. indicates that the integral is to be computed in the principle value sense. Moreover, $T^\delta$ is a zero Fourier multiplier, in the sense that $\partial_x T^\delta$ is the multiplier with symbol
\begin{align}
	\sigma(\partial_x T^\delta)=\widehat{\partial_x T^\delta}=-2\pi\xi\coth(2\pi\delta \xi).
\end{align}

The ILW equation (\ref{ILW}) describes weakly nonlinear internal wave propagation in stratified fluids of finite depth $\delta$. First derived by Kubota, Ko and Dobbs \cite{KKD}, this model captures essential nonlinear and dispersive characteristics of wave dynamics. Joseph \cite{J} subsequently formalized the derivation of the ILW equation by integrating the linear dispersion relation from \cite{P} within Whitham's nonlocal framework \cite{W}, and explicitly constructed its solitary wave solution. This represents a key advancement in establishing the integrability of the equation. Furthermore, the ILW equation itself arises in the context of the two-layer internal wave system \cite{BLS, CGK}. A rigorous derivation of its system form in one or two spatial dimensions, including the case with a free upper surface, can be found in \cite{Xu}.

In addition, the ILW equation (\ref{ILW}) establishes a fundamental connection between two canonical models in water wave theory: the Korteweg-de Vries (KdV) equation for shallow water and the Benjamin-Ono (BO) equation for deep water \cite{ABFS, AFS,HKO,SAF}, with shared initial data in appropriate Sobolev spaces. Specifically, the asymptotic behavior of the ILW solution is governed by the depth parameter $\delta$. Let $u(t,x)$ be the solution of the ILW equation (\ref{ILW}). Then, under the scaling transformation
\begin{align}
	v = \frac{3}{\delta} u\left( \frac{3}{\delta}t,x\right), \label{T1}
\end{align}
the rescaled function $v$ converges (in a suitable functional sense, typically distributional or weak convergence) to the solution of the KdV equation
\begin{align}
	u_t + 2uu_x + \frac{\delta}{3}u_{xxx} = 0, \label{KdV}
\end{align}
in the shallow-water limit ($\delta \to 0$) and in the deep-water limit ($\delta \to \infty$), (\ref{T1}) will converge to the the solution of the BO equation
\begin{align}
	u_t + 2uu_x + Hu_{xx} = 0, \label{BO}
\end{align}
where $H$ denotes the Hilbert transform, defined by the principal value integral
\begin{align*}
	(Hf)(x)=\frac{1}{\pi}\text{P.V.}\int_{\mathbb{R}}\frac{f(y)}{y-x}\mathrm{d}y.
\end{align*}

As a fundamental integrable system \cite{AC,FA,KS}, the ILW equation (\ref{ILW}) possesses an infinite-dimensional completely integrable Hamiltonian structure \cite{KAS, Sa}. This integrability is manifested in the existence of an infinite hierarchy of conservation laws and a Lax-pair formulation.
A key distinction between the ILW equation and the KdV equation lies in their dispersion operators \cite{MPS,Sa,Xu}. Specifically, the ILW equation incorporates a singular nonlocal integro-differential operator $T^\delta$, which fundamentally alters the asymptotic properties of its solutions. In particular, the soliton solutions of the ILW equation exhibit algebraic decay, in contrast to the exponential decay characteristic of KdV solitons. This difference in decay rates represents a key qualitative difference between the two models. Formally, the following quantities are conserved along the flow of the ILW equation \cite{LR}
\begin{align}
	H_0(u)&=:\int_{\mathbb{R}}u\text{d}x, \\
    H_1(u)&=:\frac{1}{2}\int_{\mathbb{R}}u^2\text{d}x, \\
	H_2(u)&=:-\int_{\mathbb{R}}\frac{1}{3}u^3+\frac{1}{2}uT^\delta u_x +\frac{u^2}{2\delta}\text{d}x,\\
	H_3(u)&=:\int_{\mathbb{R}}\frac{1}{4}u^4+\frac{3}{4}u^2T^\delta u_x+\frac{3}{8}(T^\delta u_x)^2+\frac{1}{3\delta}u^3+\frac{1}{2\delta}uT^\delta u_x+\frac{u^2}{8\delta^2}\text{d}x.
\end{align}

It is noteworthy that the energy space for the ILW equation, defined as the domain of the Hamiltonian $H_2(u)$, is $H^{\frac{1}{2}}(\mathbb{R})$.
This functional framework provides the natural setting for investigating solution properties.
In the context of weak solutions, Ginibre and Velo \cite{GV} established the existence of global solutions satisfying
$u \in L^\infty(\mathbb{R}, H^1(\mathbb{R})) \cap L^2_{\mathrm{loc}}(\mathbb{R}, H^{\frac{3}{2}}_{\mathrm{loc}}(\mathbb{R}))$.
For strong solutions, the first global well-posedness result in $H^s(\mathbb{R})$ was obtained by \cite{ABFS} for $s > \frac{3}{2}$.
Subsequent developments progressively lowered the regularity threshold: Molinet and Vento \cite{MV} extended well-posedness to $s > \frac{1}{2}$
(where unconditional uniqueness holds), while Molinet, Pilod, and Vento \cite{MPV} advanced it to $s > \frac{1}{4}$.
A significant breakthrough in the sharp low-regularity theory was achieved by Ifrim and Saut \cite{IS}, who proved global well-posedness
in $L^2(\mathbb{R})$ for small initial data. Most recently, Gassot and Laurens \cite{GAL} established sharp global well-posedness
for the ILW equation in $H^s(\mathbb{T})$ for all $s > -\frac{1}{2}$, thereby
completing the well-posedness theory on the torus. Additionally, they proved the
continuous convergence of solutions to those of the Benjamin-Ono equation in the
deep-water limit $\delta \to \infty$.

Based upon the facts above, the (\ref{ILW}) equation can naturally be viewed as a Hamiltonian system of the form \cite{KSA, LR}
\begin{align}
	u_t=\mathcal{J}\frac{\delta H_2(u)}{\delta u},
\end{align}
where $\mathcal{J}=\partial_x$, and $\frac{\delta H_2(u)}{\delta u}$ (subsequently simplified to $H_2'(u)$) refers to the variational derivative of $H_2(u)$ can be written as
\begin{align*}
	(\frac{\partial}{\partial \epsilon}H_2(u+\epsilon f))\Big|_{\epsilon=0}=\int_{\mathbb{R}}\frac{\delta H_{2}}{\delta u}(x)f(x)\text{d}x.
\end{align*}

Moreover, the ILW equation possesses a bi-Hamiltonian structure \cite{Sa}. However, unlike the KdV equation, the bi-Hamiltonian structure of the ILW equation is highly nontrivial due to its formulation involving both the spatial derivative operator $\partial_x$ and the singular integral operator $T^\delta$. This operator composition aligns the ILW equation with structural features characteristic of completely integrable systems in two spatial dimensions.

To formalize this connection, let subscripts $1, 2$ denote dependence on the independent variables $x:= x_1$ and $x_2$. For arbitrary functions $f_{12}$ and $g_{12}$, we define the associated bilinear form by
\begin{align}
	\langle f_{12},g_{12}\rangle
	&:= \int_{\mathbb{R}^{2}}f_{12}g_{12}^{*} \mathrm{d}x_{1}\mathrm{d}x_{2},\label{1.9}
\end{align}
where the asterisk superscript $^*$ denotes complex conjugation. Define operators in $L^{2}(\mathbb{R}^{2},\mathbb{C})$ (with domain $H^{1}(\mathbb{R}^{2},\mathbb{C})$)
\begin{align}
	\theta_{12}^{\pm}
	&:= u_{1}\pm u_{2} + i(\partial_{x_{1}} \mp \partial_{x_{2}}), \quad u_{j}=u(x_{j},t), \quad j=1,2.\label{1.10}
\end{align}
Then the two compatible Hamiltonian operators for the ILW equation are given by
\begin{align}
	\mathcal{J}_{12}^{(1)}
	:= \theta_{12}^{-}, \quad
	\mathcal{J}_{12}^{(2)}
	:= (i\theta_{12}^{-}T^\delta_{12} - \theta_{12}^{+})\theta_{12}^{-},
\end{align}
where $T^\delta_{12}$ is an extended operator defined as
\begin{align}
	(T^\delta_{12}f_{12})(x_{1},x_{2}) =\frac{1}{2\delta}\mathrm{P.V.}\int_{\mathbb{R}}\coth \left(\frac{\pi}{2\delta}[\xi-(x_1+x_2)]\right)F(\xi,x_1-x_2)\mathrm{d}\xi,\label{1.13}
\end{align}
and
\begin{align}
	f(x_1,x_2)=F(x_1+x_2,x_1-x_2).
\end{align}
Then the ILW hierarchy can be expressed in the form
\begin{align}\label{1.14}
	u_t&=\frac{i}{2^n}\int_{\mathbb{R}}\delta(x_1-x_2)(\mathcal{R}_{12}^\star)^n\theta_{12}^-\cdot1\mathrm{d}x_2\\\nonumber
	&=\frac{i}{2^n}\int_{\mathbb{R}}\delta(x_1-x_2)\theta_{12}^-\mathcal{R}_{12}^n\cdot1\mathrm{d}x_2=\mathcal{J}\frac{\delta H_n(u)}{\delta u},\quad \text{for all} ~n\in \mathbb{N},
\end{align}
where $\star$ denotes the adjoint with respect to the bilinear form (\ref{1.9}). The recursion operator $\mathcal{R}_{12}$ and its adjoint $\mathcal{R}_{12}^{\star}$ are defined by
\begin{equation}
	\mathcal{R}_{12} := (\mathcal{J}_{12}^{(1)})^{-1}\mathcal{J}_{12}^{(2)},\quad
	\mathcal{R}_{12}^{\star} := \mathcal{J}_{12}^{(2)}(\mathcal{J}_{12}^{(1)})^{-1} = i\theta_{12}^{-}T^\delta_{12} - \theta_{12}^{+},\label{1.15}
\end{equation}
and in view of (\ref{1.15}), they satisfy the compatibility condition
\begin{equation*}
	\mathcal{R}_{12}^{\star}\mathcal{J}_{12}^{(1)} = \mathcal{J}_{12}^{(1)}\mathcal{R}_{12}.
\end{equation*}
Consequently, the first few equations of the ILW hierarchy are
\begin{align*}
	&u_t-u_x=0, \quad \text{for}\quad n=1, \\
	&u_t+\frac{1}{\delta}u_x+2uu_x+T^\delta u_{xx}=0,\quad \text{for}\quad n=2;\\
	&u_t-\left(\frac{1}{4}({T^{\delta}}^2-1)u_{xx}+ u^3+\frac{3}{2}(uT^\delta u_x+T^\delta uu_{xx})\right)_x=0, \quad \text{for}\quad n=3.
\end{align*}

In common with the classical KdV and BO equations, between which the ILW equation (\ref{ILW}) serves as a model-theoretical bridge and admits a family of unique exact solitary-wave solutions \cite{AT,SA,SAK} of the form
\begin{align}
	u(x,t)=Q_c(x-ct-x_0), \quad Q_c(s)=\frac{a\sin(a\delta)}{\cosh(as)+\cos(a\delta)},\quad c>0,~ x_0\in \mathbb{R}, \label{1.16}
\end{align}
where $c$ represents the wave speed and $a$ is the unique solution of the transcendental equation
\begin{align*}
	a\delta \cot(a\delta)=1-c\delta, \quad a\in \left(0, \frac{\pi}{\delta}\right).
\end{align*}
Substituting (\ref{1.16}) into equation (\ref{ILW}) yields
\begin{align}
	T^\delta \partial_x Q_c +\left(\frac{1}{\delta}-c\right)Q_c+Q_c^2=0, \quad c>0.\label{1.17}
\end{align}
Beyond these fundamental solitary waves, the ILW equation (\ref{ILW}) also supports more complex solutions, including multi-solitons \cite{JE} that admit parametric representations similar to the single-soliton case \cite{AT,LP,M,PD}.
The ILW $n$-soliton solution $U^{(n)}_\mathbf{c}$ is characterized by a collection of wave speeds $\mathbf{c}=(c_1, c_2, \cdots, c_n)$ and initial positions $\mathbf{x}=(x_1, x_2, \cdots, x_n)$, where the wave speeds satisfy $c_j>0$ with $c_j\neq c_k$ for $j\neq k$ ($j,k=1, 2, \cdots, n$). Moreover, in the long-time limit, these $n$-soliton solutions decompose into a superposition of $n$ individual solitons as follows
\begin{align}
	\lim_{|t| \to +\infty} \left\| U^{(n)}(t,\cdot;\mathbf{c},\mathbf{x}) - \sum_{j=1}^{n} Q_{c_{j}}(\cdot - c_{j}t - x_{j}) \right\|_{H^{s}(\mathbb{R})} = 0, \quad s \in \mathbb{N}.\label{resolution}
\end{align}

In recent years, the stability theory for solitary waves and multi-solitons has emerged as a prominent research direction. Within soliton theory, stability concepts are systematically categorized into four distinct types based on analytical methodology and robustness guarantees: (i) linear (spectral) stability, concerning the eigenvalue distribution of linearized operators; (ii) Lyapunov (dynamical) stability, established via positive definiteness of the second variation of Lyapunov functionals at soliton solutions; (iii) orbital (nonlinear) stability, which requires that solutions remain within a neighborhood of the soliton orbit under finite-amplitude disturbances; and (iv) asymptotic stability, demanding convergence to specific soliton profiles.

As fundamental models in integrable systems, the stability of solitary waves for both KdV and BO equations has been extensively investigated. For instance, Maddocks and Sachs \cite{MS} established orbital stability of KdV multi-solitons in the energy space defined by conservation laws. Killip and Visan \cite{KV} proved orbital stability of KdV multi-solitons in $H^{-1}(\mathbb{R})$ using low-regularity conservation laws in \cite{BP}. For the BO equation, Wang \cite{LW} and Matsuno \cite{Mat} studied dynamical stability of multi-soliton solutions. Additionally, Badreddine, Killip and Visan \cite{BKP} proved that the multi-soliton solutions to the BO equation are uniformly orbitally stable in $H^s(\mathbb{R})$ with $-\frac{1}{2}<s<\frac{1}{2}$.

In contrast to the extensive results for KdV and BO equations, stability analysis for solitary wave solutions of the ILW equation remains comparatively underdeveloped. Existing studies primarily focus on orbital stability. Albert and Bona \cite{AB} employed a rigorous variational approach combined with spectral analysis to establish orbital stability of ILW solitary waves in its energy space $\mathcal{X} = \{g \in L^2(\mathbb{R})| \int_\mathbb{R}(1+\alpha(k))|\hat{g}|^2\mathrm{d}k < \infty\}$. Subsequent work \cite{A} extended these results to spaces $H^{\frac{1}{2}}(\mathbb{R})$, while \cite{PCA} addressed linear and orbital stability of periodic wave solutions in periodic Sobolev spaces.

Considering the growing importance of understanding coherent structures in nonlocal dispersive equations, a systematic analysis of soliton and multi-soliton stability for the ILW equation represents a significant open research direction. Hence, in this work, we aim to establish dynamical stability results of $n$-soliton solutions to the ILW equation (\ref{ILW}) in an appropriate Sobolev space.

Our strategy is to adapt the methods of Maddocks and Sachs \cite{MS}.
In accordance with the ideas in \cite{MS}, we construct an appropriate  Lyapunov functional $\mathcal{S}_{n}$ of the ILW $n$-solitons is given by
\begin{align}
	\mathcal{S}_{n}(u) = H_{n+1}(u) + \sum_{m=1}^{n} \mu_{m} H_{m}(u),\label{lagrange}
\end{align}
and $\mu_{m}$ are Lagrange multipliers which will be expressed in terms of the elementary symmetric functions of $c_{1},c_{2},\dots,c_{n}$. Using (\ref{lagrange}), this condition can be written as the following Euler-Lagrange equation
\begin{align}
	\frac{\delta H_{n+1}(u)}{\delta u} + \sum_{m=1}^{n} \mu_{m} \frac{\delta H_{m}(u)}{\delta u} = 0,\quad \text{at } u = U^{(n)}.\label{1.20}
\end{align}
The dynamical stability of $U^{(n)}$ is implied by the fact that $U^{(n)}(x)$ is a minimizer of the functional $H_{n+1}$ under the following $n$ constraints
\begin{align}
	H_{m}(u) = H_{m}\left(U^{(n)}\right),\quad m = 1,2,\dots,n,
\end{align}
which requires that the self-adjoint second variation of the operator $\mathcal{S}_n$,
\begin{align}
	\mathcal{L}_n:=\mathcal{S}''_{n}(U^{(n)}), \label{VL}
\end{align}
is strictly positive if one modulates the directions given by the constraints.

The demonstration of dynamical stability for the ILW equation relies on the recursion operators and the construction of a Lyapunov functional. This approach utilizes the trace formula derived from the inverse scattering transform (IST) framework to forge a link with conservation laws, in a manner analogous to techniques applied in the study of the Camassa-Holm equation \cite{AYP, CGI, DKS}. The stability proof centers on a spectral analysis of the second variation $\mathcal{L}_{n} = \mathcal{S}_{n}''(U^{(n)})$ evaluated at a smooth $n$-soliton $U^{(n)}$, coupled with an eigenvalue computation for the Hessian matrix $D = \left\{\frac{\partial^2 \mathcal{S}_{n}}{\partial \mu_{i} \partial \mu_{j}} \right\}$.

The spectral characterization of the linearized operator $\mathcal{L}_{n}$ around the $n$-soliton $U^{(n)}$ is central to this endeavor. In the case of a single soliton, the operator $L_n = H''_{n+1}(Q_c) + c H''_{n}(Q_c)$ is analyzed. The inertia index $\mathrm{in}(L_n)$ is determined by constructing the derivative operators $\mathcal{J}L_n$ and $L_n\mathcal{J}$ and leveraging IST theory, specifically employing Jost solutions and square eigenfunctions. For the general $n$-soliton, the recursive operator $\mathcal{R}$ originating from the bi-Hamiltonian structure is applied to compute $\mathrm{in}(\mathcal{L}_n)$.

Finally, we conclude by proving the orbital stability of double solitons. This final result is obtained through a spectral analysis of the linearized operator $\mathcal{L}_{2}$ around the double soliton $U^{(2)}$, supported by a contradictory argument. Our analysis shows that $\mathcal{L}_{2}$ has one simple negative eigenvalue and one double eigenvalue at zero.

In what follows, we will present our main results. The first is the dynamical stability of the multi-solitons to the ILW equation (\ref{ILW}).

To formalize this connection, let subscripts $1, 2$ denote dependence on the independent variables $x:= x_1$ and $x_2$. For arbitrary functions $f_{12}$ and $g_{12}$, we define the associated bilinear form by
\begin{align}
 \langle f_{12},g_{12}\rangle
 &:= \int_{\mathbb{R}^{2}}f_{12}g_{12}^{*} \mathrm{d}x_{1}\mathrm{d}x_{2},\label{1.9}
\end{align}
where the asterisk superscript $^*$ denotes complex conjugation. Define operators in $L^{2}(\mathbb{R}^{2},\mathbb{C})$ (with domain $H^{1}(\mathbb{R}^{2},\mathbb{C})$)
\begin{align}
 \theta_{12}^{\pm}
 &:= u_{1}\pm u_{2} + i(\partial_{x_{1}} \mp \partial_{x_{2}}), \quad u_{j}=u(x_{j},t), \quad j=1,2.\label{1.10}
\end{align}
Then the two compatible Hamiltonian operators for the ILW equation are given by
\begin{align}
 \mathcal{J}_{12}^{(1)}
 := \theta_{12}^{-}, \quad
 \mathcal{J}_{12}^{(2)}
 := (i\theta_{12}^{-}T^\delta_{12} - \theta_{12}^{+})\theta_{12}^{-},
\end{align}
where $T^\delta_{12}$ is an extended operator defined as
\begin{align}
 (T^\delta_{12}f_{12})(x_{1},x_{2}) =\frac{1}{2\delta}\mathrm{P.V.}\int_{\mathbb{R}}\coth \left(\frac{\pi}{2\delta}[\xi-(x_1+x_2)]\right)F(\xi,x_1-x_2)\mathrm{d}\xi,\label{1.13}
\end{align}
and
\begin{align}
 f(x_1,x_2)=F(x_1+x_2,x_1-x_2).
\end{align}
Then the ILW hierarchy can be expressed in the form
\begin{align}\label{1.14}
 u_t&=\frac{i}{2^n}\int_{\mathbb{R}}\delta(x_1-x_2)(\mathcal{R}_{12}^\star)^n\theta_{12}^-\cdot1\mathrm{d}x_2\\\nonumber
 &=\frac{i}{2^n}\int_{\mathbb{R}}\delta(x_1-x_2)\theta_{12}^-\mathcal{R}_{12}^n\cdot1\mathrm{d}x_2=\mathcal{J}\frac{\delta H_n(u)}{\delta u},\quad \text{for all} ~n\in \mathbb{N},
\end{align}
where $\star$ denotes the adjoint with respect to the bilinear form (\ref{1.9}). The recursion operator $\mathcal{R}_{12}$ and its adjoint $\mathcal{R}_{12}^{\star}$ are defined by
\begin{equation}
 \mathcal{R}_{12} := (\mathcal{J}_{12}^{(1)})^{-1}\mathcal{J}_{12}^{(2)},\quad
 \mathcal{R}_{12}^{\star} := \mathcal{J}_{12}^{(2)}(\mathcal{J}_{12}^{(1)})^{-1} = i\theta_{12}^{-}T^\delta_{12} - \theta_{12}^{+},\label{1.15}
\end{equation}
and in view of (\ref{1.15}), they satisfy the compatibility condition
\begin{equation*}
 \mathcal{R}_{12}^{\star}\mathcal{J}_{12}^{(1)} = \mathcal{J}_{12}^{(1)}\mathcal{R}_{12}.
\end{equation*}
Consequently, the first few equations of the ILW hierarchy are
\begin{align*}
 &u_t-u_x=0, \quad \text{for}\quad n=1, \\
 &u_t+\frac{1}{\delta}u_x+2uu_x+T^\delta u_{xx}=0,\quad \text{for}\quad n=2;\\
 &u_t-\left(\frac{1}{4}({T^{\delta}}^2-1)u_{xx}+ u^3+\frac{3}{2}(uT^\delta u_x+T^\delta uu_{xx})\right)_x=0, \quad \text{for}\quad n=3.
\end{align*}

In common with the classical KdV and BO equations, between which the ILW equation (\ref{ILW}) serves as a model-theoretical bridge and admits a family of unique exact solitary-wave solutions \cite{AT,SA,SAK} of the form
\begin{align}
 u(x,t)=Q_c(x-ct-x_0), \quad Q_c(s)=\frac{a\sin(a\delta)}{\cosh(as)+\cos(a\delta)},\quad c>0,~ x_0\in \mathbb{R}, \label{1.16}
\end{align}
where $c$ represents the wave speed and $a$ is the unique solution of the transcendental equation
\begin{align*}
 a\delta \cot(a\delta)=1-c\delta, \quad a\in \left(0, \frac{\pi}{\delta}\right).
\end{align*}
Substituting (\ref{1.16}) into equation (\ref{ILW}) yields
\begin{align}
 T^\delta \partial_x Q_c +\left(\frac{1}{\delta}-c\right)Q_c+Q_c^2=0, \quad c>0.\label{1.17}
\end{align}
Beyond these fundamental solitary waves, the ILW equation (\ref{ILW}) also supports more complex solutions, including multi-solitons \cite{JE} that admit parametric representations similar to the single-soliton case \cite{AT,LP,M,PD}.
The ILW $n$-soliton solution $U^{(n)}_\mathbf{c}$ is characterized by a collection of wave speeds $\mathbf{c}=(c_1, c_2, \cdots, c_n)$ and initial positions $\mathbf{x}=(x_1, x_2, \cdots, x_n)$, where the wave speeds satisfy $c_j>0$ with $c_j\neq c_k$ for $j\neq k$ ($j,k=1, 2, \cdots, n$). Moreover, in the long-time limit, these $n$-soliton solutions decompose into a superposition of $n$ individual solitons as follows
\begin{align}
 \lim_{|t| \to +\infty} \left\| U^{(n)}(t,\cdot;\mathbf{c},\mathbf{x}) - \sum_{j=1}^{n} Q_{c_{j}}(\cdot - c_{j}t - x_{j}) \right\|_{H^{s}(\mathbb{R})} = 0, \quad s \in \mathbb{N}.\label{resolution}
\end{align}

Our strategy is to adapt the methods of Maddocks and Sachs \cite{MS}.
In accordance with the ideas in \cite{MS}, we construct an appropriate Lyapunov functional $\mathcal{S}_{n}$ of the ILW $n$-solitons is given by
\begin{align}
 \mathcal{S}_{n}(u) = H_{n+1}(u) + \sum_{m=1}^{n} \mu_{m} H_{m}(u),\label{lagrange}
\end{align}
and $\mu_{m}$ are Lagrange multipliers which will be expressed in terms of the elementary symmetric functions of $c_{1},c_{2},\dots,c_{n}$. Using (\ref{lagrange}), this condition can be written as the following Euler-Lagrange equation
\begin{align}
 \frac{\delta H_{n+1}(u)}{\delta u} + \sum_{m=1}^{n} \mu_{m} \frac{\delta H_{m}(u)}{\delta u} = 0,\quad \text{at } u = U^{(n)}.\label{1.20}
\end{align}
The dynamical stability of $U^{(n)}$ is implied by the fact that $U^{(n)}(x)$ is a minimizer of the functional $H_{n+1}$ under the following $n$ constraints
\begin{align}
 H_{m}(u) = H_{m}\left(U^{(n)}\right),\quad m = 1,2,\dots,n,
\end{align}
which requires that the self-adjoint second variation of the operator $\mathcal{S}_n$,
\begin{align}
 \mathcal{L}_n:=\mathcal{S}''_{n}(U^{(n)}), \label{VL}
\end{align}
is strictly positive if one modulates the directions given by the constraints.

Finally, we conclude by proving the orbital stability of double solitons. This final result is obtained through a spectral analysis of the linearized operator $\mathcal{L}_{2}$ around the double soliton $U^{(2)}$, supported by a contradictory argument. Our analysis shows that $\mathcal{L}_{2}$ has one simple negative eigenvalue and one double eigenvalue at zero.

In what follows, we will present our main results. The first is the dynamical stability of the multi-solitons to the ILW equation (\ref{ILW}).

Then we will provide a more precise description of the orbital stability of the double solitons to the ILW equation.

\begin{theorem}[\textbf{Orbital stability of double solitons}]\label{thm1.2}
 The double solitons $U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)$ of ILW equation (\ref{ILW}) with $0<c_1<c_2$ are orbitally stable in $H^1(\mathbb{R})$ in the following sense.
There exist parameters $\epsilon_0$ and $A_0$, depending on $c_1$ and $c_2$, if there exists $\epsilon \in (0,\epsilon_0)$ such that for any $u_0\in H^1(\mathbb{R})$,
\begin{align}\label{u02}
 \bigl\lVert u_0 - U^{(2)}_{c_1,c_2}(0;0,0) \bigr\rVert_{H^1(\mathbb{R})} < \epsilon,
\end{align}
then there exist $x_1(t), x_2(t) \in \mathbb{R}$, such that the corresponding solution $u(t,x)$ of the ILW equation (\ref{ILW}) with the initial data $u(0)=u_0$ satisfies $u_t \in C([0,\infty),H^1(\mathbb{R}))$ and
\begin{align}\label{1.22}
 \sup_{t\in (0,\infty)}|\!|u(t)-U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)|\!|_{H^1(\mathbb{R})}<A_0\epsilon,
\end{align}
with
\begin{align}
 \sup_{t\in (0,\infty)}(|x_1'(t)|+|x_2'(t)|)\leq CA_0\epsilon.
\end{align}
\end{theorem}

\begin{align*}
 (\mathcal{R}^\star(u))^m u_x &= \mathcal{J} (\mathcal{R} (u))^m H_1'(u) = \mathcal{J} (\mathcal{R} (u))^m u, \quad m \in \mathbb{N}.
\end{align*}
For subsequent applications, establishing the uniqueness and differentiability of $\mathcal{R}(u)$ becomes essential. In contrast, the Korteweg-de Vries system ($\delta = 3$) exhibits explicit structure. For Schwartz-class functions $u \in \mathcal{S}(\mathbb{R})$, its recursion operator is $\mathcal{R}_K(u) = -\partial_x^2 - \tfrac{2}{3}u - \tfrac{2}{3}\partial_x^{-1} u \partial_x$, then $\mathcal{R}_K'(u) = -\tfrac{2}{3} - \tfrac{2}{3}\partial_x^{-1}(\cdot\partial_x)$.
\begin{prop}\label{prop2.2}
 Given $u \in H^{k+1}(\mathbb{R})$ with $k \geq 0$, there exists a unique linear operator
 \[
 \mathcal{R}(u): H^{k+1}(\mathbb{R}) \to H^{k}(\mathbb{R}),
 \]
 such that relations (\ref{2.39}) and (\ref{2.41}) hold true. Moreover, $\mathcal{R}(u)$ is differentiable with respect to $u$.
\end{prop}
\begin{proof}
 The fundamental approach involves establishing a connection between the recursion operators $\mathcal{R}(u)$ and $\mathcal{R}_{12}$ (\ref{1.15}). Assuming $u \in \mathcal{S}(\mathbb{R})$, it follows from (\ref{1.14}) and (\ref{2.39}) that
 \begin{align*}
 \mathcal{S}(\mathbb{R}) \ni H_{m+1}^{\prime}(u) & = \frac{i}{2^{m+1}}\mathcal{J}^{-1}\int_{\mathbb{R}} \delta(x_{1} - x_{2}) \theta_{12}^{-} \mathcal{R}_{12}^{m+1} \cdot 1 \mathrm{d}x_{2} \\
 & = \mathcal{R}(u)H_{m}^{\prime}(u) \\
 & = \mathcal{R}(u) \frac{i}{2^m}\mathcal{J}^{-1}\int_{\mathbb{R}} \delta(x_{1} - x_{2}) \theta_{12}^{-} \mathcal{R}_{12}^{m} \cdot 1 \mathrm{d}x_{2}.
 \end{align*}
 The uniqueness of $\mathcal{R}(u)$ follows by an induction argument over $m$. Furthermore, the asymptotic behavior of the operator is characterized by
 \[
 \mathcal{R}(u) \sim -T^\delta\partial_{x} + L(u),
 \]
 where $L(u): \mathcal{S}(\mathbb{R}) \mapsto \mathcal{S}(\mathbb{R})$ represents a differentiable higher-order remainder term. By standard density argument, $\mathcal{R}(u)$ itself is differentiable and satisfies $\mathcal{R}^{\prime}(u) \sim L^{\prime}(u)$.
\end{proof}
\begin{re}
The propositions above will be shown in Section \ref{sec.3} that understanding the spectral information of the (adjoint) recursion operators $\mathcal{R} (u)$ and $\mathcal{R}^\star(u)$ is essential in proving the (spectral) stability of the ILW multi-solitons.
\end{re}

Next, we will proceed to prove Theorem \ref{th1.3}.
\subsection{The negative eigenvalues of $\mathcal{L}_n$ :Proof of Theorem \ref{th1.3}}
In this subsection, our goal is to prove the Theorem \ref{th1.3}.
\begin{proof}[\textbf{Proof of Theorem \ref{th1.3}}]
 The linearized operators around the $n$-solitons $\mathcal{L}_{n} = S_{n}^{\prime\prime}(U^{(n)})$ possess $\left[\frac{n+1}{2}\right]$ negative eigenvalues, as previously verified in (\ref{3.47}). We now prove the expression (\ref{negetive}) for these eigenvalues. As $t\to \infty$, the spectrum $\sigma(\mathcal{L}_{n}(t))$ converges to the union of the spectra $\sigma(\mathcal{L}_{n,j})$ where $\mathcal{L}_{n,j} = S_{n}^{\prime\prime}(Q_{c_{j}})$, that is
 \begin{align}
 \sigma(\mathcal{L}_{n}(t)) \rightarrow \bigcup_{j=1}^{n} \sigma(\mathcal{L}_{n,j}), \quad \text{as} \quad t \rightarrow +\infty.
 \end{align}
 Since the operators $\mathcal{L}_{n}(t)$ are isoinertial for each $n$, the spectrum $\sigma(\mathcal{L}_{n}(t))$ is independent of $t$. Thus, the negative eigenvalues of $\mathcal{L}_{n}$ match those of $\mathcal{L}_{n,j}$ for all $j = 1, 2, \dots, n$. According to Lemma \ref{lem3.5}, $\mathcal{L}_{n,j}$ has negative eigenvalues when $j = 2k-1$ and $1 \leq k \leq \left[\frac{n+1}{2}\right]$. Specially, when $n=1$, one has $\mathcal{L}_1=L_1$. Therefore, (\ref{n=1}) indicates that $\mathcal{L}_1$ has the negative $\lambda_1<0$. With the methods in \cite{BBSSB}, we could compute $\lambda_1$, which is
 \begin{align}\label{1.30}
 \lambda_1=-\frac{c-\frac{1}{\delta}}{2a^2\sin^2(a\delta)}<0,
 \end{align}
 Then for $n=2$, we have
 \begin{align}\label{lamb}
 \mathcal{L}_{2,1}=(\mathcal{R}(Q_{c_1})+c_2)\mathcal{S}_1''(Q_{c_1}).
 \end{align}
In terms of Lemma \ref{lem3.2}, the operator $(\mathcal{R}(Q_{c_1}) + c_{2})$ has an eigenvalue $c_{2} - c_{1} > 0$, with continuous spectrum $(c_2, +\infty)$ whose generalized eigenfunctions are not in $L^{2}(\mathbb{R})$. Therefore, the unique negative eigenvalues of $\mathcal{L}_{2,1}$ is $(c_2-c_1)\lambda_1<0$.\\
Assuming the result holds for $n = p$, namely, the $\left[\frac{p+1}{2}\right]$-th negative eigenvalue of $\mathcal{L}_{p}$ is
 \begin{align}\label{4.48}
 \lambda_{k}^{(p)} = -M^\delta (c_{2k-1}-\frac{1}{\delta}) \prod_{j \neq 2k-1}^{p} (c_{j} - c_{2k-1}), \quad k = 1, 2, \dots, \left[\frac{p+1}{2}\right].
 \end{align}
Then for $n = p+1$ is even, we have $\left[\frac{p+1}{2}\right] = \left[\frac{p+2}{2}\right]$ and for $k = 1, 2, \dots, \left[\frac{p+1}{2}\right]$, we have
 \begin{align}
 \mathcal{L}_{p+1,2k-1} =(\mathcal{R}(Q_{c_{2k-1}}) + c_{p+1}) \mathcal{S}_{p}^{\prime\prime}(Q_{c_{2k-1}}).
 \end{align}
 By Lemma \ref{lem3.2}, the operator $(\mathcal{R}(Q_{c_{2k-1}}) + c_{p+1})$ has an eigenvalue $c_{p+1} - c_{2k-1} > 0$, with continuous spectrum $(c_{p+1}, +\infty)$ whose generalized eigenfunctions are not in $L^{2}(\mathbb{R})$. Therefore, the $\left[\frac{p+2}{2}\right]$-th negative eigenvalues of $\mathcal{L}_{p+1,2k-1}$ is
 \begin{align}\label{4.50}
 \lambda_{k}^{(p+1)}= -M^\delta (c_{2k-1}-\frac{1}{\delta}) \prod_{j \neq 2k-1}^{p+1} (c_{j} - c_{2k-1}), \quad k = 1, 2, \dots, \left[\frac{p+1}{2}\right].
 \end{align}
For the case where $n = p+1$ is odd, we have $\left[\frac{p+1}{2}\right] + 1 = \left[\frac{p+2}{2}\right]$. The first $\left[\frac{p+1}{2}\right]$ negative eigenvalues of $\mathcal{L}_{p+1}$ are given by expression (\ref{4.50}). We now compute the final negative eigenvalue. Consider
\begin{align}
 \mathcal{L}_{p+1,p+1} =\left( \mathcal{R}(Q_{c_{p+1}}) + c_{j} \right) \tilde{S}_{p}^{\prime\prime}(Q_{c_{p+1}})
\end{align}
where $\tilde{S}_{p}$ denotes the action with wave speed $c_j$ in $S_p$ replaced by $c_{p+1}$ for some $1 \leq j \leq p$. By assumption (\ref{4.48}), the discrete eigenvalue of $\tilde{S}_{p}^{\prime\prime}(Q_{c_{p+1}})$ is
\begin{align}
 -M^\delta (c_{p+1}-\frac{1}{\delta})\prod_{l \neq j}^{p} (c_l - c_{p+1}).
\end{align}
The operator $\mathcal{R}(Q_{c_{p+1}}) + c_j$ has eigenvalue $c_j - c_{p+1} < 0$ by Lemma \ref{lem3.2}, with continuous spectrum $(c_j, +\infty)$ whose generalized eigenfunctions are not in $L^2(\mathbb{R})$. Therefore, the last negative eigenvalue of $\mathcal{L}_{p+1}$ is
\begin{align*}
 \lambda_{\left[\frac{p+2}{2}\right]}^{(p+1)} = -M^\delta (c_{p+1}-\frac{1}{\delta}) \prod_{l=1}^{p} (c_l - c_{p+1}).
\end{align*}
The proof of Theorem \ref{th1.3} is completed by combining the eigenvalue expressions.
\end{proof}

Since the stability result in \cite{MS} coincides with orbital stability theorem in \cite{GSS} for $n=1$, we obtain the following corollary regarding the orbital stability of ILW soliton solutions.
\begin{cor}
The single soliton solution of the ILW equation is orbital stable in $H^{\frac{1}{2}}(\mathbb{R})$.
\end{cor}
\begin{re}
For $n \geq 2$, it should be noted that the stability result in \cite{MS} aligns with the notion of orbital stability in \cite{GSS} when the stability framework of \cite{MS} is suitably extended. In Hamiltonian systems, invariant functionals (i.e., integrals of motion) generate Hamiltonian flows that commute with the original time evolution.
\end{re}

Then we will provide a more precise description of the orbital stability of the double solitons to the ILW equation.

\begin{theorem}[\textbf{Orbital stability of double solitons}]\label{thm1.2}
	The double solitons $U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)$ of ILW equation (\ref{ILW}) with $0<c_1<c_2$ are orbitally stable in $H^1(\mathbb{R})$ in the following sense.
There exist parameters $\epsilon_0$ and $A_0$, depending on $c_1$ and $c_2$, if there exists $\epsilon \in (0,\epsilon_0)$ such that for any $u_0\in H^1(\mathbb{R})$,
\begin{align}\label{u02}
	\bigl\lVert u_0 - U^{(2)}_{c_1,c_2}(0;0,0) \bigr\rVert_{H^1(\mathbb{R})} < \epsilon,
\end{align}
then there exist $x_1(t), x_2(t) \in \mathbb{R}$, such that the corresponding solution $u(t,x)$ of the ILW equation (\ref{ILW}) with the initial data $u(0)=u_0$ satisfies $u_t \in C([0,\infty),H^1(\mathbb{R}))$ and
\begin{align}\label{1.22}
	\sup_{t\in (0,\infty)}|\!|u(t)-U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)|\!|_{H^1(\mathbb{R})}<A_0\epsilon,
\end{align}
with
\begin{align}
	\sup_{t\in (0,\infty)}(|x_1'(t)|+|x_2'(t)|)\leq CA_0\epsilon.
\end{align}
\end{theorem}

Moreover, something is interesting is that one can express the negative eigenvalues of the isoinertial operator $\mathcal{L}_n$ in terms of the wave speeds $\{c_j\}^n_{j=1}$ as follows.
\begin{theorem}[\textbf{The negative eigenvalues of $\mathcal{L}_n$}]\label{th1.3}
	The linearized operator $\mathcal{L}_{n}$ about the $n$-soliton solution possesses exactly $\left\lfloor\frac{n+1}{2}\right\rfloor$ negative eigenvalues $\lambda_{k}$, where $k=1,2,\cdots,\left\lfloor\frac{n+1}{2}\right\rfloor$ and $\lfloor x\rfloor$ denotes the floor function. Furthermore, the negative eigenvalues of $\mathcal{L}_n(n\geq 1$)  have the following expression
	\begin{align}\label{negetive}
		\lambda_{k} = -M^\delta_k (c_{2k-1}-\frac{1}{\delta})\prod_{ j\neq 2k-1}^{n}(c_{j}-c_{2k-1}),\quad k=1,2,\cdots,\left\lfloor\frac{n+1}{2}\right\rfloor,
	\end{align}
	where $M^\delta_k$ are positive constants independent of the wave speeds $c_1,\cdots, c_n$.
\end{theorem}
\begin{re}
	Since the ILW equation serves as the bridge connecting KdV equation and BO equation. The stability results above are also valid for both equations. The specific results can be found in \cite{WA} and \cite{LW}, respectively.
\end{re}

The proof of our main stability result, Theorem \ref{thm1.1}, proceeds by transforming the stability problem into a spectral analysis of an explicitly constructed linearized operator. Through the IST method, we establish a fundamental connection between the Hamiltonians and scattering data. Similarly, the proof of Theorem \ref{thm1.2} on orbital stability is reduced to a detailed spectral analysis of the linearized operator $\mathcal{L}_n$ arising from the second variation of the Lyapunov functional. Finally, the proof of Theorem \ref{th1.3} relies on the application of spectral analysis results. The proof strategy unfolds as follows.

In Section \ref{sec.2}, we begin by establishing the properties of the singular integral operators $T^\delta$ and $T^\delta_{12}$. We then introduce the IST framework for the ILW equation, from which the $n$-soliton solutions are formally derived.

Section \ref{sec.3} is devoted to a detailed spectral analysis of the linearized operator $\mathcal{L}_n = \mathcal{S}_n''(U^{(n)})$. We demonstrate that its inertia index is $\left( \left\lfloor \frac{n+1}{2} \right\rfloor, n \right)$, thereby satisfying the crucial stability condition.

In Section \ref{sec.4}, we apply the framework of constrained variational principles to establish the dynamical stability result in Theorem \ref{thm1.1}. We then prove Theorem \ref{thm1.2} on orbital stability of double solitons ($n=2$) through a refined spectral analysis of the linearized operator $\mathcal{L}_2$, yielding a direct proof of orbital stability. Finally, we establish Theorem \ref{th1.3} based on the spectral analysis results from Section \ref{sec.3}.

The proof of Theorem~\ref{thm1.1} “proceeds by transforming the stability problem into a spectral analysis of an explicitly constructed linearized operator.” “Through the IST method, we establish a fundamental connection between the Hamiltonians and scattering data.”

More concretely, the stated strategy is organized by sections as follows:
\begin{itemize}
\item \emph{Section~\ref{sec.2}:} establish “the properties of the singular integral operators $T^\delta$ and $T^\delta_{12}$,” and “introduce the IST framework for the ILW equation, from which the $n$-soliton solutions are formally derived.”
\item \emph{Section~\ref{sec.3}:} carry out “a detailed spectral analysis of the linearized operator $\mathcal{L}_n = \mathcal{S}_n''(U^{(n)})$,” showing “its inertia index is $\left( \left\lfloor \frac{n+1}{2} \right\rfloor, n \right)$, thereby satisfying the crucial stability condition.”
\item \emph{Section~\ref{sec.4}:} “apply the framework of constrained variational principles to establish the dynamical stability result in Theorem~\ref{thm1.1}.”
\end{itemize}

(Additionally, the introduction notes that the orbital stability theorem (Theorem~\ref{thm1.2}) is “reduced to a detailed spectral analysis of the linearized operator $\mathcal{L}_n$ arising from the second variation of the Lyapunov functional,” and that Theorem~\ref{th1.3} “relies on the application of spectral analysis results.”)

The proof of the main theorem proceeds by transforming the stability problem into a spectral analysis of an explicitly constructed linearized operator. Through the IST method, we establish a fundamental connection between the Hamiltonians and scattering data.

More concretely, the stated strategy is organized by sections as follows:
\begin{itemize}
\item \emph{Section~\ref{sec.2}:} establish “the properties of the singular integral operators $T^\delta$ and $T^\delta_{12}$,” and “introduce the IST framework for the ILW equation, from which the $n$-soliton solutions are formally derived.”
\item \emph{Section~\ref{sec.3}:} carry out “a detailed spectral analysis of the linearized operator $\mathcal{L}_n = \mathcal{S}_n''(U^{(n)})$,” showing “its inertia index is $\left( \left\lfloor \frac{n+1}{2} \right\rfloor, n \right)$, thereby satisfying the crucial stability condition.”
\item \emph{Section~\ref{sec.4}:} “apply the framework of constrained variational principles to establish the dynamical stability result in the main theorem.”
\end{itemize}

Additionally, the introduction notes that the orbital stability theorem
\begin{theorem}[\textbf{Orbital stability of double solitons}]\label{thm1.2}
	The double solitons $U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)$ of ILW equation (\ref{ILW}) with $0<c_1<c_2$ are orbitally stable in $H^1(\mathbb{R})$ in the following sense.
There exist parameters $\epsilon_0$ and $A_0$, depending on $c_1$ and $c_2$, if there exists $\epsilon \in (0,\epsilon_0)$ such that for any $u_0\in H^1(\mathbb{R})$,
\begin{align}\label{u02}
	\bigl\lVert u_0 - U^{(2)}_{c_1,c_2}(0;0,0) \bigr\rVert_{H^1(\mathbb{R})} < \epsilon,
\end{align}
then there exist $x_1(t), x_2(t) \in \mathbb{R}$, such that the corresponding solution $u(t,x)$ of the ILW equation (\ref{ILW}) with the initial data $u(0)=u_0$ satisfies $u_t \in C([0,\infty),H^1(\mathbb{R}))$ and
\begin{align}\label{1.22}
	\sup_{t\in (0,\infty)}|\!|u(t)-U^{(2)}_{c_1,c_2}(t,x;x_1,x_2)|\!|_{H^1(\mathbb{R})}<A_0\epsilon,
\end{align}
with
\begin{align}
	\sup_{t\in (0,\infty)}(|x_1'(t)|+|x_2'(t)|)\leq CA_0\epsilon.
\end{align}
\end{theorem}

is “reduced to a detailed spectral analysis of the linearized operator $\mathcal{L}_n$ arising from the second variation of the Lyapunov functional,” and that the following theorem “relies on the application of spectral analysis results”:
\begin{theorem}[\textbf{The negative eigenvalues of $\mathcal{L}_n$}]\label{th1.3}
	The linearized operator $\mathcal{L}_{n}$ about the $n$-soliton solution possesses exactly $\left\lfloor\frac{n+1}{2}\right\rfloor$ negative eigenvalues $\lambda_{k}$, where $k=1,2,\cdots,\left\lfloor\frac{n+1}{2}\right\rfloor$ and $\lfloor x\rfloor$ denotes the floor function. Furthermore, the negative eigenvalues of $\mathcal{L}_n(n\geq 1$)  have the following expression
	\begin{align}\label{negetive}
		\lambda_{k} = -M^\delta_k (c_{2k-1}-\frac{1}{\delta})\prod_{ j\neq 2k-1}^{n}(c_{j}-c_{2k-1}),\quad k=1,2,\cdots,\left\lfloor\frac{n+1}{2}\right\rfloor,
	\end{align}
	where $M^\delta_k$ are positive constants independent of the wave speeds $c_1,\cdots, c_n$.
\end{theorem}

[\textbf{Dynamical stability of $n$-solitons}]\label{thm1.1}
Given $ n\in \mathbb{N} $, $ n\geq 1 $, a collection of wave speeds $\textbf{c} = (c_{1},\dots,c_{n})$ with $ 0 < c_{1} < \dots < c_{n} $ and a collection of space transitions $\textbf{x} = (x_{1},\dots,x_{n})\in \mathbb{R}^{n}$, let $ U^{(n)}(\cdot,\cdot;\textbf{c},\textbf{x}) $ be the corresponding $n$-soliton solutions of
\begin{align}
\tu_t+\frac{1}{\delta}u_x+2uu_x+T^\delta u_{xx}=0,\quad  (t,x)\in \mathbb{R} \times \mathbb{R},\label{ILW}
\end{align}
Then for any $ \epsilon > 0 $, there exists $ \delta > 0 $ such that for any $ u_{0}\in H^{\frac{n}{2}}(\mathbb{R}) $, the following stability property holds, if

\begin{align}
	&\bigl\lVert u_{0} - U^{(n)}(0,\cdot;\textbf{c},\textbf{x}) \bigr\rVert_{H^{\frac{n}{2}}} < \delta,
\end{align}
then for any $t \in \mathbb{R}$, the corresponding solution of the equation above verifies
\begin{align}
\inf_{\tau \in \mathbb{R},\  \textbf{y} \in \mathbb{R}^{n}} \bigl\lVert u(t) - U^{(n)}(\tau,\cdot;\textbf{c},\textbf{y}) \bigr\rVert_{H^{\frac{n}{2}}} < \epsilon.
\end{align}

Question:
Let \(n\in\mathbb N\) with \(n\ge 1\), let \(\mathbf c=(c_1,\dots,c_n)\) satisfy \(0<c_1<\cdots<c_n\), and let \(\mathbf x=(x_1,\dots,x_n)\in\mathbb R^n\). Let \(U^{(n)}(t,x;\mathbf c,\mathbf x)\) be the corresponding \(n\)-soliton solution of the ILW equation
\[
u_t+\frac{1}{\delta}u_x+2uu_x+T^\delta u_{xx}=0,\qquad (t,x)\in\mathbb R\times\mathbb R,
\]
where \(T^\delta\) is the ILW nonlocal operator. Suppose \(u(t)\) is the solution with initial data \(u(0)=u_0\in H^{n/2}(\mathbb R)\). Which of the following conclusions holds about the orbit of the \(n\)-soliton family \(\{U^{(n)}(\tau,\cdot;\mathbf c,\mathbf y):\tau\in\mathbb R,\ \mathbf y=(y_1,\dots,y_n)\in\mathbb R^n\}\) in the \(H^{n/2}(\mathbb R)\) norm?

For every \(\epsilon>0\), there exists \(\eta>0\) such that if
\[
\bigl\lVert u_0-U^{(n)}(0,\cdot;\mathbf c,\mathbf x)\bigr\rVert_{H^{n/2}}<\eta,
\]
then the corresponding solution satisfies
\[
\inf_{\tau\in\mathbb R,\ \mathbf y\in\mathbb R^n}\bigl\lVert u(t)-U^{(n)}(\tau,\cdot;\mathbf c,\mathbf y)\bigr\rVert_{H^{n/2}}<\epsilon
\quad\text{for every } t\in\mathbb R.
\]
Thus the \(n\)-soliton family is orbitally stable in \(H^{n/2}(\mathbb R)\).

Choices:
B.
For every \(\epsilon>0\), there exists \(\eta>0\) such that if
\[
\bigl\lVert u_0-U^{(n)}(0,\cdot;\mathbf c,\mathbf x)\bigr\rVert_{H^{n/2}}<\eta,
\]
then the corresponding solution satisfies
\[
\inf_{\mathbf y\in\mathbb R^n}\bigl\lVert u(t)-U^{(n)}(t,\cdot;\mathbf c,\mathbf y)\bigr\rVert_{H^{n/2}}<\epsilon
\quad\text{for every } t\in\mathbb R.
\]
Thus the \(n\)-soliton family is orbitally stable in \(H^{n/2}(\mathbb R)\) with the time parameter fixed.
C.
For every \(\epsilon>0\), there exists \(\eta>0\) such that if
\[
\bigl\lVert u_0-U^{(n)}(0,\cdot;\mathbf c,\mathbf x)\bigr\rVert_{H^{n/2}}<\eta,
\]
then the corresponding solution satisfies
\[
\inf_{\mathbf y\in\mathbb R^n}\bigl\lVert u(t)-U^{(n)}(t,\cdot;\mathbf c,\mathbf y)\bigr\rVert_{H^{n/2}}<\epsilon
\quad\text{for every } t\in\mathbb R.
\]
In particular, the orbit remains close to the \(n\)-soliton family in \(H^{n/2}(\mathbb R)\).
D.
There exists \(\eta>0\) such that for every \(\epsilon>0\), if
\[
\bigl\lVert u_0-U^{(n)}(0,\cdot;\mathbf c,\mathbf x)\bigr\rVert_{H^{n/2}}<\eta,
\]
then the corresponding solution satisfies
\[
\inf_{\tau\in\mathbb R,\ \mathbf y\in\mathbb R^n}\bigl\lVert u(t)-U^{(n)}(\tau,\cdot;\mathbf c,\mathbf y)\bigr\rVert_{H^{n/2}}<\epsilon
\quad\text{for every } t\in\mathbb R.
\]
Thus the \(n\)-soliton family is orbitally stable in \(H^{n/2}(\mathbb R)\) with a uniform stability radius independent of \(\epsilon\).
E.
For every \(\epsilon>0\), there exists \(\eta>0\) such that if
\[
\bigl\lVert u_0-U^{(n)}(0,\cdot;\mathbf c,\mathbf x)\bigr\rVert_{H^{n/2}}<\eta,
\]
then the corresponding solution satisfies
\[
\bigl\lVert u(t)-U^{(n)}(t,\cdot;\mathbf c,\mathbf x)\bigr\rVert_{H^{n/2}}<\epsilon
\quad\text{for every } t\in\mathbb R.
\]
Thus the given \(n\)-soliton solution is asymptotically stable in \(H^{n/2}(\mathbb R)\).

quantifier order between \(\epsilon\) and neighborhood size

orbital stability modulo symmetries replaced by fixed-profile/asymptotic stability

The stem does not explicitly state the correct option. It asks for the valid uniform orbital stability formulation, and the answer is not trivially recoverable without knowing or reasoning through the role of the modulation parameters and quantifiers.

The item is close to a theorem-recall question: the correct choice is essentially the precise stability statement. However, it is not a pure restatement because the options vary in meaningful ways (time shift, quantifier order, fixed versus time-dependent orbit).

Some reasoning is required to distinguish the strongest valid orbital stability claim from nearby false or weaker variants, especially regarding the extra time-shift parameter and quantifier structure. Still, it mainly tests recognition of the exact theorem form rather than deeper derivation.

The distractors are strong: they reflect realistic mathematical confusions such as removing a modulation parameter, weakening to a pointwise-in-time existence statement, reversing quantifiers, or demanding an unrealistically fixed orbit for all time.

A solid but theorem-centered MCQ: it avoids obvious leakage and has high-quality distractors, but it leans more toward precise statement recognition than genuinely generative mathematical reasoning.