\begin{theorem}\label{Thm for E}
    If $E$ is uniformly weakly RC-positive over a compact K\"ahler manifold $X$, then $S^kE\otimes \det E$ is uniformly RC-positive for any $k\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large.
\end{theorem}

\begin{lemma}\label{lemma abc}
  Assume the curvature $\Theta$ of $h$ is positive on every fiber $\mathcal{X}_t$. For $1\leq k\leq m$, the following are equivalent.
\begin{enumerate}[label=\Alph*.]
    \item\label{A} The curvature $\Theta$ has at least $n+k$ positive eigenvalues at every point in $\mathcal{X}$.
    \item\label{B} The horizontal component $\Theta_\mathcal{H}$ has at least $k$ positive eigenvalues at every point in $\mathcal{X}$.
    \item\label{C}  There exists a positive $\beta\in C^{\infty}(\mathcal{X}, p^*(\wedge^{1,1}T^*Y))$ such that the sum of any $m-k+1$ eigenvalues of $\Theta_\mathcal{H}$ with respect to $\beta$ is positive.
\end{enumerate}
Moreover, when $k=1$, statement C can be rephrased as $\Theta_\mathcal{H}\wedge \beta^{m-1}>0$ or equivalently $\Theta^{n+1}\wedge \beta^{m-1}>0$.
\end{lemma}

In \cite{YangCamb}, Yang introduces the notion of RC-positivity as a differential geometric counterpart of rational connectedness.  RC-positivity plays a crucial role in Yang's proof of a conjecture of Yau: If a compact K\"ahler manifold has positive holomorphic sectional curvature, then the manifold is projective and rationally connected. A stronger notion called uniform RC-positivity is introduced by Yang in \cite{YangForum} which also can be used to prove the same conjecture of  Yau. For the semipositive case of Yau's conjecture, see \cite{HeierWong,MatsumuraPAMQ,MatsumuraAJM}.

Let us recall the definition of RC-positivity and uniform RC-positivity. Let $E$ be a holomorphic vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. Given a Hermitian metric $H$ on $E$, we denote the Chern curvature of $H$ by $\Theta^H$, which is an $\text{End}E$-valued $(1,1)$-form. We denote by $TX$ the holomorphic tangent bundle of $X$. For a vector $u\in E_t$ and a tangent vector $v\in T_tX$ with $t\in X$, we define the expression $$H(\Theta^H u,u)(v,\bar{v})$$ to be  $\sum_{j,k} H(\Theta^H_{j\bar{k}}u,u)v_j \bar{v}_k$ locally where we write the curvature $\Theta^H=\sum_{j,k}\Theta^H_{j\bar{k}}dt_j\wedge d\bar{t}_k$ and  $v=\sum_j v_j\partial/\partial t_j$.

\begin{definition}

A Hermitian metric $H$ on a holomorphic vector bundle $E\to X$ is called RC-positive if for any $t\in X$ and any nonzero $u\in E_t$, there is a nonzero tangent vector $v\in T_tX$ such that $H(\Theta^Hu,u)(v,\bar v)>0$. On the other hand, a Hermitian metric $H$ is called uniformly RC-positive if for any $t\in X$, there is a nonzero tangent vector $v\in T_tX$ such that for any nonzero $u\in E_t$, we have $H(\Theta^H u,u)(v,\bar v)>0$.

A holomorphic vector bundle $E\to X$ is called (uniformly) RC-positive if it admits a (uniformly) RC-positive Hermitian metric.
\end{definition}
It is clear that uniform RC-positivity implies RC-positivity. To motivate the definition of (uniform) weak RC-positivity, let us consider a Hermitian metric $H$ on $E$ and the induced metric $h$ on the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$. By a standard computation (for example, see \cite[Formula (4.5)]{YangCamb}), we know that if $(E,H)$ is RC-positive, then the curvature $\Theta$ of $h$ is positive on every fiber and has at least $r$ positive eigenvalues at every point in $P(E^*)$. The existence of such a metric $h$ on $O_{P(E^*)}(1)$ is called weak RC-positivity of $E$  (\cite[Definition 3.3]{YangCamb}).

Similarly, using the same computation, we see that if $(E,H)$ is uniformly RC-positive, then the curvature $\Theta$ of the induced metric $h$ satisfies
\begin{enumerate}[label=\alph*.]
    \item\label{a} $\Theta$ is positive on every fiber.
    \item $\Theta$ has at least $r$ positive eigenvalues at every point in $P(E^*)$.
    \item For any point $t\in X$, there exists a nonzero tangent vector $v\in T_tX$, such that $\Theta(\Tilde{v},\bar{\Tilde{v}})|_{(t,[\zeta])}>0$ for any lift $\Tilde{v}$ of $v$ to $T_{(t,[\zeta])}P(E^*)$.
\end{enumerate}
Following Yang, we call the existence of such a metric $h$ on $O_{P(E^*)}(1)$ uniform weak RC-positivity of $E$. Note that in the third condition, we consider the lifts to the tangent space $T_{(t,[\zeta])}P(E^*)$ for any point $[\zeta]$ in the fiber $P(E_t^*)$ not just one point $[\zeta]$. The second condition is implied by the first and the third, so we will omit it later on. Let us summarize the definition.
\begin{definition} The bundle $E$ is called weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and b. The bundle $E$ is called uniformly weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and c.
\end{definition}
In Yang's solution to Yau's conjecture, two main theorems \cite[Theroem 1.3 and Theorem 1.4]{YangCamb}, although formulated in terms of RC-positivity,  hold under weak RC-positivity. So, it is natural to ask if weak RC-positivity of $E$ implies RC-positivity of $E$ (\cite[Question 7.11]{YangCamb} and \cite[Problem 13]{Inayama}). This question has the same flavor as a conjecture of Griffiths \cite{Griff69}: If $E$ is ample, then $E$ is Griffiths positive. For the developments on the Griffiths conjecture, see \cite{Umemura,CampanaFlenner,Berndtsson09,MourouganeTaka,positivityandvanishingthmliu,liu2014curvatures,FengLiuWan,demailly2020hermitianyangmills,pingali2021note,finski2020monge,Finskichara,wu_2022,wupositivelyII,wuIII,Mazhang,lempert2024two,murakami2025analytic,wu2025mean}.

In this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.

In \cite{YangCamb}, Yang introduces the notion of RC-positivity as a differential geometric counterpart of rational connectedness. RC-positivity plays a crucial role in Yang's proof of a conjecture of Yau: If a compact K\"ahler manifold has positive holomorphic sectional curvature, then the manifold is projective and rationally connected. A stronger notion called uniform RC-positivity is introduced by Yang in \cite{YangForum} which also can be used to prove the same conjecture of Yau. For the semipositive case of Yau's conjecture, see \cite{HeierWong,MatsumuraPAMQ,MatsumuraAJM}.

Let us recall the definition of RC-positivity and uniform RC-positivity. Let $E$ be a holomorphic vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. Given a Hermitian metric $H$ on $E$, we denote the Chern curvature of $H$ by $\Theta^H$, which is an $\text{End}E$-valued $(1,1)$-form. We denote by $TX$ the holomorphic tangent bundle of $X$. For a vector $u\in E_t$ and a tangent vector $v\in T_tX$ with $t\in X$, we define the expression $$H(\Theta^H u,u)(v,\bar{v})$$ to be $\sum_{j,k} H(\Theta^H_{j\bar{k}}u,u)v_j \bar{v}_k$ locally where we write the curvature $\Theta^H=\sum_{j,k}\Theta^H_{j\bar{k}}dt_j\wedge d\bar{t}_k$ and $v=\sum_j v_j\partial/\partial t_j$.

A Hermitian metric $H$ on a holomorphic vector bundle $E\to X$ is called RC-positive if for any $t\in X$ and any nonzero $u\in E_t$, there is a nonzero tangent vector $v\in T_tX$ such that $H(\Theta^Hu,u)(v,\bar v)>0$. On the other hand, a Hermitian metric $H$ is called uniformly RC-positive if for any $t\in X$, there is a nonzero tangent vector $v\in T_tX$ such that for any nonzero $u\in E_t$, we have $H(\Theta^H u,u)(v,\bar v)>0$.

A holomorphic vector bundle $E\to X$ is called (uniformly) RC-positive if it admits a (uniformly) RC-positive Hermitian metric.
\end{definition}
It is clear that uniform RC-positivity implies RC-positivity. To motivate the definition of (uniform) weak RC-positivity, let us consider a Hermitian metric $H$ on $E$ and the induced metric $h$ on the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$. By a standard computation (for example, see \cite[Formula (4.5)]{YangCamb}), we know that if $(E,H)$ is RC-positive, then the curvature $\Theta$ of $h$ is positive on every fiber and has at least $r$ positive eigenvalues at every point in $P(E^*)$. The existence of such a metric $h$ on $O_{P(E^*)}(1)$ is called weak RC-positivity of $E$ (\cite[Definition 3.3]{YangCamb}).

Similarly, using the same computation, we see that if $(E,H)$ is uniformly RC-positive, then the curvature $\Theta$ of the induced metric $h$ satisfies
\begin{enumerate}[label=\alph*.]
 \item\label{a} $\Theta$ is positive on every fiber.
 \item $\Theta$ has at least $r$ positive eigenvalues at every point in $P(E^*)$.
 \item For any point $t\in X$, there exists a nonzero tangent vector $v\in T_tX$, such that $\Theta(\Tilde{v},\bar{\Tilde{v}})|_{(t,[\zeta])}>0$ for any lift $\Tilde{v}$ of $v$ to $T_{(t,[\zeta])}P(E^*)$.
\end{enumerate}
Following Yang, we call the existence of such a metric $h$ on $O_{P(E^*)}(1)$ uniform weak RC-positivity of $E$. Note that in the third condition, we consider the lifts to the tangent space $T_{(t,[\zeta])}P(E^*)$ for any point $[\zeta]$ in the fiber $P(E_t^*)$ not just one point $[\zeta]$. The second condition is implied by the first and the third, so we will omit it later on. Let us summarize the definition.
\begin{definition} The bundle $E$ is called weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and b. The bundle $E$ is called uniformly weakly RC-positive if there exists a metric $h$ on $O_{P(E^*)}(1)$ with properties a and c.
\end{definition}
In Yang's solution to Yau's conjecture, two main theorems \cite[Theroem 1.3 and Theorem 1.4]{YangCamb}, although formulated in terms of RC-positivity, hold under weak RC-positivity. So, it is natural to ask if weak RC-positivity of $E$ implies RC-positivity of $E$ (\cite[Question 7.11]{YangCamb} and \cite[Problem 13]{Inayama}). This question has the same flavor as a conjecture of Griffiths \cite{Griff69}: If $E$ is ample, then $E$ is Griffiths positive. For the developments on the Griffiths conjecture, see \cite{Umemura,CampanaFlenner,Berndtsson09,MourouganeTaka,positivityandvanishingthmliu,liu2014curvatures,FengLiuWan,demailly2020hermitianyangmills,pingali2021note,finski2020monge,Finskichara,wu_2022,wupositivelyII,wuIII,Mazhang,lempert2024two,murakami2025analytic,wu2025mean}.

In this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.

In this paper, we obtain results in this direction. In particular, we show that if a vector bundle $E$ is uniformly weakly RC-positive, then $S^kE\otimes \det E$ is uniformly RC-positive for any $k\geq 0$, and $S^kE$ is uniformly RC-positive for $k$ large. We also discuss an approach that might lead to a solution to the question of whether weak RC-positivity of $E$ implies RC-positivity of $E$.

In this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.

Another motivation for establishing Theorem \ref{Thm for E} is to understand the relation between rational connectedness of $X$ and (uniform) RC-positivity of the holomorphic tangent bundle $TX$. According to Yang \cite[Theorem 1.4]{YangCamb} and \cite[Theorem 1.3]{YangForum}, for a compact K\"ahler manifold $X^n$, if one of the following is true, then $X$ is projective and rationally connected.
\begin{enumerate}
 \item The holomorphic tangent bundle $TX$ is uniformly RC-positive.
 \item The exterior power $\wedge^p TX$ is RC-positive for $1\leq p\leq n$.
 \end{enumerate}
 One can ask if the converse is true (\cite[Problem 4.15]{YangForum}). A partial converse is proved in \cite[Theorem 1.4]{YangForum}: if $X$ is projective and rationally connected, then the line bundle $O_{\wedge^p TX}(-1)$ is RC-positive for $1\leq p \leq n$. So, Theorem \ref{Thm for E} can be viewed as a step towards this converse problem: constructing uniformly RC-positive Hermitian metrics out of metrics on the line bundle $O_{P(E^*)}(1)$.

For the proof of Theorem \ref{Thm for E}, instead of the fibration $p:P(E^*)\to X$, we will work on a more general fibration and prove a general theorem which contains Theorem \ref{Thm for E}
as a special case. We consider a proper holomorphic surjection $p:\mathcal{X}^{n+m}\to Y^m$ between two complex manifolds with $\mathcal{X}$ K\"ahler, $Y$ compact, and the differential $dp$ surjective at every point. We denote the fibers $p^{-1}(t)$ by $\mathcal{X}_t$ for $t\in Y$. Let $(L,h)$ be a Hermitian line bundle over $\mathcal{X}$. Let $$V_t=H^0(\mathcal{X}_t, L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}).$$ We assume that $\dim V_t$ is independent of $t\in Y$. So, the direct image of the sheaf of sections of $L\otimes K_{\mathcal{X}/Y}$ is locally free by Grauert's direct image theorem, where $K_{\mathcal{X}/Y}$ is the relative canonical bundle. We denote by $V$ the associated vector bundle over $Y$. There is a naturally defined Hermitian metric $H$ on $V$. For $u$ in $V_t$ with $t\in Y$,
\begin{equation}\label{metric}
 H(u,u):=\int_{\mathcal{X}_t}h(u,u).
\end{equation}
We extend the metric $h$ to act on sections $u$ of $L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}$ so that $h(u,u)$ is an $(n,n)$-form on $\mathcal{X}_t$. In terms of local coordinates, if $u=u'\otimes e$ with $u'$ an $(n,0)$-form and $e$ a frame of $L|_{\mathcal{X}_t}$, then $h(u,u)=c_n u' \wedge \overline{u'} h(e,e)$ where $c_n=i^{n^2}$. Under this more general fibration, we can show
\begin{theorem}\label{Thm 1}
 If the curvature $\Theta$ of $h$ is positive on every fiber, and for any point $t\in Y$, there exists a nonzero tangent vector $v\in T_tY$, such that $\Theta(\Tilde{v},\bar{\Tilde{v}})|_{(t,z)}>0$ for any lift $\Tilde{v}$ of $v$ to $T_{(t,z)}\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive.
\end{theorem}

Now, we consider the fibration $p:P(E^*)\to X$, and we assume $X$ is K\"ahler to make sure $P(E^*)$ is K\"ahler (see \cite[Subsection 5.2]{wu2025mean}). Therefore, by Theorem \ref{Thm 1}, we have Theorem \ref{Thm for E}. Indeed, the vector bundle $V$ in Theorem \ref{Thm 1} is associated with the direct image of $L\otimes K_{\mathcal{X}/Y}$. In the present situation, the relative canonical bundle
$K_{P(E^*)/X}$ is isomorphic to $O_{P(E^*)}(-r)\otimes p^*\det E$. If we choose $O_{P(E^*)}(r+k)$ for the line bundle $L$, then $V$ is $S^k\otimes \det E$. On the other hand, if we choose $O_{P(E^*)}(k)\otimes K^{-1}_{P(E^*)/X}$ for $L$, then $V$ is $S^kE$ (we use an arbitrary metric $g$ on $K^{-1}_{P(E^*)/X}$, and the effect of $g$ can be absorbed by taking $k$ large).

We choose a coordinate system $(t_1,\ldots,t_m)$ around the point $t_0$ in $Y$ such that $v_0=\partial/\partial t_1$ at $t_0$. Consider a fixed $u_0\neq 0$ in $V_{t_0}$. A standard argument allows us to extend $u_0$ to a local holomorphic section $u$ of $V$ such that $D'u=0$ at $t_0$ and $u(t_0)=u_0\neq0$. A straightforward computation gives
\begin{equation}\label{standard} \partial \bar{\partial} H(u,u)=- H(\Theta^V u,u) \text{ at } t_0.
\end{equation}
On the other hand, if we let $\mathbf{u}$ be a representative of $u$ and write $\mathbf{u}=u'\otimes e$ with $u'$ an $(n,0)$-form and $e$ some local frame of $L$, then
\begin{equation}
 H(u,u)=p_*(c_n u'\wedge \overline{u'} e^{-\phi})
\end{equation}
where $e^{-\phi}=h(e,e)$ and $c_n=i^{n^2}$. According to \cite[Proposition 4.2]{Berndtsson09}, we can choose a representative $\mathbf{u}$ such that in $\bar{\partial}\mathbf{u}= \sum \eta_j\wedge dt_j$, the $\eta_j$ is primitive on $\mathcal{X}_{t_0}$. Moreover,
$\partial^\phi u'=0$ at $t_0$. After using such a representative, we obtain
\begin{equation}\label{4.4}
\partial\bar{\partial}H(u,u)
 =
 -c_n p_* ( u'\wedge\overline{u'}\wedge \partial\bar{\partial} \phi e^{-\phi})
 +
 (-1)^n c_np_* (\bar{\partial}u'\wedge \overline{\bar{\partial}u'}e^{-\phi}) \text{ at } t_0.
\end{equation}
We apply the above $(1,1)$-form to the tangent vector $v_0=\partial/\partial t_1$ and get
\begin{equation}\label{4.6}
\partial\bar{\partial}H(u,u)(v_0,\bar{v}_0)
 =
 -c_n p_* ( u'\wedge\overline{u'}\wedge \partial\bar{\partial} \phi e^{-\phi})(v_0,\bar{v}_0)
 +
 (-1)^n c_np_* (\bar{\partial}u'\wedge \overline{\bar{\partial}u'}e^{-\phi}) (v_0,\bar{v}_0)\text{ at } t_0.
\end{equation}
Because $\bar{\partial}\mathbf{u}= \sum \eta_j\wedge dt_j$ and $\mathbf{u}=u'\otimes e$, we see $\sum \eta_j\wedge dt_j=\bar{\partial}\mathbf{u}=\bar{\partial}u'\otimes e $. If we write $\eta_j=\eta_j'\otimes e$, then $\bar{\partial}u'=\sum \eta_j'\wedge dt_j$. So the last term in (\ref{4.6}) is equal to \begin{equation}\label{4.7}
 (-1)^n c_n\int_{\mathcal{X}_{t_0}}(-1)^{n}\sum \eta'_j\wedge
\overline{\eta'}_k \wedge dt_j\wedge d\bar{t}_k e^{-\phi}(v_0,\bar{v}_0)= c_n \int_{\mathcal{X}_{t_0}} \eta'_1\wedge
\overline{\eta'}_1 e^{-\phi}\leq 0;
\end{equation}
the last inequality is by the fact that the $\eta_1$ is primitive on $\mathcal{X}_{t_0}$.

The case we care about most in this paper is when $k=1$ in Lemma \ref{lemma abc} because it corresponds to weak RC-positivity. The difficulty in proving a theorem like Theorem \ref{Thm for E} or Theorem \ref{Thm 1} for weak RC-positivity is that the $\beta$ in Lemma \ref{lemma abc} is on $\mathcal{X}$, so it does not quite fit into Berndtsson's computation, especially formula (\ref{4.11}).
So, we raise the question:
\begin{question}
 Is it possible to choose $\beta$ in Lemma \ref{lemma abc} so that $\beta=p^*\alpha$ for some Hermitian metric $\alpha$ on $Y$?
\end{question}
This question is somewhat bold because if it is possible to choose $\beta=p^*\alpha$, then we can use \cite[Theorem 4]{wu2025mean} to deduce that if $E$ is weakly RC-positive, then $S^k\otimes \det E$ has positive mean curvature for $k\geq 0$. Moreover, it is even possible to use \cite[Theorem 5]{wu2025mean} to deduce that if $E$ is weakly RC-positive, then $E$ has positive mean curvature. Since positive mean curvature implies RC-positivity (\cite[Theorem 3.6]{YangCamb}), this would mean that RC-positivity, weak RC-positivity, and mean curvature positivity are all equivalent. Such an equivalence is conjectured for tangent bundle $TX$ in
\cite[Problems 4.15 and 4.17]{YangForum}.

Another motivation for establishing Theorem \ref{Thm for E} is to understand the relation between rational connectedness of $X$ and (uniform) RC-positivity of the holomorphic tangent bundle $TX$. According to Yang \cite[Theorem 1.4]{YangCamb} and \cite[Theorem 1.3]{YangForum}, for a compact K\"ahler manifold $X^n$, if one of the following is true, then $X$ is projective and rationally connected.
\begin{enumerate}
     \item The holomorphic tangent bundle $TX$ is uniformly RC-positive.
     \item The exterior power $\wedge^p TX$ is RC-positive for $1\leq p\leq n$.
 \end{enumerate}
 One can ask if the converse is true (\cite[Problem 4.15]{YangForum}). A partial converse is proved in \cite[Theorem 1.4]{YangForum}: if $X$ is projective and rationally connected, then the line bundle $O_{\wedge^p TX}(-1)$ is RC-positive for $1\leq p \leq n$. So, Theorem \ref{Thm for E} can be viewed as a step towards this converse problem: constructing uniformly RC-positive Hermitian metrics out of metrics on the line bundle $O_{P(E^*)}(1)$.

We also prove a lemma (Lemma \ref{lemma abc} in Section \ref{section ?}) and discuss how a variant of this lemma might lead to a solution to the original question of Yang, namely, weak RC-positivity of $E$ implying RC positivity of $E$.

For the proof of Theorem \ref{Thm for E}, instead of the fibration $p:P(E^*)\to X$, we will work on a more general fibration and prove a general theorem which contains Theorem \ref{Thm for E}
as a special case. We consider a proper holomorphic surjection $p:\mathcal{X}^{n+m}\to Y^m$ between two complex manifolds with $\mathcal{X}$ K\"ahler, $Y$ compact, and the differential $dp$ surjective at every point. We denote the fibers $p^{-1}(t)$ by $\mathcal{X}_t$ for $t\in Y$. Let $(L,h)$ be a Hermitian line bundle over $\mathcal{X}$. Let $$V_t=H^0(\mathcal{X}_t, L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}).$$ We assume that $\dim V_t$ is independent of $t\in Y$. So, the direct image of the sheaf of sections of $L\otimes K_{\mathcal{X}/Y}$ is locally free by Grauert's direct image theorem, where $K_{\mathcal{X}/Y}$ is the relative canonical bundle. We denote by $V$ the associated vector bundle over $Y$. There is a naturally defined Hermitian metric $H$ on $V$. For $u$ in $V_t$ with $t\in Y$,
\begin{equation}\label{metric}
  H(u,u):=\int_{\mathcal{X}_t}h(u,u).
\end{equation}
We extend the metric $h$ to act on sections $u$ of $L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}$ so that $h(u,u)$ is an $(n,n)$-form on $\mathcal{X}_t$. In terms of local coordinates, if $u=u'\otimes e$ with $u'$ an $(n,0)$-form and $e$ a frame of $L|_{\mathcal{X}_t}$, then $h(u,u)=c_n u' \wedge \overline{u'} h(e,e)$ where $c_n=i^{n^2}$. Under this more general fibration, we can show
\begin{theorem}\label{Thm 1}
    If the curvature $\Theta$ of $h$ is positive on every fiber, and for any point $t\in Y$, there exists a nonzero tangent vector $v\in T_tY$, such that $\Theta(\Tilde{v},\bar{\Tilde{v}})|_{(t,z)}>0$ for any lift $\Tilde{v}$ of $v$ to $T_{(t,z)}\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive.
\end{theorem}

Actually, the precise statement we prove in Theorem \ref{Thm 1} is: For a fixed point $t_0 \in Y$, if the curvature $\Theta$ of $h$ is positive on the fiber $\mathcal{X}_{t_0}$ , and  there exists a nonzero tangent vector $v\in T_{t_0}Y$, such that $\Theta(\Tilde{v},\bar{\Tilde{v}})|_{(t_0,z)}>0$ for any lift $\Tilde{v}$ of $v$ to $T_{(t_0,z)}\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive at $t_0$.

Now, we consider the fibration $p:P(E^*)\to X$, and we assume $X$ is K\"ahler to make sure $P(E^*)$ is K\"ahler (see \cite[Subsection 5.2]{wu2025mean}). Therefore, by Theorem \ref{Thm 1}, we have Theorem \ref{Thm for E}. Indeed, the vector bundle $V$ in Theorem \ref{Thm 1} is associated with the direct image of $L\otimes K_{\mathcal{X}/Y}$. In the present situation, the relative canonical bundle
$K_{P(E^*)/X}$ is isomorphic to $O_{P(E^*)}(-r)\otimes p^*\det E$. If we choose $O_{P(E^*)}(r+k)$ for the line bundle $L$, then $V$ is $S^k\otimes \det E$. On the other hand, if we choose $O_{P(E^*)}(k)\otimes K^{-1}_{P(E^*)/X}$ for $L$, then $V$ is $S^kE$ (we use an arbitrary metric $g$ on $K^{-1}_{P(E^*)/X}$, and the effect of $g$ can be absorbed by taking $k$ large).

The proof of Theorem \ref{Thm 1} is an adaptation of \cite[Section 3]{wu2025mean}, but we still include the details for completeness  (the original argument is due to Berndtsson in \cite[Section 4]{Berndtsson09} and \cite[Section 2]{BoMathz}. See also \cite{CampanaCaoMihai}).

This paper is organized as follows. In Section \ref{section prelim}, we give a local expression for the assumption (uniform weak RC-positivity) in Theorem \ref{Thm 1} which will be used in the proof of the main theorem. In Section \ref{section proof}, we prove Theorem \ref{Thm 1}. In Section \ref{section ?}, we discuss
a characterization of weak RC-positivity and its possible application.

I would like to thank Shin-ichi Matsumura for bringing to my attention the question of RC-positivity and weak RC-positivity. I am grateful to L\'aszl\'o Lempert, Siarhei Finski, and Xiaokui Yang for their interest in the paper. Thanks are also due to the Erd\H{o}s Center, Budapest for the support.

To prove Theorem~\ref{Thm for E}, the text says that “instead of the fibration $p:P(E^*)\to X$, we will work on a more general fibration and prove a general theorem which contains Theorem~\ref{Thm for E}$ as a special case.” Concretely, one considers a proper holomorphic submersion $p:\mathcal{X}^{n+m}\to Y^m$ with $\mathcal{X}$ K\"ahler and $Y$ compact, a Hermitian line bundle $(L,h)$ on $\mathcal{X}$, and the direct-image bundle $V$ with fibers
\[
V_t=H^0(\mathcal{X}_t,\, L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}),
\]
endowed with the “naturally defined Hermitian metric”
\[
H(u,u):=\int_{\mathcal{X}_t} h(u,u).\tag{\ref{metric}}
\]
Under the hypothesis (in Theorem~\ref{Thm 1}) that “the curvature $\Theta$ of $h$ is positive on every fiber” and that for each $t\in Y$ there is a nonzero $v\in T_tY$ with “$\Theta(\widetilde v,\overline{\widetilde v})|_{(t,z)}>0$ for any lift $\widetilde v$ of $v$,” one concludes that “the Hermitian bundle $(V,H)$ is uniformly RC-positive.”

Then Theorem~\ref{Thm for E} is obtained by applying Theorem~\ref{Thm 1} to $p:P(E^*)\to X$ (with $X$ K\"ahler so $P(E^*)$ is K\"ahler). In this specialization, one uses that
\[
K_{P(E^*)/X}\simeq O_{P(E^*)}(-r)\otimes p^*\det E.
\]
Choosing $L=O_{P(E^*)}(r+k)$ gives that the resulting direct image bundle $V$ is “$S^kE\otimes\det E$.” Alternatively, choosing $L=O_{P(E^*)}(k)\otimes K^{-1}_{P(E^*)/X}$ gives $V=S^kE$, and “the effect of [a metric on $K^{-1}_{P(E^*)/X}$] can be absorbed by taking $k$ large.”

Finally, the proof of Theorem~\ref{Thm 1} is described as “an adaptation of \cite[Section 3]{wu2025mean}” and attributed to Berndtsson (with references).

To prove the main theorem, the text says that “instead of the fibration $p:P(E^*)\to X$, we will work on a more general fibration and prove a general theorem which contains the main theorem as a special case.” Concretely, one considers a proper holomorphic submersion $p:\mathcal{X}^{n+m}\to Y^m$ with $\mathcal{X}$ K\"ahler and $Y$ compact, a Hermitian line bundle $(L,h)$ on $\mathcal{X}$, and the direct-image bundle $V$ with fibers
\[
V_t=H^0(\mathcal{X}_t,\, L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}),
\]
endowed with the “naturally defined Hermitian metric”
\[
H(u,u):=\int_{\mathcal{X}_t} h(u,u).\tag{\ref{metric}}
\]
We first prove the following theorem.
\begin{theorem}\label{Thm 1}
    If the curvature $\Theta$ of $h$ is positive on every fiber, and for any point $t\in Y$, there exists a nonzero tangent vector $v\in T_tY$, such that $\Theta(\Tilde{v},\bar{\Tilde{v}})|_{(t,z)}>0$ for any lift $\Tilde{v}$ of $v$ to $T_{(t,z)}\mathcal{X}$, then the Hermitian bundle $(V,H)$ is uniformly RC-positive.
\end{theorem}
Under these hypotheses, one concludes that “the Hermitian bundle $(V,H)$ is uniformly RC-positive.”

Then the main theorem is obtained by applying the preceding theorem to $p:P(E^*)\to X$ (with $X$ K\"ahler so $P(E^*)$ is K\"ahler). In this specialization, one uses that
\[
K_{P(E^*)/X}\simeq O_{P(E^*)}(-r)\otimes p^*\det E.
\]
Choosing $L=O_{P(E^*)}(r+k)$ gives that the resulting direct image bundle $V$ is “$S^kE\otimes\det E$.” Alternatively, choosing $L=O_{P(E^*)}(k)\otimes K^{-1}_{P(E^*)/X}$ gives $V=S^kE$, and “the effect of [a metric on $K^{-1}_{P(E^*)/X}$] can be absorbed by taking $k$ large.”

Finally, the proof of the preceding theorem is described as “an adaptation of \cite[Section 3]{wu2025mean}” and attributed to Berndtsson (with references).

Question:
Let $X$ be a compact K\"ahler manifold and let $E\to X$ be a holomorphic vector bundle. Write $P(E^*)$ for the projectivized dual bundle and $O_{P(E^*)}(1)$ for its tautological line bundle. Say that $E$ is uniformly weakly RC-positive if there exists a Hermitian metric $h$ on $O_{P(E^*)}(1)$ whose curvature form $\Theta$ satisfies: (i) $\Theta$ is positive on every fiber of $P(E^*)\to X$; and (ii) for every $t\in X$, there exists a nonzero tangent vector $v\in T_tX$ such that for every point $[\zeta]\in P(E_t^*)$ and every lift $\widetilde v\in T_{(t,[\zeta])}P(E^*)$ of $v$, one has $\Theta(\widetilde v,\overline{\widetilde v})>0$. Also, a holomorphic vector bundle $F\to X$ is uniformly RC-positive if it admits a Hermitian metric $H$ such that for every $t\in X$, there exists a nonzero tangent vector $v\in T_tX$ with $H(\Theta^H u,u)(v,\bar v)>0$ for every nonzero $u\in F_t$. Here $S^kE$ denotes the $k$th symmetric power of $E$, and $\det E$ its determinant line bundle.

Which statement holds for every uniformly weakly RC-positive holomorphic vector bundle $E$ over a compact K\"ahler manifold $X$?

For every integer $k\ge 0$, the bundle $S^kE\otimes \det E$ is uniformly RC-positive, and the bundle $S^kE$ is uniformly RC-positive for all sufficiently large integers $k$.

Choices:
B. For every integer $k\ge 0$, the bundle $S^kE\otimes \det E$ is RC-positive, and the bundle $S^kE$ is uniformly RC-positive for all sufficiently large integers $k$.
C. For every integer $k\ge 0$, the bundle $S^kE\otimes \det E$ is uniformly RC-positive.
D. For every integer $k\ge 0$, both $S^kE\otimes \det E$ and $S^kE$ are uniformly RC-positive.
E. There exists an integer $k_0\ge 0$, depending only on $\operatorname{rank}(E)$, such that for every integer $k\ge k_0$, both $S^kE\otimes \det E$ and $S^kE$ are uniformly RC-positive.

necessity of the large-$k$ absorption step for $K^{-1}_{P(E^*)/X}$

non-uniform dependence of the large-$k$ threshold

The stem gives only definitions and hypotheses; it does not reveal the conclusion about symmetric powers or determinant twists. The correct option is not signposted by wording in the stem.

This is not a restatement of a definition. It asks for a substantive theorem-level consequence involving symmetric powers, determinant twisting, and an eventual-large-k conclusion, so the respondent must distinguish among competing claims.

The item requires some reasoning about theorem strength, quantifiers, and positivity notions, especially to separate 'for all k' from 'for sufficiently large k' and 'RC-positive' from 'uniformly RC-positive.' However, it is still largely theorem recall/discrimination rather than deep derivation.

The distractors are plausible and mathematically targeted: they vary the positivity strength, remove the large-k caveat, or introduce an unjustified uniform threshold depending only on rank. These reflect common overgeneralization and quantifier errors.

A strong MCQ with no answer leakage and well-designed distractors. It tests careful discrimination of a nontrivial theorem, though it leans more toward precise recall than fully generative reasoning.


In recent years the pluriclosed flow \cite{PCF, PCFReg} and generalized K\"ahler-Ricci flow \cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\"ahler, manifolds \cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}. A natural class of non-K\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \cite{inoue1974surfaces}. In \cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense. Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton. It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \ref{c:OT_PCF}). In this work we confirm some aspects of this conjecture.

The first main result is to establish the global existence of the flow:

\begin{conj} \label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\omega_0$, the normalized pluriclosed flow \ref{e:PCF_normalized} with initial metric $\omega_0$ exists on $[0, \infty)$, and converges to $(\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.
\end{conj}

The first main result is to establish the global existence of the flow:

The expected qualitative behavior of the normalized pluriclosed flow on OT manifolds is captured by solutions with initial data the model metrics $\gw_{h}^{a,b}$. Straightforward computations show that the normalized pluriclosed flow with this initial data is
\begin{align}\label{e:normalized equation}
 \omega^{a,b}_h(t) = \sum_{i=1}^s \sqrt{-1}((1-e^{-t})\frac{3}{4}+e^{-t}a_i)\frac{1}{(\operatorname{Im}w_i)^2}dw_i\wedge d\bar{w}_i + \sqrt{-1}e^{-t}b_i\operatorname{Im}w_i dz_i\wedge d\bar{z}_i.
\end{align}
Later, we shall denote the time-dependent normalized model metric starting with $\omega^{a,b}_h$ as $\omega^{a,b}_h(t)$. We shall write $\omega_h(t)$ for short if $a_i$, $b_i=1$, for all $1\leq i\leq s$.
Observe that for these model solutions the Chern torsion $T$ is uniformly bounded in time, i.e., $|T(t)|\leq C$. Moreover, it follows that
\begin{align*}
 \frac{\omega_h^{a,b}(t)}{t+1}\to \sum_{i=1}^s \frac{3}{4(\operatorname{Im}w_i)^2}dw_i\wedge d\bar{w}_i,
\end{align*}
which can be considered as a degenerate metric on $X(K, U)$. As explained in \cite{fusi2024pluriclosed}, the blowdown manifolds converge in Gromov-Hausdorff sense to a torus $\mathbb T^s$ with a canonical flat metric $d(K, U)$ depending only on the algebraic field $K$ and the rank $s$ subgroup $U$. We recall the result here, noting that OT manifolds are compact solvmanifolds and the model metrics are left-invariant:
\begin{thm} \label{t:homogOT} (\cite{fusi2024pluriclosed}) Let $\omega_0$ be a left-invariant pluriclosed metric on an OT manifold $M$, then the normalized pluriclosed flow starting with $\omega_0$ converges to $(\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.
\end{thm}

\noindent It is reasonable to conjecture that this behavior holds in the general case:
\begin{conj} \label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\omega_0$, the normalized pluriclosed flow \ref{e:PCF_normalized} with initial metric $\omega_0$ exists on $[0, \infty)$, and converges to $(\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.
\end{conj}

We shall prove the first part of the conjecture in Section \ref{s:long time section}. For the second half of the conjecture, we have the following sufficiency condition.
\begin{prop}\label{p:GH convergence argument}
 Let $\omega(t)$ be the normalized pluriclosed flow solution on the OT manifold $X(K,U)$. Suppose that there exists a constant $C>0$ such that
 \begin{itemize}
 \item $C^{-1}\leq \operatorname{tr}_{\omega(t)}\omega_h(t)\leq C$,
 \item $\lim\limits_{t\to\infty}\operatorname{tr}_{\omega(t)}\omega_h(t) = 1$ for all $x\in X(K,U)$.
 \end{itemize}
 Then, we have
 \begin{align*}
 (X(K,U),\omega(t))\to (\mathbb{T}^s,d(K, U)),
\end{align*}
to the flat metric $d(K,U)$ in the Gromov-Hausdorff sense.
\end{prop}
\begin{proof} We give a brief sketch as the result is already essentially contained in e.g. \cite{fusi2024pluriclosed}, \cite{ZhengOT}. Note that for OT manifolds, there is a canonical fibration map:
 \begin{align*}
 F:X(K,U)\to \mathbb{T}^s
 \end{align*}
 where the fiber is diffeomorphic to $\mathbb{T}^{3s}$. In particular, $E_\mathbb{C}$ will be in the kernel of $dF$. Since the quotient of $\{z\}\times \mathbb{C}^s$ is dense in the $\mathbb{T}^{3s}$ fiber (See Section 2 of \cite{VerbiOT}), the degenerate metric $\sum_{i=1}^s \frac{3}{4(\operatorname{Im}w_i)^2}dw_i\wedge d\bar{w}_i$ will induce a metric on the base $\mathbb{T}^s$. Let $d(K,U)$ be the metric on $\mathbb{T}^s$ induced from $\sum_{i=1}^s \frac{3}{4(\operatorname{Im}w_i)^2}dw_i\wedge d\bar{w}_i$, which is flat.

Secondly, when $s = 1$, then $E_\mathbb{C}$ and $E_\mathbb{H}$ are holomorphic line bundles. Thus, the AM-GM inequality becomes equality in this case, and we can have the following lower bound of $\dot{\phi}$.
\begin{lemma}\label{l:potential derivative lower bound}
 On Inoue surface $S_M$, for the potential $\phi$, there exists a constant $C>0$, such that:
 \begin{align*}
 \dot{\phi}\geq -C.
 \end{align*}
\end{lemma}
\begin{proof}
 Note $s=1$. Choose a large constant $\Lambda>0$, consider the quantity $\dot{\phi} + \phi+\frac{1}{\Lambda}\phi$. Then
 \begin{align*}
 (\frac{\partial}{\partial t} - \Delta)(\dot{\phi}+\phi+\frac{1}{\Lambda}\phi) = 1 +\frac{1}{\Lambda}\dot{\phi}+ \frac{1}{\Lambda}e^{-t}\operatorname{tr}_{g_\mathbb{C}}g^0_\mathbb{C} - \frac{1}{\Lambda}e^{-t}\operatorname{tr}_{g_\mathbb{H}}g^0_\mathbb{H} - (\frac{1}{\Lambda}(1-e^{-t})+1)\frac{3}{4}\operatorname{tr}_{g_\mathbb{H}}h_\mathbb{H}.
 \end{align*}
 Now, for $A>0$, a very large constant, such that $\dot{\phi}+\phi +\frac{1}{\Lambda}\phi < -A$. By \eqref{e:potential eq}, notice that in this case $s=1$, we know:
 \begin{align*}
 \dot{\phi} +\phi = \log\frac{\det g_\mathbb{C}/c_1}{\det e^{-t}h_\mathbb{C}} - \log\frac{\det g_\mathbb{H}}{\det \frac{3}{4}h_\mathbb{H}} = -\log(c_1e^{-t}\operatorname{tr}_{g_\mathbb{C}}h_\mathbb{C}) + \log (\frac{3}{4}\operatorname{tr}_{g_\mathbb{H}}h_\mathbb{H}).
 \end{align*}
 By Corollary \ref{c:Inoue C0}, in particular $|\phi|\leq C_0$. Then, there is a large constant $A' = A- C_0>0$ such that $\dot{\phi}\leq -A'$, and
 \begin{align*}
 \operatorname{tr}_{g_\mathbb{H}}h_\mathbb{H}\leq \frac{4}{3}e^{\frac{C_0}{\Lambda}-A}e^{-t}c_1 \operatorname{tr}_{g_\mathbb{C}}h_\mathbb{C}\leq \frac{4}{3}e^{\frac{C_0}{\Lambda}-A}c_0e^{-t}\operatorname{tr}_{g_\mathbb{C}}g^0_\mathbb{C},
 \end{align*}
 for some constant $c_0$, only depends on the initial metric.
 By choosing $\Lambda$ large, such that $1-\frac{C_0}{\Lambda}>\frac{2}{3}$, fixed, when $A$ is large enough, $c_0e^{\frac{C_0}{\Lambda}-A} \leq \frac{1}{\Lambda}$. Thus, choose an $A = \frac{\Lambda}{2}$ large enough, such that $\dot{\phi}(t)\geq -\frac{1}{2}A$, for $t\in[0,T]$, where $T$ is large enough. Then for the first time $\dot{\phi}+\phi +\frac{1}{\Lambda}\phi = -A$, we have:
 \begin{align*}
 (\frac{\partial}{\partial t}-\Delta)(\dot{\phi} + \phi +\frac{1}{\Lambda}\phi)&\geq (\frac{1}{\Lambda } - \frac{C_1e^{-t}}{\Lambda}-c_0e^{\frac{C_0}{\Lambda}-A})e^{-t}\operatorname{tr}_{g_\mathbb{C}}h_\mathbb{C} + 1+ \frac{1}{\Lambda}\dot{\phi}\\
 &>0.
 \end{align*}
 By the maximum principle, the result follows.
\end{proof}
Combine with the potential derivative upper bound (Lemma \ref{l:potential derivative upper bound}), we have the following metric lower bound.
\begin{thm}\label{t:Inoue metric lower bound}
 On an Inoue surface $S_M$, for generalized K\"ahler metric $\omega_0$, there is a constant $C$, such that the normalized pluriclosed flow solution $\omega(t)$with initial data $\gw_0$ will satisfy
 \begin{align*}
 \omega(t)\geq C\omega_h(t),
 \end{align*}
 where $\omega_h(t)$ is the model flow.
\end{thm}
\begin{proof}
 By Lemma \ref{l:potential derivative upper bound} and Lemma \ref{l:potential derivative lower bound}, we have that:
 \begin{align*}
 -C\leq\dot{\phi} +\phi \leq C
 \end{align*}
 for some constant $C$. Now, from the equation \eqref{e:potential eq} and $s=1$, we have:
 \begin{align*}
 -C\leq \operatorname{tr}_{g_\mathbb{H}}h_\mathbb{H}/\operatorname{tr}_{g_\mathbb{C}}e^{-t}h_\mathbb{C}\leq C.
 \end{align*}
 Note that in the normalized pluriclosed flow model case, \eqref{e:normalized equation}, the $E_\mathbb{C}$ part will shrink at the rate of $e^{-t}$, while the $E_\mathbb{H}$ part will be equivalent to $h_\mathbb{H}$. Thus, by Corollary \ref{c:lower z}, we have the desired lower bound for $E_\mathbb{C}$.
\end{proof}
\begin{proof} [Proof of Theorem \ref{t:mainthm2}]
Combine Lemma \ref{l:C0 bound}, Corollary \ref{c:imporved C0}, Lemma \ref{l:potential derivative upper bound}, Corollary \ref{c:Inoue C0}, Lemma \ref{l:potential derivative lower bound}, and Theorem \ref{t:Inoue metric lower bound}, the results follow.
\end{proof}

\noindent The proof exploits natural class of background metrics arising from the homogeneous structure on OT-manifolds.  In the case of Inoue surfaces these are known as Tricerri metrics \cite{tricerri1982some}.  In the next theorem we give some refined estimates in the special case of generalized K\"ahler-Ricci flow (GKRF) \cite{GKRF}.  Oeljeklaus-Toma manifolds admit natural classes of generalized K\"ahler structures with split tangent bundle, and for such metrics the GKRF reduces to a scalar parabolic flow of Monge-Amp\`ere type \cite{StreetsSTB}.

\begin{thm} \label{t:mainthm2} For generalized K\"ahler-Ricci flow on an Oeljeklaus-Toma manifold
\begin{enumerate}
\item The scalar potential $\phi$ satisfies
\begin{align*}
    - C \leq \phi \leq C e^{-t}(1 + t).
\end{align*}
\item Assuming there exists a Tricerri-type metric in $[\gw_0]$, we have
\begin{align*}
    - C e^{-t}(1 + t) \leq \phi \leq C e^{-t}(1 + t).
\end{align*}
\item On Inoue surfaces of type $S_M$, the estimate of item (2) holds. In addition:
\begin{align*}
    -C\leq \dot{\phi}\leq C.
\end{align*}
\item On Inoue surfaces of type $S_M$, we have:
\begin{align*}
        \omega(t)\geq C\omega_h(t),
\end{align*}
where $\omega_h(t)$ is the model flow with initial data the Tricerri metric $h$.
\end{enumerate}
\end{thm}

The proof of Theorem~\ref{t:mainthm1} is described only at a high level: it “exploits [a] natural class of background metrics arising from the homogeneous structure on OT-manifolds.” In the case of Inoue surfaces, “these are known as Tricerri metrics.”

The proof of To prove the main theorem, is described only at a high level: it “exploits [a] natural class of background metrics arising from the homogeneous structure on OT-manifolds.” In the case of Inoue surfaces, “these are known as Tricerri metrics.”

\label{t:mainthm1} Fix $(M^{2n}, J)$ an OT manifold and $g_0$ a pluriclosed metric on $M$. The solution to pluriclosed flow with initial data $g_0$ exists on $[0, \infty)$.,

Question: Which statement holds for every Oeljeklaus--Toma (OT) complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$ (that is, the associated fundamental $(1,1)$-form $\omega_0$ satisfies $\partial\bar\partial\omega_0=0$)?

The pluriclosed flow with initial metric $g_0$ exists for all forward time; equivalently, its solution is defined on the entire interval $[0,\infty)$.

Choices:
B. The pluriclosed flow with initial metric $g_0$ exists for all forward time and, for every such initial metric, converges as $t\to\infty$ in the Gromov--Hausdorff sense to a flat torus canonically associated to $M$.
C. The pluriclosed flow with initial metric $g_0$ exists on some maximal time interval $[0,T)$ with $T>0$.
D. There exists a time $T=T(M,J)>0$, depending only on the OT complex manifold, such that for every pluriclosed Hermitian metric $g_0$ on $M$, the pluriclosed flow with initial metric $g_0$ exists on $[0,T]$.
E. For every OT complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$, the normalized pluriclosed flow with initial metric $g_0$ exists on $[0,\infty)$ and is generated by one of the homogeneous background metrics arising from the OT-manifold structure.

adds asymptotic Gromov--Hausdorff convergence not asserted in the theorem

dropped the global-in-time conclusion $[0,\infty)$ to mere short-time existence on a maximal interval

replaces infinite-time existence by a uniform manifold-dependent finite existence time

confuses use of homogeneous background metrics in the proof with the actual initial-data/general-flow statement

The stem does not state the global-in-time conclusion outright; it only asks which existence statement is valid. The equivalence defining pluriclosed is background, not a hint toward the correct option.

The item is close to a theorem-recall question: it asks for the correct existence statement attached to a specific geometric setting. However, it is not a pure verbatim restatement because the options vary in maximal-time behavior, quantifiers, and added convergence claims.

Some reasoning is needed to distinguish global existence from merely local existence and from stronger unsupported claims. Still, the question mainly tests recognition of the theorem rather than substantial derivation or synthesis.

The distractors are plausible and mathematically meaningful: finite-time maximal existence, merely local existence, incorrect quantifier dependence, and an overstrong convergence/uniqueness statement. They reflect common failure modes in interpreting PDE existence theorems.

A solid theorem-identification MCQ with good distractors and little answer leakage, but it leans more toward recall of a known result than genuine generative reasoning.


Question: Which statement holds for every Oeljeklaus--Toma (OT) complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$ (that is, the associated fundamental $(1,1)$-form $\omega_0$ satisfies $\partial\bar\partial\omega_0=0$)?

Choices:
B. The pluriclosed flow with initial metric $g_0$ exists for all forward time and, for every such initial metric, converges as $t\to\infty$ in the Gromov--Hausdorff sense to a flat torus canonically associated to $M$.
C. The pluriclosed flow with initial metric $g_0$ exists on some maximal time interval $[0,T)$ with $T>0$.
D. There exists a time $T=T(M,J)>0$, depending only on the OT complex manifold, such that for every pluriclosed Hermitian metric $g_0$ on $M$, the pluriclosed flow with initial metric $g_0$ exists on $[0,T]$.
E. For every OT complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$, the normalized pluriclosed flow with initial metric $g_0$ exists on $[0,\infty)$ and is generated by one of the homogeneous background metrics arising from the OT-manifold structure.

The stem does not explicitly reveal the correct conclusion, and it does not contain strong linguistic cues that single out choice A. The correct answer must be identified from the mathematical content of the options.

The item is close to a direct theorem-recall question: choice A is essentially the theorem-level statement. However, the presence of stronger, weaker, and proof-confusion alternatives makes it more than a pure verbatim restatement.

Some reasoning is needed to distinguish the exact asserted result from stronger claims (B, E) and weaker or altered existence statements (C, D). Still, the question primarily tests recognition of the theorem rather than substantial generative mathematical reasoning.

The distractors are plausible and mathematically targeted: B is an overstrengthening, C is a weaker true statement, D alters the time range in a subtle way, and E confuses proof machinery with the theorem statement. These reflect realistic failure modes.

A solid MCQ with strong distractors and no answer leakage, but it mainly tests theorem recall/precision rather than deeper generative reasoning.


The study of zero sets of smooth functions and ideals in the ring
$\mathcal{E}(\Omega)$ of infinitely differentiable functions on an open set
$\Omega \subset \mathbb{R}^n$ lies at the intersection of real analytic geometry,
singularity theory, and differential topology.
A central tool in this context is the {\L}ojasiewicz inequality, which provides a
quantitative relation between a function and its gradient, and plays a fundamental
role in the analysis of subanalytic sets, stratifications, and resolution of singularities.

A finitely generated ideal $I = (f_1, \ldots, f_k)\mathcal{E}(\Omega)$ is said
to be a {\L}ojasiewicz ideal if there exists a function $g \in I$ that satisfies
a {\L}ojasiewicz inequality with respect to
$
Z(I) := \{ x \in \Omega \mid f_1(x) = f_2(x) = \cdots = f_k(x) = 0 \}.
$
See the definition below.\\
A classical result of Ren\'e Thom \cite{thom1967some} asserts that if $I$ is a finitely generated Lojasiewicz ideal,
then its zero set $Z(I)$ contains an open dense subset of smooth points.\\ In analytic geometry this is expected: analytic ideals naturally enforce
geometric regularity on their zero sets.
However, in the smooth category such regularity properties are more subtle, since
arbitrary smooth functions may exhibit behavior impossible in the analytic setting.

This raises a natural question:
\emph{does the converse hold?}
Does the presence of an open dense set of smooth points in $Z(I)$ imply that
the ideal $I$ is \L{}ojasiewicz?
The purpose of the present work is to examine this question and to identify
mechanisms by which such a converse can fail.\\
To illustrate this phenomenon, we focus on a classical topological example:
the \emph{Hawaiian earring}, defined as the union of the circles
\[
C_n = \left\{ (x_1,x_2) \in \mathbb{R}^2 :
\left(x_1 - \frac{1}{n}\right)^2 + x_2^2 = \frac{1}{n^2} \right\}, \qquad n \ge 1.
\]
This compact set consists of infinitely many circles tangent at the origin and
accumulating there.
It is well known that the Hawaiian earring has highly pathological local
topology at the origin: it is not locally contractible, not semianalytic, and
cannot arise as the zero set of any nontrivial real analytic function.

We consider smooth functions with real values $f \in \mathcal{E}^\infty(\mathbb{R}^2)$ whose zero
set $Z(f)$ is \emph{exactly} the Hawaiian earring.
Such functions exist by Whitney's extension theorem, see theorem 3.1, Ch IV, \cite{Tougeron1972}, yet they exhibit extremely
degenerate behavior at the origin.
Our first main result states that any such function must vanish to infinite order.

This raises a natural question:
\emph{does the converse hold?}
Does the presence of an open dense set of smooth points in $Z(I)$ imply that
the ideal $I$ is \L{}ojasiewicz?
The purpose of the present work is to examine this question and to identify
mechanisms by which such a converse can fail.\\
To illustrate this phenomenon, we focus on a classical topological example:
the \emph{Hawaiian earring}, defined as the union of the circles
\[
C_n = \left\{ (x_1,x_2) \in \mathbb{R}^2 :
\left(x_1 - \frac{1}{n}\right)^2 + x_2^2 = \frac{1}{n^2} \right\}, \qquad n \ge 1.
\]
This compact set consists of infinitely many circles tangent at the origin and
accumulating there.
It is well known that the Hawaiian earring has highly pathological local
topology at the origin: it is not locally contractible, not semianalytic, and
cannot arise as the zero set of any nontrivial real analytic function.

We then formulate a general geometric criterion for flatness, valid for arbitrary
collections of smooth arcs tangent at a point with unbounded curvature or
infinitely varying radius of osculation.
This leads to a \emph{degenerate \L{}ojasiewicz inequality} adapted to situations
where the classical inequality necessarily fails.

\begin{theo}
Let $\Gamma = \bigcup\limits_{n \ge 1} \gamma_n$ be a union of smooth embedded arcs
meeting at a common point $p$ with curvature tending to $\infty$ or oscillating
without bound.
If $f \in \mathcal{E}^\infty(\Omega)$ vanishes on $\Gamma$, then $f$ satisfies a flatness
estimate of the form
\[
|f(x)| \le C_N\, d(x,\Gamma)^N,
\qquad \text{for all } N \ge 1 \text{ and all } x \text{ near } p,
\]
for suitable constants $C_N>0$.
In particular, $f$ is flat at $p$.
\end{theo}

\begin{rem}
In this case, for any system of generators $g_1,\ldots,g_p$ of $I$,
the functions $\sum\limits_{j=1}^p g_j^2$ and $\sum\limits_{j=1}^p |g_j|$ both satisfy
the {\L}ojasiewicz inequality with respect to $Z(I)$.
\end{rem}
It should be noted that the property of an ideal to be {\L}ojasiewicz is a
local one: if an ideal is {\L}ojasiewicz on an open set $U$, then the induced
ideal on any smaller open subset is also {\L}ojasiewicz.\\
Any analytic ideal is {\L}ojasiewicz, as this follows from the fundamental
{\L}ojasiewicz inequality for analytic functions \cite{Lojasiewicz1965}. Moreover, any finitely
generated ideal that is closed in $\mathcal{E}^\infty(\Omega)$ is also a
{\L}ojasiewicz ideal, see Corollary 4.4, Ch V, \cite{Tougeron1972}. Indeed, by Whitney's spectral theorem, a smooth function
belongs to a closed finitely generated ideal $I=(f_1,\ldots,f_k)$ if and only if,
for every point $x\in\Omega$, its Taylor expansion at $x$ lies in the ideal
generated by the Taylor expansions $T_x f_1,\ldots,T_x f_k$ in the formal power
series algebra $\mathbb{R}[[X-x]]$.
However, it should be emphasized that there exist {\L}ojasiewicz ideals
which are not closed, in dimension $n >1$, see Examples 4.8, Ch V, \cite{Tougeron1972}.
\subsection{Smooth points of the locus of zeros}
\begin{defn}[Smooth point]
Let $E \subset \mathbb{R}^n$ be closed set.
A point $x \in E$ is said to be smooth if, in a neighborhood of $x$,
the set $E$ coincides with a $k$-dimensional embedded submanifold of $\mathbb{R}^n$,
where $k$ is its local dimension at $x$ and may vary with $x$.
\end{defn}
\begin{Ex}
 The \emph{Hawaiian earring},
 defined as the union of circles
\[
\bigcup_{n=1}^{\infty} \left\{(x,y)\in \mathbb{R}^2 : \left(x-\frac{1}{n}\right)^2 + y^2 = \frac{1}{n^2} \right\},
\]
is smooth at every point except the origin, where infinitely many circles accumulate.
Hence, the set of smooth points of the Hawaiian earring is dense in $E= \bigcup_{n=1}^{\infty} \left\{(x,y)\in \mathbb{R}^2 : \left(x-\frac{1}{n}\right)^2 + y^2 = \frac{1}{n^2} \right\}$.

\begin{theo}
Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^\infty$ function.
Assume that $f$ vanishes on the Hawaiian earring
\[
\mathcal H=\bigcup_{n\ge1} C_n, \qquad
C_n=\left\{(x,y):\left(x-\tfrac1n\right)^2+y^2=\tfrac1{n^2}\right\}.
\]
Then $f$ is flat at the origin:
\[
\frac{\partial^{\alpha_1 +\alpha_2 f}}{\partial x_1^{\alpha_1} \partial x^{\alpha_2}_2}(0,0)
 = 0
\qquad \text{for all } \alpha =(\alpha_1,\alpha_2) \in\N^2.
\]
\end{theo}

Thus there is no nonzero homogeneous term in the Taylor expansion of $f$ at $(0,0)$,
so all derivatives of $f$ at the origin vanish.
Hence $f$ is flat at $(0,0)$.
\section{Łojasiewicz Inequality and the Hawaiian Earring}
\begin{theo}
Let $f \in C^\infty(\mathbb{R}^2)$ and assume its zero set is the Hawaiian earring
\[
\mathcal H:= \bigcup_{k=1}^{\infty} \left\{ (x,y) \in \mathbb{R}^2 : \left(x - \frac{1}{k}\right)^2 + y^2 = \frac{1}{k^2} \right\}.
\]
Then $f$ is flat at the origin and does \emph{not} satisfy a Łojasiewicz inequality with respect to $H$ at $0$; that is, there exist no constants $C>0$ and $\theta>0$ such that
\[
|f(x)| \ge C \, \mathrm{dist}(x,H)^\theta
\]
for all $x$ in a neighborhood of $0$.
\end{theo}
{\bf{Proof.}}
Since $f \in C^\infty$ vanishes on the Hawaiian earring, which has tangent directions at $0$ dense in the unit circle $S^1$, Theorem 4 implies that $f$ is flat at the origin:
\[
D^\alpha f(0) = 0 \quad \text{for all multi-indices } \alpha.
\]
Let $x \in \mathbb{R}^2$ be close to $0$. Consider the sequence of circles in $H$ with radii $r_k = 1/k$ and centers $c_k = (1/k,0)$.
Choosing $k \sim 1/\|x\|$, the circle $C_k$ closest to $x$ has radius $r_k \sim \|x\|$.
Hence, for points near $0$,
\[
\mathrm{dist}(x,H) \sim \|x\|.
\]
Assume, for contradiction, that there exist constants $C>0$ and $\theta>0$ such that
\[
|f(x)| \ge C \, \mathrm{dist}(x,H)^\theta
\]
for all $x$ near $0$.

Choosing $N > \theta$ gives
\[
|f(x)| \ll \mathrm{dist}(x,H)^\theta
\]
for $x$ sufficiently close to $0$, which contradicts the assumed Łojasiewicz inequality.
Therefore, the ideal generated by
 the function $f$ in the the ring $\mathcal{E}^\infty(\R^2)$ is not a
 Lojasiewicz ideal.
\section{{Jet-Determination of Smooth Functions via Families of Arcs}}
\begin{theo}
Let $\mathcal A = \{\gamma_i\}_{i \in I}$ be a family of smooth arcs
\[
\gamma_i : (-\varepsilon,\varepsilon) \to \mathbb{R}^n, \qquad \gamma_i(0)=0, \quad \gamma_i \not\equiv \text{constant}.
\]
For each arc, denote its $m$-jet at $0$ by
\[
j^m \gamma_i(0) = (\gamma_i'(0), \gamma_i''(0), \dots, \gamma_i^{(m)}(0)).
\]
The following assertions are equivalent:
\begin{enumerate}
\item[(A)] \textbf{Jet–Determination Property:} For every function $f \in C^\infty(\mathbb{R}^n)$,
\[
f(\gamma_i(s)) \equiv 0 \ \text{for all } i \quad \Longrightarrow \quad D^\alpha f(0)=0 \ \text{for all multi-indices } \alpha.
\]
\item[(B)] \textbf{Jet–Nondegeneracy Condition:} For every integer $m \ge 1$, there exists no nonzero homogeneous polynomial $P_m$ of degree $m$ such that
\[
P_m(j^m \gamma_i(0))=0 \qquad \text{for all } i \in I.
\]
\end{enumerate}
\end{theo}
{\bf{Proof.}}
\textbf{(A) $\implies$ (B).} \\Suppose (A) holds and, for contradiction, that (B) fails for some $m$.
Then there exists a nonzero homogeneous polynomial $P_m$ of degree $m$ such that
\[
P_m(j^m \gamma_i(0)) = 0 \quad \text{for all } i.
\]

Then $E_n$ is jet-determining at $0$: if $f \in C^\infty(\mathbb{R}^n)$ vanishes on $E_n$, then $f$ is flat at the origin:
\[
D^\alpha f(0) = 0 \quad \text{for all multi-indices } \alpha.
\]
\end{cor}

The proof relies on purely geometric considerations: any nonzero term of the
Taylor expansion of $f$ at $(0,0)$ would define an algebraic curve of finite
order, and no such curve can contain infinitely many smooth arcs with tangency
and curvature behavior comparable to the circles $C_n$.
Thus the geometry of the Hawaiian earring forces infinite-order vanishing.

We then formulate a general geometric criterion for flatness, valid for arbitrary
collections of smooth arcs tangent at a point with unbounded curvature or
infinitely varying radius of osculation.
This leads to a \emph{degenerate \L{}ojasiewicz inequality} adapted to situations
where the classical inequality necessarily fails.

\begin{theo}
Let $\Gamma = \bigcup\limits_{n \ge 1} \gamma_n$ be a union of smooth embedded arcs
meeting at a common point $p$ with curvature tending to $\infty$ or oscillating
without bound.
If $f \in \mathcal{E}^\infty(\Omega)$ vanishes on $\Gamma$, then $f$ satisfies a flatness
estimate of the form
\[
|f(x)| \le C_N\, d(x,\Gamma)^N,
\qquad \text{for all } N \ge 1 \text{ and all } x \text{ near } p,
\]
for suitable constants $C_N>0$.
In particular, $f$ is flat at $p$.
\end{theo}

This result demonstrates that certain pathological smooth zero sets necessarily
force infinite-order degeneracy, and therefore cannot arise from \L{}ojasiewicz
ideals.
In particular, the ideal generated by a function defining the Hawaiian earring is
never \L{}ojasiewicz, despite the fact that its zero set is smooth away from a
single point.

The proof is described as relying on “purely geometric considerations”: if the Taylor expansion of $f$ at $(0,0)$ had “any nonzero term,” it would “define an algebraic curve of finite order,” but “no such curve can contain infinitely many smooth arcs with tangency and curvature behavior comparable to the circles $C_n$.” Hence, “the geometry of the Hawaiian earring forces infinite-order vanishing,” i.e., flatness at the origin.

The introduction then says it “formulate[s] a general geometric criterion for flatness” for “collections of smooth arcs tangent at a point with unbounded curvature or infinitely varying radius of osculation,” yielding a “degenerate \L{}ojasiewicz inequality” (a flatness estimate $|f(x)|\le C_N d(x,\Gamma)^N$) in settings “where the classical inequality necessarily fails.”

Let $f \in \mathcal{E}^\infty(\mathbb{R}^2)$ be such that $Z(f)$ is the Hawaiian earring. Then $f$ is flat at the origin:
\[
\frac{\partial^{\alpha +\beta}f}{\partial x_1^\alpha \partial x_2^\beta}(0,0)
 = 0
\qquad \text{for all } \alpha,\beta \in\mathbb{N}.
\],

Question:
Let
\[
\mathcal H:= \bigcup_{k=1}^{\infty} \left\{ (x,y) \in \mathbb{R}^2 : \left(x-\frac{1}{k}\right)^2+y^2=\frac{1}{k^2} \right\}
\]
be the Hawaiian earring, and let \(Z(f)=\{(x,y)\in\mathbb R^2: f(x,y)=0\}\) denote the zero set of a smooth function \(f\in \mathcal E^{\infty}(\mathbb R^2)=C^{\infty}(\mathbb R^2)\). Which statement holds for every smooth function \(f\) such that \(Z(f)=\mathcal H\)?

The function \(f\) is flat at the origin; equivalently, for every pair of integers \(\alpha,\beta\ge 0\),
\[
\frac{\partial^{\alpha+\beta}f}{\partial x_1^{\alpha}\partial x_2^{\beta}}(0,0)=0.
\]

Choices:
B.
There exists an integer \(N\ge 1\) such that \(f\) vanishes to order at least \(N\) at the origin; equivalently,
\[
\frac{\partial^{\alpha+\beta}f}{\partial x_1^{\alpha}\partial x_2^{\beta}}(0,0)=0
\qquad\text{for all }\alpha,\beta\in\mathbb N\text{ with }\alpha+\beta\le N,
\]
but not necessarily for all orders.
C.
For every pair of integers \(\alpha,\beta\ge 0\), if \(\alpha+\beta=1\) then
\[
\frac{\partial^{\alpha+\beta}f}{\partial x_1^{\alpha}\partial x_2^{\beta}}(0,0)=0.
\]
D.
There exist constants \(C>0\) and \(\theta>0\) such that, for all \((x,y)\) sufficiently close to \((0,0)\),
\[
|f(x,y)|\ge C\,\mathrm{dist}\bigl((x,y),\mathcal H\bigr)^{\theta}.
\]
E. The function \(f\) is real-analytic in a neighborhood of the origin; equivalently, if its Taylor expansion at \((0,0)\) has any nonzero term, then that finite-order algebraic curve contains the arcs of \(\mathcal H\) near the origin.

infinite-order vanishing replaced by finite-order vanishing

dropped all higher-order derivative vanishing, keeping only first-order consequences

classical Lojasiewicz lower bound in a setting where only degenerate flatness estimates are compatible

finite-order algebraic-curve obstruction turned into analyticity near the origin

The stem does not explicitly state or strongly hint that flatness at the origin is the required conclusion. It only specifies the Hawaiian earring as the zero set and asks for a universal consequence.

The question is close to a theorem-recall item: the correct choice is essentially the main conclusion associated with this zero-set configuration. However, it is not completely tautological because the options include weaker true and plausible false alternatives that must be distinguished.

The item demands substantial reasoning or theorem-level understanding: one must infer infinite-order vanishing from the geometry of the Hawaiian earring and reject tempting alternatives such as finite-order vanishing, mere first-order vanishing, a Łojasiewicz bound, or analyticity.

The distractors are strong and mathematically meaningful. B is a near-miss, C is a weaker true statement, D reflects a common confusion with analytic/geometric inequalities, and E exploits a plausible but false analyticity intuition.

A strong MCQ with little answer leakage and excellent distractors. Its main weakness is that the correct option closely mirrors the underlying theorem, so it leans somewhat toward theorem recognition rather than fully novel inference.


\begin{thm}\label{Pointwise Semi-Classical OT} There exists $C>0$ and $p_1\in \N$ such that for any $p\geq p_1$, any $x\in X$, and any $v\in (A^p)_x$,  one can find a global holomorphic section $a=a_{p,x,v}\in H^0(X,A^p)$ with
\begin{equation}\label{OT bound}
    a(x)=v,\quad |v|_{(h_0^A)^p}\leq \sup_X|a|_{(h_0^A)^p}\leq \left(1+\frac{C}{p}\right)|v|_{(h_0^A)^p},
\end{equation}
where $C=C(n,X,\omega)$ depends only on dimension $n$, manifold $X$, and K\"ahler form $\omega=c_1(A,h_0^A)$.
\end{thm}

\begin{thm}\label{quantization of MA energy} If $L$ is a big and semipositive line bundle on $X$, then for any  $u\in \PSH(X,\theta)\cap L^\infty$, we have
\[
\lim_{k\to\infty}E_k(u)=E_\theta(u).
\]
\end{thm}

\begin{equation}\label{MA envelope}
		P_\theta(f):=usc(\sup\{u\in \PSH(X,\theta):u\leq f\})
	\end{equation}

Given an ample line bundle \(L\) over a complex projective manifold \(X\), endowed with the Kähler form \(\omega\) polarized by \(L\), a central theme in Kähler geometry—first articulated by Yau \cite[p.~139]{Yau}—is the approximation of transcendental geometric objects on \((X,\omega)\) by asymptotic algebraic data coming from the powers \(L^k:=L^{\otimes k}\). This is the philosophy of \emph{Kähler quantization}.

The initial proposal is to approximate the infinite-dimensional space
\[
   \mathcal{H}_\omega := \{\,\phi\in C^\infty(X) : \omega + dd^c \phi > 0\,\}
\]
of Kähler potentials by the finite-dimensional spaces \(\mathcal{H}_k\) of positive Hermitian forms on \(H^0(X,L^k)\). Via the Kodaira embedding, elements of \(\mathcal{H}_k\) correspond to Fubini–Study metrics and thus form a distinguished  subspace of \(\mathcal{H}_\omega\). This was established in the seminal works of Tian \cite{Tian}, Bouche \cite{Bouche}, Ruan \cite{Ruan}, Catlin \cite{Catlin}, Zelditch \cite{Zelditch}, Lu \cite{Lu}, among many others, using asymptotics of the Bergman kernel (cf. Theorem~\ref{on diagonal Bergman kernel}).

Donaldson later proposed that not only the metrics but also the \emph{geometry} of \(\mathcal{H}_\omega\) should be captured by the finite-dimensional geometries of \(\mathcal{H}_k\) \cite[p.~483]{DonaldsonI}. This principle has since been realized in various settings and has had deep impacts on problems concerning canonical metrics and stability; see, for example, \cite{PS,SongZelditchqu,CS,BerndtssonProb,DLR,DonaldsonI,DonaldsonScalII,RTZ,Zhang} among a rapidly expanding body of literature. We refer to \cite{Mama} for a detailed exposition of some classical developments in this direction.

A natural question is whether this quantization picture extends beyond the ample case. This has become increasingly relevant in view of recent progress on canonical metrics in degenerate or singular settings \cite{BBJ,LTW,Li,DZ24,Dervan,Xu,PT25-1,PTT23,PT25-2}, where the positivity of \(L\)  only holds in a weak sense.  The present note fits into this broader direction, establishing a quantization result for Monge--Amp\`ere energy in big and semipositive setting.

\subsection*{Statement of Main Result}
We now describe our results, referring to  Section~\ref{finite energy space and weak geodesic} and Section~\ref{Bergman Measures, Positivity of Direct Images, and Equilibrium} for details. Let $L$ be holomorphic line bundle over a complex projective manifold $X$ of dimension $n$. We assume that $L$ is big and semipositive.  Let $h_0^L$ be a smooth metric on $L$ and let $\theta:=c_1(L,h_0^L)$ denote the Chern curvature of $h_0^L$ so that $\theta$ is a smooth semipositive $(1,1)$-form.

One of the difficulties in the degenerate setting is the lack of smooth potentials. Instead, we work with  the set of $\theta$-plurisubharmonic functions on $X$, denoted by $\PSH(X,\theta)$. This parametrizes singular hermitian metrics on $L$ whose curvature $\theta_u:=\theta+dd^cu$ is positive in the current sense. Following the pioneering work of Boucksom et al. \cite{BEGZ}, for $u\in \PSH(X,\theta)$, one can define the \textit{non-pluripolar Monge--Amp\`ere measure} $\theta_u^n$, which is a Borel measure on $X$ that puts no mass on pluripolar sets. Moreover, we denote by
\[
E_\theta:\PSH(X,\theta)\to [-\infty,\infty)
\]
the \textit{Monge--Amp\`ere energy} or \textit{Aubin--Yau energy}. It is the primitive of the Monge--Amp\`ere operator and plays an important role in the variational approach to canonical K\"ahler metrics.

On the other hand, let $K_X:=\Omega^{n,0}_X$ be the canonical bundle on $X$. For $k\in\mathbb N$, we consider the adjoint line bundle $L^k\otimes K_X$. For global sections $s_1,s_2\in H^0(X,L^k\otimes K_X)$, locally written  as
\[
s_j=\sigma_j\, dz_1\wedge \cdots \wedge dz_n,\quad j=1,2,
\]
where $\sigma_j$ are local holomorphic sections of $L^k$, we set
\[
\langle s_1,s_2\ra_{(h_0^L)^k}:=\langle \sigma_1,\sigma_2\ra_{(h_0^L)^k}\left(\frac{i}{2}\right)^n
dz_1\wedge d\bar{z}_1\wedge \cdots \wedge dz_n\wedge d\bar{z}_n,
\]
yielding  a smooth volume form on $X$. For a function $u:X\to [-\infty,\infty)$, we define the \textit{Hilbert norm}
\begin{equation}\label{Hilbert norm}
		(s_1,s_2)_{Hilb_k(u)}:=\int_X \langle s_1,s_2\ra_{(h_0^L)^k}e^{-ku},\quad k\in \mathbb N,
\end{equation}
for sections $s_1,s_2\in H^0(X,L^k\otimes K_X)$ such that \eqref{Hilbert norm} is finite. We say $u$ is \textit{admissible} if $Hilb_k(u)$ is finite on the whole space $H^0(X,L^k\otimes K_X)$ for any $k\in \mathbb N$. Clearly, any bounded function is admissible. For an admissible function $u$, we define the \textit{quantized Monge--Amp\`ere energy} by
\begin{equation}\label{quantized MA-intro}
    E_k(u):=-\frac{1}{kN_k}\log\left(\frac{\det Hilb_k(u)}{\det Hilb_k(0)}\right),
\end{equation}
which is first introduced by Donaldson \cite{DonaldsonScalII} (see also Berman–Boucksom \cite{BB10}).

Our main result is the  quantization of Monge--Amp\`ere energy in adjoint twisting.

The initial proposal is to approximate the infinite-dimensional space
\[
 \mathcal{H}_\omega := \{\,\phi\in C^\infty(X) : \omega + dd^c \phi > 0\,\}
\]
of Kähler potentials by the finite-dimensional spaces \(\mathcal{H}_k\) of positive Hermitian forms on \(H^0(X,L^k)\). Via the Kodaira embedding, elements of \(\mathcal{H}_k\) correspond to Fubini–Study metrics and thus form a distinguished subspace of \(\mathcal{H}_\omega\). This was established in the seminal works of Tian \cite{Tian}, Bouche \cite{Bouche}, Ruan \cite{Ruan}, Catlin \cite{Catlin}, Zelditch \cite{Zelditch}, Lu \cite{Lu}, among many others, using asymptotics of the Bergman kernel (cf. Theorem~\ref{on diagonal Bergman kernel}).

A natural question is whether this quantization picture extends beyond the ample case. This has become increasingly relevant in view of recent progress on canonical metrics in degenerate or singular settings \cite{BBJ,LTW,Li,DZ24,Dervan,Xu,PT25-1,PTT23,PT25-2}, where the positivity of \(L\) only holds in a weak sense. The present note fits into this broader direction, establishing a quantization result for Monge--Amp\`ere energy in big and semipositive setting.

\subsection*{Statement of Main Result}
We now describe our results, referring to Section~\ref{finite energy space and weak geodesic} and Section~\ref{Bergman Measures, Positivity of Direct Images, and Equilibrium} for details. Let $L$ be holomorphic line bundle over a complex projective manifold $X$ of dimension $n$. We assume that $L$ is big and semipositive. Let $h_0^L$ be a smooth metric on $L$ and let $\theta:=c_1(L,h_0^L)$ denote the Chern curvature of $h_0^L$ so that $\theta$ is a smooth semipositive $(1,1)$-form.

One of the difficulties in the degenerate setting is the lack of smooth potentials. Instead, we work with the set of $\theta$-plurisubharmonic functions on $X$, denoted by $\PSH(X,\theta)$. This parametrizes singular hermitian metrics on $L$ whose curvature $\theta_u:=\theta+dd^cu$ is positive in the current sense. Following the pioneering work of Boucksom et al. \cite{BEGZ}, for $u\in \PSH(X,\theta)$, one can define the \textit{non-pluripolar Monge--Amp\`ere measure} $\theta_u^n$, which is a Borel measure on $X$ that puts no mass on pluripolar sets. Moreover, we denote by
\[
E_\theta:\PSH(X,\theta)\to [-\infty,\infty)
\]
the \textit{Monge--Amp\`ere energy} or \textit{Aubin--Yau energy}. It is the primitive of the Monge--Amp\`ere operator and plays an important role in the variational approach to canonical K\"ahler metrics.

On the other hand, let $K_X:=\Omega^{n,0}_X$ be the canonical bundle on $X$. For $k\in\mathbb N$, we consider the adjoint line bundle $L^k\otimes K_X$. For global sections $s_1,s_2\in H^0(X,L^k\otimes K_X)$, locally written as
\[
s_j=\sigma_j\, dz_1\wedge \cdots \wedge dz_n,\quad j=1,2,
\]
where $\sigma_j$ are local holomorphic sections of $L^k$, we set
\[
\langle s_1,s_2\ra_{(h_0^L)^k}:=\langle \sigma_1,\sigma_2\ra_{(h_0^L)^k}\left(\frac{i}{2}\right)^n
dz_1\wedge d\bar{z}_1\wedge \cdots \wedge dz_n\wedge d\bar{z}_n,
\]
yielding a smooth volume form on $X$. For a function $u:X\to [-\infty,\infty)$, we define the \textit{Hilbert norm}
\begin{equation}\label{Hilbert norm}
 (s_1,s_2)_{Hilb_k(u)}:=\int_X \langle s_1,s_2\ra_{(h_0^L)^k}e^{-ku},\quad k\in \mathbb N,
\end{equation}
for sections $s_1,s_2\in H^0(X,L^k\otimes K_X)$ such that \eqref{Hilbert norm} is finite. We say $u$ is \textit{admissible} if $Hilb_k(u)$ is finite on the whole space $H^0(X,L^k\otimes K_X)$ for any $k\in \mathbb N$. Clearly, any bounded function is admissible. For an admissible function $u$, we define the \textit{quantized Monge--Amp\`ere energy} by
\begin{equation}\label{quantized MA-intro}
 E_k(u):=-\frac{1}{kN_k}\log\left(\frac{\det Hilb_k(u)}{\det Hilb_k(0)}\right),
\end{equation}
which is first introduced by Donaldson \cite{DonaldsonScalII} (see also Berman–Boucksom \cite{BB10}).

Our main result is the quantization of Monge--Amp\`ere energy in adjoint twisting.

\begin{equation}
    \frac{d}{dt}\Big|_{t=0}E_\theta(P_\theta(u+tf))=\frac{1}{\vol(\{\theta\})}\int_Xf\theta_u^n.
\end{equation}
\end{lem}
\section{Bergman Measures, Positivity of Direct Images, and Equilibrium}\label{Bergman Measures, Positivity of Direct Images, and Equilibrium}
\subsection*{Bergman Measure and Quantized Monge--Amp\`ere Energy}
Let $V$ be a finite-dimensional complex vector space of dimension $N$. We denote $\H_V$ by the space of positive-definite hermitian inner products on $V$. For $H,H'\in \H_V$, there exists an $H$-hermitian endomorphism $A\in \End(V)$ such that
\begin{equation}\label{transfer map}
(s_1,s_2)_{H'}=(e^As_1,s_2)_H,\quad s_1,s_2\in V.
\end{equation}

\begin{equation}\label{Hilbert norm}
		(s_1,s_2)_{Hilb_k(u)}:=\int_X \la s_1,s_2\ra_{(h_0^L)^k}e^{-ku},\quad k\in \N,
\end{equation}

\begin{equation}\label{volume of a line bundle}
		\vol(\{L\})=\limsup_{k\to\infty}\frac{\dim_\C H^0(X,L^k)}{k^n/n!},
	\end{equation}
	which we  denote by $\vol(L)$. Hence, $L$ is big iff the right-hand side of \eqref{volume of a line bundle} is positive and furthermore $\limsup$ in \eqref{volume of a line bundle} is actually a limit by a Theorem of Fujita \cite{Fujita} (see also  \cite[\S 2.2C]{LazarsfeldI}).

\subsection*{Finite Energy Space and Weak Geodesics}\label{finite energy space and weak geodesic}
	Let $\theta$ be a closed, smooth $(1,1)$-form so that $\{\theta\}$ is big. If $u\in\PSH(X,\theta)$ has minimal singularity type, then we define \textit{Monge--Amp\`ere energy} by
	\begin{equation}\label{MA energy}
	E_\theta(u):=\frac{1}{(n+1)\vol(\{\theta\})}\sum_{j=0}^n\int_X (u-V_\theta) \theta_u^j\wedge \theta^{n-j}_{V_\theta}
	\end{equation}

\begin{equation}\label{on diagonal Bergman kernel}
K_p(x):=K_p(x,x)=\sum_{j=1}^{d_p}|S_j(x)|^2_{(h_0^A)^p}.
\end{equation}

Rescaling $w=\sqrt{k}z$ on the standard shrinking polydisc gives the uniform approximation
\[
k\Phi_\epsilon(w/\sqrt{k})
=\Phi_\epsilon^0(w)+\rho_k+\epsilon\widetilde\rho_k,
\qquad
\rho_k,\widetilde\rho_k\to0.
\]
Repeating the submean-value argument used for Theorem~\ref{Berman's local Morse inequality} then yields
\[
\frac{n!|s|^2(x)}{k^n\|s\|^2}
\le
e^{\rho_k+\epsilon\widetilde\rho_k}
\frac{n!dV_n}{(2\pi)^n\prod_{j=1}^n\int_0^{\log k} e^{-(\lambda_j+\epsilon)r^2}r\,dr}.
\]
Since the quadratic model satisfies $\lambda_j+\epsilon\ge\epsilon>0$, the Gaussian integrals in the denominator behave exactly as in the proof of Theorem~\ref{Berman's local Morse inequality}, with $\lambda_j$ replaced by $\lambda_j+\epsilon$.
Using the identity
\[
(\theta+\epsilon\omega)^n
= \frac{n!}{\pi^n}\!\left(\prod_{j=1}^n (\lambda_j+\epsilon)\right)dV_n,
\]
the same computation as in the untwisted case yields
\[
\frac{n!|s|^2(x)}{k^n\|s\|^2}
\;\le\;
e^{\rho_k+\epsilon\widetilde\rho_k}
\Biggl(\prod_{j=1}^n
 \frac{1}{1-e^{-(\lambda_j+\epsilon)(\log k)^2}}\Biggr)
(\theta+\epsilon\omega)^n(x)\leq {(1-e^{-\epsilon(\log k)^2})^n}(\theta+\epsilon\omega)^n(x),
\]
since $\lambda_j(x)+\epsilon\ge\epsilon$ for all $j$ and $x\in X$. \eqref{epsilon twisted local holomorphic morse} then follows from extremal characterization \eqref{extremal for twisted Bergman measure}.
\end{proof}
\section{Proof of Theorem \ref{quantization of MA energy} and Corollary \ref{convergence of Bergman measures}}
Recall that $X$ is a compact projective manifold of dimension $n$, $L$ be a big and semipositive line bundle over $X$ with fixed smooth metric $h_0^L$ on $L$ with Chern curvature form $\theta:=c_1(L,h^L_0)\geq 0$. Let us first deduce Corollary \ref{convergence of Bergman measures} from Theorem \ref{quantization of MA energy}.
\begin{proof}[Proof of Corollary \ref{convergence of Bergman measures}] For $f\in C^0(X)$ and $t\in \R$, $u+tf$ is bounded and thus is admissible. We set
\[
f_k(t):=E_k(u+tf),\quad g(t):=E_\theta(P_\theta(u+tf)).
\]
Notice that $u+tf\geq P_\theta(u+tf)\geq m:=\inf_X(u+tf)>-\infty$ by definition of envelope and thus, $P_\theta(u+tf)\in \PSH(X,\theta)\cap L^\infty$. By Theorem \ref{quantization of MA energy}, we obtain that for any $t\in \R$,
\[
\liminf_{k\to\infty}f_k(t)\geq \lim E_k(P_\theta(u+tf))=g(t),\quad \lim_{k\to\infty}f_k(0)=\lim_{k\to\infty}E_k(u)=E_\theta(u)=g(0).
\]
By Lemma \ref{variational formula for QMA along affine path}, $f_k(t)$ is concave and is differentiable at $t=0$:
\[
f_k'(0)=\int_X f\beta_{L^k,u}.
\]
On the other hand, by Lemma \ref{variation formula of MA energy}, $g(t)$ is differentiable at $t=0$ and
\[
g'(0)=\int_Xf\frac{\theta_u^n}{\vol(\{\theta\})}.
\]
The result follows from the elementary lemma proved in \cite[Lemma 3.1]{BBWN}.
\end{proof}
\begin{rmk}
 In fact, the arguments in Theorem \ref{upper bound estimate} below shows that $\lim_{k\to\infty}f_k(t)=g(t)$.
\end{rmk}

We now begin with the upper bound in Theorem \ref{quantization of MA energy} which holds for any finite-energy potentials.
\begin{thm}\label{upper bound estimate} For $u\in \E^1(X,\theta)$, $\limsup_{k\to\infty}E_k(u)\leq E_\theta(u)$.
\end{thm}
\begin{proof} The proof goes exactly as in \cite[Theorem 3.5]{BF14} and \cite[Theorem A]{BB10}. Since $u$ is upper semi-continuous, we can find a sequence $\phi_j\in C^\infty(X)$ such that $\phi_j\downarrow u$. Clearly, $\phi_j$ is admissible and $E_k(u)\leq E_k(\phi_j)$ by Lemma \ref{properties of Hilb and QMA} (ii). By Corollary \ref{Quantization of MOnge-Ampere energy for smooth functions},
\[
\limsup_{k\to\infty}E_k(u)\leq \lim_{k\to\infty}E_k(\phi_j)=E_\theta(P_\theta(\phi_j)),\quad \forall j\in \N.
\]
On the other hand, by definition of envelope, $u\leq P_\theta(\phi_j)\leq \phi_j$ and thus $P_\theta(\phi_j)\downarrow u$. Then
\[
\lim_{k\to\infty}E_k(u)\leq\lim_{j\to\infty}E_\theta(P_\theta(\phi_j))=E_\theta(u),
\]
by continuity of Monge--Amp\`ere energy along decreasing sequences.
\end{proof}
\begin{rmk} The proof actually shows that Theorem \ref{upper bound estimate} holds for any big and nef line bundle $L$.
\end{rmk}
\begin{proof}[Proof of Theorem \ref{quantization of MA energy}] It remains to prove that for any $u\in \PSH(X,\theta)\cap L^\infty$, we have
\begin{equation}\label{lower bound on energy}
 \liminf_{{k\to\infty}}E_k(u)\geq E_\theta(u).
\end{equation}
Following \cite{BF14}, we consider the following functional on $\E^1(X,\theta)$:
\[
\mathcal{F}_k(u):=E_k(u)-E_\theta(u),\quad u\in \E^1(X,\theta).
\]
By Lemma \ref{Properties on Energy and E1} (iii) and Lemma \ref{properties of Hilb and QMA} (ii), $\F_k$ is continuous along decreasing sequences in $\E^1(X,\theta)$. Also, by definition \eqref{MA energy} and Lemma \ref{properties of Hilb and QMA} (iii), $\F_k(u+C)=\F_k(u)$ for $C\in \R$. For fixed $u\in \PSH(X,\theta)\cap L^\infty$, we may assume that $u\leq 0$ by addition of a constant. By Lemma \ref{minimal singular/finite energy geodesic} (iii), $\dot{u}_0\in L^\infty(X)$.

Now, for fixed $\epsilon\in \Q^+$ and an positive line bundle $(A,h_0^A)$ with curvature $\omega$, we set $\omega_\epsilon:=\theta+\epsilon\omega$. By Theorem \ref{Bergman kernel comparison} and \eqref{epsilon twisted local holomorphic morse},
\[
\frac{1}{N_k}B_{L^k,0}\leq N_k^{-1}\left(1+\frac
{C}{\epsilon k}\right)^2B_{k,\epsilon}\leq \frac{k^n/n!}{N_k}\frac{e^{\rho_k+\epsilon\td{\rho}_k}}{(1-e^{-\epsilon(\log k)^2})^n}\omega_\epsilon^n
\]
As $\dot{u}_0\leq 0$, we obtain
\begin{equation}\label{key estimate}
\F_k(u)\geq \int_X\dot{u}_0\left(\left(1+\frac{C}{\epsilon k}\right)^2\frac{k^n/n!}{N_k}\frac{e^{\rho_k+\epsilon\td{\rho}_k}}{(1-e^{-\epsilon(\log k)^2})^n}\omega_\epsilon^n-\frac{1}{\vol L}\theta^n\right).
\end{equation}
Taking $k\to \infty$, we see that for fixed $\epsilon\in \Q^+$, since $\rho_k,\td{\rho}_k\to 0$ and from \eqref{volume of a line bundle} (since $L$ is big):
\[
1+\frac{C}{\epsilon k}\to 1,\quad \frac{k^n/n!}{N_k}\to \frac{1}{\vol L},\quad\frac{e^{\rho_k+\epsilon\td{\rho}_k}}{(1-e^{-\epsilon(\log k)^2})^n}\to 1.
\]
As a result, we obtain for fixed $\epsilon\in \Q^+$,
\[
\liminf_{k\to\infty}\F_k(u)\geq \frac{1}{\vol L}\int_X \dot{u}_0\left(\omega_\epsilon^n-\theta^n\right)=\frac{1}{\vol L}\sum_{j=1}^n\int_X\dot{u}_0\theta^{n-j}\wedge (\epsilon \omega)^j.
\]
Taking $\epsilon\to 0+$, we obtain the result.
\end{proof}\begin{rmk}\label{pitfall for finite energy potentials}
The argument proving \eqref{lower bound on energy} fails for general
\(u\in \mathcal{E}^1(X,\theta)\), since one cannot in general guarantee that the
initial tangent \(\dot u_0\) of the weak geodesic \(t\mapsto u_t\) in
\(\PSH(X,\theta)\) is integrable with respect to \(\omega_\epsilon^n\) in
\eqref{key estimate}, even for arbitrarily small \(\epsilon>0\). One obstruction comes from a counterexample of Di Nezza
\cite[Example 4.5]{DN15}, which shows that for any \(\epsilon>0\),
\[
\mathcal{E}^1(X,\theta)\not\subset \mathcal{E}^1(X,\omega_\epsilon),
\]
Indeed, let \(u\in \mathcal{E}^1(X,\theta)\setminus
\mathcal{E}^1(X,\omega_\epsilon)\) with \(u\le0\). From
\(\PSH(X,\theta)\subset \PSH(X,\omega_\epsilon)\) and envelope definition of the weak geodesic, the weak geodesic
\(v_t\) joining \(0\) and \(u\) in \(\PSH(X,\omega_\epsilon)\) satisfies
\[
v_t \le u_t,\qquad t\in[0,1],
\]
by the envelope construction. Since \(u\notin \mathcal{E}^1(X,\omega_\epsilon)\), one can shows that
\[
\int_X \dot v_0\,\omega_\epsilon^n = -\infty.
\]
As \(u_t\) and \(v_t\) agree at the endpoints,
we get $
\dot u_0 \le \dot v_0 \le 0$. This implies that $\dot{u}_0\notin L^1(X,\omega_\epsilon^n)$ since
\[
\int_X \dot u_0\,\omega_\epsilon^n
 \;\le\; \int_X \dot v_0\,\omega_\epsilon^n
 \;=\; -\infty,
\]
Thus, a necessary condition for the integrability of \(\dot u_0\) is that
\(u\in \mathcal{E}^1(X,\omega_\epsilon)\). To author's current knowledge, we do not aware of any general sufficient condition.
\end{rmk}
\printbibliography

A result such as Theorem~\ref{quantization of MA energy} was raised as an open problem in Berman–Freixas i Montplet \cite[Remark~3.6]{BF14}, and our theorem provides a partial answer in the big and semipositive setting.
When $L$ is ample and no adjoint twisting is present, Donaldson \cite{DonaldsonScalII} first established the convergence for smooth Kähler potentials; this was later extended to continuous potentials by Berman–Boucksom \cite[Theorem~A]{BB10} and to all finite-energy potentials by Darvas–Lu–Rubinstein \cite{DLR}.
In the adjoint setting, Berman–Freixas i Montplet \cite[Theorem~3.5]{BF14} proved the quantization of energy for finite-energy potentials, and subsequently Darvas–Xia \cite{DX22} obtained the result for arbitrary twisting.
In the big, untwisted case, Berman–Boucksom \cite[Theorem~A]{BB10} showed convergence for all continuous metrics.
More recently, Darvas–Xia \cite{DX24} proved a “partial quantization’’ theorem for continuous metrics under arbitrary twisting on a pseudoeffective line bundle.

Theorem \ref{quantization of MA energy} has an immediate application to quantization of non-pluripolar measure. For an admissible $u$, we fix a basis $\{S_1,\dots,S_{N_k}\}$ of $H^0(X,L^k\otimes K_X)$ with $N_k=\dim_\mathbb C H^0(X,L^k\otimes K_X)$. Then we define \textit{Bergman measure} and its normalization by
	\begin{equation}\label{Bergman measure}
	B_{L^k,u}(x)=\sum_{j=1}^{N_k}|S_j(x)|^2_{(h_u^L)^k}=\sum_{j=1}^{N_k}|S_j(x)|^2_{(h_0^L)^k}e^{-ku(x)},\quad \beta_{L^k,u}=\frac{1}{N_k}B_{L^k,u}.
	\end{equation}
Coupling with  a standard trick in Berman--Boucksom--Witt Nystr\"om \cite{BBWN}, Theorem \ref{quantization of MA energy} yields the following weak convergence of normalized Bergman measures to non-pluripolar measures
\begin{cor}\label{convergence of Bergman measures}  Let $u\in \PSH(X,\theta)\cap L^\infty(X)$. Then $\beta_{L^k,u}$ converges weakly to non-pluripolar measure $\frac{1}{\vol L}\theta_{u}^n$ as $k\to\infty$, where $\vol L$ is the volume of the line bundle $L$, see \eqref{volume of a line bundle}.
\end{cor}
Our approach refines the arguments in \cite{BF14} as follows. Fix an ample line bundle $A$ on $X$ and $\varepsilon\in \mathbb Q^+$. We prove a comparison Theorem for Bergman measure to $L$ and to small ample twist $L\otimes A^\varepsilon$ with almost optimal asymptotic bound (Theorem \ref{Bergman kernel comparison}). This allows us to apply Berman's local Morse inequality \cite[Proposition 2.5]{BF14} to the twisted bundle with explicit dependence on $\varepsilon$, see  \eqref{epsilon twisted local holomorphic morse}. Combining this with the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures, and the convexity of the quantized Monge–Ampère energy along weak geodesics,  we can carry out the proof of Theorem \ref{quantization of MA energy} for bounded potentials following standard argument as in \cite[Theorem A]{BB10} and \cite[Theorem 3.5]{BF14}. However, as we will explain in Remark \ref{pitfall for finite energy potentials}, our method does not presently extend to arbitrary finite-energy potentials, due to integrability issues for initial tangent vectors of weak geodesics.

As this work was being finalized, Siarhei Finski kindly informed us of his related work \cite{FinskiBig}, in which he quantizes the Darvas–Mabuchi $d_p$ distances on big line bundles and for potentials of the type $P_\theta(f)$ (defined below \ref{MA envelope}), where $f$ is continuous. In particular, for $p=1$ and $\theta$ semipositive, his result implies the version of Theorem~\ref{quantization of MA energy} for such potentials,  a subset of the class $\PSH(X,\theta)\cap L^\infty$. Our setting and approach differ from his in two main aspects. His quantization is formulated using the untwisted section spaces $H^0(X,L^k)$, whereas we work in the adjoint setting. This allows us to apply the Berndtsson–P\u{a}un positivity theorem directly. Additionally, he reduces the problem to the ample case via passing to certain birational models over $X$, while our reduction is by twisting $L^k\otimes K_X$ by an ample line bundle on $X$ as explained above.

\subsection*{Organization}
We briefly describe the structure of the paper.

In Section~2 we collect the necessary background from pluripotential theory,  following \cite{BEGZ,BBGZ,DDL18,DDL18L1}. We also recall the basic variational formulas for \(E_\theta\).

Section~3 is devoted to Bergman measures and quantized Monge--Amp\`ere energies in the adjoint setting. We prove basic properties of $B_{L^k,u}$ and $E_k(u)$ and discuss their variational formula. Next, we recall the positivity of direct images à la Berndtsson–P\u{a}un \cite{BerndtssonPositivity1,BerndtssonPositivity2,BP08,PT18} and the relation with convexity of $E_k$ along weak geodesics. Finally, we recall Berman’s local holomorphic Morse inequalities and the convergence of Bergman measures at equilibrium, adapting the arguments of \cite{Ber09,BB10} to our adjoint framework.

Section 4 is the new input in our argument.  We prove the the comparison Theorem  (Theorem \ref{Bergman kernel comparison}).    This follows from choosing peak sections with asymptotically optimal $L^\infty$-control from a pointwise semi-classical Ohsawa--Takegoshi type extension theorem for powers of an ample line bundle (Theorem \ref{Pointwise Semi-Classical OT}). We also discuss the relation of Theorem \ref{Pointwise Semi-Classical OT} with recent work of Finski \cite[Theorem 1.10]{Finski1}. We also present a  local Morse inequality in ample twisting setting.

Finally, in Section~5 we combine these ingredients to prove Theorems~\ref{quantization of MA energy} and Corollary  \ref{convergence of Bergman measures} and conclude with obstacles to aribtrary finite-energy potentials in Remark \ref{pitfall for finite energy potentials}.
\subsection*{Acknowledgment}  This work was supported by NSF Grant DMS-2405274 and the Hauptman Fellowship. The author is deeply grateful to his advisor, Tamás Darvas, for his constant support and guidance. The author also wishes to thank Siarhei Finski for his enlightening lecture and discussions at the summer school at the Rényi Institute, Budapest, and for kindly sharing his preprint \cite{FinskiBig} along with related insights.

Fix an ample line bundle $A$ on $X$ and $\varepsilon\in\mathbb Q^+$. The approach is to (i) prove a comparison theorem for Bergman measures for $L$ versus the small ample twist $L\otimes A^\varepsilon$ with an almost optimal asymptotic bound (Theorem~\ref{Bergman kernel comparison}); (ii) use this to apply Berman's local Morse inequality to the twisted bundle with explicit dependence on $\varepsilon$ (see \eqref{epsilon twisted local holomorphic morse}); (iii) combine this with “the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures,” together with “the convexity of the quantized Monge–Ampère energy along weak geodesics,” to “carry out the proof of Theorem~\ref{quantization of MA energy} for bounded potentials following standard argument as in \cite[Theorem A]{BB10} and \cite[Theorem 3.5]{BF14}.” The method is noted not to extend to arbitrary finite-energy potentials because of “integrability issues for initial tangent vectors of weak geodesics” (Remark~\ref{pitfall for finite energy potentials}).

Fix an ample line bundle $A$ on $X$ and $\varepsilon\in\mathbb Q^+$. The approach is to (i) prove a comparison theorem for Bergman measures for $L$ versus the small ample twist $L\otimes A^\varepsilon$ with an almost optimal asymptotic bound. We first prove the following theorem.

\begin{thm}\label{Bergman kernel comparison} For $\epsilon\in \Q^+$ and admissible $u$, and for sufficiently divisible $k\gg 1$, we have
\[
B_{L^k,u}\leq \left(1+\frac{C}{\epsilon k}\right)^2B_{(L\otimes A^{\epsilon})^k,u},
\]
where $C=C(n,X,\omega)$ depends only on dimension $n$, manifold $X$, and K\"ahler form $\omega=c_1(A,h_0^A)$.
\end{thm}

(ii) use this to apply Berman's local Morse inequality to the twisted bundle with explicit dependence on $\varepsilon$, namely
\begin{equation}\label{epsilon twisted local holomorphic morse}
    \frac{n!}{k^n}B_{k,\epsilon}\leq \frac{e^{\rho_k+\epsilon\td{\rho}_k}}{(1-e^{-\epsilon(\log k)^2})^n}(\theta+\epsilon\omega)^n,
\end{equation}

(iii) combine this with “the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures,” together with “the convexity of the quantized Monge–Ampère energy along weak geodesics,” to carry out the proof of the main theorem for bounded potentials following the standard argument as in Berman–Boucksom, Theorem A in \cite[Theorem A]{BB10} and Berman–Fujita, Theorem 3.5 in \cite[Theorem 3.5]{BF14}.

The method is noted not to extend to arbitrary finite-energy potentials because of “integrability issues for initial tangent vectors of weak geodesics.” More precisely:

\begin{rmk}\label{pitfall for finite energy potentials}
The argument proving \eqref{lower bound on energy} fails for general
\(u\in \mathcal{E}^1(X,\theta)\), since one cannot in general guarantee that the
initial tangent \(\dot u_0\) of the weak geodesic \(t\mapsto u_t\) in
\(\PSH(X,\theta)\) is integrable with respect to \(\omega_\epsilon^n\) in
\eqref{key estimate}, even for arbitrarily small \(\epsilon>0\). One obstruction comes from a counterexample of Di Nezza
\cite[Example 4.5]{DN15}, which shows that for any \(\epsilon>0\),
\[
\mathcal{E}^1(X,\theta)\not\subset \mathcal{E}^1(X,\omega_\epsilon),
\]
Indeed, let \(u\in \mathcal{E}^1(X,\theta)\setminus
\mathcal{E}^1(X,\omega_\epsilon)\) with \(u\le0\). From
\(\PSH(X,\theta)\subset \PSH(X,\omega_\epsilon)\) and envelope definition of the weak geodesic, the weak geodesic
\(v_t\) joining \(0\) and \(u\) in \(\PSH(X,\omega_\epsilon)\) satisfies
\[
v_t \le u_t,\qquad t\in[0,1],
\]
by the envelope construction. Since \(u\notin \mathcal{E}^1(X,\omega_\epsilon)\), one can  shows that
\[
\int_X \dot v_0\,\omega_\epsilon^n = -\infty.
\]
As \(u_t\) and \(v_t\) agree at the endpoints,
we get $
\dot u_0 \le \dot v_0 \le 0$. This implies that $\dot{u}_0\notin L^1(X,\omega_\epsilon^n)$ since
\[
\int_X \dot u_0\,\omega_\epsilon^n
   \;\le\; \int_X \dot v_0\,\omega_\epsilon^n
   \;=\; -\infty,
\]
Thus, a necessary condition for the integrability of \(\dot u_0\) is that
\(u\in \mathcal{E}^1(X,\omega_\epsilon)\). To author's current knowledge, we do not aware of any  general sufficient condition.
\end{rmk}

\label{quantization of MA energy} If $L$ is a big and semipositive line bundle on $X$, then for any  $u\in \PSH(X,\theta)\cap L^\infty$, we have
\[
\lim_{k\to\infty}E_k(u)=E_\theta(u).
\],

Question:
Let $X$ be a complex projective manifold, let $L$ be a big and semipositive holomorphic line bundle on $X$, and choose a smooth Hermitian metric $h_0^L$ on $L$ with semipositive Chern curvature $\theta:=c_1(L,h_0^L)$. For $u\in \PSH(X,\theta)\cap L^\infty(X)$, where $\PSH(X,\theta)$ denotes the $\theta$-plurisubharmonic functions (so $\theta+dd^c u\ge 0$ in the sense of currents), define for each $k\in\mathbb N$ the Hilbert norm on $H^0(X,L^k\otimes K_X)$ by
\[
(s_1,s_2)_{\mathrm{Hilb}_k(u)}:=\int_X \langle s_1,s_2\rangle_{(h_0^L)^k}e^{-ku},
\]
and the quantized Monge--Amp\`ere energy by
\[
E_k(u):=-\frac{1}{kN_k}\log\left(\frac{\det \mathrm{Hilb}_k(u)}{\det \mathrm{Hilb}_k(0)}\right),
\]
where $N_k=\dim_\mathbb C H^0(X,L^k\otimes K_X)$. Let $E_\theta(u)$ denote the Monge--Amp\`ere (Aubin--Yau) energy on $\PSH(X,\theta)$, i.e. the primitive of the non-pluripolar Monge--Amp\`ere operator. Under these assumptions, which statement about $E_k(u)$ holds as $k\to\infty$?

One has
\[
\lim_{k\to\infty} E_k(u)=E_\theta(u).
\]

Choices:
B.
One has
\[
\limsup_{k\to\infty} E_k(u)\le E_\theta(u),
\]
and in fact equality holds only when the semipositive form \(\theta\) is K\"ahler.
C.
One has
\[
\limsup_{k\to\infty} E_k(u)\le E_\theta(u).
\]
D.
For every \(u\in \PSH(X,\theta)\cap L^\infty(X)\), there exists a subsequence \(k_j\to\infty\) such that
\[
\lim_{j\to\infty} E_{k_j}(u)=E_\theta(u),
\]
and the convergence is uniform in \(u\) over bounded subsets of \(\PSH(X,\theta)\cap L^\infty(X)\).
\]
E.
If \(L\) is big and semipositive, then for any \(u\in \PSH(X,\theta)\cap \mathcal E^1(X,\theta)\), one has
\[
\lim_{k\to\infty} E_k(u)=E_\theta(u).
\]

full limit equality weakened to upper-bound asymptotics

pointwise convergence upgraded to uniform-in-u subsequential convergence

boundedness hypothesis replaced by finite-energy class

The stem gives definitions and setup but does not reveal the asymptotic conclusion. The correct convergence statement is not stated or strongly hinted at.

The correct option is essentially the standard theorem statement in this setting, so the item is close to theorem recall. However, it is not a pure restatement because the alternatives include stronger, weaker, and misdirected variants.

Some reasoning is needed to distinguish the exact theorem from nearby false or weaker claims, especially regarding boundedness, identification of the limit, and uniform rates. Still, the item primarily tests recognition of the precise statement rather than deeper derivation.

The distractors are mathematically plausible and target common errors: overgeneralizing the class of potentials, settling for a weaker convergence claim, asserting an unjustified uniform O(1/k) rate, or confusing the limiting energy functional.

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