\label{thm: A}
Let $W$ be a complement-exact subdomain in a convex symplectic domain $V$. Assume that $c_1(V)|_{\pi_2(V)} = 0$ and $\pi_1(\partial W) = 0$. There exists a $\Z$-graded algebra homomorphism
$$
\Phi: \SH^*(V; \Lda_V) \rightarrow \SH^*(W; \Lda_W)
$$
under a canonical identification $\Lda_V = \Lda_W$. Moreover, it fits the following commutative diagram with quantum cohomology rings.
\[\begin{tikzcd}
        \SH^*(V) \arrow{r}{\Phi}  & \SH^*(W)  \\
        \QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*}& \QH^*(W) \arrow[swap]{u}{c^*}
    \end{tikzcd}
\]
where the vertical maps are the canonical $c^*$-map.

\begin{example}\label{ex: neglinebdl2}
Following Example \ref{ex: neglinebdl}, the total space $E$ can be seen as a (completed) convex symplectic manifold, and the total space of a disk bundle $\pi: D(E) \rightarrow B$ now serves as a complement-exact subdomain. Assume that
\begin{itemize}
\item $[\ow] = c_1(B)$ and $[\ow]$ is primitive in $\Ho_2(B; \Z);$
\item $B$ is simply-connected.
\end{itemize}
Then $\pi_1(S(E)) = 0$ and $c_1(E) =0$ as in Theorem \ref{thm: A}, where $S(E)$ is the corresponding circle bundle.

For a more concrete example, one takes the blow-up $B : = \CP^2 \# k \overline{\CP}^2$ of $\CP^2$ at $k$ generic points, with $0 < k < 9$. This admits a symplectic form $\ow_k$ whose Poincar\'e dual is given by $3H - E_1 - \cdots -E_k$ where $H$ is the class of a line in $\CP^2$ and $E_i$ is the exceptional curve derived from $i$-th blow-up. In particular $[\ow_k] = c_1(B)$ is primitive in $H_2(B;\Z)$ and $B$ is simply-connected.
\end{example}

\begin{lemma}\label{lem: actionorbits}
For 1-orbits $x$ in the region \emph{(I)} and \emph{(II)}, we have $\mathcal{F}(x) \leq 0$. In the other regions, we have $\mathcal{F}(x) > 0$.
\end{lemma}

\begin{example}\label{ex: resolution}
    Let $X$ be a $\Q$-factorial variety with a unique singularity at the origin $O \in X$ and assume that the complement $X \setminus \{O\}$ admits an exact K\"ahler form $\ow_X$. Then by \cite[Lemma 3.2]{McRi23}, any resolution $\pi: Y \rightarrow X$ admits a K\"ahler form $\ow_Y$ such that $\pi^* \ow_X = \ow_Y$. In particular, $Y$ is a convex symplectic domain, and a neighborhood of $\pi^{-1}(\{O\}) \subset Y$ serves as a complement-exact subdomain.
\end{example}

\begin{remark}\label{rem: usualactionfctl}
For an admissible Hamiltonian $H$, a standard action functional $\mathcal{A}_H : \widetilde{\mathcal{L}_0 V} \rightarrow \R$ is given by
$$
\mathcal{A}_H(x, v) =  -\int_{D^2} v^*\ow + \int_{S^1} H(x(t)) dt.
$$
The class $(x, v)$ is a critical point of $\mathcal{A}_H$ if $x$ is a $1$-periodic Hamiltonian orbit of $H$. Here, we use the convention $\ow(\cdot, X_H)  = d H$ for the definition of the Hamiltonian vector field $X_H$.
\end{remark}

Transfer maps in Floer theory, introduced by Viterbo \cite{Vit}, have played a central role in fundamental questions of symplectic topology. For example, it serves as a quite nontrivial obstruction to the existence of certain Lagrangian submanifolds \cite{Vit}, and it is also an essential ingredient for constructing symplectic capacities from Floer theory, as e.g. in \cite{FHW, GH18}.

Roughly speaking, for a symplectic manifold $V$ and a codimension zero embedding $W \hookrightarrow V$, transfer maps appear as natural homomorphisms between various flavors of Floer (co)homology of $V$ and $W$.
To our knowledge, it is only constructed when the ambient symplectic manifold $V$ is globally exact in the literature, and
the purpose of this article is to give a construction of transfer maps with favorable functorial properties in symplectic cohomology for possibly non-exact symplectic manifolds. We do this when the complement of a subdomain is exact.

Let $V$ be a convex symplectic domain, that is, a compact symplectic manifold which is exact near the boundary $\partial V$ and the Liouville vector field points outward along $\partial V$. Under some additional assumptions depending on the context,
the symplectic cohomology $\SH^*(V)$ is defined as a $\Z$-graded algebra over the Novikov field $\Lda_{V}$. A standard construction of $\SH^*(V)$ is briefly reviewed in Section \ref{sec: symcohomology}; we refer the reader to \cite{BeRi20} for an intensive study of the symplectic cohomology of convex symplectic domains.

Let $W$ be a codimension zero subdomain of $V$ such that the symplectic form is exact in the complement $V \setminus W$; we call it a \emph{complement-exact} subdomain. When the boundary $\partial W$ is simply-connected, the Novikov field $\Lda_W$ associated with $W$ can be canonically identified with the one $\Lda_V$ of the ambient domain $V$; see Theorem \ref{thm: nonnegandW}. Under this identification, the main result of this article is to establish an algebra homomorphism from $\SH^*(V)$ to $\SH^*(W)$ that has a functorial property with respect to the restriction map $i^*: \QH^*(V) \rightarrow \QH^*(W)$ on the quantum cohomology rings.

\begin{theorem}\label{thm: nonnegandW}
Let $W$ be a complement-exact subdomain in a convex symplectic domain $V$ such that $\pi_1(\p W) = 0$ and $c_1(V)|_{\pi_2(V)} = 0$. Then
\begin{enumerate}
\item The inclusion $i: W \rightarrow V$ induces an isomorphism between the Novikov fields $\Lda_W$ and $\Lda_V$.
\item Under the above identification $\Lda_V \cong \Lda_W$, we have a canonical vector space isomorphism
$$
\SH^*_{\leq 0}(V; \Lda_V) \cong \SH^*(W;\Lda_W).
$$
\end{enumerate}
\end{theorem}

Let $W$ be a codimension zero subdomain of $V$ such that the symplectic form is exact in the complement $V \setminus W$; we call it a \emph{complement-exact} subdomain. When the boundary $\p W$ is simply-connected, the Novikov field $\Lda_W$ associated with $W$ can be canonically identified with the one $\Lda_V$ of the ambient domain $V$; see Theorem \ref{thm: nonnegandW}. Under this identification, the main result of this article is to establish an algebra homomorphism from $\SH^*(V)$ to $\SH^*(W)$ that has a functorial property with respect to the restriction map $i^*: \QH^*(V) \rightarrow \QH^*(W)$ on the quantum cohomology rings.

The $c^*$-map above is induced by a natural inclusion involved in the definition of symplectic cohomology; see \cite[Section 4.1]{BeRi20} and \cite[Section 5]{Rit}.

\begin{example}\label{ex: neglinebdl2}
Following Example \ref{ex: neglinebdl}, the total space $E$ can be seen as a (completed) convex symplectic manifold, and the total space of a disk bundle $\pi: D(E) \rightarrow B$ now serves as a complement-exact subdomain. Assume that
\begin{itemize}
\item $[\ow] = c_1(B)$ and $[\ow]$ is primitive in $\Ho_2(B; \Z);$
\item $B$ is simply-connected.
\end{itemize}
Then $\pi_1(S(E)) = 0$ and $c_1(E) =0$ as in Theorem \ref{thm: A}, where $S(E)$ is the corresponding circle bundle.

\subsubsection{Admissible Hamiltonians} \label{sec: admHam} Let $(V, \ow, \lda)$ be a convex symplectic domain. We assume that the first Chern class $c_1(V)$ vanishes on the second homotopy group $\pi_2(V)$
A Hamiltonian $H: \widehat V \rightarrow \R$ is called \emph{admissible} if
\begin{itemize}
\item there exists $r_0 \in [1, \infty)$ such that $H$ depends only on the radial coordinate $r$ for $r \geq r_0$, say $H(z) = h(r)$, and $h'(r) \geq 0$;
\item $h(r) = \kappa r + b$ for sufficiently large $r$ where the \emph{slope} $\kappa$ is not the period of the Reeb orbit on the contact boundary $(\p V, \alpha)$.
\end{itemize}

\subsubsection{Ring structure} The symplectic cohomology $\SH^*(V)$ admits a ring structure by the pair-of-pants product.
Let $\mathcal{S}$ be the Riemann sphere with two positive punctures and one negative puncture; this means that $\mathcal{S}$ admits a parametrization $[0, \infty) \times S^1$ and $(-\infty, 0] \times S^1$, respectively, near the punctures. Given three admissible Hamiltonians $H_1, H_2, H_3$ on $\widehat V$, we take an $\mathcal{S}$-parametrized Hamiltonian $H_{\mathcal{S}}$ which coincides with $H_1, H_2$ near the positive punctures and with $H_3$ near the negative puncture. We likewise take an $\mathcal{S}$-parameterized admissible almost complex structure $J_{\mathcal{S}}$. We define a product
\[
\HF^k(H_1) \otimes \HF^{\ell}(H_2) \rightarrow \HF^{k+\ell}(H_3), \quad x_1 \otimes x_2 \mapsto x_3
\]
by counting the solutions $u: \mathcal{S} \rightarrow \widehat V$ of the Floer equation
\begin{equation} \label{eq: Floereq}
 (du - X_{H_{\mathcal{S}}} \otimes \beta)^{0, 1} = 0
\end{equation}
which converges to $x_1, x_2$ at the positive punctures and converges to $x_3$ at the negative puncture. Here, $\beta \in \Omega^1(\mathcal{S})$ is a one-form that agrees with $dt$ near the punctures (where $t$ denotes the $S^1$-coordinates of the parameterizations). As is well-known, the product is compatible with the continuation maps and hence induces a product on the symplectic cohomology as
\[
\SH^k(V) \otimes \SH^{\ell}(V) \rightarrow \SH^{k+\ell}(V).
\]
With this product, the symplectic cohomology $\SH^*(V)$ is now a unital $\Z$-graded algebra over the Novikov field $\Lda_V$. For more details on the construction of the ring structure, we refer the reader to \cite[Section 6]{Rit} and \cite[Section 4.7]{BeRi20}.

\begin{theorem}\label{thm: nonnegandW}
Let $W$ be a complement-exact subdomain in a convex symplectic domain $V$ such that $\pi_1(\p W) = 0$ and $c_1(V)|_{\pi_2(V)} = 0$. Then
\begin{enumerate}
\item The inclusion $i: W \rightarrow V$ induces an isomorphism between the Novikov fields $\Lda_W$ and $\Lda_V$.
\item Under the above identification $\Lda_V \cong \Lda_W$, we have a canonical vector space isomorphism
$$
\SH^*_{\leq 0}(V; \Lda_V) \cong \SH^*(W;\Lda_W).
$$
\end{enumerate}
\end{theorem}

\subsection{Transfer maps} Transfer-admissible Hamiltonians $H: \widehat V \rightarrow \R$ form a cofinal family of admissible Hamiltonians. The symplectic cohomology $\SH^*(V; \Lda_V)$ is the direct limit of Hamiltonian Floer cohomology $\HF^*(H)$ along the slope $\kappa$ of transfer-admissible Hamiltonians $H$. Consequently, the action functional $\mathcal{F}$ defines a natural map into the quotient
\begin{equation}\label{eq: fulltononpositive}
 \SH^*(V; \Lda_V) \rightarrow \SH_{\leq 0}^*(V; \Lda_V),
\end{equation}
and by the discussion in Section \ref{sec: ring on nonneg}, this map preserves the ring structure. Now, the composition with the identification $\SH^*_{\leq 0}(V; \Lda_V) \cong \SH^*(W; \Lda_W)$ by Corollary \ref{cor: algebraisom} yields an algebra homomorphism
$$
\Phi: \SH^*(V; \Lda_V) \rightarrow \SH^*(W, \Lda_W).
$$
over $\Lda_V = \Lda_W$.
\begin{proof}[Proof of Theorem \ref{thm: A}]
It only remains to explain about the commutative diagram
\[
\begin{tikzcd}
 \SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
 \QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*}& \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
For a transfer-admissible Hamiltonian $H^0$ with the slope $\kappa$ (and hence $\frac{1}{2}\kappa$) sufficiently small, 1-periodic orbits of $H^0$ are precisely those of constant orbits corresponding to Morse critical points on $W$ and $V$ respectively. Moreover, the standard Floer theory, e.g. \cite{Fl88}, tells us that they recover the quantum cohomology $\QH^*(V)$ and $\QH^*(W)$ including the ring structure \cite[Lemma 13]{Ri14}. Now the restriction of the Hamiltonian $H^0: \widehat{V} \rightarrow \R$ to $H^0_W: \widehat{W}\rightarrow \R$, as in the proof of Theorem \ref{thm: nonnegandW}, corresponds to the natural restriction map $i^*: \QH^*(V) \rightarrow \QH^*(W)$. In other words, we have a commutative diagram:
\[
\begin{tikzcd}
 \SH^*(V) \arrow{r} & \SH_{\leq 0}^*(V) \\
 \QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*}& \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
Here, the upper horizontal map is the homomorphism in \eqref{eq: fulltononpositive}, and the canonical $c^*$-map is defined by the inclusion; see \cite[Section 5]{Rit}. Now, applying the identification $\SH^*_{\leq 0}(V) \cong \SH^*(W)$ in Corollary \ref{cor: algebraisom}, we obtain the desired diagram.
\end{proof}

\begin{remark}\label{rem: Weinattach}
 Given a convex symplectic domain $W$ with $\pi_1(\p W) = 0$ and $c_1(W)|_{\pi_2(W)} = 0$, attach the Weinstein handle $\mathcal{H}$ to $W$ along the boundary $\p W$. Denote the resulting convex domain by $V$. Note that $W$ is now a complement-exact subdomain of $V$ which meets the assumptions of Theorem \ref{thm: A}. As an analogue of the seminal result of Cieliebak \cite{Ci02}, it is likely that the transfer map $\Phi: \SH^*(V) \rightarrow \SH^*(W)$ gives rise to an algebra isomorphism when the handle $\mathcal{H}$ is subcritical.
\end{remark}

\begin{corollary}\label{cor: algebraisom}
    Under the assumptions of Theorem \ref{thm: nonnegandW}, we have a canonical $\Lda$-algebra isomorphism
    \[
    \SH^*_{ \leq 0}(V) \cong \SH^*(W).
    \]
\end{corollary}

\begin{theorem}\label{thm: A}
Let $W$ be a complement-exact subdomain in a convex symplectic domain $V$. Assume that $c_1(V)|_{\pi_2(V)} = 0$ and $\pi_1(\p W) = 0$. There exists a $\Z$-graded algebra homomorphism
$$
\Phi: \SH^*(V; \Lda_V) \rightarrow \SH^*(W; \Lda_W)
$$
under a canonical identification $\Lda_V = \Lda_W$. Moreover, it fits the following commutative diagram with quantum cohomology rings.
\[\begin{tikzcd}
        \SH^*(V) \arrow{r}{\Phi}  & \SH^*(W)  \\
        \QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*}& \QH^*(W) \arrow[swap]{u}{c^*}
    \end{tikzcd}
\]
where the vertical maps are the canonical $c^*$-map.
\end{theorem}

The $c^*$-map above is induced by a natural inclusion involved in the definition of symplectic cohomology; see \cite[Section 4.1]{BeRi20} and \cite[Section 5]{Rit}.

For construction, we basically follow McLean \cite[Section 10.2]{Mclean_extension} and Ritter \cite[Section 9]{Rit} where transfer maps are constructed in symplectic (co)homology for exact domains. An essential ingredient is to use a special type of admissible Hamiltonians which is adapted to the embedding $W \hookrightarrow V$, see Figure \ref{fig: tranfer_Ham}, and also properly interacts with an action filtration on the symplectic (co)homology. In particular, a careful estimate of the action values of the generators of the corresponding Hamiltonian Floer cochain complex shows that the part $\SH^*_{\leq 0}(V)$ whose generators have non-positive action is canonically isomorphic to the symplectic cohomology $\SH^*(W)$ of the subdomain $W$. Then the natural quotient map $\SH^*(V) \rightarrow \SH^*_{\leq 0}(V)$ modding out by the generators with positive action produces the desired homomorphism $\SH^*(V) \rightarrow \SH^*(W)$. This was shown to be a ring homomorphism \cite[Section 10.2]{Mclean_extension} and more generally compatible with TQFT operations \cite[Section 9]{Rit} for exact domains.

The main technical point of the current non-exact setup is that the standard action functional, as in Remark \ref{rem: usualactionfctl}, now depends not only on loops but on their capping disks, and this makes the action value estimates delicate. To handle this, we introduce a new action functional, inspired by the work of McLean--Ritter \cite{McRi23},
which is designed to deal with non-exact convex symplectic domains. The basic idea is to impose action values to be zero in the non-exact region so that only the exact part contributes to the action values, and moreover, they can be effectively understood by Reeb periods of the contact boundary.

In Section \ref{sec: newfiltration}, we define an action functional $\mathcal{F}$ and give the relevant action value estimates in Lemma \ref{lem: actionorbits}. In particular, we show that the non-positive part $\SH^*_{\leq 0}(V)$ with respect to the action filtration induced by $\mathcal{F}$ again corresponds to the symplectic cohomology $\SH^*(W)$ of the subdomain $W$; see Theorem \ref{thm: nonnegandW}. It is also shown in Corollary \ref{cor: algebraisom} that the correspondence is compatible with the respective ring structure. The commutative diagram in Theorem \ref{thm: A} then follows from a standard argument.

There are various interesting examples of convex symplectic domains with complement-exact subdomains for potential applications. We consider negative line bundles in Example \ref{ex: neglinebdl2} and resolutions of isolated singularities in Example \ref{ex: resolution}. For a given convex domain $W$, attaching an exact cobordism along the boundary produces a larger convex domain $V$, and $W$ is complement-exact in $V$; See Remark \ref{rem: Weinattach}.

For construction, the authors “basically follow McLean … and Ritter … where transfer maps are constructed in symplectic (co)homology for exact domains.” The key ingredient is “a special type of admissible Hamiltonians which is adapted to the embedding $W\hookrightarrow V$ … and also properly interacts with an action filtration.” A “careful estimate of the action values of the generators” shows that “the part $\SH^*_{\le 0}(V)$ whose generators have non-positive action is canonically isomorphic to the symplectic cohomology $\SH^*(W)$.” Then “the natural quotient map $\SH^*(V)\to \SH^*_{\le 0}(V)$ modding out by the generators with positive action produces the desired homomorphism $\SH^*(V)\to \SH^*(W)$,” and in the exact case this “was shown to be a ring homomorphism … and more generally compatible with TQFT operations.”

In the “current non-exact setup,” the “main technical point” is that the “standard action functional … now depends not only on loops but on their capping disks,” making estimates delicate. To handle this they “introduce a new action functional, inspired by … McLean--Ritter,” whose “basic idea is to impose action values to be zero in the non-exact region so that only the exact part contributes,” and the values “can be effectively understood by Reeb periods.” They then “define an action functional $\mathcal{F}$ and give the relevant action value estimates,” proving that “the non-positive part $\SH^*_{\le 0}(V)$ with respect to the action filtration induced by $\mathcal{F}$ again corresponds to $\SH^*(W)$,” and that “the correspondence is compatible with the respective ring structure.” “The commutative diagram in Theorem~\ref{thm: A} then follows from a standard argument.”

For construction, the authors “basically follow McLean … and Ritter … where transfer maps are constructed in symplectic (co)homology for exact domains.” The key ingredient is “a special type of admissible Hamiltonians which is adapted to the embedding $W\hookrightarrow V$ … and also properly interacts with an action filtration.” A “careful estimate of the action values of the generators” shows that “the part $\SH^*_{\le 0}(V)$ whose generators have non-positive action is canonically isomorphic to the symplectic cohomology $\SH^*(W)$.” Then “the natural quotient map $\SH^*(V)\to \SH^*_{\le 0}(V)$ modding out by the generators with positive action produces the desired homomorphism $\SH^*(V)\to \SH^*(W)$,” and in the exact case this “was shown to be a ring homomorphism … and more generally compatible with TQFT operations.”

In the “current non-exact setup,” the “main technical point” is that the “standard action functional … now depends not only on loops but on their capping disks,” making estimates delicate. To handle this they “introduce a new action functional, inspired by … McLean--Ritter,” whose “basic idea is to impose action values to be zero in the non-exact region so that only the exact part contributes,” and the values “can be effectively understood by Reeb periods.” They then “define an action functional $\mathcal{F}$ and give the relevant action value estimates,” proving that “the non-positive part $\SH^*_{\le 0}(V)$ with respect to the action filtration induced by $\mathcal{F}$ again corresponds to $\SH^*(W)$,” and that “the correspondence is compatible with the respective ring structure.” The commutative diagram in establishing the main theorem then follows from a standard argument.”

Question: Let $V$ be a convex symplectic domain, meaning a compact symplectic manifold $(V,\omega)$ that is exact near $\partial V$ and whose Liouville vector field points outward along $\partial V$. Let $W\subset V$ be a codimension-zero subdomain that is complement-exact, i.e. $\omega$ is exact on $V\setminus W$. Assume moreover that $c_1(V)|_{\pi_2(V)}=0$ and $\pi_1(\partial W)=0$. Write $\Lambda_V$ and $\Lambda_W$ for the Novikov fields associated to $V$ and $W$, $\SH^*$ for symplectic cohomology, $\QH^*$ for quantum cohomology, $i^*:\QH^*(V)\to\QH^*(W)$ for the restriction map induced by the inclusion $i:W\hookrightarrow V$, and $c^*$ for the canonical map from quantum cohomology to symplectic cohomology. Under these assumptions, which statement holds?

Under the canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded algebra homomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\to\SH^*(W;\Lambda_W)
$$
such that the diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commutes.

Choices:
B.
Under the canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded algebra isomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\xrightarrow{\cong}\SH^*(W;\Lambda_W)
$$
such that the diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commutes.
C.
Under the canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded homomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\to\SH^*(W;\Lambda_W)
$$
such that the diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commutes.
D.
Under the canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded algebra homomorphism
$$
\Phi:\SH^*(W;\Lambda_W)\to\SH^*(V;\Lambda_V)
$$
such that the diagram
\[
\begin{tikzcd}
\SH^*(W) \arrow{r}{\Phi} & \SH^*(V) \\
\QH^*(W) \arrow{r}{i_*} \arrow{u}{c^*} & \QH^*(V) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commutes.
E.
Under the canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded algebra homomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\to\SH^*(W;\Lambda_W)
$$
such that the diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commutes, and moreover $\Phi$ is induced by restricting Hamiltonians from $V$ to $W$ on the full symplectic cohomology complex, without passing to any action filtration or quotient.

direction of transfer and induced map on quantum cohomology

need for action filtration and quotient by positive-action generators

The stem does not state the correct conclusion outright, but it preloads nearly all objects appearing in the true answer (the maps i^* and c^*, the Novikov-field identification context, and the focus on an existence statement), which strongly hints that the correct option will be a transfer-map theorem with a commutative diagram.

This is close to theorem restatement/recognition: the task is essentially to identify the precise formal statement of a known result under the listed hypotheses. The nearby variants prevent it from being completely tautological, but it is still largely a reformulation of the theorem.

Some reasoning is required to distinguish the exact theorem from weaker true or stronger false variants (mere existence vs commutative diagram vs isomorphism vs extra positive-action claim). However, the problem mainly tests precise recall/discrimination rather than substantial generative mathematical reasoning.

The distractors are strong and mathematically meaningful: one is a weaker true statement, one is an unjustified strengthening to an isomorphism, and others add plausible but false extra properties about positive-action summands or construction details. These reflect common theorem-distortion failure modes.

A reasonably well-constructed theorem-discrimination MCQ with strong distractors, but it leans heavily on recognizing the exact theorem statement and gives substantial structural hints in the stem rather than demanding deeper generative reasoning.


Question: Let $V$ be a convex symplectic domain, meaning a compact symplectic manifold $(V,\omega)$ that is exact near $\partial V$ and whose Liouville vector field points outward along $\partial V$. Let $W\subset V$ be a codimension-zero subdomain that is complement-exact, i.e. $\omega$ is exact on $V\setminus W$. Assume moreover that $c_1(V)|_{\pi_2(V)}=0$ and $\pi_1(\partial W)=0$. Write $\Lambda_V$ and $\Lambda_W$ for the Novikov fields associated to $V$ and $W$, $\SH^*$ for symplectic cohomology, $\QH^*$ for quantum cohomology, $i^*:\QH^*(V)\to\QH^*(W)$ for the restriction map induced by the inclusion $i:W\hookrightarrow V$, and $c^*$ for the canonical map from quantum cohomology to symplectic cohomology. Under these assumptions, which statement holds?

Choices:
B.
Under the canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded algebra isomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\xrightarrow{\cong}\SH^*(W;\Lambda_W)
$$
such that the diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commutes.
C.
Under the canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded homomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\to\SH^*(W;\Lambda_W)
$$
such that the diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commutes.
D.
Under the canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded algebra homomorphism
$$
\Phi:\SH^*(W;\Lambda_W)\to\SH^*(V;\Lambda_V)
$$
such that the diagram
\[
\begin{tikzcd}
\SH^*(W) \arrow{r}{\Phi} & \SH^*(V) \\
\QH^*(W) \arrow{r}{i_*} \arrow{u}{c^*} & \QH^*(V) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commutes.
E.
Under the canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded algebra homomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\to\SH^*(W;\Lambda_W)
$$
such that the diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commutes, and moreover $\Phi$ is induced by restricting Hamiltonians from $V$ to $W$ on the full symplectic cohomology complex, without passing to any action filtration or quotient.

The stem does not explicitly reveal the correct choice. It sets up hypotheses and asks for the valid conclusion, without giving away the direction, strength, or algebraic properties of the map.

This is largely a theorem-recall item: the stem lists the assumptions and asks which conclusion holds. The options do introduce nearby variants (isomorphism, wrong direction, weaker structure, extra unsupported claim), so it is not a pure verbatim restatement, but it is still close to one.

There is some reasoning pressure because the solver must distinguish between existence vs. isomorphism, algebra homomorphism vs. plain homomorphism, and correct functorial direction. However, the task mainly tests recognition of the precise theorem statement rather than substantial generative mathematical reasoning.

Several distractors are plausible and reflect natural failure modes: overstrengthening to an isomorphism, reversing the transfer direction, and adding an unjustified chain-level claim. But choice C is a weakly true statement if A is true, so the options are not cleanly exclusive, which weakens the distractor set.

Moderate-quality MCQ: it avoids answer leakage and has mostly plausible distractors, but it is still close to theorem restatement and is weakened by overlap between the correct answer and the weaker option C.


\label{vbint-}
Let $0<r\leq\infty$.
Let $M$ be a domain in ${\bf C}^n$ with $\partial M\in C^2$. Assume that   the Levi form of $\partial M$ on $T^{(1,0)}\partial M$ has either $n-1$ positive or at least three negative eigenvalues at  $\zeta_0\in\partial M$.
Let $E$ be a complex vector bundle of class $Lip$ $($resp. $\Lambda^{r+1})$ over $\overline M$. Assume that a connection  $D\colon Lip(M,E)\to L_{(1)}^\infty(M,E)$  $($resp. $\Lambda^{r+1}(M,E)\to \Lambda^{r}_{(1)}(M,E))$ satisfies $\Omega^{(0,2)}=0$ on $ M$ in the sense of distributions. Then $E|_U$ has a    holomorphic vector bundle structure, where $U$ is some open subset of $\overline M$ and $\zeta_0\in U$. Furthermore, the holomorphic structure is of class $C^{1/2}$ $($resp. $\Lambda^{r+1/2})$ with respect to the original $Lip$ $($resp. $\Lambda^{r+1})$ structure of $E|_U$.

\begin{thm}\label{vbint-}
Let $0<r\leq\infty$.
Let $M$ be a domain in $\cc^n$ with $\pd M\in C^2$. Assume that   the Levi form of $\pd M$ on $T^{(1,0)}\pd M$ has either $n-1$ positive or at least three negative eigenvalues at  $\zeta_0\in\pd M$.
Let $E$ be a complex vector bundle of class $Lip$ $($resp. $\Lambda^{r+1})$ over $\ov M$. Assume that a connection  $D\colon Lip(M,E)\to L_{(1)}^\infty(M,E)$  $($resp. $\Lambda^{r+1}(M,E)\to \Lambda^{r}_{(1)}(M,E))$ satisfies $\Om^{(0,2)}=0$ on $ M$ in the sense of distributions. Then $E|_U$ has a    holomorphic vector bundle structure, where $U$ is some open subset of $\ov M$ and $\zeta_0\in U$. Furthermore, the holomorphic structure is of class $C^{1/2}$ $($resp. $\Lambda^{r+1/2})$ with respect to the original $Lip$ $($resp. $\Lambda^{r+1})$ structure of $E|_U$.
\end{thm}

\label{sec1}
 Let $M\subset{\bf C}^n$ be a relatively compact domain with boundary $\partial M\in C^2$. Let $E$ be a smooth (i.e. $C^\infty$) complex vector bundle over the closure $\overline M$. Denote by $C^\infty(\overline M, E), C_{(k)}^\infty(E,\overline M)$ the spaces of smooth sections of $E$ and $E$-valued smooth $k$-forms on $\overline M$, respectively.
A Koszul {\it connection} $D$ in $E$ of is a ${\bf C}$-linear map
from $
 C_{(k)}^\infty(\overline M,E)$ to $ C^\infty_{(k+1)}(\overline M,E)
$
satisfying, for any $f\in C_{(k)}^\infty(\overline M)$ and $s\in C^\infty(\overline M,E)$,
$$
D(fs)=df\otimes s +(-1)^kf\wedge Ds.
$$ The curvature $R$ of $D$ is defined as
$
R=D\circ D\colon C^\infty_{(k)}(\overline M,E)\to C^\infty_{(k+2)}(\overline M,E).
$
 Let $e=(e_1,\dots, e_{k_0})^t$ be a smooth local frame of $E$. Then
$$
De_i=\sum \omega^j_i\otimes e_j, \ De=\omega e;\quad D^2e_i=\sum \Omega^j_i\otimes e_j, \ D^2e=\Omega e,
$$
where $\omega$ and $\Omega$ are the connection and curvature forms.
If $\hat e=Ae$ is another frame, then
\begin{equation}\label{hatomom}
\hat \omega =(dA)A^{-1} +A\omega A^{-1}, \quad \widehat\Omega=A\Omega A^{-1}.
\end{equation}
Decomposing into types, $\omega=\omega^{(1,0)}+\omega^{(0,1)}$ and
$\Omega=\Omega^{(2,0)}+\Omega^{(1,1)}+\Omega^{(0,2)}$, yields
\begin{equation}\label{om02}
 \Omega^{(0,2)}=\dbar\omega^{(0,1)}-\omega^{(0,1)}\wedge\omega^{(0,1)}.
\end{equation}

The {\it integrability problem} studied in this paper is to find a frame $\hat e=Ae$ for which $\hat\omega^{(0,1)}=0$, which is equivalent to solving
\begin{equation}\label{dbAA}
A^{-1}\dbar A+\omega^{(0,1)}=0.
\end{equation}
If such frames $\hat e_k$ exist on an open covering $\{U_k\}$ of $\overline M$, then on overlap $U_j\cap U_k$ their transition matrices satisfy $\dbar A_{kj}=0$. In this case we say that $(E,\overline M,D)$ carries a holomorphic vector bundle structure. By (\ref{hatomom})-(\ref{om02}), the solvability of (\ref{dbAA}) requires the {\it formal} integrability condition
\begin{equation}\label{Om02=0}
\dbar\omega^{(0,1)}=\omega^{(0,1)}\wedge\omega^{(0,1)},\quad \text{i.e.}\quad \Omega^{(0,2)}=0.
\end{equation}
Koszul and Malgrange~~\cite{MR131882} proved that when the boundary is not considered, the formal integrability condition is sufficient for the integrality; see also Kobayashi~~\cite{MR0909698}*{Prop.~3.7, p.~17} for a proof using the Newlander-Nirenberg theorem.

Our main result establishes integrability and optimal boundary regularity for the H\"older-Zygmund class (see Section 3 for definition) under the formal integrability condition on $M$.

\begin{thm}[convex configuration estimate]\label{conv-est} Let $r>0$ and $1\leq q\leq n-1$.
Let $(D^1,D_{r_2}^2)$ be an $(n-q)$-convex configuration. Let $D^{12}_r=D^1\cap D^2_r$.
The homotopy operators $H_q, H_{q+1}$ in \rta{hf-c} satisfies
\aln{}\|H_s\var\|_{\Lambda^{r+1/2}(D^{1 2}_{(1-\theta)r_2})}&\leq \f{C_r( \nabla\rho^1,\nabla^2\rho^1)}{\theta^{3r+2r_0}}\|\var\|_{\Lambda^r(D^{1 2}_{r_2})},\quad \forall\theta\in(0,1).
\end{align*}
Moreover, $C_r(\nabla\rho^1,\nabla^2\rho^1)$ is stable under small $C^2$ perturbations of $\rho^1$.
\end{thm}
\begin{proof}Recall from \re{hq1}-\re{nhq2-} that $H_s =H^{(1)}_s + H_s^{(2)}$ with
\aln{}
 H^{(1)}_s f&:=R_{ D^{2}; s-1}^0 \cL E_{D^{12}} f+R_{U^1;s-1 }^{01}[\db,\cL E_{D^{12}}] f,\\
H^{(2)}_sf&=L_{1^+;{s-1}
}^{01} \cL E_{D^{12}} f +L_{2;s-1}^{02} f+L_{12;s-1}^{012}f.
\end{align*}
Take a cut-off function $\chi\in C^\infty_0(D^{2}_{(1-\theta/3)r_2})$ with $\chi=1$ on $D^{2}_{(1-\theta/2)r_2}$, and decompose
$
 H^{(1)}_s f= H^{(1)}_s(\chi f)+ H^{(1)}_s( (1-\chi)f).
$
Then the estimate in~\cite{gong-shi-nn} says that
$$
\|H_s^{(1)}(\chi \var)\|_{\Lambda^{r+1/2}(D^{1 2}_{r_2})}\leq C_r \|\chi\var\|_{\Lambda^r(D^{1 2}_{r_2})}.
$$
We can choose $\chi$ such that $\|\chi\|_{C^r}\leq {C_r}{\theta^{-r-1}}$. Thus,
$$
\|\chi\var\|_{\Lambda^r(D^{12}_{r_2})}\leq {C_r }{\theta^{-r-r_0}}\|\var\|_{\Lambda^r(D^{12}_{r_2})}.
$$
Since $(1-\chi)\var$ vanishes on $D_{(1-\theta/2)r}^{12}$ and
\ga{}\label{giest}
|g^i(z,\zeta)\cdot(\zeta-z)|\geq |\zeta-z|^2/C,\quad z\in D^{1}\cap D^2,\zeta\in D^2\setminus D^1, i=0,1,2,
\end{gather}
then we can obtain $$
\|H_s^{(1)}((1-\chi) \var)\|_{\Lambda^{r+1/2}(D^{12}_{(1-\theta)r_2})}\leq C_r\theta^{-2r-r_0} \|(1-\chi)\var\|_{\Lambda^r(D^{12}_{r_2})}.
$$
The $H_s^{(2)}$ is given by boundary integrals on $S^1_+, S^2$ and $ S^{12}$ respectively. The estimate \re{giest} also yields
$\|H_s^{(2)}\var\|_{\Lambda^{r+1/2}(D^{12}_{(1-\theta)r_2})}\leq C_r \theta^{-2r-r_0} \|\var\|_{\Lambda^r(D^{12}_{r_2})}.
$
We have obtained the desired estimate of $H_s$.
\end{proof}
We now derive estimates for the concave case. We still have $H_s=H_s^{(1)}+H_s^{(2)}$, where $H_s^{(1)}$ has the same form as in the convex case. However,
$$
H_{s}^{(2)}f=\sum_{i=1}^2L_{i3,s}^{0i3}f- L^{01}_{1^+,s-1}\cL Ef +L_{12;s-1}^{123}f-\hat T_{ D^{23}_{(1-\theta)r}, s}L^{23}_{12;s} f.
$$
By \re{g3theta}, we have $
\RE\{g^3(z,\zeta)\cdot(\zeta-z)\} \geq\theta r_3^2$ for $z\in D^{23}_{(1-\theta)r}$ and $\zeta\in S^{13}\cup S^{23}\cup S^{12}\subset\pd D^{123}_r$. Some kernels in $H^{(2)}_s$ involve first-order $z$-derivatives of $\nabla\rho^1(z)$.
As in the convex case, we can obtain
$$\|H^{(2)}_sf\|_{\Lambda^{r+1/2}(D^{123}_{(1-\theta)r})}\leq {C_r(\nabla\rho^1 )}{\theta^{-r-r_0}}\|\rho^1\|_{\Lambda^{3/2}}\|f\|_{\Lambda^\e}.
$$The kernels of $H^{(1)}f$ involve the second-order $z$-derivatives $ \nabla\rho^2(z)$.
Then by ~\cite{gong-shi-nn}*{Thm. 5.12}, we have
$$\|H^{(1)}_sf
\|_{\Lambda^{r+\del}(D^{123}_{(1-\theta)r})}\leq {C_r }{\theta^{-r-r_0}}\left(\|\rho^1\|_{\Lambda^{r+2+\del}}\|f\|_{\Lambda^\e}+ \|f\|_{\Lambda^r}\right).
$$Therefore, we have obtained part $(a)$ below.
\begin{thm}[concave configuration estimate]\label{concave-est} Let $\e>0,r>0$ and $1\leq q\leq n-3$.
Assume that $\rho^1\in C^{2+\e}$. The following hold.
\bpp
\item The operators $H_q, H_{q+1}$ in Theorem~$\ref{cchf}$ satisfy
\al{}
\label{hqfr12}
\|H_sf\|_{\Lambda^{r+\del}(D^{123}_{(1-\theta)r})}&\leq \f{C_{r,\e}}{\theta^{3r+r_0}} (\|\rho^1\|_{\Lambda^{r+2+\del}}\|f\|_{\Lambda^\epsilon({D_{r}^{123}})}+ \|f\|_{\Lambda^r(D^{123}_{r})})
\end{align}
for $0\leq\delta\leq1/2$ and $0<\theta<1$. Furthermore, $C_{r,\e}=C_{r,\e}(\nabla\rho^1,\nabla^2\rho^1)$ is stable under small $C^2$ perturbations of $\rho^1$;
 the same estimate holds for the $\db$ solution operator $H_q$ in \rta{cchf-closed}.
 \item
The homotopy formula \rea{tsqf+-cv} in Theorem~$\ref{cchf}$ holds for $f\in \Lambda^{\e}_{(0,q)}(D^{123})$ when $\db f\in \Lambda^{\e}_{(0,q)}(D^{123})$, and the $\db$-solution operator $H_q$ in \rta{cchf-closed} is valid for all $\db$-closed $f\in\Lambda^{\e}_{(0,q)}(D^{123})$ and satisfies \rea{hqfr12}.
\epp
\end{thm}
\begin{proof}We verify $(b)$ using $(a)$. We know that $f=\db H_qf+H_{q+1}\db f$ when $f\in C^{1+\e}$. Let us write $H_q$ as $H_{D^{123}_r}$ to indicate the dependent on the domain. Now assume $f\in \Lambda^r_{(0,q)}(D^{123})$ with $\db f\in\Lambda^r$. We find a sequence $f_j\in C^{\infty}_{(0,q)}(D^{123})$ such that $f_j $ and $\db f_j $ converge to $f,\db f$ in $\Lambda^{r'}(D^{123})$ as $j\to\infty$ for any $r'<r$. Furthermore, $$\|f_j\|_{\Lambda^{r}(D^{123})}\leq C_r\|f\|_{\Lambda^{r}(D^{123})}, \quad \|\db f_j\|_{\Lambda^{r}(D^{123})}\leq C_r\|\db f\|_{\Lambda^{r}(D^{123})}.$$
Then $H_qf=\lim H_qf_j$ and $H_{q+1}\db f=\lim H_{q+1}\db f_j$ gives us \rea{tsqf+-cv}.
Assume now $f$ is $\db$-closed. Set $L_t(z)=z+tie_{q+2}$ with $t>0$, where $e_{q+2}$ is the $(q+2)$-th unit vector of $\cc^n$. Fix $\theta\in(0,1/2)$. When $0<t<t_\theta$, we have $L_t(D^{123}_{(1-\theta)r})\subset D^{123}_r$. Using a mollifier, we find $\db$-closed $f_j\in C^\infty(D^{123}_{(1-\theta)r})$ such that $f_j$ converges to $f$ in $\Lambda^{r'}(D^{123}_{(1-\theta)r})$-norm for any $r'<r$, whereas $\|f_j\|_{\Lambda^{r}(D^{123}_{(1-\theta)r})}\leq C_r\|f\|_{\Lambda^{r}(D^{123}_{r})}$. We have $f_j=\db H_{D^{123}_{(1-\theta')r},q}f_j$ on $D^{123}_{(1-\theta)r}$ for $0<\theta'<\theta/2$.
Letting $j\to\infty$, we obtain $f=\db H_{D^{123}_{(1-\theta')r},q}f$ on $D^{123}_{(1-\theta)r}$ and
\eq{Hth''}
\|H_{D^{123}_{(1-\theta')r},q}f\|_{\Lambda^{r+\del}(D^{123}_{(1-\theta)r})}\leq \f{C_{r,\e}}{\theta^{3r+r_0}} (\|\rho^1\|_{\Lambda^{r+2+\del}}\|f\|_{\Lambda^\epsilon }+ \|f\|_{\Lambda^r }).
\eeq
By the Arzel\`{a}-Ascoli theorem, $H_{D^{123}_{(1-\theta'_j)r},q}f$ converges to $H_{D^{123}_{r},q}f$ in $\Lambda^{r'+\delta}(D^{123}_{(1-\theta)r})$ norm for a sequence $\theta'_j\to0$ for any $r'<r$. Then \re{Hth''} yield
\re{hqfr12} for $H_q$.
\end{proof}

\setcounter{thm}{0}\setcounter{equation}{0}
\section{Integrability of complex vector bundles}\label{sec:NM}
In Section 2, we have seen that \rt{vbint-} follows from the following result.
\begin{thm}\label{vbint}
Let $M$ be a domain in $\mathbb C^n$ with $\pd M\in C^2$. Assume that the Levi-form of $M$ has
either $(n-1)$ positive or at least $3$ negative Levi eigenvalues at $\zeta_0\in\pd M$. Let $U$ be an open set in $\ov M$ and $\zeta_0\in U$. Assume that $\om$ is a $k_0\times k_0$ matrix of $(0,1)$ forms on $U$ such that $\db\om=\om\wedge\om$ holds on $U\setminus\pd M$ in the sense of distributions. Assume that $\om$ is in $L^\infty(U)$ $($resp. $\Lambda^r(U)$ with $r>0)$. Then there is an invertible matrix $A$ of class $\Lambda^{1/2}$ $($resp. $\Lambda^{r+1/2})$ on some neighborhood $U'$ of $\zeta_0$ in $\ov M$ such that
$
\db A+A\om=0
$
in the sense of distributions on $U'\setminus\pd M$.
\end{thm}
\begin{proof}The proof is based on a KAM-type iteration method. Such an approach was developed by Webster~\ci{MR999729} for an interior version of Newlander-Nirenberg theorem. It was also used in~\cites{MR2742034,MR2829316} to establish the integrability of CR vector bundles on strictly pseudoconvex hypersurfaces in $\cc^n$ for $n\geq4$.

Our theorem concerns the integrability of $(E,D)$ in the H\"older-Zygmund spaces $\Lambda^r$, which is the standard H\"older spaces $C^r$ when $r$ is not an integer; see Section 5 for definition. The theorem also treats  a Koszul connection $D$ that maps the space $ Lip(M,E)$ of Lipschitz sections
 into  $ L^\infty_{(1)}( M,E)$, the space  of $E$-valued $1$-forms of class $L^\infty$ on $M$. Such a connection extends trivially to a Koszul connection on $E$ over $\overline M$.   The theory of $L^\infty$ connections, which is of independent interest, is required  even when the original connections are $C^\infty$ for the (strictly) $3$-concave case. Theorem~\ref{vbint-}  is reduced to Theorem~\ref{vbint}, and the reduction requires a careful discussion in Section~\ref{sec:int-formal} of   $L^\infty$ connections on Lipschitz bundles  and of  {\it admissible} frame changes for which the transformation law (\ref{hatomom}) remains valid; see Proposition~\ref{d2=0}.

\medskip

We now outline our approach. Equation (\ref{dbAA}) may be viewed as a nonlinear $\dbar$-equation, and the integrability condition (\ref{Om02=0}) is also nonlinear.
 We   apply homotopy formulas to the study of (\ref{dbAA}).
The use of integral representations to establish regularity of $\overline\partial$-solutions on strongly pseudoconvex domains in ${\bf C}^n$ has a long history. Sup-norm estimate for $\dbar$-solutions to $(0,1)$-forms was proved by  Grauert and Lieb~\cite{MR273057} and  Henkin~\cite{MR0249660}.   Kerzman~\cite{MR0281944} obtained  $L^p$ and $C^{\beta}$ estimates of $\dbar$-solutions for $(0,1)$-forms and all $\beta<1/2$.
  Lieb~~\cite{MR283235} obtained the $L^\infty$ and the $C^\beta$ estimates of $\dbar$-solutions for $(0,q)$-forms.
Henkin and Romanov~\cite{MR0293121}  proved the sharp $C^{1/2}$ estimate of
$\dbar$-solutions for continuous $(0,1)$-forms.

We recall several important developments on integral representation formulas for  $\overline\partial$   that lead to derivative estimates on strongly pseudoconvex domains $D\subset{\bf C}^n$. When $\partial D$ is of class $ C^{k+2}$,   Siu~\cite{MR330515} showed that the Henkin solution operator satisfies a $C^{k+1/2}$ estimate  for  $\dbar$-closed $(0,1)$ forms of class $C^k$, and for  $\dbar$-closed $(0,q)$-forms   with $q\geq1$,  Lieb and Range~\cite{MR597825} constructed a new  $\dbar$ solution operator   and established  the same estimate. When $\partial D$ is only $ C^2$,  a homotopy formula was constructed  in~\cite{MR3961327} using the Stein extension, producing   homotopy operators with $\Lambda^{r+1/2}$ estimate for $r>1$.    Shi and Yao~\cites{MR4688544,MR4861589} later developed   a homotopy formula   employing the Rychkov extension and obtained $\Lambda^{r+1/2}$ estimates for all $r>0$, as well as $H^{s+1/2,p}$ estimates  for $1<p<\infty$ and $s>1/p$ when $\partial D\in C^2$; they also showed that the  estimates hold  for all $s\in{\bf R}$ when $\partial D$ is sufficiently smooth.  Very recently, Yao~~\cite{yaoc2} further reduced the boundary smoothness assumption to $ C^2$ for all $s\in{\bf R}$. We also refer the reader to references cited in~\cites{MR3961327,MR4688544,MR4861589,yaoc2}.

We also note that   Range and Siu~~\cite{MR338450} proved the $C^\beta$ estimate for all $\beta<1/2$ for $\dbar$-solutions for continuous $(0,q)$-forms on the (real) transversal intersection of
  strictly pseudoconvex domains. Whether the optimal gain of $1/2$  derivative   estimates  holds for $\dbar$-solutions on such transversal intersections remains an open problem.
  Higher-order derivative estimates in this setting were obtained by Brinkmann~~\cite{Br84}, Michel~~\cite{MR928297}, and Michel-Perotti~~\cite{MR1038709}.
 Peters~~\cite{MR1135535}  constructed a homotopy formula for the weakly transversal intersection of strictly pseudoconvex domains and established higher-order estimates, though with a loss of derivatives.
 It would be interesting to determine whether the results mentioned above can be applied to the integrability problem for intersections of strictly pseudoconvex domains. We also remark that the integrability of almost CR vector bundles on strongly pseudoconvex hypersurfaces in $\mathbb C^n$ with $n\geq4$ was proved by Webster~~\cite{MR1128608}, and sharper regularity results in the finitely smooth case were obtained by Gong-Webster~\cites{MR2742034,MR2829316}.
It remains an open question whether Theorem~\ref{vbint-} holds if $\partial M$ has two negative Levi eigenvalues. This appears to be related to the unsettled embedding problem for strongly pseudoconvex local CR structures in ${\bf R}^5$, investigated by Webster~\cite{MR0995504} and Gong-Webster~~\cite{MR2868966}.

We organize the paper as follows. In Section 2, we examine the formal integrability condition in detail for connections on Lipschitz complex vector bundles and introduce the notion of admissible frame changes. Sections 3 and 4 reformulate the local homotopy formulas in~\cites{MR986248,MR4866351}
 for shrinking domains. In Section 5, we recall  estimates established in \cites{MR986248,gong-shi-nn} and derive  new estimates on shrinking domains. Finally, in Section 6, we complete the proof of Theorem~\ref{vbint-}  by establishing Theorem~\ref{vbint}.

\setcounter{thm}{0}\setcounter{equation}{0}

Theorem~\ref{vbint-} is reduced to Theorem~\ref{vbint}, and the reduction requires a careful discussion in Section~\ref{sec:int-formal} of $L^\infty$ connections on Lipschitz bundles and of \textit{admissible} frame changes for which the transformation law (\ref{hatomom}) remains valid; see Proposition~\ref{d2=0}.

“We now outline our approach. Equation (\ref{dbAA}) may be viewed as a nonlinear $\dbar$-equation, and the integrability condition (\ref{Om02=0}) is also nonlinear. We apply homotopy formulas to the study of (\ref{dbAA}).”

The proof is then completed as follows: Sections 3 and 4 “reformulate the local homotopy formulas … for shrinking domains,” Section 5 recalls and derives “estimates … on shrinking domains,” and “Finally, in Section 6, we complete the proof of Theorem~\ref{vbint-} by establishing Theorem~\ref{vbint}.”

To prove the main theorem, we reduce it to the following theorem.

\begin{thm}\label{vbint}
Let $M$ be a domain in $\mathbb C^n$ with $\pd M\in C^2$. Assume that    the Levi-form of $M$ has
either $(n-1)$ positive or at least $3$ negative Levi eigenvalues at $\zeta_0\in\pd M$. Let $U$ be an open set in $\ov M$ and $\zeta_0\in U$.    Assume that $\om$ is a $k_0\times k_0$ matrix of $(0,1)$ forms on $U$ such that $\db\om=\om\wedge\om$ holds on $U\setminus\pd M$ in the sense of distributions. Assume that $\om$ is in $L^\infty(U)$ $($resp. $\Lambda^r(U)$ with $r>0)$. Then there is an invertible matrix $A$   of class $\Lambda^{1/2}$ $($resp. $\Lambda^{r+1/2})$ on some neighborhood $U'$ of $\zeta_0$ in $\ov M$ such that
$
\db A+A\om=0
$
in the sense of distributions on $U'\setminus\pd M$.
\end{thm}

The reduction requires a careful discussion of $L^\infty$ connections on Lipschitz bundles and of \textit{admissible} frame changes for which the transformation law (\ref{hatomom}) remains valid; for this we use the following proposition.

\begin{prop}\label{d2=0}
Let $D$ be a connection for a complex vector bundle $E$. Let $e=(e_1,\dots, e_{k_0})^t$ be a local frame field of $E$ over $U$. Assume that $De=\om \otimes e$ and $\om\in L^2_{loc}(U)$ and $d\om\in L^1_{loc}(U)$. If $A$ is admissible, then $\hat\om=(dA)A^{-1}+A\om A^{-1}$ is still in $L^2_{loc}(U)$ and  the curvature  forms    satisfy $A\Om A^{-1}=\hat\Om$.
\end{prop}

“We now outline our approach. Equation (\ref{dbAA}) may be viewed as a nonlinear $\dbar$-equation, and the integrability condition (\ref{Om02=0}) is also nonlinear. We apply homotopy formulas to the study of (\ref{dbAA}).”

The proof is then completed as follows: Sections 3 and 4 “reformulate the local homotopy formulas … for shrinking domains,” Section 5 recalls and derives “estimates … on shrinking domains,” and finally we complete the proof of the main theorem by establishing the theorem above.

Question:
Let $0<r\le \infty$, and let $M\subset \mathbb C^n$ be a domain with $\partial M\in C^2$. Fix $\zeta_0\in \partial M$, and assume that the Levi form of $\partial M$ on $T^{(1,0)}\partial M$ has either $n-1$ positive eigenvalues or at least three negative eigenvalues at $\zeta_0$. Let $E$ be a complex vector bundle over $\overline M$ of class $Lip$ (respectively of class $\Lambda^{r+1}$). Suppose $D$ is a connection with mapping property
$$
D: Lip(M,E)\to L_{(1)}^\infty(M,E)
\qquad \text{(respectively } D:\Lambda^{r+1}(M,E)\to \Lambda^r_{(1)}(M,E)\text{)},
$$
and let $\Omega=D^2$ be its curvature. Assume that the $(0,2)$-component of the curvature vanishes on $M$ in the sense of distributions, i.e. $\Omega^{(0,2)}=0$. A holomorphic vector bundle structure here means that locally one can choose bundle frames in which the $(0,1)$-part of the connection vanishes (equivalently, in local frames with connection matrix $\omega$, one can solve $\bar\partial A + A\omega=0$ for an invertible change-of-frame matrix $A$). Under these hypotheses, which conclusion holds near $\zeta_0$?

There exists an open subset $U\subset \overline M$ with $\zeta_0\in U$ such that $E|_U$ admits a holomorphic vector bundle structure. Moreover, relative to the original $Lip$ structure on $E|_U$ this holomorphic structure is of class $C^{1/2}$, and relative to the original $\Lambda^{r+1}$ structure on $E|_U$ it is of class $\Lambda^{r+1/2}$.

Choices:
B. There exists an open subset $U\subset \overline M$ with $\zeta_0\in U$ such that $E|_U$ admits a holomorphic vector bundle structure. Moreover, relative to the original $Lip$ structure on $E|_U$ this holomorphic structure is of class $C^{1}$, and relative to the original $\Lambda^{r+1}$ structure on $E|_U$ it is of class $\Lambda^{r+1}$.
C. There exists an open subset $U\subset \overline M$ with $\zeta_0\in U$ such that $E|_U$ admits a holomorphic vector bundle structure.
D. There exists an open neighborhood $U\subset \overline M$ of $\zeta_0$ and an invertible change-of-frame matrix $A$ on $U$ such that $\bar\partial A+A\omega=0$ on $U\setminus \partial M$, where $A$ is of class $Lip$ in the $Lip$ case and of class $\Lambda^{r+1}$ in the $\Lambda^{r+1}$ case; hence $E|_U$ carries a holomorphic vector bundle structure with the same regularity as the original bundle structure.
E. There exists an open subset $U\subset \overline M$ with $\zeta_0\in U$ such that $E|_U$ admits a holomorphic vector bundle structure, provided the Levi form at $\zeta_0$ has either $n-1$ positive eigenvalues or at least two negative eigenvalues. Moreover, relative to the original $Lip$ structure on $E|_U$ this holomorphic structure is of class $C^{1/2}$, and relative to the original $\Lambda^{r+1}$ structure on $E|_U$ it is of class $\Lambda^{r+1/2}$.

dropped boundary-regularity conclusion for the induced holomorphic structure

regularity of the gauge matrix and induced structure

Levi-signature threshold: at least three negative eigenvalues

The stem gives hypotheses and definitions but does not explicitly state the conclusion. The correct option is not leaked directly; the examinee must know or infer the precise local consequence and regularity statement.

The item is very close to a theorem-recall question: it presents the exact hypotheses and asks for the conclusion. However, it is not a pure verbatim restatement because the choices vary in strength, regularity, and Levi-signature threshold.

Some reasoning is required to distinguish the exact conclusion from stronger, weaker, or slightly altered variants, especially the half-derivative regularity loss and the 'at least three negative eigenvalues' condition. Still, the pressure is mainly on precise theorem recall rather than substantial derivation.

The distractors are plausible and mathematically targeted: they test overclaiming regularity, accepting a weaker-but-true statement, confusing gauge regularity with induced holomorphic regularity, and misremembering the Levi-signature hypothesis.

A solid but theorem-centric MCQ. It avoids direct answer leakage and has strong distractors, but it mainly tests precise recall of a stated result rather than deeper generative reasoning.


\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.