\begin{theorem}\label{thm:mainthm}
Let $S$ be a complex algebraic variety admitting a trim resolution. Then
\begin{itemize}
\item
For $s\in S$, the local Euler obstruction $\Eu_S(s)$ equals the Euler characteristic of
the fiber over $s$ in any trim resolution of $S$.
\item
The {\em stringy\/} Chern class of $S$ equals its Chern-Mather class:
$\cstr(S)=\cma(S)$.
\item
The characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.
\end{itemize}
\end{theorem}

\begin{equation}\label{eq:strMa}
f_*(c(TY)\cap [Y]) = \cma(S)
\end{equation}
in the Chow group of $S$. In general, stringy Chern classes are defined for
normal varieties with a $\Qbb$-Cartier canonical divisor and at worst log-terminal
singularities, and take value in the Chow group with rational coefficients. These
additional stipulations are not needed for the notion considered in this note; we can
adopt the left-hand side of~\eqref{eq:strMa} as the definition of stringy Chern class
in the (integral) Chow group of $S$, compatibly with the more general definition.
The fact that the class is independent of the choice of trim resolution is
also a consequence of Theorem~\ref{thm:mainthm}.

We note that Theorem~\ref{thm:mainthm} implies that Batyrev's {\em stringy Euler
number\/} (\cite{MR2001j:14018}) of a variety admitting a trim resolution equals
its Euler characteristic weighted by the local Euler obstruction.

Theorem~\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's
formalism of {\em conical Lagrangian cycles,\/} to which local Euler obstructions
and Chern-Mather classes relate directly (see~\S\ref{Lagrapf}). Concerning intersection
cohomology, recall that if $\pi:Y \to S$ is a small resolution, then the
intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant
sheaf on~$Y$ (\cite[\S6.2]{MR696691}). We evaluate the corresponding push-forward
at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.
More precisely, we prove (Proposition~\ref{prop:Sabpf}) that, for trim resolutions, the
Lagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle
$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this
more basic observation, as we show in~\S\ref{thmpf}.

There is considerable interest in conditions implying that the characteristic cycle of
the intersection cohomology sheaf is irreducible. Lusztig (\cite[13.7, p.~414]{MR1088333})
expressed the `hope' that this may be the case for Schubert varieties in flag manifolds of
type A, D, E. A counterexample was constructed by Kashiwara and Saito for type A in
$F\ell(8)$ (\cite{MR1458969}, \cite{MR1896039}), while it holds in $F\ell(n)$ for $n\le 7$.
Irreducibility is also known for all Schubert
varieties in the standard Grassmannian (\cite{MR1084458}), and more generally
for all Schubert varieties in cominuscule Grassmannians of
types A, D, and E, see~\cite{MR1451256, MihalceaSingh}.

We hope that Theorem~\ref{thm:mainthm} may help in streamlining such verifications.
For instance, since the Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic
curve is trim, the irreducibility of the $\IC$ characteristic cycle in this case (first established
in \cite{MR1642745}) follows directly from~Theorem~\ref{thm:mainthm}.

The connection between the irreducibility of the $\IC$ characteristic cycle and
the equality of Chern-Mather and stringy Chern classes was pointed out by B.~Jones
in~\cite[Remark 3.3.2]{MR2628830}, ultimately as an application of the microlocal
index formula of Dubson and Kashiwara.
We freely borrow ideas from~\cite{MR2628830}
in~\S\ref{thmpf}.

A condition on fibers of small resolutions is considered in~\cite{graham}, including a proof
that the condition implies the irreducibility of the IC characteristic cycles.\medskip

{\em Acknowledgments.} This work was supported in part by an award from the Simons
Foundation, SFI-MPS-TSM-00013681. The author also thanks David Massey and
Leonardo Mihalcea for useful conversations, and Caltech for hospitality as most of this
work was carried out.

\section{Lagrangian push-forward}\label{Lagrapf}
Let $X$ be a nonsingular variety and denote by $T^* X$ the cotangent bundle of $X$.
The {\em conormal variety\/} $T^*_W X$ of a closed subvariety $W\subseteq X$ is the
closure in $T^* X$ of the conormal variety to the nonsingular part $W^\circ$ of $V$:
$T^*_W X = \overline{T^*_{W^\circ} X}$. The conormal variety of $X$ itself is the
zero-section $T^*_XX$ of the cotangent bundle. All conormal varieties have dimension
$\dim X$; they determine conormal {\em cycles\/} in $Z_{\dim X} T^* X$.
It will also be convenient to take the projective completion of these constructions:
we will denote by $\Tbb^* X$ the projective completion $\Pbb(T^* X\oplus \one)$
of $T^* X$ (here, $\Pbb$ denotes the projective bundle of lines) and by~$\Tbb^*_WX$
the closure of the conormal variety in $\Tbb^* X$.

For a nonsingular variety $X$, we denote by $\cL (X)$ the free abelian group of
{\em conical Lagrangian cycles\/} in the cotangent bundle $T^* X$.
Conormal cycles are conical Lagrangian, and in fact (cf.~\cite[Lemma~3]{MR1063344})
$\cL (X)$ may be realized as the free abelian group on conormal cycles.
For a closed (and possibly singular) subvariety $V\subseteq X$, we denote by
$\cL(V)$ the subgroup of $\cL(X)$ generated by the conormal cycles $T^*_W X$
with $W\subseteq V$. Thus, elements of~$\cL (V)$ may be viewed as finite integer linear
combinations $\sum_W m_W T^*_WX$ ranging over closed subvarieties $W$ of $V$.
Clearly $\cL(V)$ is isomorphic to the group of algebraic cycles of~$V$, and in particular
it is independent of the ambient nonsingular variety $X$.

Important invariants of $V$ may be expressed directly in terms of Lagrangian cycles
by means of intersection-theoretic operations, after taking the projective completion.
As above, realize $V$ as a closed subvariety of a
nonsingular variety $X$; the results will be independent of the choice of $X$.
Let $\pi: \Tbb^*_V X\to V$ be the natural projection; and let $\cO(1)$ be the
tautological line bundle on $\Tbb^* X=\Pbb(T^* X\oplus \one)$.
\begin{itemize}
\item[---]
The {\em local Euler obstruction\/} $\Eu_V: V\to \Zbb$ is
\[
\Eu_V(p)= (-1)^{\dim X-\dim V} \int c(\pi^* TX|_V) c(\cO(1))^{-1}\cap s(\pi^{-1}(p), \Tbb^*_V X)
\]
where $s(\pi^{-1}(p), \Tbb^*_V X)$ denotes the {\em Segre class\/} in the sense
of~\cite[Chapter~4]{85k:14004};
\item[---]
The {\em Chern-Mather class\/} $\cma(V)\in A_*V$ is
\[
\cma(V)=(-1)^{\dim X-\dim V} c(TX|_V)\cap \pi_* \left(c(\cO(1))^{-1}\cap [\Tbb^*_V X]\right)\saf.
\]
\end{itemize}
Equivalent results were established by C.~Sabbah (\cite[1.2.1, 1.2.2]{MR804052}); we also
address the reader to~\cite{MR1063344} and~\cite{MR2002g:14005} for clear treatments of these
formulas.

Let $\cF(V)$ denote the abelian group of constructible functions $V\to \Zbb$.
Following~\cite[\S1]{MR2002g:14005}, the relation between conormal cycles and local
Euler obstructions is recorded by the homomorphism
\[
\CC: \cF(V) \to \cL(V)
\]
defined by prescribing
\[
\Eu_W \mapsto (-1)^{\dim W} T^*_W X
\]
for all closed subvarieties $W$ of $V$.
In fact, $\CC$ is an {\em isomorphism:\/} it simply matches a basis of $\cF(V)$ with a
basis of $\cL(V)$.

\begin{defin}\label{def:CC}
The {\em characteristic cycle\/} $\CC(\alpha)$ of a constructible function $\alpha$ on $V$
is the image of $\alpha$ in $\cL(V)$ under this isomorphism.
\qede\end{defin}

Likewise, the formula for Chern-Mather classes motivates the introduction of a homomorphism
\[
c_*:\cL(V) \to A_*(V)
\]
defined on generators by
\[
T^*_W X \mapsto (-1)^{\dim W} \cma(W)\saf.
\]
Then the composition $c_*\circ \CC:\cF(V) \to A_*(V)$ agrees with the value at $V$
of MacPherson's natural transformation $\cF \leadsto A_*$, where $\cF$ is taken as
a functor with push-forward defined by Euler characteristics of fibers;
see~\cite{MR0361141} and~\cite[Example 19.1.7]{85k:14004}.
For instance, the {\em Chern-Schwartz-MacPherson class\/} of a (possibly singular)
algebraic variety $V$ is the image $c_*(\CC(\one_V))\in A_*V$ of the characteristic
cycle in~$\cL(V)$ of the constant function $\one_V$.
Sabbah provides an alternative proof of the naturality of this assignment, which is
the main result of~\cite{MR0361141}, by defining a covariant {\em push-forward\/}
\[
\varphi_*: \cL(V') \to \cL(V'')
\]
for every proper map $\varphi: V' \to V''$, making $\cL$ into a functor, in such a way
that the above homomorphisms define natural transformations
\[
\cF \leadsto \cL\quad, \quad \cL \leadsto A_*
\]
whose composition agrees with MacPherson's natural transformation.

We are interested in explicit formulas for this {\em Lagrangian push-forward.\/}
After embedding~$V''$ in a nonsingular variety $X$ and replacing $V'$ by a resolution $Y$,
we can reduce to the case of a proper morphism $f:Y \to X$ of nonsingular varieties.
We are specifically interested in the image $f_*(T^*_YY)$ of the zero-section in this
situation.

Theorem~\ref{thm:mainthm} will be a consequence of the following result. We recall the
definition of `trim' given in the introduction.

\begin{defin}\label{def:contained}
Let $\pi:Y \to S$ be a proper birational morphism of varieties, with $Y$ nonsingular.
For $0\le d\le \dim Y$, let $Y_d$ denote the locus where the rank of the differential
$d\pi$ equals $d$. Then $\pi$ is a {\em trim resolution\/} if $\dim Y_d<d$
for all $0\le d<\dim Y$.
\qede\end{defin}

\begin{prop}\label{prop:Sabpf}
Let $f: Y\to X$ be a proper morphism of nonsingular varieties, such that $Y \to f(Y)$ is
a trim resolution. Then $f_* (T^*_Y Y)=T^*_{f(Y)} X$.
\end{prop}

\begin{remark}\label{rem:YtoS}
Let $\pi: Y\to S$ be a proper surjective morphism, with $Y$ nonsingular, and
$\iota:S\hookrightarrow X$ a closed embedding, with $X$ nonsingular; and let
$f=\iota\circ \pi: Y \to X$ be the composition.
As explained above, $T^*_{f(Y)} X=T^*_S X$ may be viewed as an element of $\cL(S)$;
Proposition~\ref{prop:Sabpf} states that if $\pi$ is trim, then this is the image $\pi_*(T^*_YY)$
under $\pi_*: \cL(Y)\to \cL(S)$.
\qede\end{remark}

The rest of this section is devoted to the proof of Proposition~\ref{prop:Sabpf}.\smallskip

{\em A priori,\/} $f_*(T^*_Y Y)$ is an integer linear combination of conormal cycles
$\sum_Z m_Z T^*_ZX$, with $Z\subseteq f(Y)$. One of the summands is $T^*_{f(Y)}X$.
The task is to show that if $Y\to f(Y)$ is trim and $Z\subsetneq f(Y)$ is a {\em proper\/}
subvariety of $f(Y)$, then the coefficient of $T^*_Z X$ in $f_* (T^*_Y Y)$ is~$0$.
Following Sabbah, we write $f$ as the composition
\[
\xymatrix{
Y \ar[r]^-\gamma & Y\times X \ar[r]^-\rho & X
}
\]
where $\gamma$ is the graph of $f$ and $\rho$ is the (smooth) projection.
Let $p: \rho^*(T^* X) \to T^* X$ be the natural mophism. Then  (cf.~\cite[\S2]{MR804052})
the components $T^*_ZX$ appearing in the decomposition of $f_*(T^*_YY)$ are the
irreducible components of the image
\begin{equation}\label{eq:pint}
p \left(\rho^*(T^* X) \cap T^*_{\gamma(Y)}(Y\times X)\right)
\end{equation}

\label{intro}
Recall that a proper surjective stratified map $\pi: Y\to S$ of complex algebraic varieties
of the same dimension is {\em small\/} if for all proper strata $Z$ of $S$, $2d(Z)< \codim_Z S$,
where $d(Z)$ denotes the (common) dimension of the fibers of $\pi$ over points of $Z$.
We say that $\pi$ is a {\em small resolution\/} if $Y$ is nonsingular and $\pi$ is birational
and small.

We introduce a strengthening of this condition, which we name `trim'
(Definition~\ref{def:contained}).
Assume $\pi:Y\to S$ is a proper birational map, with $Y$ nonsingular.
For $d\ge 0$, denote by $Y_d\subseteq Y$ the locally closed subset along which
the rank of the differential of $\pi$ is~$d$. We say that $\pi$ is {\em trim\/} if
$\dim Y_d<d$ for all $d<\dim Y$.

Trim maps are small (Proposition~\ref{prop:consma}).
If $\pi:Y\to S$ is small, and for all strata~$Z$ of~$S$ the restriction $\pi^{-1}(Z)\to Z$ is a smooth
morphism, then $\pi$ is trim (\S\ref{Furrem}).
For instance, the standard Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic
curve (which is small, cf.~\cite{MR1642745}) is trim.

Our main result is the following.

\label{Furrem}

1. In this note we have considered the notion of `trim' only for proper birational morphisms
because that is the case relevant to our application in Theorem~\ref{thm:mainthm}.
It co

\begin{defin}\label{def:contained}
Let $\pi:Y \to S$ be a proper birational morphism of varieties, with $Y$ nonsingular.
For $0\le d\le \dim Y$, let $Y_d$ denote the locus where the rank of the differential
$d\pi$ equals $d$. Then $\pi$ is a {\em trim resolution\/} if $\dim Y_d<d$
for all $0\le d<\dim Y$.
\qede\end{defin}

\begin{prop}\label{prop:consma}
Let $\pi: Y \to S$ be a trim proper birational map. Then $\pi$ is small.
\end{prop}

\begin{abstract}
We introduce {\em trim\/} resolutions of complex algebraic varieties, a strengthening
of the notion of small resolution. We prove that the characteristic cycle of the intersection
cohomology sheaf of a variety admitting a trim resolution is irreducible and that for
such varieties the stringy and Chern-Mather classes coincide.
\end{abstract}

Our main result is the following.

The definition of characteristic cycle will be recalled below (\S\ref{Lagrapf}).
`Stringy' Chern classes were defined in~\cite{MR2183846, MR2304329}
(see~\cite{MR2280127} for a lean account). If $\pi: Y\to S$ is a crepant resolution,
$\cstr(S)$ equals the push-forward of the total Chern class of the tangent bundle
of~$Y$; the equality stated in the theorem is
\begin{equation}\label{eq:strMa}
f_*(c(TY)\cap [Y]) = \cma(S)
\end{equation}
in the Chow group of $S$. In general, stringy Chern classes are defined for
normal varieties with a $\Qbb$-Cartier canonical divisor and at worst log-terminal
singularities, and take value in the Chow group with rational coefficients. These
additional stipulations are not needed for the notion considered in this note; we can
adopt the left-hand side of~\eqref{eq:strMa} as the definition of stringy Chern class
in the (integral) Chow group of $S$, compatibly with the more general definition.
The fact that the class is independent of the choice of trim resolution is
also a consequence of Theorem~\ref{thm:mainthm}.

We note that Theorem~\ref{thm:mainthm} implies that Batyrev's {\em stringy Euler
number\/} (\cite{MR2001j:14018}) of a variety admitting a trim resolution equals
its Euler characteristic weighted by the local Euler obstruction.

Theorem~\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's
formalism of {\em conical Lagrangian cycles,\/} to which local Euler obstructions
and Chern-Mather classes relate directly (see~\S\ref{Lagrapf}). Concerning intersection
cohomology, recall that if $\pi:Y \to S$ is a small resolution, then the
intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant
sheaf on~$Y$ (\cite[\S6.2]{MR696691}). We evaluate the corresponding push-forward
at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.
More precisely, we prove (Proposition~\ref{prop:Sabpf}) that, for trim resolutions, the
Lagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle
$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this
more basic observation, as we show in~\S\ref{thmpf}.

Important invariants of $V$ may be expressed directly in terms of Lagrangian cycles
by means of intersection-theoretic operations, after taking the projective completion.
As above, realize $V$ as a closed subvariety of a
nonsingular variety $X$; the results will be independent of the choice of $X$.
Let $\pi: \Tbb^*_V X\to V$ be the natural projection; and let $\cO(1)$ be the
tautological line bundle on $\Tbb^* X=\Pbb(T^* X\oplus \one)$.
\begin{itemize}
\item[---]
The {\em local Euler obstruction\/} $\Eu_V: V\to \Zbb$ is
\[
\Eu_V(p)= (-1)^{\dim X-\dim V} \int c(\pi^* TX|_V) c(\cO(1))^{-1}\cap s(\pi^{-1}(p), \Tbb^*_V X)
\]
where $s(\pi^{-1}(p), \Tbb^*_V X)$ denotes the {\em Segre class\/} in the sense
of~\cite[Chapter~4]{85k:14004};
\item[---]
The {\em Chern-Mather class\/} $\cma(V)\in A_*V$ is
\[
\cma(V)=(-1)^{\dim X-\dim V} c(TX|_V)\cap \pi_* \left(c(\cO(1))^{-1}\cap [\Tbb^*_V X]\right)\saf.
\]
\end{itemize}
Equivalent results were established by C.~Sabbah (\cite[1.2.1, 1.2.2]{MR804052}); we also
address the reader to~\cite{MR1063344} and~\cite{MR2002g:14005} for clear treatments of these
formulas.

{\em A priori,\/} $f_*(T^*_Y Y)$ is an integer linear combination of conormal cycles
$\sum_Z m_Z T^*_ZX$, with $Z\subseteq f(Y)$. One of the summands is $T^*_{f(Y)}X$.
The task is to show that if $Y\to f(Y)$ is trim and $Z\subsetneq f(Y)$ is a {\em proper\/}
subvariety of $f(Y)$, then the coefficient of $T^*_Z X$ in $f_* (T^*_Y Y)$ is~$0$.
Following Sabbah, we write $f$ as the composition
\[
\xymatrix{
Y \ar[r]^-\gamma & Y\times X \ar[r]^-\rho & X
}
\]
where $\gamma$ is the graph of $f$ and $\rho$ is the (smooth) projection.
Let $p: \rho^*(T^* X) \to T^* X$ be the natural mophism. Then (cf.~\cite[\S2]{MR804052})
the components $T^*_ZX$ appearing in the decomposition of $f_*(T^*_YY)$ are the
irreducible components of the image
\begin{equation}\label{eq:pint}
p \left(\rho^*(T^* X) \cap T^*_{\gamma(Y)}(Y\times X)\right)
\end{equation}
where we view both $\rho^*(T^* X)$ and $T^*_{\gamma(Y)}(Y\times X)$
as subschemes of $T^* (Y\times X)$.
Let $T^*_ZX$ be one such component, and let $z$ be a general point of $Z$
and $\xi$ a general covector in the fiber $(T^*_ZX)_z$. For $(z,\xi)$ to be in
the image~\eqref{eq:pint}, there must be a point $(y,z)\in \gamma(Y)$ mapping to
$z$, such that
$
(0,\xi)\in T^*_y Y\oplus T^*_z X\cong T^*_{(y,z)} (Y\times X)
$
vanishes on $T_{(y,z)}\gamma(Y)$.

By Proposition~\ref{prop:consma}, $\pi$ is small. It follows (cf.~\cite[\S6.2]{MR696691})
that the intersection cohomology sheaf $\IC^\bullet_S$ is the direct image of the intersection
cohomology sheaf of $Y$. Since $Y$ is nonsingular, the latter is a shift of the constant
sheaf. Therefore,
\[
\IC^\bullet_S = R\pi_* \Qbb_Y[\dim Y]\saf.
\]
(Concerning the shift, we follow the modern convention as in e.g.,
\cite[Remark~4.2.4]{MR2525735}.)
The following formula for the stalk Euler characteristic of $\IC^\bullet_S$ at $z\in S$
is a consequence of standard properties of (derived) direct images:
\[
\chi_z(\IC^\bullet_S) =
\chi_z(R\pi_* \Qbb_Y[\dim Y]) =
(-1)^{\dim S} \sum_i (-1)^i \dim H^i(\pi^{-1}(z);\Qbb)
= (-1)^{\dim S}\chi(\pi^{-1}(z))\saf.
\]
By the first assertion in Theorem~\ref{thm:mainthm}, this implies
\[
\chi_z(\IC^\bullet_S) = (-1)^{\dim S}\Eu_S(z)\saf.
\]
Now embed $S$ in a nonsingular variety $X$.
The characteristic cycle of $\IC^\bullet_S$ is
\[
CC(\chi_z(\IC^\bullet_S)) = CC((-1)^{\dim S}\Eu_S(z)) = T^*_SX\saf,
\]
and this verifies it is irreducible, completing the proof of
Theorem~\ref{thm:mainthm}.

The definition of characteristic cycle will be recalled below (\S\ref{Lagrapf}).
`Stringy' Chern classes were defined in~\cite{MR2183846, MR2304329}
(see~\cite{MR2280127} for a lean account). If $\pi: Y\to S$ is a crepant resolution,
$\cstr(S)$ equals the push-forward of the total Chern class of the tangent bundle
of~$Y$; the equality stated in the theorem is
\begin{equation}\label{eq:strMa}
f_*(c(TY)\cap [Y]) = \cma(S)
\end{equation}
in the Chow group of $S$. In general, stringy Chern classes are defined for
normal varieties with a $\Qbb$-Cartier canonical divisor and at worst log-terminal
singularities, and take value in the Chow group with rational coefficients. These
additional stipulations are not needed for the notion considered in this note; we can
adopt the left-hand side of~\eqref{eq:strMa} as the definition of stringy Chern class
in the (integral) Chow group of $S$, compatibly with the more general definition.
The fact that the class is independent of the choice of trim resolution is
also a consequence of Theorem~\ref{thm:mainthm}.

We note that Theorem~\ref{thm:mainthm} implies that Batyrev's {\em stringy Euler
number\/} (\cite{MR2001j:14018}) of a variety admitting a trim resolution equals
its Euler characteristic weighted by the local Euler obstruction.

Theorem~\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's
formalism of {\em conical Lagrangian cycles,\/} to which local Euler obstructions
and Chern-Mather classes relate directly (see~\S\ref{Lagrapf}). Concerning intersection
cohomology, recall that if $\pi:Y \to S$ is a small resolution, then the
intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant
sheaf on~$Y$ (\cite[\S6.2]{MR696691}). We evaluate the corresponding push-forward
at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.
More precisely, we prove (Proposition~\ref{prop:Sabpf}) that, for trim resolutions, the
Lagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle
$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this
more basic observation, as we show in~\S\ref{thmpf}.

There is considerable interest in conditions implying that the characteristic cycle of
the intersection cohomology sheaf is irreducible. Lusztig (\cite[13.7, p.~414]{MR1088333})
expressed the `hope' that this may be the case for Schubert varieties in flag manifolds of
type A, D, E. A counterexample was constructed by Kashiwara and Saito for type A in
$F\ell(8)$ (\cite{MR1458969}, \cite{MR1896039}), while it holds in $F\ell(n)$ for $n\le 7$.
Irreducibility is also known for all Schubert
varieties in the standard Grassmannian (\cite{MR1084458}), and more generally
for all Schubert varieties in cominuscule Grassmannians of
types A, D, and E, see~\cite{MR1451256, MihalceaSingh}.

We hope that Theorem~\ref{thm:mainthm} may help in streamlining such verifications.
For instance, since the Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic
curve is trim, the irreducibility of the $\IC$ characteristic cycle in this case (first established
in \cite{MR1642745}) follows directly from~Theorem~\ref{thm:mainthm}.

The connection between the irreducibility of the $\IC$ characteristic cycle and
the equality of Chern-Mather and stringy Chern classes was pointed out by B.~Jones
in~\cite[Remark 3.3.2]{MR2628830}, ultimately as an application of the microlocal
index formula of Dubson and Kashiwara.
We freely borrow ideas from~\cite{MR2628830}
in~\S\ref{thmpf}.

A condition on fibers of small resolutions is considered in~\cite{graham}, including a proof
that the condition implies the irreducibility of the IC characteristic cycles.\medskip

{\em Acknowledgments.} This work was supported in part by an award from the Simons
Foundation, SFI-MPS-TSM-00013681. The author also thanks David Massey and
Leonardo Mihalcea for useful conversations, and Caltech for hospitality as most of this
work was carried out.

Theorem~\ref{thm:mainthm} is said to be “a consequence of considerations concerning Sabbah's formalism of {\em conical Lagrangian cycles,\/}” (to which “local Euler obstructions and Chern-Mather classes relate directly”). For intersection cohomology, using that for a small resolution $\pi:Y\to S$ “the intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant sheaf on~$Y$,” the paper “evaluate[s] the corresponding push-forward at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.” More precisely, it proves (Proposition~\ref{prop:Sabpf}) that for trim resolutions “the Lagrangian push-forward of the zero-section~$T^*_Y Y$ of the cotangent bundle $T^*Y$ equals the conormal cycle $T^*_S X$.” “The theorem follows from this more basic observation,” as shown in~\S\ref{thmpf}.

To prove the main theorem, it is said to be “a consequence of considerations concerning Sabbah's formalism of {\em conical Lagrangian cycles,\/}” (to which “local Euler obstructions and Chern-Mather classes relate directly”). For intersection cohomology, using that for a small resolution $\pi:Y\to S$ “the intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant sheaf on~$Y$,” the paper “evaluate[s] the corresponding push-forward at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.” More precisely, it proves the following proposition.

\begin{prop}\label{prop:Sabpf}
Let $f: Y\to X$ be a proper morphism of nonsingular varieties, such that $Y \to f(Y)$ is
a trim resolution. Then $f_* (T^*_Y Y)=T^*_{f(Y)} X$.
\end{prop}

That is, for trim resolutions “the Lagrangian push-forward of the zero-section~$T^*_Y Y$ of the cotangent bundle $T^*Y$ equals the conormal cycle $T^*_S X$.” “The theorem follows from this more basic observation,” as shown later.
\label{thmpf}

Let $S$ be a variety admitting a trim resolution $\pi: Y\to S$; let $\iota: S\hookrightarrow X$
be an embedding in a nonsingular variety, and let $f=\iota\circ \pi:Y \to X$ be the compo

Question: Let $S$ be a complex algebraic variety that admits a trim resolution, i.e. there exists a proper birational morphism $\pi:Y\to S$ with $Y$ nonsingular such that, for each $0\le d\le \dim Y$, if $Y_d\subseteq Y$ denotes the locus where the differential $d\pi$ has rank $d$, then $\dim Y_d<d$ for every $d<\dim Y$. In this setting, write $\Eu_S(s)$ for the local Euler obstruction at $s\in S$, write $\cma(S)$ for the Chern--Mather class of $S$, and write $\cstr(S)$ for the stringy Chern class (here, for a trim resolution $\pi$, one may take $\cstr(S)=\pi_*(c(TY)\cap [Y])$). Which statement holds for every such variety $S$?

For every trim resolution $\pi:Y\to S$ and every point $s\in S$, one has $\Eu_S(s)=\chi(\pi^{-1}(s))$; moreover $\cstr(S)=\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.

Choices:
B. For every trim resolution $\pi:Y\to S$ and every point $s\in S$, one has $\Eu_S(s)=\chi(\pi^{-1}(s))$; moreover $\cstr(S)=\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is the push-forward of the zero-section $T^*_Y Y$ for any trim resolution, hence may have several irreducible components.
C. For every trim resolution $\pi:Y\to S$ and every point $s\in S$, one has $\Eu_S(s)=\chi(\pi^{-1}(s))$; moreover $\cstr(S)=\cma(S)$ in the Chow group of $S$.
D. For every proper birational morphism $\pi:Y\to S$ with $Y$ nonsingular such that $\dim Y_d\le d$ for every $d<\dim Y$, and for every point $s\in S$, one has $\Eu_S(s)=\chi(\pi^{-1}(s))$; moreover $\cstr(S)=\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.
E. For every complex algebraic variety $S$ admitting a trim resolution, there exists a trim resolution $\pi:Y\to S$ such that for every $s\in S$ one has $\Eu_S(s)=\chi(\pi^{-1}(s))$, and for this choice of $\pi$ one has $\cstr(S)=\cma(S)$ in the Chow group of $S$; however the characteristic cycle of the intersection cohomology sheaf of $S$ need not be irreducible.

The stem sets up definitions and hypotheses but does not itself state the target conclusion. There is no direct textual leakage of the correct option from the stem alone.

The item is close to a theorem-recall question: it asks which packaged conclusion holds under the stated trim-resolution hypothesis. The alternatives add quantifier and strength variations, so it is not purely tautological, but it remains a mild reformulation of a likely source theorem.

Some reasoning is required to distinguish the strongest correct statement from a weaker true-looking option and from quantifier/condition tamperings. However, for someone who knows the theorem, the answer is mostly recognition rather than substantial generative derivation.

The distractors are mathematically plausible and target realistic failure modes: dropping a clause, weakening 'for every' to 'there exists', altering the trim inequality, and modifying the characteristic-cycle conclusion. They are distinct and nontrivial.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/recognition than deep generative reasoning.


\begin{theorem}\label{main theorem}
Let $\Omega\subset\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.
If $\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \ then the Bergman metric of $\Omega$ is well-defined in a nonempty open subset of $\Omega$, denoted by $\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\Omega^*$.
\end{theorem}

For any bounded domain $D\subset \mathbb{C}^n$, its Bergman metric is an invariant Kähler metric.
Cheng and Yau \cite{CY80} proved that every bounded pseudoconvex domain in $\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\"ahler–Einstein metric (up to a scaling factor) which is also biholomorphically invariant.
Later, Mok and Yau \cite{MY80} removed the boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.

A natural problem arising from these works is to determine under what circumstances these two important invariant metrics coincide. A classical conjecture of Yau \cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain. Earlier,
Cheng \cite{C79} had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.
Cheng's conjecture was confirmed in dimension two by Fu–Wong \cite{FW97} and Nemirovski--Shafikov \cite{NS06}, and was resolved in all dimensions by Huang–Xiao \cite{HX16} based on earlier work of many authors.
Subsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \cite{HL23}, Ebenfelt–Xiao–Xu \cite{EXX22,EXX24}, and references therein.
Related variations of Cheng’s conjecture were also discussed by S. Li in his papers \cite{L1, L2, L3}.

In a more recent development, Savale and Xiao \cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded pseudoconvex domains in $\mathbb{C}^2$.
They proved that a smoothly bounded pseudoconvex domain of finite type in $\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\mathbb{C}^2$.
A prior result by Fu–Wong \cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in $\mathbb{C}^2$.

Despite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with rough boundary points. In this paper, we aim to conduct a study along these lines.
We will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \ref{def1-9-24}) is not Einstein.

\begin{definition}\label{def1-9-24}
Let $\Omega$ be a possibly unbounded domain in $\mathbb{C}^{n}$  with $n>1$ and let $p\in\partial\Omega$.
We say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\rho_{1},\dots,\rho_{m}\colon U\to\mathbb{R}$ with $m>1$  such that
$\Omega\cap U=\{z\in U:\rho_{1}(z)<0,\dots,\rho_{m}(z)<0\}$  and
$\{\partial\rho_{1}|_p,\dots,\partial\rho_{m}|_p\}$  are linearly independent over $\mathbb{C}$.
\end{definition}

One of the main tools used in the proof of Theorem \ref{main theorem} is the rescaling argument, which has been used to work on many related
problems. In particular, in connection with our present work, we
mention the papers by Wong \cite{W77},
Kim \cite{Kim}, Kim-Yu \cite{KY}, Krantz-Yu \cite{KYu}
and Boas-Straube-Yu \cite{BSY}, where the rescaling method has been used to study the boundary limit of various quantities associated with the Bergman metric. Indeed, our current work has benefited from their studies. A recent application of the rescaling method can also be found in Huang-Zhu \cite{HZ}, where it is employed in solving a CR transversality problem. Another recent application of the rescaling method was used in working on the pinched properties of a K\"ahler metric is a recent paper of Bracci-Gauthier-Zimmer \cite{BGZ}.

We next recall the definition of strongly pseudoconvex polyhedral boundary points for a domain $\Omega\subset{\mathbb C}^n$.
\begin{definition}\label{def1-9-24}
Let $\Omega$ be a possibly unbounded domain in $\mathbb{C}^{n}$ with $n>1$ and let $p\in\partial\Omega$.
We say that $p$ is a strongly pseudoconvex polyhedral boundary point if there exists a neighborhood $U$ of $p$ in $\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\rho_{1},\dots,\rho_{m}\colon U\to\mathbb{R}$ with $m>1$ such that
$\Omega\cap U=\{z\in U:\rho_{1}(z)<0,\dots,\rho_{m}(z)<0\}$ and
$\{\partial\rho_{1}|_p,\dots,\partial\rho_{m}|_p\}$ are linearly independent over $\mathbb{C}$.
\end{definition}

\begin{proposition}\label{prop1-4-3}
Let $\Omega$ be a possibly unbounded pseudoconvex domain in $\mathbb C^n$ which is stronlgy pseudoconvex polyhedral at some boundary point $p\in\partial\Omega$. Let $U$ be a neighborhood of $p$ such that $U\cap\Omega$ is connected, and on which the Bergman metric $g_{\Omega}$ is well defined. Then its Bergman metric $g_\Omega$ is K\"ahler-Einstein on $U\cap\Omega$ if and only if its Bergman canonical invariant $J_{\Omega}\equiv (n+1)^n\frac{\pi^n}{n!}$ on $U\cap\Omega$.
\end{proposition}
\begin{proof}
By Corollary \ref{coro1-3-30}, for any smooth boundary point $q\in U\cap\partial\Omega$ near which $\partial\Omega$ is strongly pseudoconvex, one has
 \begin{equation}\label{e2-3-31}
 \lim_{z\rightarrow q}J_{\Omega}(z)=\frac{(n+1)^n\pi^n}{n!}.
 \end{equation}
 The Bergman metric $g_{\Omega}$ {is} K\"ahler-Einstein if its Ricci curvature $R_{\Omega}=c g_{\Omega}$ for some constant $c$. By Corollary \ref{coro1-3-30}, one has $c=-1$. Consequently, the K\"ahler-Einstein assumption implies that $\log J_{\Omega}$ is a pluriharmonic function on $U\cap\Omega$. Now, for any attached holomorphic disk $\phi:\Delta\rightarrow\Omega$ where $\phi$ is holomorphic in $\Delta:=\{t\in {\mathbb C}: |t|<1\}$, continuous up to $\overline {\Delta}$, and $\phi(\partial\Delta)$ is contained in the smooth part of $ U\cap \partial\Omega$, we have that $\log
 J_{\Omega}(\phi(t))$ is harmonic. Since it is constant on the strongly pseudoconvex part of the boundary by \eqref{e2-3-31}, it assumes the value
\[
\log\frac{(n+1)^n\pi^n}{n!}
\]
everywhere on $\Delta$. Now, since $\partial \Omega$ is strongly pseudoconvex near $q$, the union of such disks fills up an open subset of $\partial\Omega$ near $q$. Since $\log
 J_{\Omega}$ is well defined in $U\cap\Omega$ on which it is real analytic, we conclude that $\log J_{\Omega}\equiv
 \log\frac{(n+1)^n\pi^n}{n!}$ over $U\cap\Omega$ as $U\cap\Omega$ is connected by definition.
Conversely, if $J_\Omega(z)$ takes a constant value near $p$, then the Bergman metric is obviously K\"ahler-Einstien. Thus, we have the conclusion of the proposition.
\end{proof}
\begin{remark}\label{remark1-10-21}

Note that the zero set of the Bergman kernel function, denoted by $E$, is a complex analytic variety in $\Omega$.
Thus, $J_\Omega$ is a well-defined real-analytic function on $\Omega\setminus E$. Since $\Omega\setminus E$ is connected, $J_\Omega$ is constant if and only if it is constant on some nonempty open subset of $\Omega$. In particular, when $\Omega$ contains a $C^2$-smooth strongly pseudoconvex boundary point, the Bergman metric of the domain $\Omega$ is Kähler–Einstein wherever it is well-defined if and only if $J_\Omega=c$ is a constant on a certain open subset of $\Omega\setminus E$ .
 In this case, $c=\frac{(n+1)^n\pi^n}{n!}$, and the Bergman space $A^2(\Omega)$ separates holomorphic directions at any point in $\Omega\setminus E$ and thus the Bergman metric is well-defined in $\Omega\setminus E$.
\end{remark}
\section{Stability of the Bergman kernels}\label{sec2}

\begin{proof}[Proof of Theorem \ref{main theorem}]
Let $\Omega$ be a pseudoconvex domain which is strongly pseudoconvex polyhedral at a boundary point $p\in\partial\Omega$ as defined in Definition~\ref{def1-9-24}. After shrinking $U$ if necessary, we may assume that
\begin{enumerate}
\item[(i)] there are $C^{2}$-smooth strongly plurisubharmonic functions $\{\rho_{j}\}_{j=1}^{m}$ with $m>1$ on $U$ such that
\begin{equation}\label{9-24-a2}
 U\cap\Omega=\{z\in U:\rho_{1}(z)<0,\dots,\rho_{m}(z)<0\};
\end{equation}
\item[(ii)] the vectors $\partial\rho_{1}(q),\dots,\partial\rho_{m}(q)$ are linearly independent over $\mathbb C$ for $q\in U\cap \overline\Omega$.
\end{enumerate}
By Lemma \ref{25-9-23a1}, after a suitable change of coordinates on a small neighborhood $V\Subset U$ of $p$, we may assume that $p=0$ and
\begin{equation}\label{9-24-a1}
V\cap\Omega=\{(z_1,\cdots, z_n)\in V: {\rm Im~ }z_1>\Phi_1(z, \overline z), \cdots, {\rm Im}~ z_m>\Phi_m(z, \overline z)\}
\end{equation}
where $$\Phi_1(z, \overline z)=|z|^2+R_1(z), \Phi_j(z, \overline z)=\sum a^j_{\alpha\overline\beta} z_{\alpha}\overline z_{\beta}+R_j(z), \quad 2\leq j\leq m$$
with each remainder $R_j=\mathcal O(|z|^3)$. Write $$U_0=\{z\in\mathbb C^n: |z_j|<\varepsilon_0, 1\leq j\leq n\}$$ with $\varepsilon_0\ll 1$ such that $U_0\Subset V$ and on $U_0$ one has
$$|R_j(z)|\leq \frac{A_0}{2}|z|^2, \forall z\in U_0, 1\leq j\leq m,$$
where $A_0$ is defined in (\ref{10-27-a3}).
We first construct a bounded continuous plurisubharmonic function $\psi$ in $\Omega$ where $\psi$ is strictly plurisubharmonic near $p=0$ as follows:
\begin{lemma}\label{le3-28-1}
After shrinking $U_0$ if necessary, there exists a plurisubharmonic function $\psi:\Omega\rightarrow (-\infty, 0)$ such that $$\psi (z)>-c_0, ~\left(\frac{\partial^2\psi(z)}{\partial z_j\partial\overline z_k}\right)\geq c ~I_{n}, ~~z\in U_0\cap\Omega$$for some constants $c_0>0,~c>0$.
\end{lemma}
\begin{proof}
Let $(V, z)$ be the coordinates given as in (\ref{9-24-a1}) and set $\varphi=\frac{A_0}4|z|^2-y_1$. Then $\varphi$ is strictly plurisubharmonic on $V$ and satisfies $\varphi(0)=0, \varphi(z)<0$ when $z\in U_0\cap\overline\Omega\setminus \{0\}$. Take $r>0$ such that $\overline {\mathbb B^n(0, r)}\Subset U_0$. Set $M=\max\{\varphi(z): z\in \partial \mathbb B^n(0, r)\cap\overline \Omega\}$. Then $M<0$. Now we define $\psi$ as follows:
\begin{equation}
 \psi=\begin{cases}
 \max\{\varphi(z), M\}, &z\in \mathbb B^n(0, r)\cap\Omega\\
 M, &z\in\Omega\setminus \mathbb B^n(0, r).
 \end{cases}
\end{equation}
Then $\psi$ is a bounded and continuous plurisubharmonic function on $\Omega$ with $$\psi(0)=0, ~~\psi(z)<0, \forall z\in\overline\Omega\setminus\{0\}.$$
Furthermore, $\psi$ is equal to $\varphi$ near $0$ with $$\left(\frac{\partial^2\psi}{\partial z_i\partial\overline z_j}\right)= \frac{A_0}4 I_n $$
in some neighborhood $U$ of $0$ in $\mathbb C^n$. Moreover, $\psi>-c_0$ on $U$ for some positive constant $c_0$.
\end{proof}

\begin{corollary}
Let $\Omega\subset\mathbb{C}^{n}$, with $n>1$,  be a bounded pseudoconvex domain.  If $\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point,then its  Bergman metric cannot be Einstein.
\end{corollary}

One of the main tools used in the proof of Theorem \ref{main theorem} is the rescaling argument, which has been  used  to work on many related
problems. In particular, in connection with our present work, we
mention   the papers by Wong \cite{W77},
Kim \cite{Kim},  Kim-Yu \cite{KY}, Krantz-Yu \cite{KYu}
and Boas-Straube-Yu \cite{BSY},  where the rescaling method has been used to study the boundary limit of  various quantities associated with  the Bergman metric. Indeed, our current work  has benefited   from  their studies.  A recent application of the rescaling method can also be found in Huang-Zhu \cite{HZ}, where it is employed in solving a CR transversality problem. Another recent  application of the rescaling method was used in working on the pinched  properties of a K\"ahler metric is a recent  paper of Bracci-Gauthier-Zimmer \cite{BGZ}.

The  ideas of our proof of the main theorem can be stated briefly as follows: First, we   show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball. Next we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point  tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.
To obtain such a sequence, we assign weight $2$ to one of the complex normal directions, weight $1.5$  to the other normal directions, and weight $1$  to the remaining CR directions. The main part of the paper is then devoted to showing that the Bergman invariant function of this product domain coincides with that of  $\Omega$.
A direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned  product domain, leading to a contradiction.  In this respect, our proof departs from earlier approaches to the Cheng conjecture and its generalizations, where the contradiction is derived via spherical CR geometry and the Qi-Keng Lu uniformization theorem \cite{HX20}.

The ideas of the proof of Theorem~\ref{main theorem} are stated as follows. One main tool is a “rescaling argument.” First, “we show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball.” Next, “we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.” To obtain such a sequence, “we assign weight $2$ to one of the complex normal directions, weight $1.5$ to the other normal directions, and weight $1$ to the remaining CR directions.” The main part then shows “that the Bergman invariant function of this product domain coincides with that of $\Omega$.” Finally, “a direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned product domain,” yielding “a contradiction.”

The ideas of the proof, in establishing the main theorem, are stated as follows. One main tool is a “rescaling argument.” First, “we show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball.” Next, “we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.” To obtain such a sequence, “we assign weight $2$ to one of the complex normal directions, weight $1.5$ to the other normal directions, and weight $1$ to the remaining CR directions.” The main part then shows “that the Bergman invariant function of this product domain coincides with that of $\Omega$.” Finally, “a direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned product domain,” yielding “a contradiction.”

Question:
Let \(\Omega\subset \mathbb C^n\) with \(n>1\) be a pseudoconvex domain, possibly unbounded. Assume that \(\Omega\) has a strongly pseudoconvex polyhedral boundary point \(p\in \partial\Omega\), meaning that there exist a neighborhood \(U\) of \(p\) in \(\mathbb C^n\) and \(C^2\)-smooth strongly plurisubharmonic functions \(\rho_1,\dots,\rho_m:U\to\mathbb R\) with \(m>1\) such that
\[
\Omega\cap U=\{z\in U: \rho_1(z)<0,\dots,\rho_m(z)<0\},
\]
and the complex differentials \(\{\partial\rho_1|_p,\dots,\partial\rho_m|_p\}\) are linearly independent over \(\mathbb C\). Under these assumptions, which statement about the Bergman metric of \(\Omega\) holds?

There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and the Bergman metric cannot be Einstein on any open subset of \(\Omega^*\).

Choices:
B. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and if the Bergman metric is Einstein on some open subset of \(\Omega^*\), then \(\Omega\) is biholomorphic to the unit ball.
C. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined.
D. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and the Bergman metric is not Einstein on some nonempty open subset of \(\Omega^*\).
E. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and the Bergman invariant function on \(\Omega^*\) is constant and equal to that of the unit ball.

conclusion-from-contradiction replaced by global classification

constant-ball invariant asserted unconditionally instead of under Einstein hypothesis

The stem gives hypotheses and relevant definitions but does not explicitly or implicitly reveal the correct conclusion. No option is singled out by wording in the prompt.

The item is largely a theorem-recall/application question: the correct choice appears to match the intended theorem conclusion closely. However, it is not a pure tautology because the distractors include stronger, weaker, and altered statements.

Some reasoning is needed to distinguish the exact supported conclusion from a weaker true statement and stronger false variants, but the problem mainly tests recognition of the theorem rather than deeper generative derivation.

The distractors are plausible and mathematically targeted: one is weaker-but-true, one is an overstrong rigidity claim, one reverses the quantifier structure, and one asserts an unjustified ball-like invariant. These reflect realistic failure modes.

A solid MCQ with good distractors and no answer leakage, but it remains mostly a theorem-recall item rather than a strongly generative reasoning question.


Question:
Let \(\Omega\subset \mathbb C^n\) with \(n>1\) be a pseudoconvex domain, possibly unbounded. Assume that \(\Omega\) has a strongly pseudoconvex polyhedral boundary point \(p\in \partial\Omega\), meaning that there exist a neighborhood \(U\) of \(p\) in \(\mathbb C^n\) and \(C^2\)-smooth strongly plurisubharmonic functions \(\rho_1,\dots,\rho_m:U\to\mathbb R\) with \(m>1\) such that
\[
\Omega\cap U=\{z\in U: \rho_1(z)<0,\dots,\rho_m(z)<0\},
\]
and the complex differentials \(\{\partial\rho_1|_p,\dots,\partial\rho_m|_p\}\) are linearly independent over \(\mathbb C\). Under these assumptions, which statement about the Bergman metric of \(\Omega\) holds?

Choices:
B. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and if the Bergman metric is Einstein on some open subset of \(\Omega^*\), then \(\Omega\) is biholomorphic to the unit ball.
C. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined.
D. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and the Bergman metric is not Einstein on some nonempty open subset of \(\Omega^*\).
E. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and the Bergman invariant function on \(\Omega^*\) is constant and equal to that of the unit ball.

The stem does not explicitly state the conclusion or otherwise reveal choice A. It gives only the hypotheses and asks for the valid consequence.

This is close to a theorem-recall item: the stem lists the hypotheses of a specific result and asks for its conclusion. However, it is not a pure verbatim restatement because the options vary in logical strength and quantification.

Some reasoning is needed to distinguish the exact conclusion from weaker true statements and stronger false variants, especially around quantifiers and rigidity claims. Still, the item mainly tests recognition of the theorem rather than substantial derivation.

The distractors are mathematically plausible and well targeted: one is weaker but true, one overstates the conclusion via ball-rigidity, one weakens the quantifier structure, and one asserts an unjustified invariant conclusion. These reflect realistic failure modes.

A solid MCQ with strong distractors and no answer leakage, but it leans heavily on theorem recall and is only moderately generative rather than deeply reasoning-based.


\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.