\label{thmA}
Let $(G,H)$ be a pair of finite groups. There exists a finite incidence geometry $\Gamma$ such that $(\operatorname{Aut}(\Gamma),\AutI(\Gamma))\cong (G,H)$.

\begin{example}\label{ex:(Sn,An) incidence systems}
   We use an inductive argument to construct an $(S_n,A_n)$-incidence system $\Gamma_n=(X_n,*,t_n,I_n)$ for $n\geq 2$ such that
    \begin{enumerate}[label={\rm (\arabic{*})}]
        \item\label{item:Ex_SnAn_1} Every vertex in the incidence graph associated to $\Gamma_n$ has degree at least $2$.
        \item\label{item:Ex_SnAn_2} All cliques in the incidence graph associated to $\Gamma_n$ have rank at most $2$.
        \item\label{item:Ex_SnAn_3} $I_n=\{0,1,\ldots,n\}$ and
        \[
        |t_n^{-1}(i)|=
        \begin{cases}
            n & \text{if $i=n,$}\\
            \frac{n!}{2\cdot (i-1)!} & \text{if $0\leq i<n$,}\\
        \end{cases}
        \]
        where $(-1)!=0!=1$. In particular, $|t_n^{-1}(n-1)|=\binom{n}{2}$.
        \item\label{item:Ex_SnAn_4} $\Aut(\Gamma_n)$ acts on $t_n^{-1}(n)$, it does so as the usual permutation action on a set of $n$ elements, and this action fully determines the action on $\Gamma_n$, i.e., the induced map $\Aut(\Gamma_n)\to \operatorname{Bijections}(t_n^{-1}(n))$ is injective.
    \end{enumerate}

    For $n=2$, we define our pre-induction object $\Gamma_2$ as the $(S_2,A_2)$-incidence system associated with the colored graph in \autoref{fig:Gamma_2},  where the label inside the vertex indicates the color. Observe that there are just two vertices with color/type $2$.

\begin{figure}[ht]
    \centering
    \begin{tikzpicture}[scale=1.5, font=\scriptsize]

\def\radius{4pt}

\foreach \i/\label in {0/2, 1/0, 2/1, 3/2} {
    \coordinate (V\i) at (\i, 0);
    \draw[thick] (V\i) circle (\radius);     \node at (V\i) {\label};             }

\foreach \i in {0,1,2} {
    \pgfmathtruncatemacro{\j}{\i+1}
    \draw[thick, shorten >=6pt, shorten <=6pt] (V\i) -- (V\j);
}
\end{tikzpicture}
\caption{The $(S_2,A_2)$-incidence system $\Gamma_2$}\label{fig:Gamma_2}
\end{figure}

Assume that, for $n\geq 3$, the $(S_{n-1},A_{n-1})$-incidence system
\[
\Gamma_{n-1}=(X_{n-1},*,t_{n-1},I_{n-1})
\]
satisfies the properties \ref{item:Ex_SnAn_1}, \ref{item:Ex_SnAn_2}, \ref{item:Ex_SnAn_3}, and \ref{item:Ex_SnAn_4} above. We define $\Gamma_n=(X_n,*,t_n,I_n)$ as the incidence system associated to the  colored graph constructed as follows: First, we consider the complete graph whose  vertices are the left cosets $A_n/A_{n-1}$. More precisely, we choose representatives $g_i\in A_n$ such that
\begin{equation}\label{equ:choice_of_representatives_An}
\text{$A_n=\sqcup_{i=1}^n g_iA_{n-1}$ and $g_n$ equals the identity element},
\end{equation}
and then define $v^n_i$ as the coset $g_iA_{n-1}$, so that $A_n/A_{n-1}=\{v^n_1,\ldots, v^n_n\}$. We note that this set of vertices is an $A_n$-(left) set and we assign color $n$ to all these vertices and define $I_n=I_{n-1}\sqcup\{n\}=\{0,1,\ldots,n\}$.

Second, we add new vertices $v^n_{i,j}=v^n_{j,i}$ for $i\ne j,$, thus splitting every original edge $\{v^n_i,v^n_j\}$ into two new edges $\{v^n_i,v^n_{i,j}\}$ and $\{v^n_{i,j},v^n_j\}$. We assign color $(n-1)$ to all these vertices $v^n_{i,j}.$

Third, we add a copy $\Gamma^n_{n-1}$of $\Gamma_{n-1}$ by identifying the vertex $v^{n-1}_i\in X_{n-1}$ (which has assigned color $(n-1)$) with the vertex $v^n_{i,n}$.

Finally, we add $(n-1)$ more copies of $\Gamma_{n-1}$ by propagating this construction to the remaining vertices $v^n_j,$ $1\leq j<n$, via the action of $A_n$. More precisely, for such an index $j$, let $g\in A_n$ satisfy that $g(v^n_n)=v^n_j$. Then we add a copy $\Gamma^j_{n-1}$ of $\Gamma_{n-1}$ in the neighborhood of the vertex $v^n_j$ by identifying the vertex $v^{n-1}_i\in X_{n-1}$ with the vertex $v^n_{g(i),j}$ for $g(v^n_i)=v^n_{g(i)}$. Note that the names of the vertices of such a copy of $\Gamma_{n-1}$ may depend on the chosen elements $g$'s but, on the contrary, their types do not depend by Equation \eqref{equ:choice_of_representatives_An} and the induction hypothesis that the automorphism group of $\Gamma_{n-1}$ is $A_{n-1}$.

By construction, it is clear that $\Gamma_n$ satisfies points \ref{item:Ex_SnAn_1} and  \ref{item:Ex_SnAn_2}. Moreover, $\Gamma_n$ has $n$ vertices of type ``$n$", so that $|t_n^{-1}(n)| = n$, we have added $n$ copies of $\Gamma_{n-1}$ in which we identify pairs of vertices of type ``$n-1$", hence $|t_n^{-1}(n-1)| = \binom{n}{2}$, and we clearly have $|t_n^{-1}(i)| = n \cdot |t_{n-1}^{-1}(i)|$ for $0\leq i \leq  n - 2$. Thus, $\Gamma_n$ also satisfies point \ref{item:Ex_SnAn_3}. Finally, $\Aut(\Gamma_n)$ acts on $t_n^{-1}(n)$ by examination of Figure \ref{fig:Gamma_3} if $n=3$, or by comparing the sizes described in point (3) if $n\geq 4$. Moreover, this action clearly determines the action of $\Aut(\Gamma_n)$ on the $\binom{n}{2}$ vertices of type $n-1$.  Then, by the induction hypothesis that point \ref{item:Ex_SnAn_4} holds for $\Gamma_{n-1}$, the action of $\Aut(\Gamma_n)$ on $t_n^{-1}(n)$ also determines the action of $\Aut(\Gamma_n)$ on the $n$ copies of $\Gamma_{n-1}$ that build up $\Gamma_n$. Thus, point \ref{item:Ex_SnAn_4} also holds for $\Gamma_n$ and, in particular,
\[
\AutI(\Gamma_n)\leq\Aut(\Gamma_n)\leq S_n.
\]

\begin{figure}[ht]
    \centering
\begin{tikzpicture}[scale=1.5, font=\scriptsize]
\def\radius{4pt}

\coordinate (A) at (90:2);    \coordinate (B) at (210:2);   \coordinate (C) at (330:2);
\foreach \point in {A,B,C} {
  \draw[thick] (\point) circle (\radius);
}
\node at (A) {3};
\node at (B) {3};
\node at (C) {3};

\path (A) -- (B) coordinate[midway] (AB);
\path (B) -- (C) coordinate[midway] (BC);
\path (C) -- (A) coordinate[midway] (CA);

\foreach \point/\label in {AB/2, BC/2, CA/2} {
  \draw[thick] (\point) circle (\radius);
  \node at (\point) {\label};
}

\path (AB) -- (BC) coordinate[pos=1/3] (ABC1);
\path (AB) -- (BC) coordinate[pos=2/3] (ABC2);
\path (BC) -- (CA) coordinate[pos=1/3] (BCA1);
\path (BC) -- (CA) coordinate[pos=2/3] (BCA2);
\path (CA) -- (AB) coordinate[pos=1/3] (CAB1);
\path (CA) -- (AB) coordinate[pos=2/3] (CAB2);

\foreach \point/\label in {ABC1/0, ABC2/1, BCA1/0, BCA2/1,CAB1/0, CAB2/1} {
  \draw[thick] (\point) circle (\radius);
  \node at (\point) {\label};
}
 \foreach \v/\w in {A/AB, AB/B, B/BC, BC/C, C/CA, CA/A, AB/ABC1, ABC1/ABC2, ABC2/BC, BC/BCA1, BCA1/BCA2, BCA2/CA, CA/CAB1, CAB1/CAB2, CAB2/AB} {
 \draw[thick, shorten >=6pt, shorten <=6pt] (\v) -- (\w);
 }
\end{tikzpicture}
\caption{The $(S_3,A_3)$-incidence system $\Gamma_3$. Observe that the inner triangle contains $3$ copies of $\Gamma_2$.}\label{fig:Gamma_3}
\end{figure}

It is left to show that $\Gamma_n$ is indeed a $(S_n,A_n)$-incidence system and we do so next. The case $n=2$ is a straightforward calculation and, for $n>2$, we observe that:
\begin{enumerate}[label={\rm (\alph{*})}]

    \item By Equation \eqref{equ:choice_of_representatives_An} and because the automorphism group of $\Gamma_{n-1}$ is $A_{n-1}$, the group of automorphisms of this incidence system acts transitively on the set of vertices of type ``$n$" and contains the group $A_n$ acting in the standard way, thus the group of correlations does so. Therefore
    $$A_n\leq \AutI(\Gamma_n)\leq\Aut(\Gamma_n)\leq S_n.$$

    \item Given a transposition $\sigma=(i,j)\in S_n$, $\sigma$ induces a non-automorphic correlation in the copies $\Gamma^k_{n-1}$ with $k\neq i,j$ as, on those copies, $\sigma$ induces a transposition on the the vertices of type $n-1$ of $\Gamma_{n-1}^k$. (Note that, as $n\geq 3$, there is such a copy $\Gamma_{n-1}^k$.) In particular, $\sigma$ induces a non-automorphic correlation in $\Gamma_n$, i.e.,
    $$A_n\leq \AutI(\Gamma_n)\lneq\Aut(\Gamma_n)= S_n.$$
\end{enumerate}
\end{example}

Finite groups are often difficult to understand, and a powerful tool for studying them is to examine their actions on various structures such as sets, vector spaces, and graphs. One may also wish to consider more geometric actions, such as those on Tits buildings \cite{Tits} or, more generally, on incidence geometries \cite{Bue1,Bue2}.

An incident geometry is, in particular, an \emph{incidence system}, that is, a set of elements, each with a given type (interpreted as ``point", ``line", ``plane", etc.), together with an incidence relation between them, subject to the condition that two elements of the same type cannot be incident. An incidence system $\Gamma$ can be represented by a simple vertex-colored graph $\mathcal{G}$, called the \emph{incidence graph}, where the vertices correspond to the elements, the colors correspond to their types, and two vertices are adjacent if and only if the corresponding elements are incident. In fact, the incidence system is totally determined by its incidence graph and many of its properties can be deduced from those of the associated graph. In this paper, we will therefore work primarily with incidence graphs.

Then, an incident system $\Gamma$ is an \emph{incident geometry} if, in the setting of its incidence graph, every complete subgraph $K$ of the corresponding incidence graph $\mathcal{G}$ containing at most one element of each type is contained in a \emph{chamber}: a complete subgraph of $\mathcal{G}$ containing exactly one element of each type. More precise definitions are provided in \autoref{sec:def incidence} and in the textbook~\cite{Bue2}.

On these geometrical structures, two different notions of automorphisms are usually considered: \emph{automorphisms} and \emph{correlations}. Both are permutations of the elements; however, while automorphisms must preserve both types and incidences, correlations preserve only incidences. These correspond, respectively, to color-preserving and non–color-preserving (colorblind) automorphisms of the associated incidence graph. In particular, every automorphism is a correlation, and the group $\AutI(\Gamma)$ of automorphisms of a incidence system $\Gamma$ defines a normal subgroup of the group $\operatorname{Aut}(\Gamma)$ of correlations of $\Gamma$.

In a recent paper \cite{LST2025}, Leeman, Stokes, and Tranchida construed incident geometries for some finite groups. More precisely, they are interested in, for a given finite group $G$, finding an incident geometry $\Gamma$ such that the automorphism group of $G$ is isomorphic to the group of correlations of $\Gamma$ so that the inner automorphisms of $G$ correspond to the automorphisms of $\Gamma$. The authors hope that the combinatorial properties of such incident geometries are deeply connected to the algebraic structure of $G$ and its automorphisms. In \cite{LST2025}, they constructed several incident geometries for classical families of groups and pave the way for the search of incident geometries for any group.

In this paper, we put this question in a more general framework and study the realizability of pair of groups as pairs correlations-automorphisms of incident geometries.

\begin{definition}\label{def:pair_of_groups}
A pair of groups is couple $(G,H)$ where $G$ is a group and $H$ is a normal subgroup of $G$. We say that two pairs $(G,H)$ and $(G',H')$ are \emph{isomorphic}, denoted $(G,H)\cong(G',H')$,  if there exists an isomorphism $\varphi\colon G\to G'$ such that $\varphi(H)=H'$.
\end{definition}

The main result of this paper is that every pair of finite groups $(G,H)$ is isomorphic to a pair $\big(\operatorname{Aut}(\Gamma),\AutI(\Gamma)\big)$ for some incident geometry $\Gamma$.

An incident geometry is, in particular, an \emph{incidence system}, that is, a set of elements, each with a given type (interpreted as ``point", ``line", ``plane", etc.), together with an incidence relation between them, subject to the condition that two elements of the same type cannot be incident. An incidence system $\Gamma$ can be represented by a simple vertex-colored graph $\mathcal{G}$, called the \emph{incidence graph}, where the vertices correspond to the elements, the colors correspond to their types, and two vertices are adjacent if and only if the corresponding elements are incident. In fact, the incidence system is totally determined by its incidence graph and many of its properties can be deduced from those of the associated graph. In this paper, we will therefore work primarily with incidence graphs.

Then, an incident system $\Gamma$ is an \emph{incident geometry} if, in the setting of its incidence graph, every complete subgraph $K$ of the corresponding incidence graph $\mathcal{G}$ containing at most one element of each type is contained in a \emph{chamber}: a complete subgraph of $\mathcal{G}$ containing exactly one element of each type. More precise definitions are provided in \autoref{sec:def incidence} and in the textbook~\cite{Bue2}.

On these geometrical structures, two different notions of automorphisms are usually considered: \emph{automorphisms} and \emph{correlations}. Both are permutations of the elements; however, while automorphisms must preserve both types and incidences, correlations preserve only incidences. These correspond, respectively, to color-preserving and non–color-preserving (colorblind) automorphisms of the associated incidence graph. In particular, every automorphism is a correlation, and the group $\AutI(\Gamma)$ of automorphisms of a incidence system $\Gamma$ defines a normal subgroup of the group $\operatorname{Aut}(\Gamma)$ of correlations of $\Gamma$.

In a recent paper \cite{LST2025}, Leeman, Stokes, and Tranchida construed incident geometries for some finite groups. More precisely, they are interested in, for a given finite group $G$, finding an incident geometry $\Gamma$ such that the automorphism group of $G$ is isomorphic to the group of correlations of $\Gamma$ so that the inner automorphisms of $G$ correspond to the automorphisms of $\Gamma$. The authors hope that the combinatorial properties of such incident geometries are deeply connected to the algebraic structure of $G$ and its automorphisms. In \cite{LST2025}, they constructed several incident geometries for classical families of groups and pave the way for the search of incident geometries for any group.

\begin{definition}\label{def:pair_of_groups}
A pair of groups is couple $(G,H)$ where $G$ is a group and $H$ is a normal subgroup of $G$. We say that two pairs $(G,H)$ and $(G',H')$ are \emph{isomorphic}, denoted $(G,H)\cong(G',H')$, if there exists an isomorphism $\varphi\colon G\to G'$ such that $\varphi(H)=H'$.
\end{definition}

The main result of this paper is that every pair of finite groups $(G,H)$ is isomorphic to a pair $\big(\operatorname{Aut}(\Gamma),\AutI(\Gamma)\big)$ for some incident geometry $\Gamma$.

\begin{abstract}
We investigate the relationship between finite groups and incidence geometries through their automorphism structures. Building upon classical results on the realizability of groups as automorphism groups of graphs, we develop a general framework to represent pairs of finite groups $(G, H)$, where $H \trianglelefteq G$, as pairs of correlation--automorphism groups of suitable incidence geometries. Specifically, we prove that for every such pair $(G, H)$, there exists a finite incidence geometry $\Gamma$ satisfying that the pair $(\operatorname{Aut}(\Gamma), \operatorname{Aut}_I(\Gamma))$ of correlation--automorphism groups of $\Gamma$ is isomorphic to $(G, H)$. Our construction proceeds in two main steps: first, we realize $(G, H)$ as the correlation and automorphism groups of an incidence system; then, we refine this system into a genuine incidence geometry preserving the same pair of automorphisms groups. We also provide explicit examples, including a family of geometries realizing $(S_n, A_n)$ for all $n \ge 2$.
\end{abstract}

\begin{definition}\label{def:pair_of_groups}
A pair of groups is couple $(G,H)$ where $G$ is a group and $H$ is a normal subgroup of $G$. We say that two pairs $(G,H)$ and $(G',H')$ are \emph{isomorphic}, denoted $(G,H)\cong(G',H')$, if there exists an isomorphism $\varphi\colon G\to G'$ such that $\varphi(H)=H'$.
\end{definition}

The main result of this paper is that every pair of finite groups $(G,H)$ is isomorphic to a pair $\big(\Aut(\Gamma),\AutI(\Gamma)\big)$ for some incident geometry $\Gamma$.

\begin{proof} Let $(G,H)$ be a pair of finite groups.
If $G$ is trivial, the incidence geometry with one single element will suffice. Assume now that $G$ is non trivial.
By \autoref{thm:realizing_inicedence_systems}, there exists a finite incidence system $\Gamma$ such that $(\Aut(\Gamma),\AutI(\Gamma))\cong (G,H)$ and such that all elements have degree at least 2, and $\Gamma$ has no flag of rank $3$ or more. Then by \autoref{thm:from incidence system to incidence geometry}, there exists an incidence geometry $\Gamma'$ such that $(\Aut(\Gamma'),\AutI(\Gamma'))\cong (\Aut(\Gamma),\AutI(\Gamma))\cong(G,H)$.
\end{proof}

The rest of the paper is divided into four sections. \autoref{sec:def incidence} gives the principal definition on incidence systems, incidence geometries and their automorphisms. In \autoref{sec:pair of incidence system} the first main step of the proof of~\autoref{thmA} is paved by giving, for every pair $(G,H)$, an incidence system $\Gamma$ with $(\Aut(\Gamma),\AutI(\Gamma))\cong (G,H)$. \autoref{sec:systems to geometries} give the remaining ingredients by explaining how to induce an incidence geometry from an incident system with the same pair correlations-automorphisms. Finally \autoref{sec:example} gives some examples to illustrate our construction and how one may compose with them in practice.

\begin{theorem}\label{thm:realizing_inicedence_systems}
 Let $G$ be a finite non trivial group and $H\leq G$ be a normal subgroup, so they define a pair $(G,H)$. Then there exists a finite incidence system $\Gamma$ such that
 \begin{enumerate}[label={\rm (\arabic{*})}]
 \item\label{thm:1_realizing_inicedence_systems} Every vertex in the incidence graph has degree at least $2$.
 \item\label{thm:2_realizing_inicedence_systems} All cliques in the incidence graph have rank at most $2$.
 \item\label{thm:3_realizing_inicedence_systems} $(\Aut(\Gamma),\AutI(\Gamma))$ is isomorphic to the pair $(G,H)$.
 \end{enumerate}
\end{theorem}
\begin{proof}
In this proof, we follow the ideas in \cite[Section 2]{Fr39}. We first introduce some notation:

\begin{theorem}\label{thm:from incidence system to incidence geometry}
Let $\Gamma=(X,\ast,t,I)$ be a finite incidence system such that all elements have degree at least two and all flags have rank at most two, then there exists a finite incidence geometry $\Gamma'=(X',\ast',t', I)$ with $(\Aut(\Gamma'),\AutI(\Gamma'))\cong (\Aut(\Gamma),\AutI(\Gamma))$.
\end{theorem}

\begin{theorem}\label{thm:improving incidence system}
For any finite incidence system $\Gamma$, there exists a finite incidence system $\Gamma'$ with $(\Aut(\Gamma'),\AutI(\Gamma'))\cong (\Aut(\Gamma),\AutI(\Gamma))$ and such that all flags of $\Gamma'$ have rank at most two and all elements of $\Gamma'$ have degree at least two.
\end{theorem}
\begin{proof}
Let $\mathcal{G}=(V,E,I,t)$ be the incidence graph corresponding to $\Gamma$. Starting from $\mathcal{G}$, we will construct an incidence graph $\mathcal{G}'$ such that, if $\Gamma'$ is the incidence system associated to $\mathcal{G}'$, then $\Gamma'$ satisfies the conclusions of the statement.

\begin{theorem}\label{thm:realizing_inicedence_systems}
    Let $G$ be a finite non trivial group and $H\leq G$ be a normal subgroup, so they define a pair $(G,H)$. Then there exists a finite incidence system $\Gamma$ such that
    \begin{enumerate}[label={\rm (\arabic{*})}]
        \item\label{thm:1_realizing_inicedence_systems} Every vertex in the incidence graph has degree at least $2$.
        \item\label{thm:2_realizing_inicedence_systems} All cliques in the incidence graph have rank at most $2$.
        \item\label{thm:3_realizing_inicedence_systems} $(\Aut(\Gamma),\AutI(\Gamma))$ is isomorphic to the pair $(G,H)$.
    \end{enumerate}
\end{theorem}

\begin{thmx}\label{thmA}
Let $(G,H)$ be a pair of finite groups. There exists a finite incidence geometry $\Gamma$ such that $(\Aut(\Gamma),\AutI(\Gamma))\cong (G,H)$.
\end{thmx}

\begin{proof} Let $(G,H)$ be a pair of finite groups.
If $G$ is trivial, the incidence geometry with one single element will suffice. Assume now that $G$ is non trivial.
By \autoref{thm:realizing_inicedence_systems}, there exists a finite incidence system $\Gamma$ such that $(\operatorname{Aut}(\Gamma),\AutI(\Gamma))\cong (G,H)$ and such that all elements have degree at least 2, and $\Gamma$ has no flag of rank $3$ or more. Then by \autoref{thm:from incidence system to incidence geometry}, there exists an incidence geometry $\Gamma'$ such that $(\operatorname{Aut}(\Gamma'),\AutI(\Gamma'))\cong (\operatorname{Aut}(\Gamma),\AutI(\Gamma))\cong(G,H)$.
\end{proof}

The proof of \autoref{thmA} is inspired by the realizability of finite groups as groups of automorphisms of graphs~\cite{Fr39}. It is based on the construction given in \autoref{thm:realizing_inicedence_systems}, which provides an incidence system with the necessary properties to apply \autoref{thm:from incidence system to incidence geometry}. This latter theorem constructs, from an incident system satisfying some technical conditions on degrees and flags in the associated incident graph, an incident geometry with the same pair correlations-automorphisms.

However, one may want to work with its own preferred incidence system which may not satisfy the hypotheses of \autoref{thm:from incidence system to incidence geometry}. One may then use \autoref{thm:improving incidence system} to refine the given incidence system so that it satisfies the desired hypotheses. As an example we construct, in Example \ref{ex:(Sn,An) incidence systems}, an incident geometry for the pair $(S_n,A_n)$ for every $n\geq 2$, where $S_n$ and $A_n$ are the symmetric group and alternating group on $n$ letters. This gives an example of an incident geometry for the automorphism group of the alternating group of $A_n$ as desired in \cite{LST2025}.

The rest of the paper is divided into four sections. \autoref{sec:def incidence} gives the principal definition on incidence systems, incidence geometries and their automorphisms. In \autoref{sec:pair of incidence system} the first main step of the proof of~\autoref{thmA} is paved by giving, for every pair $(G,H)$, an incidence system $\Gamma$ with $(\operatorname{Aut}(\Gamma),\AutI(\Gamma))\cong (G,H)$. \autoref{sec:systems to geometries} give the remaining ingredients by explaining how to induce an incidence geometry from an incident system with the same pair correlations-automorphisms. Finally \autoref{sec:example} gives some examples to illustrate our construction and how one may compose with them in practice.

\subsection*{Acknowledges}
The second author was partially supported by IRN MaDeF (CNRS). The third author is partially supported by grant PID2023-149804NB-I00 funded by MCIN/AEI/
10.13039/501100011033. All the authors gratefully acknowledge the financial support and the stimulating research environment provided by Institut Fourier at Grenoble, which made it possible to develop the results presented in this paper during their stay at the institution.

To prove \autoref{thmA}: if $G$ is trivial, “the incidence geometry with one single element will suffice.” If $G$ is non trivial, “by \autoref{thm:realizing_inicedence_systems}, there exists a finite incidence system $\Gamma$ such that $(\operatorname{Aut}(\Gamma),\AutI(\Gamma))\cong (G,H)$ and such that all elements have degree at least 2, and $\Gamma$ has no flag of rank $3$ or more.” Then “by \autoref{thm:from incidence system to incidence geometry}, there exists an incidence geometry $\Gamma'$ such that $(\operatorname{Aut}(\Gamma'),\AutI(\Gamma'))\cong (\operatorname{Aut}(\Gamma),\AutI(\Gamma))\cong(G,H)$.” The introduction adds that the proof “is based on the construction given in \autoref{thm:realizing_inicedence_systems},” producing an incidence system with the “necessary properties to apply \autoref{thm:from incidence system to incidence geometry},” and that this latter theorem constructs, from an incidence system “satisfying some technical conditions on degrees and flags in the associated incident graph, an incident geometry with the same pair correlations-automorphisms.”

To prove the main theorem: if $G$ is trivial, “the incidence geometry with one single element will suffice.” If $G$ is non trivial, we first prove the following theorem.
\begin{theorem}\label{thm:realizing_inicedence_systems}
    Let $G$ be a finite non trivial group and $H\leq G$ be a normal subgroup, so they define a pair $(G,H)$. Then there exists a finite incidence system $\Gamma$ such that
    \begin{enumerate}[label={\rm (\arabic{*})}]
        \item\label{thm:1_realizing_inicedence_systems} Every vertex in the incidence graph has degree at least $2$.
        \item\label{thm:2_realizing_inicedence_systems} All cliques in the incidence graph have rank at most $2$.
        \item\label{thm:3_realizing_inicedence_systems} $(\Aut(\Gamma),\AutI(\Gamma))$ is isomorphic to the pair $(G,H)$.
    \end{enumerate}
\end{theorem}
In particular, there exists a finite incidence system $\Gamma$ such that $(\operatorname{Aut}(\Gamma),\AutI(\Gamma))\cong (G,H)$ and such that all elements have degree at least $2$, and $\Gamma$ has no flag of rank $3$ or more. Then we use the following theorem.
\begin{theorem}\label{thm:from incidence system to incidence geometry}
Let $\Gamma=(X,\ast,t,I)$ be a finite incidence system such that all elements have degree at least two and all flags have rank at most two, then there exists a finite incidence geometry $\Gamma'=(X',\ast',t', I)$ with $(\Aut(\Gamma'),\AutI(\Gamma'))\cong (\Aut(\Gamma),\AutI(\Gamma))$.
\end{theorem}
Hence there exists an incidence geometry $\Gamma'$ such that $(\operatorname{Aut}(\Gamma'),\AutI(\Gamma'))\cong (\operatorname{Aut}(\Gamma),\AutI(\Gamma))\cong(G,H)$. The introduction adds that the proof “is based on the construction given in the preceding theorem,” producing an incidence system with the “necessary properties to apply the theorem above,” and that this latter theorem constructs, from an incidence system “satisfying some technical conditions on degrees and flags in the associated incident graph, an incident geometry with the same pair correlations-automorphisms.”

Question: Let $(G,H)$ be a pair of finite groups, meaning that $G$ is a finite group and $H\trianglelefteq G$ is a normal subgroup. An incidence system consists of elements with assigned types and an incidence relation such that two elements of the same type are not incident; an incidence geometry is an incidence system in which every complete subgraph of the incidence graph containing at most one element of each type is contained in a chamber, that is, a complete subgraph containing exactly one element of each type. For an incidence geometry $\Gamma$, let $\operatorname{Aut}(\Gamma)$ denote its group of correlations (all incidence-preserving permutations of its elements) and let $\operatorname{Aut}_I(\Gamma)$ denote its group of automorphisms (the type-preserving incidence-preserving permutations). Two pairs $(A,B)$ and $(G,H)$ are called isomorphic if there is a group isomorphism $\varphi:A\to G$ with $\varphi(B)=H$. Under these assumptions, which existence statement holds?

There exists a finite incidence geometry $\Gamma$ such that $\big(\operatorname{Aut}(\Gamma),\operatorname{Aut}_I(\Gamma)\big)\cong (G,H)$.

Choices:
B. There exists a finite incidence geometry $\Gamma$ such that $\big(\operatorname{Aut}(\Gamma),\operatorname{Aut}_I(\Gamma)\big)= (G,H)$.
C. There exists a finite incidence geometry $\Gamma$ such that $\operatorname{Aut}(\Gamma)\cong G$ and $\operatorname{Aut}_I(\Gamma)\cong H$.
D. There exists a finite incidence system $\Gamma$ such that $\big(\operatorname{Aut}(\Gamma),\operatorname{Aut}_I(\Gamma)\big)\cong (G,H)$.
E. If $G$ is nontrivial, then there exists a finite incidence geometry $\Gamma$ such that $\big(\operatorname{Aut}(\Gamma),\operatorname{Aut}_I(\Gamma)\big)\cong (G,H)$.

pair-isomorphism replaced by literal equality

dropped isomorphism of pairs/subgroup correspondence, keeping only separate group isomorphisms

stops at incidence system before the refinement to an incidence geometry

omits the trivial-group case handled separately in the proof


\label{thm:main}
For $s >1 $ and $\delta>0$ there exist $c_s>0$ and $\ep_0>0$ with the following. For any $0< \epsilon \leq \ep_0$ and any zero-mean initial data $\om_{in} \in H^s(\mathbb{T} \times \mathbb{R})$ such that
\begin{equation}\label{eq thm main 1}
\| \om_{in}\|_{H^{s} }  \leq \epsilon \quad \text{and} \quad \| y \om_{in}\|_{L^{2} }  \leq \epsilon ,
\end{equation}
define the lifespan
\begin{equation}\label{eq thm main 2}
T_\epsilon: = c_s \epsilon^{-\delta_s }  \quad \text{with}  \quad  \delta_s =
    \begin{cases}
  \frac{1}{4-s}  \quad\text{if $1<s<2 $  } &\\
\frac{1}{2+\delta}    \quad\text{ if $s=2 $  } &\\
\frac12 \quad\text{if $s > 2 $  }. &
\end{cases}
\end{equation}

 Then the (unique) solution to \eqref{eq:eu_eq} satisfies the following.

\begin{itemize}

	\item (Regularity):
	The re-normalized vorticity $W( t, X,Y): = \omega(t,X + Yt, Y)$ satisfies
	\begin{eqnarray}\label{eq:reg intro}
		\| W \|_{L^\infty_t ([0,T_\epsilon]; H^s)} \leq 3 \epsilon .
	\end{eqnarray}

	\item (Invisicid damping): The velocity field $u = (u^x, u^y )$ satisfies
	\begin{equation}\label{eq:damping intro}
		\begin{aligned}
			\|P_{\neq 0} u^x (t)\|_{L^2} & \lesssim \epsilon \langle t \rangle^{-1} \\
			\|  u^y (t) \|_{L^2} & \lesssim \epsilon \langle t \rangle^{- \min \{ 2,s\}}
		\end{aligned}
		\quad \text{for all $t \in [0, T_\epsilon] $ }
	\end{equation}
	where $P_{\neq 0}f : = f -\int_{\mathbb{T} } f(x,y) \, dx $ denotes the projection of removing the $x$-mean.

\end{itemize}

\label{sec:intro}

In this note, we consider the 2D Euler equation on $\mathbb{T} \times \mathbb{R}$ near the Couette flow $(y,0)$. The vorticity perturbation $\omega : \mathbb{T} \times \mathbb{R} \to \mathbb{R}$ satisfies
\begin{equation}\label{eq:eu_eq}
\begin{cases}
	\p_t \omega + y\p_x \omega + u \cdot \nabla \omega = 0 &\\
	u = \nabla^\perp \psi, \quad \Delta \psi = \omega  &\\
	\omega|_{t= 0 } = \om_{in} &
\end{cases}
\end{equation}
where    $ \nabla^\perp = (- \p_y , \p_x)$ and $\om_{in}$ is the initial data.

We are concerned with the long-time behavior of solutions to this system, specifically the phenomenon of inviscid damping~\cite{Kelvin1887,Rayleigh1879,Orr1907}--the decay of velocity perturbations in the absence of viscosity.

For the Couette flow, the linearized dynamics are well understood: phase mixing transfers energy to small scales in the vorticity, leading to algebraic decay of the velocity.

 At the nonlinear level, the situation is subtler. The breakthrough work of Bedrossian and Masmoudi \cite{MR3415068} established nonlinear inviscid damping for small perturbations in Gevrey class regularity. Important generalizations to more complex shear flows were obtained in subsequent works \cite{MR4740211,MR4628607}.

In the studies \cite{MR3415068,MR4076093,MR4740211,MR4628607}, inviscid damping follows from asymptotic stability, which requires Gevrey-2  regularity. This assumption appears to be sharp-- below Gevrey-2, instability can occur~\cite{MR4630602}.

 For Sobolev regularity, global inviscid damping is believed to fail. In particular, there exist steady states~\cite{MR2796139} and traveling waves~\cite{MR4595614} arbitrarily close to Couette flow in $H^{3/2-}$, which prohibits the global damping in such a regime.

\subsection{Main result}

The purpose of this note is to show that although global-in-time inviscid damping near Couette does not hold for Sobolev perturbations in $H^{3/2-}$, a \emph{finite but long-time} version of damping remains valid.

In this note, we consider the 2D Euler equation on $\mathbb{T} \times \mathbb{R}$ near the Couette flow $(y,0)$. The vorticity perturbation $\omega : \mathbb{T} \times \mathbb{R} \to \mathbb{R}$ satisfies
\begin{equation}\label{eq:eu_eq}
\begin{cases}
 \p_t \omega + y\p_x \omega + u \cdot \nabla \omega = 0 &\\
 u = \nabla^\perp \psi, \quad \Delta \psi = \omega &\\
 \omega|_{t= 0 } = \om_{in} &
\end{cases}
\end{equation}
where $ \nabla^\perp = (- \p_y , \p_x)$ and $\om_{in}$ is the initial data.

We are concerned with the long-time behavior of solutions to this system, specifically the phenomenon of inviscid damping~\cite{Kelvin1887,Rayleigh1879,Orr1907}--the decay of velocity perturbations in the absence of viscosity.

At the nonlinear level, the situation is subtler. The breakthrough work of Bedrossian and Masmoudi \cite{MR3415068} established nonlinear inviscid damping for small perturbations in Gevrey class regularity. Important generalizations to more complex shear flows were obtained in subsequent works \cite{MR4740211,MR4628607}.

For Sobolev regularity, global inviscid damping is believed to fail. In particular, there exist steady states~\cite{MR2796139} and traveling waves~\cite{MR4595614} arbitrarily close to Couette flow in $H^{3/2-}$, which prohibits the global damping in such a regime.

\subsection{Main result}

The purpose of this note is to show that although global-in-time inviscid damping near Couette does not hold for Sobolev perturbations in $H^{3/2-}$, a \emph{finite but long-time} version of damping remains valid.

\begin{equation}\label{eq:eu_eq}
\begin{cases}
	\p_t \om + y\p_x \om + u \cdot \nabla \om = 0 &\\
	u = \nabla^\perp \psi, \quad \Delta \psi = \om  &\\
	\om|_{t= 0 } = \om_{in} &
\end{cases}
\end{equation}

The purpose of this note is to show that although global-in-time inviscid damping near Couette does not hold for Sobolev perturbations in $H^{3/2-}$, a \emph{finite but long-time} version of damping remains valid.

\item (Invisicid damping): The velocity field $u = (u^x, u^y )$ satisfies
 \begin{equation}\label{eq:damping intro}
 \begin{aligned}
 \|P_{\neq 0} u^x (t)\|_{L^2} & \lesssim \ep \langle t \rangle^{-1} \\
 \| u^y (t) \|_{L^2} & \lesssim \ep \langle t \rangle^{- \min \{ 2,s\}}
 \end{aligned}
 \quad \text{for all $t \in [0, T_\ep] $ }
 \end{equation}
 where $P_{\neq 0}f : = f -\int_{\T } f(x,y) \, dx $ denotes the projection of removing the $x$-mean.

\begin{remark}
\hfill
\begin{enumerate}
 \item Our nonlinear estimates remain valid down to $H^{1+}$ matching the well-posedness threshold of 2D Euler. There is no contradiction with steady states\footnote{Note that the examples in \cite{MR2796139} are on the periodic channel $\T \times [-1,1]$, where the stationary structure appears near $y=0$. Nevertheless, the result in this paper is likely to hold on $\T \times [-1 ,1]$ when restricting away from the boundary. }~\cite{MR2796139} or traveling waves~\cite{MR4595614} constructions, since the initial velocities in those examples are already smaller than the decay rate predicted by \eqref{eq:damping intro} at $t=T_\ep$.

\begin{lemma}\label{lemma:Hs a priori}
Let $s>1$. The system \eqref{eq:eu_XY} satisfies the a priori estimates:
\begin{equation}
\frac{ d \| W(t) \|_{H^s} }{dt} \lesssim \| \nabla U \|_{L^\infty } \| W \|_{H^s} +
\begin{cases}
\| J^{ 2} U \|_{ L^2 } \| W \|_{H^s} & \quad \text{1<s<2}\\
\| J^{ 2+\delta} U \|_{ L^2 } \| W \|_{H^s} &\quad \text{ s=2} \\
\| J^{ s} U \|_{ L^2 } \| W \|_{H^s} &\quad \text{ s>2}.
\end{cases}
\end{equation}

If $\int_{\T \times \R} \om_{in} = 0 $ and $ y \om_{in} \in L^2 $, then the system \eqref{eq:eu_XY} admits the following a priori estimates:
\begin{equation}\label{eq prop:apriori_yw}
\frac{ d \| W (t) \|_{ \bar{H}^s } }{dt} \lesssim \langle t\rangle^{\beta_s} \|W \|_{\bar{H}^s}^2 ,
\end{equation}
where the exponent $ \beta_s =3-s $ if $1<s<2 $, $ \beta_s =1+\delta $ for arbitrary $\delta>0$ if $s=2 $, and $ \beta_s =1 $ if $s > 2 $.

Combining \eqref{eq aux apriori 1b}, \eqref{eq aux apriori 5} and \eqref{eq aux apriori 6} with Lemma \ref{lemma:Hs a priori}, we have:
\begin{equation*}
\frac{ d \| W(t) \|_{\bar H^s} }{dt}
\begin{cases}
\lesssim \langle t\rangle^{3- s} \| W(t) \|_{\bar H^s}^2 \quad & \text{if $1<s<2$}\\
\lesssim_\delta \langle t\rangle^{1+\delta} \| W(t) \|_{\bar H^s}^2 \quad & \text{if $ s =2$}.
\end{cases}
\end{equation*}

We conclude that the decay estimate \eqref{eq:damping intro} follows from Lemma \ref{lemma:damping}; for $u^x$,
\begin{align*}
\| P_{\neq 0 } u^x (t)\|_{L^2} \lesssim \| P_{\neq 0 } \om \|_{\dot H^{-1}}
 \lesssim \langle t\rangle^{-1} \|W \|_{L^\infty_t([0,T_\ep]; H^1)}
\end{align*}
while for $u^y$ (applying Lemma \ref{lemma:damping} for $\p_x \om $ and $\p_X W$),
\begin{align*}
\| u^y (t)\|_{L^2} = \| P_{\neq 0 } \Delta^{-1}_t ( \p_X W) \|_{L^2} \lesssim \langle t \rangle^{- \min \{ 2,s\}} \|W \|_{L^\infty_t([0,T_\ep];H^s) }
\end{align*}
for any $s>1$ since the extra $\p_X$ can be absorbed in $H^s$. We have thus proven Theorem \ref{thm:main}

By standard duality, it suffices to bound
\begin{equation}
 \| P_{ \neq 0 } \om \|_{\dot H^{-1} } = \sup_{
 |\varphi \|_{\dot H^1} \leq 1 ; P_{=0} \varphi = 0} \int \om \varphi \, dx dy.
\end{equation}
So we fix such a smooth test function $\varphi \in C^\infty_c(\T \times \R)$ with zero $x$-mean and consider
\begin{equation}\label{eq:om_int}
 \int \om \varphi \, dx dy =\int W (X, Y )\varphi( X+ t Y, Y) \,dX dY.
\end{equation}
Because $\varphi$ has zero x-mean and periodic in $x$, there exists a unique zero x-mean function $ \phi= \p_x^{-1} \varphi$.
Then we use the identity
\begin{equation}\label{eq:om_int 2}
 \varphi(\cdot) = \frac{1}{t}\p_Y \left( \phi(\cdot) \right) - \frac{1}{t}\p_y \phi(\cdot)
\end{equation}
where the argument $(\cdot) = (X+ t Y, Y )$ is omitted.
Inserting this into \eqref{eq:om_int} gives
\begin{align}
 \int \om \varphi \, dx dy & =-\frac{1}{t} \int \p_Y W \phi(\cdot) \,dX dY \nonumber -\frac{1}{t} \int W \p_y \phi(\cdot) \,dX dY \nonumber \\
 & \leq t^{-1} \| \p_Y W \|_{L^2 } \| \phi \|_{L^2} + t^{-1} \| W \|_{L^2 } \| \p_y \phi \|_{L^2 } . \label{eq:aux10}
\end{align}
Note that $\left| \widehat{\phi}(k,\xi) \right|=\left| \frac{\widehat{\varphi}(k,\xi)}{k} \right|$, we have $\| \phi \|_{H^1} \lesssim \| \varphi \|_{\dot H^1} $, which gives
\begin{align}
 \Big|\int \om \varphi \, dx dy \Big| & \lesssim \frac{1}{t} \| W \|_{L^2_X H^1_Y} \| \varphi \|_{\dot H^1 } \nonumber .
\end{align}

\begin{equation}\label{eq:damping intro}
		\begin{aligned}
			\|P_{\neq 0} u^x (t)\|_{L^2} & \lesssim \ep \langle t \rangle^{-1} \\
			\|  u^y (t) \|_{L^2} & \lesssim \ep \langle t \rangle^{- \min \{ 2,s\}}
		\end{aligned}
		\quad \text{for all $t \in [0, T_\ep] $ }
	\end{equation}

\begin{equation}\label{eq:eu_eq}
\begin{cases}
	\p_t \om + y\p_x \om + u \cdot \nabla \om = 0 &\\
	u = \nabla^\perp \psi, \quad \Delta \psi = \om  &\\
	\om|_{t= 0 } = \om_{in} &
\end{cases}
\end{equation}

\begin{remark}
\hfill
\begin{enumerate}
    \item Our nonlinear estimates remain valid down to $H^{1+}$ matching the well-posedness threshold of 2D Euler.  There is no contradiction with steady states\footnote{Note that the examples in \cite{MR2796139} are on the periodic channel $\mathbb{T} \times [-1,1]$, where the stationary structure appears near $y=0$. Nevertheless, the result in this paper is likely to hold on $\mathbb{T} \times [-1 ,1]$  when restricting away from the boundary.   }~\cite{MR2796139} or traveling waves~\cite{MR4595614} constructions, since the initial velocities in those examples are already smaller than the decay rate predicted by \eqref{eq:damping intro} at $t=T_\epsilon$.

    \item While for  $s> 2$ the decaying rates match with those of the linearized problem, it is not clear to us what would be the optimal life span $T_\epsilon =\epsilon^{-\delta_s}$.

    \item The estimates are most effective for initial data of the form $\omega_{in} = \epsilon \omega_0$ for some fixed profile $ \omega_0$. In the limit $\epsilon \to 0+$, one sees that initially $ u  \sim  \omega  \sim \epsilon $ and  $ u $ exhibits the algebraic decay over the time interval $[0,T_\epsilon]$.

    \item The damping estimate \eqref{eq:damping intro} follows from the regularity \eqref{eq:reg intro}. Whether nonlinear inviscid damping can persist globally without asymptotic stability remains open, see~\cite{MR4302767} for results in the linear case.

    \item The instability construction in \cite{MR4630602} also satisfies our assumptions and estimates, with an eventual breakdown of \eqref{eq:reg intro} occurring long after $t=T_\epsilon$. A definite counterexample to inviscid damping for low Gevrey regularity is also open.
\end{enumerate}

\end{remark}

\subsection{Discussion}
The literature on stability/inviscid damping is vast; see, for example, the works  \cite{MR3710645,MR3772399,MR3987441,MR4400903} on general shears/vortices as well as \cite{MR3974608,MR4121130,MR4412070,MR4848788,MR4951440,2510.16378}  on related viscous problems and boundary effects.

In contrast, the study of nonlinear inviscid damping of 2D Euler equations in the Sobolev regime remains largely open.   The present work demonstrates that the inviscid damping remains effective on a long, but finite, time interval, which is much shorter than the nonlinear resonance breakdown time found in recent instability constructions~\cite{MR4630602}.

Our proof is short and elementary. We work in simplified shear-back coordinates of Bedrossian-Masmoudi~\cite{MR3415068,MR4076093,MR4740211,MR4628607} and exploit the fact that the principle of ``\emph{decay costs regularity}'' can, in some sense, be reversed. In these coordinates, the new velocity field remains one derivative smoother than the re-normalized vorticity, at the cost of growing time factors.

 It is desired to find the optimal exponent $T_\epsilon\sim  \epsilon^{-\delta_s}$. However, whether the lifespan obtained here can be extended, with or without additional assumptions, is unclear to us. Another important open question is whether nonlinear inviscid damping persists without the regularity of re-normalized vorticity \eqref{eq:reg intro}. See \cite{MR4302767} for the study of a linear case.

Our proof is described as "short and elementary." To prove Theorem~\ref{thm:main}, the authors "work in simplified shear-back coordinates of Bedrossian-Masmoudi" and "exploit the fact that the principle of `\emph{decay costs regularity}' can, in some sense, be reversed." In these coordinates, they use that "the new velocity field remains one derivative smoother than the re-normalized vorticity, at the cost of growing time factors," which yields the damping estimates (indeed, they also remark that "The damping estimate \eqref{eq:damping intro} follows from the regularity \eqref{eq:reg intro}.")

Our proof is described as "short and elementary." To prove the main theorem, the authors "work in simplified shear-back coordinates of Bedrossian-Masmoudi" and "exploit the fact that the principle of `\emph{decay costs regularity}' can, in some sense, be reversed." In these coordinates, they use that "the new velocity field remains one derivative smoother than the re-normalized vorticity, at the cost of growing time factors," which yields the damping estimates
\begin{equation}\label{eq:damping intro}
		\begin{aligned}
			\|P_{\neq 0} u^x (t)\|_{L^2} & \lesssim \ep \langle t \rangle^{-1} \\
			\|  u^y (t) \|_{L^2} & \lesssim \ep \langle t \rangle^{- \min \{ 2,s\}}
		\end{aligned}
		\quad \text{for all $t \in [0, T_\ep] $ }
	\end{equation}
(indeed, they also remark that the damping estimate above follows from the regularity
\begin{eqnarray}\label{eq:reg intro}
		\| W \|_{L^\infty_t ([0,T_\ep]; H^s)} \leq 3 \ep .
	\end{eqnarray}
).

\label{thm:main}
For $s >1 $ and $\delta>0$ there exist $c_s>0$ and $\ep_0>0$ with the following. For any $0< \epsilon \leq \ep_0$ and any zero-mean initial data $\om_{in} \in H^s(\mathbb{T} \times \mathbb{R})$ such that
\begin{equation}\label{eq thm main 1}
\| \om_{in}\|_{H^{s} }  \leq \epsilon \quad \text{and} \quad \| y \om_{in}\|_{L^{2} }  \leq \epsilon ,
\end{equation}
define the lifespan
\begin{equation}\label{eq thm main 2}
T_\epsilon: = c_s \epsilon^{-\delta_s }  \quad \text{with}  \quad  \delta_s =
    \begin{cases}
  \frac{1}{4-s}  \quad\text{if $1<s<2 $  } &\\
\frac{1}{2+\delta}    \quad\text{ if $s=2 $  } &\\
\frac12 \quad\text{if $s > 2 $  }. &
\end{cases}
\end{equation}

 Then the (unique) solution to
\begin{equation}\label{eq:eu_eq}
\begin{cases}
	\p_t \om + y\p_x \om + u \cdot \nabla \om = 0 &\\
	u = \nabla^\perp \psi, \quad \Delta \psi = \om  &\\
	\om|_{t= 0 } = \om_{in} &
\end{cases}
\end{equation}
satisfies the following.

\begin{itemize}

	\item (Regularity):
	The re-normalized vorticity $W( t, X,Y): = \omega(t,X + Yt, Y)$ satisfies
	\begin{eqnarray}\label{eq:reg intro}
		\| W \|_{L^\infty_t ([0,T_\epsilon]; H^s)} \leq 3 \epsilon .
	\end{eqnarray}

	\item (Invisicid damping): The velocity field $u = (u^x, u^y )$ satisfies
	\begin{equation}\label{eq:damping intro}
		\begin{aligned}
			\|P_{\neq 0} u^x (t)\|_{L^2} & \lesssim \epsilon \langle t \rangle^{-1} \\
			\|  u^y (t) \|_{L^2} & \lesssim \epsilon \langle t \rangle^{- \min \{ 2,s\}}
		\end{aligned}
		\quad \text{for all $t \in [0, T_\epsilon] $ }
	\end{equation}
	where $P_{\neq 0}f : = f -\int_{\mathbb{T} } f(x,y) \, dx $ denotes the projection of removing the $x$-mean.

\end{itemize}

Existential–Universal
Inequality or Bound

Question:
Let \(s>1\) and \(\delta>0\). Consider the 2D Euler vorticity equation on \(\mathbb T\times\mathbb R\):
\[
\partial_t\omega+y\partial_x\omega+u\cdot\nabla\omega=0,\qquad u=\nabla^\perp\psi,\qquad \Delta\psi=\omega,\qquad \omega|_{t=0}=\omega_{in},
\]
where \(\nabla^\perp=(-\partial_y,\partial_x)\). Define the renormalized vorticity by \(W(t,X,Y):=\omega(t,X+Yt,Y)\), the projection removing the \(x\)-mean by \(P_{\neq 0}f:=f-\int_{\mathbb T}f(x,y)\,dx\), and \(\langle t\rangle=(1+t^2)^{1/2}\). Which explicit uniform small-data statement is valid for this problem?

There exist \(c_s>0\) and \(\epsilon_0>0\) such that for every \(0<\epsilon\le \epsilon_0\) and every zero-mean initial datum \(\omega_{in}\in H^s(\mathbb T\times\mathbb R)\) satisfying
\[
\int_{\mathbb T\times\mathbb R}\omega_{in}\,dx\,dy=0,\qquad \|\omega_{in}\|_{H^s}\le \epsilon,\qquad \|y\omega_{in}\|_{L^2}\le \epsilon,
\]
if
\[
T_\epsilon:=c_s\epsilon^{-\delta_s},\qquad
\delta_s=
\begin{cases}
\dfrac{1}{4-s}, & \text{if } 1<s<2,\\[1mm]
\dfrac{1}{2+\delta}, & \text{if } s=2,\\[1mm]
\dfrac{1}{2}, & \text{if } s>2,
\end{cases}
\]
then the unique solution of the above initial-value problem satisfies
\[
\|W\|_{L_t^\infty([0,T_\epsilon];H^s)}\le 3\epsilon,
\]
and, for all \(t\in[0,T_\epsilon]\), with \(u=(u^x,u^y)\),
\[
\|P_{\neq 0}u^x(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-1},
\qquad
\|u^y(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-\min\{2,s\}}.
\]

Choices:
B.
There exist \(c_s>0\) and \(\epsilon_0>0\) such that for every \(0<\epsilon\le \epsilon_0\) and every zero-mean initial datum \(\omega_{in}\in H^s(\mathbb T\times\mathbb R)\) satisfying
\[
\int_{\mathbb T\times\mathbb R}\omega_{in}\,dx\,dy=0,\qquad \|\omega_{in}\|_{H^s}\le \epsilon,\qquad \|y\omega_{in}\|_{L^2}\le \epsilon,
\]
if
\[
T_\epsilon:=c_s\epsilon^{-\tilde\delta_s},\qquad
\tilde\delta_s=
\begin{cases}
\dfrac{1}{3-s}, & \text{if } 1<s<2,\\[1mm]
\dfrac{1}{1+\delta}, & \text{if } s=2,\\[1mm]
1, & \text{if } s>2,
\end{cases}
\]
then the unique solution of the above initial-value problem satisfies
\[
\|W\|_{L_t^\infty([0,T_\epsilon];H^s)}\le 3\epsilon,
\]
and, for all \(t\in[0,T_\epsilon]\), with \(u=(u^x,u^y)\),
\[
\|P_{\neq 0}u^x(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-1},
\qquad
\|u^y(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-\min\{2,s\}}.
\]
C.
There exist \(c_s>0\) and \(\epsilon_0>0\) such that for every \(0<\epsilon\le \epsilon_0\) and every zero-mean initial datum \(\omega_{in}\in H^s(\mathbb T\times\mathbb R)\) satisfying
\[
\int_{\mathbb T\times\mathbb R}\omega_{in}\,dx\,dy=0,\qquad \|\omega_{in}\|_{H^s}\le \epsilon,\qquad \|y\omega_{in}\|_{L^2}\le \epsilon,
\]
if
\[
T_\epsilon:=c_s\epsilon^{-\delta_s},\qquad
\delta_s=
\begin{cases}
\dfrac{1}{4-s}, & \text{if } 1<s<2,\\[1mm]
\dfrac{1}{2+\delta}, & \text{if } s=2,\\[1mm]
\dfrac{1}{2}, & \text{if } s>2,
\end{cases}
\]
then the unique solution of the above initial-value problem satisfies
\[
\|W\|_{L_t^\infty([0,T_\epsilon];H^s)}\le 3\epsilon,
\]
and, for all \(t\in[0,T_\epsilon]\), with \(u=(u^x,u^y)\),
\[
\|P_{\neq 0}u^x(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-1}.
\]
D.
There exist \(c_s>0\) and \(\epsilon_0>0\) such that for every \(0<\epsilon\le \epsilon_0\) and every zero-mean initial datum \(\omega_{in}\in H^s(\mathbb T\times\mathbb R)\) satisfying
\[
\int_{\mathbb T\times\mathbb R}\omega_{in}\,dx\,dy=0,\qquad \|\omega_{in}\|_{H^s}\le \epsilon,
\]
if
\[
T_\epsilon:=c_s\epsilon^{-\delta_s},\qquad
\delta_s=
\begin{cases}
\dfrac{1}{4-s}, & \text{if } 1<s<2,\\[1mm]
\dfrac{1}{2+\delta}, & \text{if } s=2,\\[1mm]
\dfrac{1}{2}, & \text{if } s>2,
\end{cases}
\]
then the unique solution of the above initial-value problem satisfies
\[
\|W\|_{L_t^\infty([0,T_\epsilon];H^s)}\le 3\epsilon,
\]
and, for all \(t\in[0,T_\epsilon]\), with \(u=(u^x,u^y)\),
\[
\|P_{\neq 0}u^x(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-1},
\qquad
\|u^y(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-\min\{2,s\}}.
\]
E.
There exist \(c_s>0\) and \(\epsilon_0>0\) such that for every \(0<\epsilon\le \epsilon_0\) and every zero-mean initial datum \(\omega_{in}\in H^s(\mathbb T\times\mathbb R)\) satisfying
\[
\int_{\mathbb T\times\mathbb R}\omega_{in}\,dx\,dy=0,\qquad \|\omega_{in}\|_{H^s}\le \epsilon,\qquad \|y\omega_{in}\|_{L^2}\le \epsilon,
\]
if
\[
T_\epsilon:=c_s\epsilon^{-\delta_s},\qquad
\delta_s=
\begin{cases}
\dfrac{1}{4-s}, & \text{if } 1<s<2,\\[1mm]
\dfrac{1}{2+\delta}, & \text{if } s=2,\\[1mm]
\dfrac{1}{2}, & \text{if } s>2,
\end{cases}
\]
then the unique solution of the above initial-value problem satisfies
\[
\|W\|_{L_t^\infty([0,T_\epsilon];H^{s+1})}\le 3\epsilon,
\]
and, for all \(t\in[0,T_\epsilon]\), with \(u=(u^x,u^y)\),
\[
\|P_{\neq 0}u^x(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-1},
\qquad
\|u^y(t)\|_{L^2}\lesssim \epsilon\langle t\rangle^{-\min\{2,s+1\}}.
\]

lifespan exponent derived from time-growth in shear coordinates

dropped the \(u^y\) damping conclusion while keeping the same hypotheses and lifespan

weighted moment assumption \(\|y\omega_{in}\|_{L^2}\le \epsilon\)

one-derivative smoothing used for velocity upgraded to an \(H^{s+1}\) bound for \(W\)


\label{main1}
Let $X$ be a 1-nodal quadric threefold  and $\pi \colon \widetilde{X} \longrightarrow X$ be its blow-up.

Then there exist a stability condition $\sigma_{\mathcal{K}u (X)}=(Z_\mathcal{A}, \mathcal{A})$ on $\mathcal{K}u (X)$,
and a weak stability condition $\sigma_{\widetilde{\mathcal{D}}'} = (Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})$ on the categorical resolution $\widetilde{\mathcal{D}}'$ of $\mathcal{K}u(X)$,
such that they are related as follows.
\begin{enumerate}
    \item[(1)] $Z_{\widetilde{\mathcal{A}}} = Z_\mathcal{A} \circ \pi_\ast$;
    \item[(2)] $\mathbf{R}\pi_\ast \colon \widetilde{\mathcal{A}}\longrightarrow \mathcal{A}$ is an exact functor.
\end{enumerate}

Bridgeland stability conditions on triangulated categories, introduced in \cite{Bri07},
have become a central tool in the study of derived categories.
They are now known to exist on curves \cite{Mac07} and on
surfaces \cite{AB13}.
In dimension three, the tilt-stability approach of Bayer--Macr\`i--Toda
\cite{BMT14} and its refinements provides a general framework for constructing
Bridgeland stability conditions,
and has led to existence results in a number of cases (see for instance \cite{BMS16}).

On the other hand, Kuznetsov studied the semiorthogonal decompositions of Fano threefolds and introduced Kuznetsov components
\cite{Kuz04,Kuz05,Kuz09},
which have played a central role in the study of their derived categories and geometry.
It is therefore natural to investigate stability conditions on Kuznetsov components
of Fano threefolds.
In the smooth case this has been carried out in
\cite{BLMS} for all smooth Fano threefolds of Picard rank one.
This leads to the question of what happens in the presence of mild singularities.

In this article, we extend our previous result in \cite{Cho25} to singular threefolds via Kuznetsov's categorical resolution \cite{Kuz08b},
and we first address the simplest such case, namely a quadric threefold with a single ordinary double point.

Our main result is the following:

\begin{abstract}
Let $X \subset \mathbb{P}^4$ be a quadric threefold with a single ordinary double point, and let $\mathcal{K}u(X)$ be its Kuznetsov component.
In this paper, we construct a weak stability condition $\sigma_{\widetilde{\mathcal{D}}'}$ on its categorical resolution $\widetilde{\mathcal{D}}' \subset \mathrm{D^b}(\widetilde{X})$,
which is compatible with the Verdier localization $\mathbf{R}\pi_\ast$ and descends to a Bridgeland stability condition on $\mathcal{K}u(X)$.
This can be viewed as a three-dimensional analogue of our previous result in \cite{Cho25}.

We describe the geometry of the blow-up $\pi \colon \widetilde{X} \longrightarrow X$ and obtain two semiorthogonal decompositions of $\mathrm{D^b}(\widetilde{X})$,
arising from the projective bundle structure of $\widetilde{X}$ and from Kuznetsov's categorical resolution.
Comparing them, we isolate an admissible subcategory $\widetilde{\mathcal{D}}' \subseteq \mathrm{D^b}(\widetilde{X})$ resolving $\mathcal{K}u(X)$ and show that it admits a full Ext-exceptional collection,
from which we construct $\sigma_{\widetilde{\mathcal{D}}'}$.
\end{abstract}

In this article, we extend our previous result in \cite{Cho25} to singular threefolds via Kuznetsov's categorical resolution \cite{Kuz08b},
and we first address the simplest such case, namely a quadric threefold with a single ordinary double point.

In order to construct stability conditions on the derived category of a singular threefold,
it is natural to first work on an appropriate ``resolution''.
For a nodal quadric threefold $X$, Kuznetsov's theory of categorical resolutions \cite{Kuz08b}
together with the construction of Kuznetsov--Shinder \cite{KS24} produce an admissible subcategory
$\widetilde{\mathcal{D}}\subset\mathrm{D^b}(\widetilde{X})$,
equipped with a functor
$\mathbf{R}\pi_\ast \colon \widetilde{\mathcal{D}} \longrightarrow \mathrm{D^b}(X)$ which is a Verdier localization,
with kernel generated by a single spherical object.

\begin{definition}\label{stability}
A weak stability condition (resp. Bridgeland stability condition) on $\mathcal{C}$ is a pair $\sigma = (Z,\mathcal{H})$ consisting of a group homomorphism (called the central charge of $\sigma$) $Z \colon \Lambda \longrightarrow \mathbb{C}$ and a heart $\mathcal{H}$ of a bounded t-structure on $\mathcal{C}$, such that the following conditions hold:
\begin{enumerate}
 \item[(a)] The composition $Z \circ v \colon {\rm K}(\mathcal{H})={\rm K}(\mathcal{C}) \longrightarrow \Lambda \longrightarrow \mathbb{C}$ is a weak stability function (resp. stability function) on $\mathcal{H}$. This gives a notion of slope: for any $E \in \mathcal{H}$, we set
 \begin{equation*}
 \mu_\sigma(E)=\mu_Z(E):=
 \begin{cases}
 \frac{-\Re Z(E)}{\Im Z(E)}, & \text{if } \Im Z(E)>0, \\
 +\infty, & \text{if } \Im Z(E)=0.
 \end{cases}
 \end{equation*}
 We say that an object $0\neq E \in \mathcal{H}$ is $\sigma$-semistable (resp. $\sigma$-stable) if for every nonzero proper subobject $F \subset E$, we have $\mu_\sigma (F) \leq \mu_\sigma (E)$ (resp. $\mu_\sigma (F) < \mu_\sigma (E)$).

\begin{theorem}[{\normalfont\cite[Theorem 5.8]{KS24}}]\label{KS24}
Let $X$ be a variety of dimension $n \geq 2$ over an algebraically closed
field $k$ of characteristic not equal to $2$ with an ordinary double point
$x_0$ and no other singularities. Let
\[
 \pi : \widetilde{X} = \operatorname{Bl}_{x_0}(X) \longrightarrow X
\]
be the blowup of the singular point, let
$\epsilon \colon E \hookrightarrow \widetilde{X}$ be the embedding of the
exceptional divisor over $x_0$, and let $S$ be a spinor bundle on $E$.
Then the subcategory
$\widetilde{\mathcal{D}}:= \{F \in \mathrm{D^b}(\widetilde{X})|\epsilon^\ast F \in \langle S,\mathcal{O}_{E}\rangle\}$
is admissible in $\mathrm{D^b}(\widetilde{X})$. Moreover,
\begin{enumerate}
 \item the induced functor
 $\mathbf{R}\pi_{*} : \widetilde{\mathcal{D}} \to \mathrm{D^b}(X)$
 is a crepant Verdier localization;
 \item the kernel $\ker(\pi_\ast|_{\widetilde{{\mathcal{D}}}})$ is generated by a single spherical object $K \in \widetilde{\mathcal{D}}$;
 \item when $\dim(X)$ is even, $K=\epsilon_\ast S$ is $2$-spherical,
 and when $\dim(X)$ is odd, $K$ is $3$-spherical and fits into the distinguished triangle $K \longrightarrow \epsilon_\ast S \longrightarrow \epsilon_\ast S'[2]$, where $S'$ is another spinor bundle on $E$.
\end{enumerate}
\end{theorem}

Assume that $\mathcal{N}$ is generated by a single object $A \in\mathcal{C}$ and that $\mathcal{Q}\cap\mathcal{N}$ is a Serre subcategory of $\mathcal{Q}$ (i.e. closed under subobjects, quotients, and extensions).
Then the following are equivalent:
\begin{enumerate}
\item $\mathcal{Q}$ descends to the heart $\mathcal{Q}/(\mathcal{Q}\cap\mathcal{N})$ of a bounded $t$-structure on $\mathcal{C}/\mathcal{N}$.
\item There exists an object $B\in\mathcal{Q}$ such that
$\mathcal{N} = \langle B\rangle$.
\end{enumerate}
In this case, $q\colon \mathcal{Q}\longrightarrow \mathcal{Q}/(\mathcal{Q}\cap\mathcal{N})$ is exact.
\end{lemma}

\begin{proposition}
There exist a weak stability condition $\sigma_{\widetilde{\mathcal{D}}'} = (Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})$ on $\widetilde{\mathcal{D}}'$, satisfying the support property with respect to the quotient lattice $\mathrm{K_{num}}(\widetilde{\mathcal{D}}')/\ker (\pi_\ast|_{\widetilde{\mathcal{D}}'})$,
and a Bridgeland stability condition $\sigma_{\mathcal{K}u(X)} = (Z_{\mathcal{A}},\mathcal{A})$ on $\mathcal{K}u(X)$,
such that
\begin{enumerate}
 \item $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{D}}' \longrightarrow \mathcal{K}u(X)$ is compatible with the central charges, i.e. $Z_{\widetilde{\mathcal{A}}} = Z_{\mathcal{A}} \circ \pi_\ast$.
 \item $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{A}} \longrightarrow \mathcal{A}$ is an exact functor.

In order to construct stability conditions on the derived category of a singular threefold,
it is natural to first work on an appropriate ``resolution''.
For a nodal quadric threefold $X$, Kuznetsov's theory of categorical resolutions \cite{Kuz08b}
together with the construction of Kuznetsov--Shinder \cite{KS24} produce an admissible subcategory
$\widetilde{\mathcal{D}}\subset\mathrm{D^b}(\widetilde{X})$,
equipped with a functor
$\mathbf{R}\pi_\ast \colon \widetilde{\mathcal{D}} \longrightarrow \mathrm{D^b}(X)$ which is a Verdier localization,
with kernel generated by a single spherical object.

The category $\widetilde{\mathcal{D}}$ admits an explicit description thanks to the projective bundle structure of $\widetilde{X}$,
and it will serve as the starting point of our construction.

First, we exploit two natural semiorthogonal decompositions of $\mathrm{D^b}(\widetilde{X})$.
The first arises from Orlov’s projective bundle formula for $\tilde{X}\cong \mathbb{P}_E(\mathcal{O}_E \oplus \mathcal{O}_E(-1,-1))$,
which yields a semiorthogonal decomposition of $\mathrm{D^b}(\widetilde{X})$ given by eight exceptional line bundles.
The second decomposition comes from Kuznetsov’s categorical resolution of the nodal quadric.

The key point is that, by performing a sequence of mutations relating these two decompositions,
we exhibit a full Ext-exceptional collection of length $3$ in the categorical resolution
$\widetilde{\mathcal{D}}' \subseteq \widetilde{\mathcal{D}}$ of $\mathcal{K}u(X)$,
which greatly simplifies the construction of stability conditions.

More precisely, we have:
\begin{theorem}
The categorical resolution  $\widetilde{\mathcal{D}}'$ of $\mathcal{K}u(X)$ admits a full Ext-exceptional collection of length $3$.
\end{theorem}

Their extension closure immediately gives a heart $\mathcal{B}$ of bounded $t$-structure.
Nevertheless, it can be seen that no weak stability condition with heart $\mathcal{B}$ descends to $\mathcal{K}u(X)$,
and we therefore find a suitable tilt to construct a new heart $\widetilde{\mathcal{A}}\subset \widetilde{\mathcal{D}}'$,
so that its pushforward $\mathcal{A}:=\mathbf{R}\pi_\ast(\widetilde{\mathcal{A}})$ is a heart on $\mathcal{K}u(X)$ and we can then define a localization-compatible central charge $Z_{\widetilde{\mathcal{A}}}$
and induce a stability condition  $\sigma_{\mathcal{A}}=(Z_{\mathcal{A}},\mathcal{A})$ on $\mathcal{K}u(X)$.
\subsection{Related work}
Our result fits into recent developments on Bridgeland stability conditions and moduli on Kuznetsov components of Fano varieties.

In the smooth case, stability conditions on Fano threefolds of Picard rank one and their Kuznetsov components are constructed in \cite{Li19} and \cite{BLMS}, respectively.

On the singular side, Kuznetsov and Shinder introduced the notion of categorical absorption of singularities and constructed, under natural hypotheses,
categorical absorptions for projective varieties with isolated ordinary double points \cite{KS24}.
They also studied the derived categories of 1-nodal prime Fano threefolds in \cite{KS25}.

Independently, in \cite{CGL+} the authors studied kernels of categorical resolutions of nodal singularities and proved that for nodal varieties the kernel is generated by a single spherical object, with applications to nodal cubic fourfolds.

Our weak stability condition on the categorical resolution also fits into the framework of partial compactifications of stability spaces by massless objects developed in \cite{BPPW22}. In their terminology it can be viewed as a lax stability condition lying on a boundary stratum,
with massless subcategory corresponding to the  kernel of the Verdier localization.

Finally, in our previous work \cite{Cho25} we constructed Bridgeland stability conditions on a singular surface and its resolution,
and the present paper can be viewed as a threefold counterpart.

\subsection{Organization of the paper}
In Section 2 we review background on stability conditions, hearts of $t$-structures, and tilting.

Section 3 is devoted to the geometry of the nodal quadric threefold $X$ and its resolution $\widetilde{X}$; we describe the projective bundle structure of $\widetilde{X}$ and derive a semiorthogonal decomposition of $\mathrm{D^b}(\widetilde{X})$ by Orlov's formula.

In Section 4, we introduce Kuznetsov’s categorical resolution $\widetilde{\mathcal{D}}$ of $\mathrm{D^b}(X)$ and isolate the admissible subcategory $\widetilde{\mathcal{D}}'$ which resolves the Kuznetsov component $\mathcal{K}u(X)$. We then perform a sequence of mutations, using the projective bundle decomposition from Section 3, to exhibit a full exceptional triple generating $\widetilde{\mathcal{D}}'$.

In Section 5, we use this exceptional collection to construct a new heart $\widetilde{\mathcal{A}}$ on $\widetilde{\mathcal{D}}'$ and prove that it descends to a heart $\mathcal{A}$ on $\mathcal{K}u(X)$.
We also carry out the construction of the stability condition on $\mathcal{K}u(X)$ and prove the main result.

Throughout the paper, we work over the complex numbers $\mathbb{C}$.

In order to prove Theorem~\ref{main1}, the construction proceeds by working first on an appropriate categorical ``resolution''. For a nodal quadric threefold $X$, one uses Kuznetsov's theory of categorical resolutions together with the construction of Kuznetsov--Shinder to obtain an admissible subcategory $\widetilde{\mathcal{D}}\subset\mathrm{D^b}(\widetilde{X})$ equipped with a functor $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{D}}\to\mathrm{D^b}(X)$ which is a Verdier localization, with kernel generated by a single spherical object.

The category $\widetilde{\mathcal{D}}$ is described explicitly using the projective bundle structure of $\widetilde{X}$, and the construction starts by exploiting two semiorthogonal decompositions of $\mathrm{D^b}(\widetilde{X})$: one from Orlov’s projective bundle formula (giving eight exceptional line bundles) and one from Kuznetsov’s categorical resolution. By performing a sequence of mutations relating these decompositions, one obtains a full Ext-exceptional collection of length $3$ in the categorical resolution $\widetilde{\mathcal{D}}'\subseteq\widetilde{\mathcal{D}}$ of $\mathcal{K}u(X)$, which simplifies the stability construction.

The extension closure of this exceptional collection gives a heart $\mathcal{B}$, but “no weak stability condition with heart $\mathcal{B}$ descends to $\mathcal{K}u(X)$”, so one performs “a suitable tilt” to produce a new heart $\widetilde{\mathcal{A}}\subset \widetilde{\mathcal{D}}'$. This is arranged so that its pushforward $\mathcal{A}:=\mathbf{R}\pi_\ast(\widetilde{\mathcal{A}})$ is a heart on $\mathcal{K}u(X)$ and $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{A}}\to\mathcal{A}$ is exact. Finally, one defines a “localization-compatible” central charge $Z_{\widetilde{\mathcal{A}}}$ and induces a stability condition $\sigma_{\mathcal{A}}=(Z_{\mathcal{A}},\mathcal{A})$ on $\mathcal{K}u(X)$ with $Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ\pi_\ast$.

In order to prove the main theorem, the construction proceeds by working first on an appropriate categorical ``resolution''. For a nodal quadric threefold $X$, one uses Kuznetsov's theory of categorical resolutions together with the construction of Kuznetsov--Shinder to obtain an admissible subcategory $\widetilde{\mathcal{D}}\subset\mathrm{D^b}(\widetilde{X})$ equipped with a functor $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{D}}\to\mathrm{D^b}(X)$ which is a Verdier localization, with kernel generated by a single spherical object.

The category $\widetilde{\mathcal{D}}$ is described explicitly using the projective bundle structure of $\widetilde{X}$, and the construction starts by exploiting two semiorthogonal decompositions of $\mathrm{D^b}(\widetilde{X})$: one from Orlov’s projective bundle formula (giving eight exceptional line bundles) and one from Kuznetsov’s categorical resolution. By performing a sequence of mutations relating these decompositions, one obtains a full Ext-exceptional collection of length $3$ in the categorical resolution $\widetilde{\mathcal{D}}'\subseteq\widetilde{\mathcal{D}}$ of $\mathcal{K}u(X)$, which simplifies the stability construction.

The extension closure of this exceptional collection gives a heart $\mathcal{B}$, but “no weak stability condition with heart $\mathcal{B}$ descends to $\mathcal{K}u(X)$”, so one performs “a suitable tilt” to produce a new heart $\widetilde{\mathcal{A}}\subset \widetilde{\mathcal{D}}'$. This is arranged so that its pushforward $\mathcal{A}:=\mathbf{R}\pi_\ast(\widetilde{\mathcal{A}})$ is a heart on $\mathcal{K}u(X)$ and $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{A}}\to\mathcal{A}$ is exact. Finally, one defines a “localization-compatible” central charge $Z_{\widetilde{\mathcal{A}}}$ and induces a stability condition $\sigma_{\mathcal{A}}=(Z_{\mathcal{A}},\mathcal{A})$ on $\mathcal{K}u(X)$ with $Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ\pi_\ast$.

Question: Let $X$ be a quadric threefold with exactly one ordinary double point (a 1-nodal quadric threefold), and let $\pi\colon \widetilde{X}\to X$ be the blow-up of the node. Let $\mathcal{K}u(X)$ be the Kuznetsov component of $X$, and let $\widetilde{\mathcal{D}}'$ be its categorical resolution, viewed as an admissible subcategory of $\mathrm{D}^\mathrm{b}(\widetilde{X})$ with the induced pushforward functor $\mathbf{R}\pi_*\colon \widetilde{\mathcal{D}}'\to \mathcal{K}u(X)$. Which statement holds?

There exist a Bridgeland stability condition $\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal A},\mathcal A)$ on $\mathcal{K}u(X)$ and a weak stability condition $\sigma_{\widetilde{\mathcal D}'}=(Z_{\widetilde{\mathcal A}},\widetilde{\mathcal A})$ on $\widetilde{\mathcal D}'$ such that $Z_{\widetilde{\mathcal A}}=Z_{\mathcal A}\circ \pi_*$ and the functor $\mathbf{R}\pi_*\colon \widetilde{\mathcal A}\to \mathcal A$ is exact.

Choices:
B. There exist Bridgeland stability conditions $\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal A},\mathcal A)$ on $\mathcal{K}u(X)$ and $\sigma_{\widetilde{\mathcal D}'}=(Z_{\widetilde{\mathcal A}},\widetilde{\mathcal A})$ on $\widetilde{\mathcal D}'$ such that $Z_{\widetilde{\mathcal A}}=Z_{\mathcal A}\circ \pi_*$ and the functor $\mathbf{R}\pi_*\colon \widetilde{\mathcal A}\to \mathcal A$ is exact.
C. There exist a Bridgeland stability condition $\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal A},\mathcal A)$ on $\mathcal{K}u(X)$ and a weak stability condition $\sigma_{\widetilde{\mathcal D}'}=(Z_{\widetilde{\mathcal A}},\widetilde{\mathcal A})$ on $\widetilde{\mathcal D}'$ such that $Z_{\widetilde{\mathcal A}}=Z_{\mathcal A}\circ \pi_*.$
D. There exist a Bridgeland stability condition $\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal A},\mathcal A)$ on $\mathcal{K}u(X)$ and a weak stability condition $\sigma_{\widetilde{\mathcal D}'}=(Z_{\widetilde{\mathcal A}},\widetilde{\mathcal A})$ on $\widetilde{\mathcal D}'$ such that $Z_{\mathcal A}=Z_{\widetilde{\mathcal A}}\circ \pi_*$ and the functor $\mathbf{R}\pi_*\colon \widetilde{\mathcal A}\to \mathcal A$ is exact.
E. There exist a Bridgeland stability condition $\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal A},\mathcal A)$ on $\mathcal{K}u(X)$ and a weak stability condition $\sigma_{\widetilde{\mathcal D}'}=(Z_{\widetilde{\mathcal A}},\widetilde{\mathcal A})$ on $\widetilde{\mathcal D}'$ such that $Z_{\widetilde{\mathcal A}}=Z_{\mathcal A}\circ \pi_*$ and the functor $\mathbf{R}\pi_*\colon \widetilde{\mathcal D}'\to \mathcal{K}u(X)$ is exact.

weak-vs-Bridgeland status on the resolution

exactness of the induced functor on hearts

direction/domain-codomain of central-charge compatibility through pushforward

exactness only after restricting to the constructed hearts, not on the whole categories


\begin{theorem}\label{FYZ}
Let $(M,\omega)$ be a compact K\"ahler manifold and $\chi$ a  closed real (1,1)-form such that the principal argument of  $\int_{M}(\chi+\sqrt{-1} \omega)^n$ is $\pi$. If there exists a smooth function $\underline{u}$ satisfying \eqref{subsolution},
then the critical LYZ equation \eqref{LYZ1} admits a unique smooth solution u with $\sup_M u=0$.
\end{theorem}

\begin{align}\label{kHessian}
    \sigma_{k}(\alpha) := C_{n}^{k} \frac{\alpha^k \wedge \omega^{n-k}}{\omega^n}.
\end{align}

\begin{remark}\label{vlaplacebound}
    The condition $\Delta v \le C$ in the weak sense means that there exists a constant $C > 0$ such that for every nonnegative test function $f \in C_c^{\infty}(\mathbb{C}^n)$ with $\int_{\mathbb{C}^n} f(z) \, \mathcal{E}^n(z) = 1$, we have
    \[
    \int_{\mathbb{C}^n} v \, \sqrt{-1}\partial\bar{\partial} f \wedge \mathcal{E}^{n-1} \le C.
    \]
    From this condition, it follows immediately that the regularization $[v]_r$ has bounded Laplacian and hence bounded complex Hessian.
\end{remark}

In K\"ahler geometry, the LYZ (Leung-Yau-Zaslow) equation, also known as the dHYM (deformed Hermitian Yang-Mills) equation, is a fully nonlinear elliptic PDE that arises from mirror symmetry \cite{LYZ1999,MMMS2000}. This equation is of current interest in both mathematics and physics.

Let $(M,\omega)$ be a compact K\"ahler manifold and $\chi$ a closed real $(1,1)$-form.
Suppose that $\mathcal{Z}=\int_{M}(\chi+\sqrt{-1}\omega)^n\neq 0$ and let $\hat\theta$ be the principal argument of $\mathcal{Z}$.
 The LYZ equation seeks a smooth function $u: M\rightarrow\mathbb{R}$ such that
 \begin{equation}\label{LYZe}
 \begin{aligned}
 \mathrm{Im}\big(e^{-\sqrt{-1}\hat\theta}(\chi_u+\sqrt{-1}\omega)^n\big)=0.
 \end{aligned}
 \end{equation}
 Let $\lambda=(\lambda_1,\cdots,\lambda_n)$ be the eigenvalues of $\chi_u$ with respect to $\omega$ and define the special Lagrangian operator as
\[
\theta_{\omega}(\chi_u):=\sum\limits_{i=1}^n\arctan \lambda_i.
\]
 Then equation (\ref{LYZe}) can be rewritten in the form
 \begin{equation}\label{LYZ2}
 \begin{aligned}
 \theta_{\omega}(\chi_u)=\theta
 \end{aligned}
 \end{equation}
 where $\theta=n\frac{\pi}{2}-\hat \theta$. We refer to (\ref{LYZ2}) as the \emph{supercritical} LYZ equation if $\theta\in\big( (n-2)\frac{\pi}{2}, n\frac{\pi}{2}\big)$, and as the \emph{critical} LYZ equation if $\theta=(n-2)\frac{\pi}{2}$, i.e., $\hat \theta=\pi$.

A key requirement for solving the LYZ equation is the existence of a {\sl subsolution}:
A smooth function $\underline{u}$ satisfying
 \begin{align}\label{subsolutiontheta}
 \min\limits_{1\le j\le n}\sum\limits_{\substack{i=1 \\ i\neq j}}^n\mathrm{arctan}\lambda_i(\chi_{\underline {u}})>\theta-\frac{\pi}{2}.
 \end{align}
This notion of subsolution is equivalent to that introduced by Guan \cite{guan2014} and Sz\'ekelyhidi \cite{szek2018}.
\subsection{Previous Results}

The 2-dimensional case of the LYZ equation was solved by Jacob-Yau \cite{jacobyau2017},
who also obtained partial results in higher dimensions. Collins-Jacob-Yau \cite{cjy2020} then solved the supercritical case under the assumption of a subsolution $\underline{u}$ and an additional condition $\theta_{\omega}(\chi_{\underline{u}})>(n-2)\frac{\pi}{2}$. The authors \cite{fyz2024} introduced a geometric flow to solve the equation under the same conditions. The extra condition was
subsequently removed by Pingali \cite{Pingali2022APDE} for $n=3$, by Lin \cite{lin2023adv} for $n=3,4$, and finally
by Lin \cite{lin2023} in full generality.
An alternative proof, based on Chen's significant result \cite{Chen2021Invent}, was given by Sun \cite{sun2024}.
For further recent developments, we refer to
\cite{ChanJacob2023arxiv, ChuCollinsLee2021GT,ChuLee2023Crelle, clt2024jdg, CollinsXieYau2018,CollinsYau2021APDE,HanJin2021CVPDE,HanJin2023Tran,HanJin2024Manu,HuangZhangZhang2022SCM,Jacob2021PAMQ, jacob2022arxiv,JacobSheu2022AsianJM,KhalidDyrefelt2024imrn,Lin2024MRL,Takahashi2021ijm}.

\subsection{Our main results}
Collins-Jacob-Yau \cite{cjy2020} suggested studying the LYZ equation when $\theta\le (n-2)\frac{\pi}{2}$. This was later explicitly posed as a question by Li \cite{YangLi2022arxiv}, as resolving it is essential for the LYZ equation to induce a Bridgeland stability condition on $D^b \text{Coh}(M)$, the bounded derived category of coherent sheaves on $M$.

In this paper, we solve the critical case; that is,
we establish the existence of smooth solutions for the \textit{critical} LYZ equation:
\begin{align}\label{LYZ1}
\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}.
\end{align}
In this case, the principal argument of $\int_{M}(\chi+\sqrt{-1} \omega)^n$ is $\pi$,
and the subsolution condition becomes
\begin{align}\label{subsolution}
 \min\limits_{1\le j\le n}\sum\limits_{\substack{i=1 \\ i\neq j}}^n\mathrm{arctan}\lambda_i(\chi_{\underline {u}})>{(n-3)\frac \pi 2}.
\end{align}

\begin{equation}\label{LYZ2}
 	\begin{aligned}
 		\theta_{\omega}(\chi_u)=\theta
 	\end{aligned}
 \end{equation}

\begin{equation}\label{LYZe}
 	\begin{aligned}
 		\mathrm{Im}\big(e^{-\sqrt{-1}\hat\theta}(\chi_u+\sqrt{-1}\omega)^n\big)=0.
 	\end{aligned}
 \end{equation}

\begin{align}\label{subsolution}
	\min\limits_{1\le j\le n}\sum\limits_{\substack{i=1 \\ i\neq j}}^n\mathrm{arctan}\lambda_i(\chi_{\underline {u}})>{(n-3)\frac \pi 2}.
\end{align}

Let $(M,\omega)$ be a compact K\"ahler manifold and $\chi$ a closed real $(1,1)$-form.
Suppose that $\mathcal{Z}=\int_{M}(\chi+\sqrt{-1}\omega)^n\neq 0$ and let $\hat\theta$ be the principal argument of $\mathcal{Z}$.
 The LYZ equation seeks a smooth function $u: M\rightarrow\mathbb{R}$ such that
 \begin{equation}\label{LYZe}
 \begin{aligned}
 \mathrm{Im}\big(e^{-\sqrt{-1}\hat\theta}(\chi_u+\sqrt{-1}\omega)^n\big)=0.
 \end{aligned}
 \end{equation}
 Let $\lambda=(\lambda_1,\cdots,\lambda_n)$ be the eigenvalues of $\chi_u$ with respect to $\omega$ and define the special Lagrangian operator as
\[
\theta_{\omega}(\chi_u):=\sum\limits_{i=1}^n\arctan \lambda_i.
\]
 Then equation (\ref{LYZe}) can be rewritten in the form
 \begin{equation}\label{LYZ2}
 \begin{aligned}
 \theta_{\omega}(\chi_u)=\theta
 \end{aligned}
 \end{equation}
 where $\theta=n\frac{\pi}{2}-\hat \theta$. We refer to (\ref{LYZ2}) as the \emph{supercritical} LYZ equation if $\theta\in\big( (n-2)\frac{\pi}{2}, n\frac{\pi}{2}\big)$, and as the \emph{critical} LYZ equation if $\theta=(n-2)\frac{\pi}{2}$, i.e., $\hat \theta=\pi$.

In this paper, we solve the critical case; that is,
we establish the existence of smooth solutions for the \textit{critical} LYZ equation:
\begin{align}\label{LYZ1}
\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}.
\end{align}
In this case, the principal argument of $\int_{M}(\chi+\sqrt{-1} \omega)^n$ is $\pi$,
and the subsolution condition becomes
\begin{align}\label{subsolution}
 \min\limits_{1\le j\le n}\sum\limits_{\substack{i=1 \\ i\neq j}}^n\mathrm{arctan}\lambda_i(\chi_{\underline {u}})>{(n-3)\frac \pi 2}.
\end{align}

\begin{remark}
For the special Lagrangian equation in $\mathbb{R}^n$, interior Hessian estimates for critical and supercritical cases were proved by
Warren-Yuan \cite{WarrenYuan2010} and Wang-Yuan \cite{WangYuan2014}.
For $\theta<(n-2)\frac{\pi}{2}$,
$C^{1,\alpha}$ solutions were constructed by Nadirashvili-Vl\u{a}du{t} \cite{NV2010IHP} and Wang-Yuan \cite{wangyuan2013ajm}. Very recently, Mooney-Savin \cite{mooneysavin2023duke} obtained a Lipschitz solution that is not $C^1$.
\end{remark}

To solve the critical LYZ equation \eqref{LYZ1},
we consider the family of equations:
\begin{align}\label{LYZt}
 \theta_{\omega}(\chi_{u^{t}}+t\omega)=\theta(t).
\end{align}
We verify that there exists a uniform constant $C>0$ such that
$$\theta(t)\in ((n-2)\frac{\pi}{2}+Ct,
(n-2)\frac{\pi}{2}+2Ct)$$ for small $t>0$ (see Lemma \ref{thetatrange} in Section 2). Hence equations \eqref{LYZt} are the supercritical LYZ equations. Then we prove the subsolution $\underline{u}$ of the critical LYZ equation \eqref{LYZ1} is also a subsolution of equations \eqref{LYZt} for small $t>0$. Thus we can apply Lin's result \cite{lin2023} (see also Sun \cite{sun2024} ) to obtain the smooth solution $u^t$ to equation \eqref{LYZt} with $\sup_M u^{t}=0$.

When $n=4$, the critical LYZ equation is the Hessian quotient equation
$$
\sigma_3(\chi_u)=\sigma_1(\chi_u).
$$
Another consequence of our result is to solve the above Hessian quotient equation in dimension 4 under weaker conditions than those by Sun \cite{sun2017cpam} and Sz\'{e}kelyhidi \cite{szek2018}.
 \begin{corollary}\label{4dHQ}
 Let $(M, \omega)$ be a compact K\"ahler manifold with dimension $n=4$.
 Let $\chi$ be a closed real $(1,1)$-form satisfying $3\chi^2\wedge\omega-\omega^3>0\ \text{as a positive $(3,3)$-form}$ and
 \begin{align*}
 \int_M\chi^3\wedge\omega=&\int_M\chi\wedge\omega^3,\\
 \int_M\chi\wedge\omega^{3}>&0,\
 \mathrm{Re}\int_M (\chi+\sqrt{-1}\omega)^4<0.\label{4dRe}\notag
 \end{align*}
 Then there exists a unique
 smooth function $u$ solving
 \begin{align}
 \sigma_3(\chi_u)=\sigma_1(\chi_u),\\
 \chi_u\in \Gamma_3(M), \ \sup_{M} u=0.\notag
 \end{align}
\end{corollary}

\subsection{$C^0$-estimates}
Let $\hat \theta(t)$ denote the principal argument of the complex number $\int_M (\chi + t\omega + \sqrt{-1}\omega)^n$, and set $\theta(t) = (n-2)\frac{\pi}{2} - \hat \theta(t)$. We first show that $\hat \theta(t)$ is close to $\pi$ when $t>0$ is sufficiently small.
\begin{lemma}\label{thetatrange}
 Assume that $\underline u$ is a subsolution of the critical LYZ equation \eqref{LYZ1}. Then there exist uniform positive constants $t_0$ and $C$, depending only on $\chi$ and $\omega$, such that for all $t \in (0, t_0)$,
 \[
 \hat \theta(t) \in \bigl( \pi - 2Ct,\; \pi - Ct \bigr).
 \]
\end{lemma}

\begin{theorem}\label{thm: Hessian estimate}
 Let $\underline{u}$ be a subsolution of the critical LYZ equation \eqref{LYZ1} and let $u^t$ be the solution of the LYZ equation \eqref{LYZt} with $\sup_M u^t = 0$.
 Then there exist uniform positive constants $t_0$ and $C$, independent of $\bigl(\theta(t)-(n-2)\frac{\pi}{2}\bigr)^{-1}$, such that for any $t \in (0, t_0)$,
 \begin{align}\label{eq: Hessian estimate}
 \sup_M |\sqrt{-1}\partial\bar{\partial} u^t|_{\omega} \le C \bigl(1 + \sup_M |\nabla u^t|_{\omega}^2 \bigr).
 \end{align}
\end{theorem}

We consider a family of LYZ equations \eqref{LYZt}
 \begin{align*}
 3(\chi+t\omega+\sqrt{-1}\partial\bar\partial u^t)^2\wedge\omega=&
 \tan\hat\theta(t)(\chi+t\omega+\sqrt{-1}\partial\bar\partial u^t)^3\\
 &-3\tan\hat\theta(t)(\chi+t\omega+\sqrt{-1}\partial\bar\partial u^t)\wedge\omega^2+\omega^3.
 \end{align*}
 As a corollary of our theorem, we solve the 2-Hessian equation under weaker conditions instead of the usual condition $\chi\in \Gamma_2(M)$.
 \begin{corollary}
 Let $(M,\omega)$ be a compact K\"ahler manifold with dimension $n=3$ and $\chi$ be a real $(1,1)$-form satisfying the following conditions:
 \begin{enumerate}[(i)]
 \item
 $3\int_M\chi^2\wedge\omega=\int_M\omega^3$,
 $\int_M\chi^3<3\int_M \chi\wedge\omega^2\label{re<0}$;
 \item
 the subsolution condition
 \begin{align}
 \chi\wedge\omega>0\ \ \text{as a $(2,2)$-form}.
 \end{align}
 \end{enumerate}
 Then there exists a unique smooth solution $u$ solving
 \begin{align}
 \sigma_2(\chi_u)=1,
 \\ \chi_u\in \Gamma_{2}(M), \ \sup_M u=0.
 \end{align}
 \end{corollary}

\begin{equation}\label{LYZ2}
 	\begin{aligned}
 		\theta_{\omega}(\chi_u)=\theta
 	\end{aligned}
 \end{equation}

\begin{equation}\label{LYZe}
 	\begin{aligned}
 		\mathrm{Im}\big(e^{-\sqrt{-1}\hat\theta}(\chi_u+\sqrt{-1}\omega)^n\big)=0.
 	\end{aligned}
 \end{equation}

\begin{align}\label{LYZt}
	\theta_{\omega}(\chi_{u^{t}}+t\omega)=\theta(t).
\end{align}

\begin{align}\label{subsolution}
	\min\limits_{1\le j\le n}\sum\limits_{\substack{i=1 \\ i\neq j}}^n\mathrm{arctan}\lambda_i(\chi_{\underline {u}})>{(n-3)\frac \pi 2}.
\end{align}

\begin{lemma}\label{thetatrange}
    Assume that $\underline u$ is a subsolution of the critical LYZ equation \eqref{LYZ1}. Then there exist uniform positive constants $t_0$ and $C$, depending only on $\chi$ and $\omega$, such that for all $t \in (0, t_0)$,
    \[
    \hat \theta(t) \in \bigl( \pi - 2Ct,\; \pi - Ct \bigr).
    \]
\end{lemma}

\begin{remark}
For the special Lagrangian equation in $\mathbb{R}^n$, interior Hessian estimates for  critical and supercritical cases were proved by
Warren-Yuan \cite{WarrenYuan2010} and Wang-Yuan \cite{WangYuan2014}.
For $\theta<(n-2)\frac{\pi}{2}$,
$C^{1,\alpha}$ solutions were constructed by Nadirashvili-Vl\u{a}du{t} \cite{NV2010IHP} and Wang-Yuan \cite{wangyuan2013ajm}. Very recently, Mooney-Savin \cite{mooneysavin2023duke} obtained a Lipschitz solution that is not $C^1$.
\end{remark}

To solve the critical LYZ equation \eqref{LYZ1},
we  consider the family of  equations:
\begin{align}\label{LYZt}
	\theta_{\omega}(\chi_{u^{t}}+t\omega)=\theta(t).
\end{align}
We  verify that there exists a uniform constant $C>0$ such that
$$\theta(t)\in ((n-2)\frac{\pi}{2}+Ct,
(n-2)\frac{\pi}{2}+2Ct)$$ for small $t>0$ (see  Lemma \ref{thetatrange}  in Section 2). Hence equations \eqref{LYZt} are the supercritical LYZ equations.  Then we  prove the subsolution $\underline{u}$ of the critical LYZ equation \eqref{LYZ1} is also a subsolution of equations \eqref{LYZt} for small $t>0$.  Thus we can apply  Lin's result \cite{lin2023} (see also Sun \cite{sun2024} ) to obtain  the  smooth solution $u^t$ to  equation \eqref{LYZt} with $\sup_M u^{t}=0$.

To prove Theorem \ref{FYZ},
we establish uniform $C^{2,\alpha}$ estimates  for $u^{t}$ that are independent of $t$ and $(\theta(t)-(n-2)\frac{\pi}{2})^{-1}$.

\begin{theorem}\label{FYZfamilly}
    Assume the same conditions as in Theorem \ref{FYZ}.
    Let $u^t$ be the smooth solution of \eqref{LYZt} with $\sup_M u^t=0$.
    Then there exist uniform constants $C$ and $t_0$, depending only on $(M, \omega, \chi)$ but independent of $t$, such that for any $t \in (0, t_0)$,
    \begin{align}
        |u^t|_{C^{2,\alpha}(M)} \le C.
    \end{align}
\end{theorem}

According to Collins--Jacob--Yau \cite{cjy2020}, $\sup_{M} |u^t|$ is uniformly bounded.
Their results imply that the $C^{2,\alpha}$ estimates can be reduced to proving uniform complex Hessian estimates.
Our main contribution is the proof of the uniform complex Hessian estimate of Hou--Ma--Wu type,
\begin{align}\label{HMW}
    \sup\limits_{M} |\sqrt{-1}\partial\bar{\partial} u|_{\omega} \le C\bigl(1 + \sup\limits_{M} |\partial u|_{\omega}^2 \bigr),
\end{align}
and the uniform gradient estimate via the Dinew--Kolodziej blow-up argument \cite{dk2017ajm}.

However, the situation here is more subtle.
If the gradient estimate fails, we obtain a bounded non-constant $C^1$ function $v$ on $\mathbb{C}^n$ that, in the sense of currents, is a weak solution to one of the following equations:
\[
\sigma_n = 0, \quad
\sigma_{n-1} = 0, \quad \text{or} \quad
\sigma_{n-1} + \sigma_n = 0.
\]
The last case is new and requires a corresponding Liouville-type theorem.
Note that $v + |z|^2$ is plurisubharmonic and $\Delta v$ is uniformly bounded in the weak sense (see Remark \ref{vlaplacebound}).
When following the argument in \cite{dk2017ajm,szek2018}, we find that the first case can be handled similarly.
For the second case, we need to show that the standard mollification $[v]_r$ of $v$, as well as $\frac{1}{3} [(v^\epsilon)^2]_r - \tau_0 |z|^2$, are subsolutions, where $v^\epsilon = [v]_\epsilon$ and $\tau_0 > 0$ is sufficiently small.
To address this issue, our starting observation is that a smooth function $v$ satisfies $\sigma_{n-1} + \sigma_n = 0$ (resp. $\ge 0$) on $\mathbb{C}^n$ if and only if the function $v + |z_{n+1}|^2$ satisfies $\sigma_n = 0$ (resp. $\ge 0$) on $\mathbb{C}^{n+1}$, where $z_{n+1}$ is a new complex variable. Based on this observation, we use potential theory to  obtain the required Liouville-type theorem (see Theorem \ref{LiouTh}).

Finally, we find some applications of the main result.
	In general, the \emph{critical} LYZ equation has the form
$$
\sigma_{n-1}(\chi_u)=\sigma_{n-3}(\chi_u)-\sigma_{n-5}(\chi_u)+\cdots,
$$
where $\sigma_{k}(\chi_u)$  is the $k$-Hessian operator of $\chi_u$ with respect to the metric $\omega$ (see \eqref{kHessian}).

When $n=3$, the critical  LYZ equation is just the 2-Hessian equation
$$\sigma_2(\chi_u)=1.$$
As a consequence of our result, we can solve the 2-Hessian equation in dimension 3 under weaker conditions than those by Sun \cite{sun2017cpam}.
	\begin{corollary}
	Let $(M,\omega)$ be a compact K\"ahler manifolds with dimension $n=3$ and $\chi$ be a closed real  $(1,1)$-form satisfying  $\chi\wedge\omega>0$\  \text{as a positive $(2,2)$-form} and
		\begin{align*}
			3\int_M\chi^2\wedge\omega=\int_M\omega^3,\ \
			\int_M\chi^3<3\int_M \chi\wedge\omega^2.
		\end{align*}
	Then there exists a unique smooth  solution $u$ solving
\[
	\begin{aligned}
		\sigma_2(\chi_u)=1,
		\\ \chi_u\in \Gamma_{2}(M), \ \sup_M u=0.
	\end{aligned}
	\]
\end{corollary}
Note that Hou-Ma-Wu \cite{hmw2010} proved the complex Hessian estimate of Hou-Ma-Wu type for the $k$-Hessian equation $\sigma_k(\omega_u)=f$ on a compact K\"{a}hler manifold.
Then Dinew-Kolodziej \cite{dk2017ajm}
obtained the gradient estimate by a Liouville  theorem for $\sigma_{k}=0$ in $\mathbb{C}^n$ and thus  solved the equation. The Hermitian case was solved independently  by Sz\'ekelyhidi \cite{szek2018} and Zhang \cite{zhang2017pjm}.

When $n=4$, the critical LYZ equation is the Hessian quotient equation
$$
\sigma_3(\chi_u)=\sigma_1(\chi_u).
$$
Another consequence of our result is to solve the above Hessian quotient equation in dimension 4 under weaker conditions than those by Sun \cite{sun2017cpam} and Sz\'{e}kelyhidi \cite{szek2018}.
	\begin{corollary}\label{4dHQ}
	Let $(M, \omega)$ be a compact K\"ahler manifold with dimension $n=4$.
	Let   $\chi$ be a closed real $(1,1)$-form satisfying 	$3\chi^2\wedge\omega-\omega^3>0\  \text{as a positive $(3,3)$-form}$ and
		\begin{align*}
			\int_M\chi^3\wedge\omega=&\int_M\chi\wedge\omega^3,\\
			\int_M\chi\wedge\omega^{3}>&0,\
			\mathrm{Re}\int_M (\chi+\sqrt{-1}\omega)^4<0.\label{4dRe}\notag
		\end{align*}
	Then there exists a unique
	smooth function $u$ solving
	\begin{align}
		\sigma_3(\chi_u)=\sigma_1(\chi_u),\\
		\chi_u\in \Gamma_3(M), \ \sup_{M} u=0.\notag
	\end{align}
\end{corollary}

{We remark that the Hessian quotient equation is of the form $\sigma_k(\chi_u)=c\sigma_{l}(\chi_u)$ with $0\le l<k\le n$. It reduces to the $J$-equation when $k=n$ and $l=n-1$ which was solved by Song-Weinkove \cite{songweinkove2008} using the flow method, and  the case $k=n, 1\le l\le n-1$ was solved by Fang-Lai-Ma \cite{flm2011}. The general case was subsequently solved by Sun \cite{sun2017cpam} and Sz\'{e}kelyhidi \cite{szek2018} via different continuity methods.}

	The organization of this paper is as follows. In Section 2, we  provide preliminaries on $k$-Hessian operators, the special Lagrangian operator $\theta_{\omega}$, and the approximating family $\theta(t)$. We also give the $C^0$-estimate for $u^t$ using results from \cite{cjy2020}. In Section 3, we first prove a key inequality implied by the subsolution condition, then we establish the uniform  complex Hessian estimate  of Hou-Ma-Wu-type which is independent of $(\theta(t)-(n-2)\frac{\pi}{2})^{-1}$. In Section 4, based on the complex Hessian estimate,  we prove the gradient estimate by a blow up argument. Then we prove the main theorems. In Section 5, we apply the main results to the 3-dimensional 2-Hessian equation and 4-dimensional Hessian quotient equation.

To solve the critical LYZ equation \eqref{LYZ1}, the authors “consider the family of equations”
\begin{align}\label{LYZt}
\theta_{\omega}(\chi_{u^{t}}+t\omega)=\theta(t).
\end{align}
They “verify that there exists a uniform constant $C>0$ such that $\theta(t)\in ((n-2)\frac{\pi}{2}+Ct,(n-2)\frac{\pi}{2}+2Ct)$ for small $t>0$,” so “equations \eqref{LYZt} are the supercritical LYZ equations.” Next they “prove the subsolution $\underline{u}$ of the critical LYZ equation \eqref{LYZ1} is also a subsolution of equations \eqref{LYZt} for small $t>0$,” and then “apply Lin's result \cite{lin2023} (see also Sun \cite{sun2024}) to obtain the smooth solution $u^t$ to equation \eqref{LYZt}$ with $\sup_M u^{t}=0$.”

“To prove Theorem \ref{FYZ}, we establish uniform $C^{2,\alpha}$ estimates for $u^{t}$ that are independent of $t$ and $(\theta(t)-(n-2)\frac{\pi}{2})^{-1}$,” i.e. the uniform a priori estimate stated as Theorem \ref{FYZfamilly}. Using Collins--Jacob--Yau \cite{cjy2020}, “$\sup_{M} |u^t|$ is uniformly bounded,” and their results “imply that the $C^{2,\alpha}$ estimates can be reduced to proving uniform complex Hessian estimates.” The “main contribution is the proof of the uniform complex Hessian estimate of Hou--Ma--Wu type”
\begin{align}\label{HMW}
\sup_{M} |\sqrt{-1}\partial\bar{\partial}u|_{\omega}\le C\bigl(1+\sup_{M}|\partial u|_{\omega}^2\bigr),
\end{align}
“and the uniform gradient estimate via the Dinew--Ko\l{}odziej blow-up argument \cite{dk2017ajm}.”

They explain the blow-up subtlety: if the gradient estimate fails, “we obtain a bounded non-constant $C^1$ function $v$ on $\mathbb{C}^n$” which is a weak solution of “$\sigma_n=0$, $\sigma_{n-1}=0$, or $\sigma_{n-1}+\sigma_n=0$,” where “the last case is new and requires a corresponding Liouville-type theorem.” They note that “$v+|z|^2$ is plurisubharmonic and $\Delta v$ is uniformly bounded in the weak sense,” and that while “the first case can be handled similarly,” for the second/third cases they need subsolution properties for mollifications and then use a key observation: “a smooth function $v$ satisfies $\sigma_{n-1}+\sigma_n=0$ (resp. $\ge0$) on $\mathbb{C}^n$ if and only if the function $v+|z_{n+1}|^2$ satisfies $\sigma_n=0$ (resp. $\ge0$) on $\mathbb{C}^{n+1}$.” “Based on this observation, we use potential theory to obtain the required Liouville-type theorem (see Theorem \ref{LiouTh}).”

With the uniform complex Hessian and gradient estimates, they obtain uniform $C^{2,\alpha}$ bounds for $u^t$ independent of $t$ (Theorem \ref{FYZfamilly}), and then “prove the main theorems,” yielding existence/uniqueness in Theorem \ref{FYZ}.

To solve the critical LYZ equation
\begin{align}\label{LYZ1}
\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}.
\end{align}
, the authors “consider the family of equations”
\begin{align}\label{LYZt}
\t\theta_{\omega}(\chi_{u^{t}}+t\omega)=\theta(t).
\end{align}
They “verify that there exists a uniform constant $C>0$ such that $\theta(t)\in ((n-2)\frac{\pi}{2}+Ct,(n-2)\frac{\pi}{2}+2Ct)$ for small $t>0$,” so “the equation above is the supercritical LYZ equation.” Next they “prove the subsolution $\underline{u}$ of the critical LYZ equation above is also a subsolution of the equation above for small $t>0$,” and then “apply Lin's result \cite{lin2023} (see also Sun \cite{sun2024}) to obtain the smooth solution $u^t$ to the equation above with $\sup_M u^{t}=0$.”

To prove the main theorem, “we establish uniform $C^{2,\alpha}$ estimates for $u^{t}$ that are independent of $t$ and $(\theta(t)-(n-2)\frac{\pi}{2})^{-1}$,” i.e. the uniform a priori estimate stated as Theorem \ref{FYZfamilly}. Using Collins--Jacob--Yau \cite{cjy2020}, “$\sup_{M} |u^t|$ is uniformly bounded,” and their results “imply that the $C^{2,\alpha}$ estimates can be reduced to proving uniform complex Hessian estimates.” The “main contribution is the proof of the uniform complex Hessian estimate of Hou--Ma--Wu type”
\begin{align}\label{HMW}
\sup_{M} |\sqrt{-1}\partial\bar{\partial}u|_{\omega}\le C\bigl(1+\sup_{M}|\partial u|_{\omega}^2\bigr),
\end{align}
“and the uniform gradient estimate via the Dinew--Ko\l{}odziej blow-up argument \cite{dk2017ajm}.”

They explain the blow-up subtlety: if the gradient estimate fails, “we obtain a bounded non-constant $C^1$ function $v$ on $\mathbb{C}^n$” which is a weak solution of “$\sigma_n=0$, $\sigma_{n-1}=0$, or $\sigma_{n-1}+\sigma_n=0$,” where “the last case is new and requires a corresponding Liouville-type theorem.” They note that “$v+|z|^2$ is plurisubharmonic and $\Delta v$ is uniformly bounded in the weak sense,” and that while “the first case can be handled similarly,” for the second/third cases they need subsolution properties for mollifications and then use a key observation: “a smooth function $v$ satisfies $\sigma_{n-1}+\sigma_n=0$ (resp. $\ge0$) on $\mathbb{C}^n$ if and only if the function $v+|z_{n+1}|^2$ satisfies $\sigma_n=0$ (resp. $\ge0$) on $\mathbb{C}^{n+1}$.”

We next use the following theorem.
\begin{theorem}\label{LiouTh}
    Let $v: \mathbb{C}^n \rightarrow \mathbb{R}$ be a $C^1$ function that is $(n-1)$-subharmonic and such that $v + |z|^2$ is plurisubharmonic. Assume there exists a positive constant $C$ such that $\|v\|_{C^1(\mathbb{C}^n)} \le C$ and $\Delta v \le C$ in the weak sense (see Remark~\ref{vlaplacebound} for details). If $v$ is a weak solution in the sense of currents to the equation
    \begin{align}\label{nn-1equation}
        \sigma_{n-1}(\sqrt{-1}\partial\bar{\partial} v) + \sigma_{n}(\sqrt{-1}\partial\bar{\partial} v) = 0,
    \end{align}
    then $v$ must be constant.
\end{theorem}

With the uniform complex Hessian and gradient estimates, they obtain uniform $C^{2,\alpha}$ bounds for $u^t$ independent of $t$ (Theorem \ref{FYZfamilly}), and then “prove the main theorems,” yielding existence/uniqueness in establishing the main theorem.

\label{FYZ}
Let $(M,\omega)$ be a compact K\"ahler manifold and $\chi$ a  closed real (1,1)-form such that the principal argument of  $\int_{M}(\chi+\sqrt{-1} \omega)^n$ is $\pi$. If there exists a smooth function $\underline{u}$ satisfying
\begin{align}\label{subsolution}
	\min\limits_{1\le j\le n}\sum\limits_{\substack{i=1 \\ i\neq j}}^n\mathrm{arctan}\lambda_i(\chi_{\underline {u}})>{(n-3)\frac \pi 2}.
\end{align}
then the critical LYZ equation (in establishing the main theorem) admits a unique smooth solution u with $\sup_M u=0$.

Question:
Let $(M,\omega)$ be a compact K\"ahler manifold, let $\chi$ be a closed real $(1,1)$-form, and for a smooth real-valued function $v$ write $\chi_v:=\chi+\sqrt{-1}\,\partial\bar\partial v$. If $\lambda_1(\alpha),\dots,\lambda_n(\alpha)$ are the eigenvalues of a real $(1,1)$-form $\alpha$ with respect to $\omega$, define
\[
\theta_{\omega}(\alpha):=\sum_{i=1}^n \arctan \lambda_i(\alpha).
\]
Assume that the principal argument of
\[
\int_M (\chi+\sqrt{-1}\,\omega)^n
\]
is $\pi$, and suppose there exists a smooth function $\underline u$ such that
\[
\min_{1\le j\le n}\sum_{\substack{i=1\\ i\neq j}}^n \arctan \lambda_i(\chi_{\underline u})>(n-3)\frac{\pi}{2}.
\]
Which statement holds for the critical LYZ equation
\[
\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}?
\]

There exists a unique smooth function $u:M\to\mathbb{R}$ satisfying $\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}$ and the normalization $\sup_M u=0$.

Choices:
B. There exists a unique smooth function $u:M\to\mathbb{R}$ satisfying $\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}$, and moreover the solution is obtained with uniform a priori bounds independent of $\bigl(\theta-(n-2)\frac{\pi}{2}\bigr)^{-1}$.
C. There exists a smooth function $u:M\to\mathbb{R}$ satisfying $\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}$.
D. There exists a unique smooth function $u:M\to\mathbb{R}$ satisfying $\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}$ provided one additionally assumes the stronger phase condition $\theta_{\omega}(\chi_{\underline u})>(n-2)\frac{\pi}{2}$ for the same subsolution $\underline u$, together with $\sup_M u=0$.
E. For every sufficiently small $t>0$, there exists a unique smooth function $u^t:M\to\mathbb{R}$ with $\sup_M u^t=0$ solving the perturbed supercritical equation $\theta_{\omega}(\chi_{u^t}+t\omega)=\theta(t)$, and hence the critical equation $\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}$ need not itself admit a smooth solution under the stated hypotheses.

uniform-estimate statement for approximating family incorrectly inserted into theorem conclusion

dropped uniqueness and normalization $\sup_M u=0$

reintroduced extra supercritical subsolution phase assumption removed in the critical theorem

approximation-by-$t$ family mistaken for nonexistence of the limiting critical solution