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On the positivity of direct image bundles Zhi Lia, Xiangyu Zhoubc a School of Science, Beijing University of Posts and Telecommunications, Beijing, China b Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China c Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, China
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     Dedicated to the centenary of V. S. Vladimirov § 1. IntroductionLet $H \times D \subset \mathbb{C}_{\tau}^m \times \mathbb{C}^n_z$ be a connected open subset and $\varphi$ be a function on $H \times D$ which is smooth up to the boundary. For each fixed ${\tau} \in H$, denote the weighted Bergman space by
$$
\begin{equation*}
A^2 (D, \varphi_{\tau}) := \biggl\{ f \in \mathcal{O}(D) \colon \int_D |f|^2 e^{-\varphi_{\tau}} \, dV < \infty \biggr\},
\end{equation*}
\notag
$$
where $\varphi_{\tau} (z) = \varphi ({\tau}, z)$ and $dV$ is the Lebesgue measure on $\mathbb{C}^n$. By the assumption that $\varphi$ is smooth up to the boundary, we have for all ${\tau} \in H$, $A^2 (D, \varphi_{\tau})$ are equal as vector spaces. Let $(\underline{\mathbb{C}},e^{-\varphi})\to H \times D$ denote the hermitian trivial holomorphic line bundle, and therefore, we can regard $\coprod_{{\tau} \in H} A^2 (D, \varphi_{\tau})$ as a trivial vector bundle over $H$ equipped with the natural $L^2$ metric, varies with ${\tau} \in H$, defined as for every $f \in A^2 (D, \varphi_{\tau})$,
$$
\begin{equation*}
\|f\|^2_{\tau} = \int_D |f|^2 e^{- \varphi_{\tau}}\, dV.
\end{equation*}
\notag
$$
It turns out that the bundle is a hermitian holomorphic vector bundle of infinite rank over $H$, which is called the direct image bundle over $H$ of the line bundle $(\underline{\mathbb{C}}, e^{-\varphi}) \to H \times D$. Throughout the present paper, a domain means a connected open set and the notation $\operatorname{PSH} (U)$ means the set of plurisubharmonic functions on a domain $U$. A celebrated theorem of Berndtsson [1] asserts that if $\varphi$ is (strictly) plurisubharmonic and $D$ is pseudoconvex, then the direct image bundle is (positive) semi-positive in the sense of Nakano. We consider the following setting. Let $D \subset \mathbb{C}^n$ be a bounded pseudoconvex domain and $\varphi \in \operatorname{PSH} (D) \cap C^{\infty} (\overline{D})$, for any domain $H \subset \mathbb{C}^m$ and any $\psi \in \operatorname{PSH} (H \times D) \cap C^{\infty} (\overline{H \times D})$. Take a trivial line bundle $(\underline{\mathbb{C}},e^{-k\pi^{\ast}\varphi-\psi})\to H\times D$, where $\pi \colon H \times D \to D$ is the projection to the second coordinate, then the direct image bundle is semi-positive in the sense of Nakano for any positive integer $k$, and thus, semi-positive in the sense of Griffiths. In particular, the fiberwised Bergman kernel $B_{k,\tau} (z)$ weighted by $k \pi^{\ast} \varphi ({\tau}, {\cdot}\,)+\psi ({\tau}, {\cdot}\,)$ is log-plurisubharmonic in $(\tau, z)$. The latter result follows from Berndtsson’s log-plurisubharmonicity theorem (see [2]). In general, Griffiths positivity could not imply Nakano positivity. A natural question arises from the theorem of Berndtsson as follows. Question 1.1. In the above setting, for the direct image with the natural $L^2$ metric, are Griffiths (semi-)positivity and (semi-)Nakano positivity equivalent? The first result in the present paper gives an answer to the above question by establishing a converse of the theorem of Berndtsson. Theorem 1.2. Let $D \subset \mathbb{C}^n_z$ be a bounded pseudoconvex domain and $\varphi\colon D\to [-\infty,+\infty)$ be a function such that $\varphi \in L^1_{\mathrm{loc}}$, and, for any set $E$ of measure zero, for any $z_0 \in D$. Assume further for any positive integer $k$, the Bergman kernel weighted by $k \varphi$ does not vanish on $D$. Then the following statements are equivalent: (1) $\varphi \in \operatorname{PSH}(D)$; (2) for any domain $H \subset \mathbb{C}^m_{\tau}$ and any locally bounded $\psi \in \operatorname{PSH} (H \times D)$, the Bergman kernel $B_{k, \tau} (z)$ of the direct image of the trivial line bundle $(\underline{\mathbb{C}}, e^{- k \pi^{\ast} \varphi - \psi})\to H \times D$ is log-plurisubharmonic; (3) for any domain $H \subset \mathbb{C}^m_{\tau}$ and any locally bounded $\psi \in \operatorname{PSH} (H \times D)$, the direct image of the trivial line bundle $(\underline{\mathbb{C}}, e^{- k \pi^{\ast} \varphi - \psi}) \to H \times D$ is semi-positive in the sense of Griffiths; (4) for any domain $H \subset \mathbb{C}^m_{\tau}$ and any locally bounded $\psi \in \operatorname{PSH} (H \times D)$, the direct image of the trivial line bundle $(\underline{\mathbb{C}}, e^{- k \pi^{\ast} \varphi - \psi}) \to H \times D$ is semi-positive in the sense of Nakano. Remark 1.3. (1) In the above theorem, since $\psi$ is locally bounded, the direct image of the trivial line bundle $(\underline{\mathbb{C}}, e^{- k \pi^{\ast} \varphi - \psi}) \to H \times D$ is a trivial vector bundle and locally the fibers of $F$ are equal to $A^2(D,e^{-\varphi})$ as vector spaces. (2) Usually the notion of positivity is defined for smooth hermitian metrics. In view of the recent works [1], [3]–[6], it is possible to define the positivity of Griffiths or Nakano for singular metrics and which will be explained in the next section. (3) It is usually impossible to derive Nakano positivity from Griffiths positivity even in the situation of smooth hermitian metrics, see [7]. However, Theorem 1.2 indicates that in this case, one can get Nakano positivity from a certain condition involved with Griffiths positivity. (4) The condition that $D$ is pseudoconvex can be omitted in the direction $(2)\Rightarrow (1)$. Berndtsson’s theorem was obtained in [1] by Hörmander’s $L^2$-estimate for $\overline{\partial}$-operator. Recently Deng, Ning, Wang, and Zhou in [5] found a general criterion for the Nakano positivity for holomorphic vector bundles by the $L^p$-estimate for $\overline{\partial}$-operator. With this criterion, a generalized version of Berndtsson’s theorem with a lower-bounded estimate has been obtained. As Nakano positivity implies Griffiths positivity, the direct image $F$ is also positive in the sense of Griffiths. In particular, the Bergman kernel $B_t (z)$ weighted by $\varphi_t$ is log-plurisubharmonic. This is also known as Berndtsson’s log-plurisubharmonicity theorem, see [2]. Guan–Zhou (see [8], [9]) proved Berndtsson’s log-plurisubharmonicity theorem and Griffiths positivity of the direct images by the $L^2$-extension theorem with optimal estimate. It should be noted that the $L^2$-extension theorem only with optimal estimate plays an essential role in Guan–Zhou’s argument. In [10], it is shown that a version of $L^2$-extension theorem with optimal estimate can also be derived by the theorem of Berndtsson. These results give unexpected relations between log-plurisubharmonicity, the optimal $L^2$-extension theorem, and positivity of the direct image. Later, Deng, Wang, Zhang, and Zhou [4] found a new characterization for plurisubharmonic functions, and with this result they proved the Griffiths positivity of the direct image with the usual $L^2$-extension theorem. Thus, the Griffiths positivity and the $L^2$-extension theorem with optimal estimate is somewhat equivalent. Another closely related topic is the minimal principle. Prékopa [11] found a version of the minimal principle in convex analysis. Later on, a celebrated theorem of Kiselman [12] generalizes Prékopa’s minimal principle to plurisubharmonic functions. In [13], Berndtsson found the following generalization of Kiselman’s minimal principle in an integral form. Let $\varphi (\tau, z)$ be a plurisubharmonic function on $U_\tau \times V_z \subset \mathbb{C}^m_\tau \times \mathbb{C}^n_z$, where $V_z$ is pseudoconvex. (1) Assume that $V$ is a Reinhardt domain and that $\varphi$ is independent of $\arg (z_j)$, $j = 1, 2, \dots, n$. Define $\widetilde{\varphi}$ by
$$
\begin{equation*}
e^{- \widetilde{\varphi} (\tau)} = \int_V e^{- \varphi (\tau, z)}\, dV_z,
\end{equation*}
\notag
$$
then $\widetilde{\varphi} \in \operatorname{PSH}(U)$. Moreover, the same conclusion holds if we only assume that $V$ and $\varphi (\tau, z)$ are invariant under the group action of $S^1$: $z \mapsto e^{i \theta} z$ for any $\theta \in \mathbb{R}$, provided $V$ contains the origin. (2) Assume $V$ is a tube domain and that $\varphi$ is independent of $\operatorname{Im} (z_j)$, $j = 1, 2, \dots, n$. Define $\widetilde{\varphi}$ by
$$
\begin{equation*}
e^{- \widetilde{\varphi} (\tau)} = \int_X e^{- \varphi (\tau, \mathrm{Rez})}\, dV_{\mathrm{Rez}},
\end{equation*}
\notag
$$
then $\widetilde{\varphi} \in \operatorname{PSH} (U)$. Let us consider a case of the above minimal principle. Let $D \subset \mathbb{C}^n$ be a pseudoconvex Reinhardt domain containing the origin and $\varphi \in \operatorname{PSH} (D)$. For any domain $H \subset \mathbb{C}^m_{\tau}$ and any $\psi \in \operatorname{PSH} (H \times D)$, the function $\widetilde{\psi}$, as defined by
$$
\begin{equation*}
e^{- \widetilde{\psi} (\tau)} = \int_D e^{- k \pi^{\ast} \varphi (\tau, z) - \psi(\tau, z)} \, dV_z,
\end{equation*}
\notag
$$
is plurisubharmonic for any $k > 0$, provided $\psi$ and $\varphi$ are invariant with respect to the group action of $S^1$: the maps $z \mapsto e^{i \theta} z$ for $\theta \in \mathbb{R}$. Another natural question asks whether there exists a converse of the above minimal principle. Question 1.4. Let $D \subset \mathbb{C}^n$ be a domain and $\varphi$ be a function on $D$. Can one derive the plurisubharmonicity of $\varphi$ from the plurisubharmonicity of $\widetilde{\psi}$ defined in the form of The second result of this paper provides an answer to this question by showing that a necessary condition is also sufficient for a function to be plurisubharmonic. Actually, we could have a more general version for the converse. Theorem 1.5. Let $D \subset \mathbb{C}^n$ be a bounded domain and $\varphi$ be a function on $D$ such that $\varphi \in L^1_{\mathrm{loc}}$, and for any set $E$ of measure zero, The Berndtsson’s generalization of Kiselman’s minimal principle was proved in [13] by a converse version of Hörmander’s $L^2$-estimates in dimension one and also can be easily derived from [5]. Inspired by Ball, Barthe, and Naor [14], Cordero–Erausquin [15] proved the Kiselman–Berndtsson’s minimal principle by computing the complex Hessian directly by the solution to the $L^2$-existence of $\overline{\partial}$-operator due to Hörmander [16] and Kohn (see [17]). Also Cordero–Erausquin found that the condition of invariance for $V$ and $\varphi$ under the maps $z \mapsto e^{i \theta}$, for any $\theta \in \mathbb{R}$, can be relaxed. And that is why in Theorem 1.5 the condition on the invariance for $D$, $\varphi$ or $\psi$ is omitted. Remark 1.6. The Kiselman–Berndtsson’s minimal principle also follows from the theorem of Berndtsson [1]. By invariance of $\varphi$ and $V$, the Bergman kernel $B_\tau (\zeta, 0)$ with respect to $\varphi (t, z)$ for a fixed $\tau$ is just a constant, which is equal to and it follows from the theorem of Berndtsson that is plurisubharmonic. § 2. PreliminariesIn this section we will recall some notions and results which are needed in proving Theorem 1.2 and Theorem 1.5. 2.1. $L^2$-estimates for $\overline{\partial}$-operatorThe following $L^2$-existence theorem is a fundamental result in the $L^2$-methods. Theorem 2.1 (Hörmander’s $L^2$-existence theorem, see [7], [16]). Let $D \subset \mathbb{C}^n$ be a pseudoconvex domain and $\varphi$ be a plurisubharmonic function on $D$ such that in the sense of distributions for some continuous positive $(1, 1)$-form $\theta$ on $D$. Then, for any $(n, 1)$-form $f$ on $D$ such that $\overline{\partial} f = 0$ with there exists an $(n, 0)$-form $u \in L^2 (D, e^{- \varphi})$ satisfying $\overline{\partial} u = f$ and If we write $\theta = i \sum_{j, k} \theta_{j \overline{k}}\,\, d z_j \wedge d \overline{z}_k$ and $f = \sum_j f_{\overline{j}}\,\, d z \wedge d \overline{z}_j$, $d z = d z_1 \wedge d z_2 \wedge \cdots \wedge d z_n$. Since $(\theta_{j \overline{k}})$ is positive definite, the inverse of $(\theta_{j \overline{k}})$ is well-defined and will be denoted as $(\theta^{\overline{k} j})$. Therefore, the norm of $f$ with respect to $\theta$ is defined to be
$$
\begin{equation*}
\int_D | f |^2_{\theta} e^{- \varphi} = \int_D \sum_{j, k} \theta^{\overline{k} j} f_j \overline{f}_k e^{- \varphi}.
\end{equation*}
\notag
$$
Hörmander’s $L^2$-existence theorem (for a pseudoeffective trivial line bundle $(\underline{\mathbb{C}}, e^{- \varphi}))$ was generalized by Demailly [18] (see also [19]) to a Nakano positive hermitian holomorphic vector bundle on a complete Kähler manifold. Recently in [5], the authors have introduced a notion of a hermitian holomorphic vector bundles satisfying the $L^2$-estimates for $\overline{\partial}$-operator, and established a converse version of Theorem 2.1. For the sake of completeness, we state Deng, Ning, Wang, Zhou’s results in [5] as follows. Definition 2.2 (see [5]). Let $(X, \omega)$ be a Kähler manifold of dimension $n$, which admits a positive hermitian holomorphic line bundle and $p > 0$. We say a hermitian holomorphic vector bundle $(E, h)$ (may be of infinite rank) satisfies the optimal $L^p$-estimate if for any positive holomorphic hermitian line bundle $(A, h_A)$ over $X$, for any $f \in C_{\mathrm{c}}^{\infty} \bigl(X, \bigwedge^{n, 1} T^{\ast}_X \otimes E \otimes A\bigr)$ with $\overline{\partial} f = 0$, there exists $u \in L^p \bigl(X, \bigwedge^{n, 0} T^{\ast}_X \otimes E \otimes A\bigr)$ such that and By Hörmander and Demailly’s $L^2$-existence theorem, a Nakano semi-positive hermitian holomorphic vector bundle on a complete Kähler manifold satisfies the optimal $L^2$-estimate. Remark 2.3. Although Definition 2.2 defines the notion via $L^p$-estimate condition, we only need the $L^2$-estimate condition in the present paper. Theorem 2.4 (see [5]). Let $(X, \omega)$ be a Kähler manifold of dimension $n$ which admits a positive holomorphic hermitian line bundle, $(E, h)$ be a smooth hermitian holomorphic vector bundle over $X$. If $(E, h)$ satisfies the optimal $L^2$-estimate, then $(E, h)$ is semi-positive in the sense of Nakano. 2.2. A new characterization for plurisubharmonic functionsThe extension of holomorphic functions or sections of a holomorphic vector bundle with suitable integrable conditions plays an important role in several complex variables and complex geometry. The reader is referred to [20], [21] for surveys on this topic. Theorem 2.5 (Ohsawa–Takegoshi, see [22]). Let $D\subset\mathbb{C}^n$ be a pseudoconvex domain and $\varphi$ be a plurisubharmonic function on $D$. Let $H\subset D$ be a closed submanifold. Then there exists a constant $C$ which depends only on the diameter of $D$ such that for any holomorphic function $f$ on $H$ with there exists a holomorphic function $F$ on $D$ such that $F|_H=f$ and where $d V_D$ and $d V_H$ denote the Lebesgue measure on $D$ and $H$, respectively. In [4], Deng, Wang, Zhang, and Zhou found a new characterization involving an $L^p$-extension property for holomorphic sections which can be regarded as a converse to Theorem 2.5. Definition 2.6 (Multiple $L^p$-extension property, see [4]). Let $(E, h)$ be a holomorphic vector bundle over a bounded domain $D \subset \mathbb{C}^n$ equipped with a singular Finsler metric $h$. Let $p > 0$ be a fixed constant. Assume that, for any $z \in D$, any nonzero element $a\in E_z$ with finite norm $| a|$, and any $m \geqslant 1$, there is a holomorphic section $f_m$ of $E^{\otimes m}$ on $D$ such that $f_m (z) = a^{\otimes m} $ and satisfies the following estimate: where $C_m$ are constants independent of $z$ and satisfying $1 / m \log C_m \to 0$ as $m \to \infty$. Then $(E, h)$ is said to have multiple $L^p$-extension property. With the multiple $L^p$-extension property, one can expect the Griffiths positivity. Theorem 2.7 (see [4]). Let $(E,h)$ be a holomorphic vector bundle over a bounded domain $D\subset\mathbb{C}^n$ equipped with a singular Finsler metric $h$ such that the norm of any local holomorphic section of $E^*$ is upper semi-continuous. If $(E,h)$ has multiple $L^p$-extension property for some $p>0$, then $(E,h)$ is semi-positive in the sense of Griffiths, namely, $\log|u|^2$ is plurisubharmonic for any local holomorphic section $u$ of $E^*$. Remark 2.8. In [4], in the case that $(\underline{\mathbb{C}},h)$ is a trivial line bundle with $h=e^{-\varphi}$, it is actually showed that if $h=e^{-\varphi}$ satisfies multiple $L^p$-extension property, there exists a sequence of functions $\{\varphi_m\}\subset \operatorname{PSH}(D)$ converging to $\varphi$ decreasingly. In particular, $\varphi_m\to\varphi$ in $L^1_{\mathrm{loc}}$ as $m\to \infty$. 2.3. Positivity of hermitian holomorphic vector bundlesWe first recall in this subsection some basic notions on positivity of smooth hermitian holomorphic vector bundles. Definition 2.9. Let $M$ be a complex manifold of dimension $n$ and $(E, h)\,{\to}\, M$ be a hermitian holomorphic vector bundle of finite or infinite rank. Let $\Theta =\sum_{j,k} \Theta_{j \overline{k}} \,\, d z_j \wedge d \overline{z}_k$ be the Chern curvature tensor of $E$. (1) $(E, h)$ is positive in the sense of Griffiths if, for any section $u$ of $E$ and any vector $v \in \mathbb{C}^n$, one has
$$
\begin{equation*}
\sum (\Theta_{j \overline{k}} u, u)_h v_j \overline{v}_k \geqslant \delta \| u \|^2_h | v |^2,
\end{equation*}
\notag
$$
for some $\delta > 0$. (2) $(E, h)$ is positive in the sense of Nakano if, for any $n$-tuple $(u_1, \dots, u_n)$ of sections of $E$, one has
$$
\begin{equation*}
\sum (\Theta_{j \overline{k}} u_j, u_k)_h \geqslant \delta \sum \| u_j \|^2_h,
\end{equation*}
\notag
$$
for some $\delta>0$. Semi-positivity concepts in two senses could be defined accordingly. It is easy to see that Nakano (semi-)positivity implies Griffiths (semi-)positivity. However, the converse does not hold in general, see [7]. In a more analytical way, Griffiths semi-positivity is equivalent to the fact that $\log | u |^2_h$ is plurisubharmonic for any local holomorphic section $u$ of the dual bundle $E^{\ast}$. However, such duality property fails for Nakano positivity, see [7]. For a holomorphic line bundle with a hermitian metric which can be locally written as $e^{-\varphi}$, the notions of Griffiths semi-positivity and Nakano semi-positivity coincide and being semi-positive means $\varphi \in \operatorname{PSH}$ locally. As we have mentioned, it is possible to define the positivity of Griffiths and Nakano for a holomorphic vector bundle with a (possibly) singular hermitian metric. By a singular hermitian metric we mean a measurable map from the base space to the space of non-negative hermitian forms which are finite almost everywhere on each fiber. By DNWZ’s results stated in the above Definition 2.2 and Theorem 2.4 (see [5]), it is reasonable to introduce the following notion. Definition 2.10 (see [6]). Let $M$ be a complex manifold of dimension $n$ and let $(E, h) \to M$ be a hermitian holomorphic vector bundle of finite or infinite rank with a (possibly) singular hermitian metric $h$. (1) $(E,h)$ is semi-positive in the sense of Griffiths if, for any holomorphic section $u$ of the dual bundle, $\log ||u||^2_{h^{\ast}}$ is plurisubharmonic. (2) $(E,h)$ is semi-positive in the sense of Nakano if $(E,h)$ is semi-positive in the sense of Griffiths and satisfies the optimal $L^2$-estimate as in Definition 2.2 locally. The following lemma and remark interpret the behaviour of $[i \Theta_{E, h}, \Lambda_{\omega}]$ acting on the bundle valued $(n,1)$-forms. Lemma 2.11 (see [7]). Let $(X, \omega)$ be a Kähler manifold, $(E, h) \to X$ be a hermitian vector bundle of rank $r$ which is positive (respectively, semi-positive) in the sense of Nakano. Then $[i\Theta_{E, h}, \Lambda_{\omega}]$ is positive definite (respectively, semi-positive definite) on $\bigwedge^{n, 1} T^{\ast}_X \otimes E$. Remark 2.12. Actually, if we fix a point $z_0 \in X$ and choose a local coordinate neighbourhood centered at $z_0$ such that $(\partial / \partial z_1, \partial / \partial z_2, \dots, \partial / \partial z_n)$ is an orthonormal basis of $T X$ at $z_0$. We also choose an orthonormal basis of $E_{{z_0} }$, denoted by $(e_1, e_2, \dots, e_r)$. We can write 2.4. The weighted Bergman kernelRecall the definition of the weighted Bergman kernel. Let $D \subset\mathbb{C}^n$ be a domain and $\varphi$ be an upper semi-continuous function on $D$. As usual, we denote
$$
\begin{equation*}
A^2 (D, \varphi) = \biggl\{ f \in \mathcal{O} (D) \colon \| f \|^2_{\varphi} := \int_D | f |^2 e^{- \varphi} < \infty \biggr\}.
\end{equation*}
\notag
$$
For $z \in D$, the weighted Bergman kernel is defined to be
$$
\begin{equation*}
B_{\varphi} (z) = \bigl(\inf \{ \| f \|^2_{\varphi} \colon f \in A^2 (D, \varphi),\, f(z) = 1 \}\bigr)^{- 1},
\end{equation*}
\notag
$$
if there exists $f \in A^2 (D, \varphi)$ with $f (z) \neq 0$ and otherwise $B_{\varphi}(z)$ is defined to be $0$. It is clearly that
$$
\begin{equation*}
B_{\varphi} (z) = \sup \{ | f (z) |^2 \colon f \in A^2 (D, \varphi),\, \| f \|^2_{\varphi} = 1 \}.
\end{equation*}
\notag
$$
If there is no confusion, we will call $B_{\varphi} (z)$ the Bergman kernel for short and omit the weight function $\varphi$. Proposition 2.13. Let $D\,{\subset}\, \mathbb{C}^n$ be a bounded domain, $\varphi$ be an upper semi-continuous function on $D$ and $B_{\varphi}(z)$ be the (diagonal) Bergman kernel weighted by $\varphi$. Let $z \in D$ be an arbitrary point, then $B_{\varphi}(z)>0$ if and only if there exists a holomorphic function on $D$ which is square integrable with respect to $\varphi$ such that $f (z) = 1$ and In particular, $e^{- \varphi} \in L^1_{\mathrm{loc}}$ near $z$. Proof. We assume that $B_{\varphi}(z)>0$ for some $z \in D$. By the definition of the Bergman kernel,
$$
\begin{equation*}
\infty > \frac{1}{B_{\varphi}(z)} = \inf \biggl\{ \int_D | f |^2 e^{- \varphi} \, d V \colon f \in \mathcal{O} (D), \, f (z) = 1 \biggr\}.
\end{equation*}
\notag
$$
Thus, there exists a sequence of holomorphic functions $\{ f_n \}$ such that, for every $n = 1, 2, \dots$, $f_n (z) = 1$ and and therefore, there exists a constant $C$ such that, for any $n = 1, 2, \dots$, We fixed an arbitrary $n$, since $f_n (z) = 1$, there exists a neighbourhood $U$ of $z$ such that on $U$ and and thus $e^{- \varphi}$ is locally integrable near $z$.
Again by (2.2), for any compact subset $K \subset D$, since $\varphi$ is an upper semi-continuous, there exists a constant $M_K$ such that on $K$. Therefore, for $n = 1, 2, \dots$,
$$
\begin{equation*}
e^{- M_K} \int_K | f_n |^2\, d V \leqslant \int_K | f_n |^2 e^{- \varphi}\, d V \leqslant \int_D | f_n |^2 e^{\varphi}\, d V < C,
\end{equation*}
\notag
$$
that is to say, $\{ f_n \}$ is locally $L^2$-bounded uniformly. Thus there exists a subsequence of $\{ f_n \}$, which is sill denoted by itself, converging to some holomorphic function $f$ uniformly on any compact subset of $D$. In particular, and
$$
\begin{equation*}
\int_D | f |^2 e^{- \varphi} \, d V \leqslant \liminf_{n \to \infty} \int_D | f_n |^2 e^{-\varphi} \, d V = \frac{1}{B_{\varphi}(z)}.
\end{equation*}
\notag
$$
By the above inequality, $f$ is square integrable with respect to $\varphi$ on $D$. On the other hand, one has by definition,
$$
\begin{equation*}
\frac{1}{B_{\varphi}(z)} \leqslant \int_D | f |^2 e^{- \varphi}\, d V,
\end{equation*}
\notag
$$
and so we have
$$
\begin{equation*}
\frac{1}{B_{\varphi}(z)} = \int_D | f |^2 e^{- \varphi}\, d V.
\end{equation*}
\notag
$$
The converse is obvious. $\Box$ Lemma 2.14. Let $\varphi \colon D \to [- \infty, \infty)$ be an upper semi-continuous function on a domain $D \subset \mathbb{C}^n$. Then the weighted Bergman kernel $B_{\varphi}(z)$ is continuous on $D$ and $\log B_{\varphi}(z)$ is plurisubharmonic on $D$. Proof. We first show that $B_{\varphi} (z)$ is upper semi-continuous. Let $z \in D$ be an arbitrary point and $z_j \in D$ that is convergent to $z$ as $j \to \infty$. Then, for any $\varepsilon > 0$, there exists $f_j \in A^2 (D, \varphi)$ such that $\| f_j \|^2_{\varphi} = 1$ and Since $\varphi$ is locally upper-bounded, it follows that $\{ f_j \}$ is a normal family on $D$ and thus there is a subsequence, which is still denoted by itself, converging to some $f \in \mathcal{O} (D)$ uniformly on any compact subset of $D$.
By Fatou’s lemma, one has
$$
\begin{equation*}
\| f \|^2_{\varphi} \leqslant \liminf_{j \to \infty} \| f_j \|^2_{\varphi} = 1
\end{equation*}
\notag
$$
and thus
$$
\begin{equation*}
B_{\varphi} (z) \geqslant \frac{| f (z) |^2}{\| f \|^2_{\varphi}} \geqslant | f (z) |^2 = \lim_{j \to \infty} | f_j (z_j) |^2 \geqslant \limsup_{j \to \infty} B_{\varphi} (z_j) - \varepsilon,
\end{equation*}
\notag
$$
and we conclude that $B_{\varphi} (z)$ is upper semi-continuous by letting $\varepsilon \to 0$.
Let $w \in D$ be a fixed point, then, for any $\varepsilon > 0$, there exists $f \in A^2 (D, \varphi)$ such that
$$
\begin{equation*}
\frac{| f (w) |^2}{\| f \|^2_{\varphi}} > B_{\varphi} (w) - \varepsilon.
\end{equation*}
\notag
$$
Thus
$$
\begin{equation*}
B_{\varphi} (z) \geqslant \frac{| f (z) |^2}{\| f \|^2_{\varphi}} = \lim_{w \to z} \frac{| f (w) |^2}{\| f \|^2_{\varphi}} \geqslant \liminf_{w \to z} B (w) - \varepsilon,
\end{equation*}
\notag
$$
and the lower semi-continuity follows by letting $\varepsilon \to 0$.
For the log-plurisubharmonicity, note that $\log B_{\varphi} (z)$ is upper semi-continuous and $\Box$Remark 2.15. It follows from Lemma 2.14 that the set is of measure zero since $\log B_{\varphi} (z)$ is plurisubharmonic. 2.5. MiscellaneousWe will also use the following classical results from real and complex analysis. Proposition 2.16 (see [23]). If $f \in L^1_{\mathrm{loc}}(\mathbb{R}^n)$, then almost every $x \in \mathbb{R}^n$ is a Lebesgue point of $f$. By a Lebesgue point of $f$ we mean a point $x \in \mathbb{R}^n$ which satisfies where $\sigma_n$ is the volume of $n$-dimensional unit ball. Lemma 2.17. Let $D \subset \mathbb{R}^n$ be a domain and $f$ be a measurable function on $D$ such that $f \in L^1_{\mathrm{loc}}$ near almost every $x \in D$. Then almost every $x \in D$ is a Lebesgue point of $f$. Proof. Denote by $E = \{ x \in D \colon f \notin L^1_{\mathrm{loc}} \text{ near } x\}$, then $E$ is a set of measure zero. Assume there exists $\{ x_k \} \subset E$ such that $x_k \to x$ as $k \to \infty$. If $f \in L^1_{\mathrm{loc}}$ near $x$, then there is a neighbourhood $U$ of $x$ such that $f \in L^1 (U)$. However, when $k$ is large enough, $x_k \in U$ and then $f \in L^1_{\mathrm{loc}}$ near $x_k$. Thus $E$ is a closed subset of $D$.
Since for every $x \in D \setminus E$, there exists a neighbourhood $V$ of $x$ such that $\overline{V} \subset D$ and $f \in L^1 (V)$, therefore, $f \in L^1_{\mathrm{loc}} (D\setminus E)$ and by the above proposition, almost every $x \in D \setminus E$ is a Lebesgue point of $f$. Since $E$ is a set of measure zero, almost every $x \in D$ is a Lebesgue point of $f$. $\Box$ Proposition 2.18 (see [24]). Let $D\subset\mathbb{C}^n$ be a domain and $\{u_j\}$ be a sequence of functions in $\operatorname{PSH}(D)$ which is locally uniformly bounded from above on $D$. If $\{u_j\}$ does not converge to $-\infty$ locally uniformly on $D$, then it admits a subsequence which converges to some $u\in \operatorname{PSH}(D)$ in $L^1_{\mathrm{loc}}(D)$. Proposition 2.19 (see [25]). Let $D \subset \mathbb{R}^n$ be a domain and $u \colon D \to [- \infty, \infty)$ be an upper semi-continuous function. Then $u$ is subharmonic if and only if is an increasing function of $r \in [0, \delta]$, where $c_n$ is the volume of the unit sphere in $\mathbb{R}^n$, $d V$ is the Lebesgue measure and $x_0 \in \{ x \in D\colon y \in D \textit{ if } | y - x | < \delta\}$, $\delta > 0$. In particular, if $u$ is subharmonic on $D$, where $\sigma_n$ is the volume of unit ball in $\mathbb{R}^n$. The following corollary is straightforward. Corollary 2.20. Let $D \subset \mathbb{R}^n$ be a domain and $u$ be a subharmonic function on $D$. Then, for any set $E$ of measure zero, for any $x_0\in D$. § 3. Proof of Theorem 1.2In this section, we prove Theorem 1.2. The proof is divided into several parts. Remark 3.1. Before we prove the above theorem, note that if we choose $\psi$ to be lower-bounded, then $\log B_{k\varphi}(z)\not\equiv -\infty$ implies $\log B_{k\varphi+\psi}(z) \not\equiv-\infty$. Indeed, if $\log B_{k\varphi}(z)> -\infty$ for some $z\in D$, there exists a holomorphic function $f\in A^2(D,e^{-\varphi})$ on $D$ such that $f(z)\neq 0$. Since $\psi$ is lower-bounded, there exists some $C>0$ such that and it follows that $\log B_{k\varphi+\psi}(z)>-\infty$. The following result establishes implication $(2)\,{\Rightarrow}\,(1)$ in Theorem 1.2. Proposition 3.2. With the same assumptions and notation in Theorem 1.2, if, for any domain $H \subset \mathbb{C}^m_{\tau}$ and any locally bounded $\psi \in \operatorname{PSH} (H \,{\times}\, D)$, the Bergman kernel $B_{k, \tau} (z)$ of the direct image of the trivial line bundle $(\underline{\mathbb{C}}, e^{- k \pi^{\ast} \varphi - \psi})\to H \times D$ is log-plurisubharmonic, then $\varphi$ is plurisubharmonic. Proof. Since $D$ is bounded, we may assume that the diameter of $D$ is $R$, then, for any $z, z_0 \in D$, $\log | z - z_0 |^2 < \log R^2$. Choose $\psi (\tau, z) = \max \{ \log | z - z_0 |^2 - \operatorname{Re} \tau,\, 0 \}$ and $H = \{\tau \in \mathbb{C}\colon \operatorname{Re} \tau < \log R^2 \}$, the half plane of $\mathbb{C}$. Hence $\psi$ is a plurisubharmonic function defined on $H \times D$ and depends only on $t = \operatorname{Re} \tau$.
By our hypothesis, is plurisubharmonic for any $p > 0$ and positive integer $k$, where
$$
\begin{equation*}
B_{k, \tau, p} (z) = \sup \biggl\{ \frac{1}{\int_D |f|^2 e^{- k \varphi - p \psi_{\tau}}} \colon f \in A^2 (D, k \varphi+p \psi_{\tau}),\, f (z) = 1 \biggr\},
\end{equation*}
\notag
$$
is the Bergman kernel weighted by $k \varphi+p \psi_{\tau}$.
By our assumption and Remark 3.1, it follows from $\psi\geqslant 0$ that, for every positive integer $k$, $\log B_{k, \tau, p} (z) \,{\not\equiv}\, {-}\infty$. It follows from Lemma 2.14 that $\{ z \,{\in}\, D \colon B_{k, \tau} (z) \,{=}\, 0 \}$ is a set of measure zero for every positive integer $k$ and is a set of measure zero. An appeal to Proposition 2.13 and Remark 2.15 shows that $e^{- k \varphi - p \psi_{\tau}} \in L^1_{\mathrm{loc}}$ near almost every $z \in D$ and $k \in \mathbb{N}^+$.Since $e^{- p \psi_{\tau}}$ is bounded for any fixed $\tau$, $e^{- k \varphi} \in L^1_{\mathrm{loc}}$ near almost every $z \in D$ and $k \in \mathbb{N}^+$. It then follows from Lemma 2.17 that almost every $z \in D$ is a Lebesgue point of $e^{- k \varphi}$, $k = 1, 2, \dots$ . Note that with this choice of $H$ and $\psi$, one has for any $k,p\in\mathbb{N}^+$ and $\tau\in H$,
$$
\begin{equation*}
\{ z \in D \colon B_{k, \tau,p} (z) = 0 \} \subset \{z\in D\colon B_{k,\log R^2}(z)=0\}.
\end{equation*}
\notag
$$
Let $z_0 \in D$ be an arbitrary point such that $z_0$ is a Lebesgue point of $e^{- k \varphi}$ for any $k \in \mathbb{N}^+$ and
$$
\begin{equation*}
z_0 \in \bigcap_k \{ z \in D \colon B_{k, \tau, p} (z) \neq 0 \},
\end{equation*}
\notag
$$
we may also assume that $\varphi (z_0) > - \infty$.
For any $f \in A^2 (D, k \varphi+p \psi_{\tau})$ with $f (z_0) = 1$, we have
$$
\begin{equation*}
\int_D |f|^2 e^{- k \varphi - p \psi_{\tau}}=\int_{D (\log |z - z_0 |^2 < t)} |f|^2 e^{- k \varphi}+\int_{D (\log |z - z_0 |^2 \geqslant t)} |f|^2 e^{- k \varphi - p \psi_{\tau}},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
D (\log |z - z_0 |^2 < t) = \{z \in D \colon \log |z - z_0 |^2 < t\}
\end{equation*}
\notag
$$
and similarly, for $D (\log |z - z_0 |^2 \geqslant t)$.
Since
$$
\begin{equation*}
\limsup_{z \to z_0,\, z \in D \setminus E} \varphi (z) = \varphi (z_0),
\end{equation*}
\notag
$$
for any $E \subset D$ of measure zero, it follows that
$$
\begin{equation*}
\liminf_{z \to z_0,\, z \in D} |f (z) |^2 e^{- \varphi (z)} = e^{- \varphi(z_0)}.
\end{equation*}
\notag
$$
That is to say, for any $\varepsilon > 0$, there exists $\delta \in\mathbb{R}$ such that on $D (\log |z - z_0 |^2 < \delta)$,
$$
\begin{equation*}
|f (z) |^2 e^{- k \varphi (z)} > e^{- k \varphi (z_0)} - \varepsilon,
\end{equation*}
\notag
$$
and thus if $t < \delta$,
$$
\begin{equation*}
\int_{D (\log |z - z_0 |^2 < t)} |f|^2 e^{- k \varphi} > (e^{- k \varphi(z_0)} - \varepsilon) \sigma_{2 n} e^{nt},
\end{equation*}
\notag
$$
that is,
$$
\begin{equation*}
\frac{1}{\sigma_{2 n} e^{nt}} \int_{D (\log |z - z_0 |^2 < t)} |f|^2 e^{- k \varphi} > e^{- k \varphi (z_0)} - \varepsilon,
\end{equation*}
\notag
$$
where $\sigma_{2 n}$ is the volume of $2 n$-dimensional unit ball. It follows from $B_{k, \tau, p} (z_0) > 0$ and Proposition 2.13 that there exists $f_{\tau, p} \in A^2 (D, k\varphi+p \psi_{\tau})$ with $f_{\tau, p} (z_0) = 1$ such that
$$
\begin{equation*}
\frac{1}{B_{k, \tau, p} (z_0)} = \int_D |f_{\tau, p} |^2 e^{- k \varphi - p \psi_{\tau}}.
\end{equation*}
\notag
$$
Now we have
$$
\begin{equation}
\begin{aligned} \, &\log B_{k, \tau, p} (z_0)+\log \sigma_{2 n}+nt = -\log \int_D |f_{\tau, p} |^2 e^{- k \varphi - p \psi_{\tau}}+\log \sigma_{2 n}+nt \nonumber \\ &\quad =- \log \biggl( \int_{D (\log |z - z_0 |^2 < t)} |f_{\tau, p} |^2 e^{-k \varphi } +\int_{D (\log |z - z_0 |^2 \geqslant t)} |f_{\tau, p} |^2 e^{- k \varphi - p \psi_{\tau}} \biggr) \nonumber \\ &\quad\qquad +\log \sigma_{2 n}+nt \nonumber \\ &\quad \leqslant-\log \biggl( \int_{D (\log |z - z_0 |^2 < t)} |f_{\tau, p}|^2 e^{- k \varphi } \biggr)+\log \sigma_{2 n}+nt \nonumber \\ &\quad= \log \biggl( \frac{1}{\sigma_{2 n} e^{nt}} \int_{D (\log |z - z_0|^2 < t)} |f_{\tau, p} |^2 e^{- k \varphi } \biggr)^{-1} \leqslant \log (e^{- k \varphi (z_0)} - \varepsilon)^{- 1}, \end{aligned}
\end{equation}
\tag{3.1}
$$
for any $p$ and $t < 0$ small enough.
On the other hand, since $B_{k, \tau, p}$ depends only on $t = \operatorname{Re} \tau$, is a convex function. It follows from (3.1) that $\log B_{k, \tau, p} (z_0)+\log \sigma_{2 n}+nt$ is upper-bounded as $t\to-\infty$ and hence an increasing function. Hence, for any $t < \log R^2$, we have
$$
\begin{equation}
\sigma_{2 n} e^{n \log R^2} B_{k, \log R^2} (z_0) = \sigma_{2 n} e^{n \log R^2} B_{k, \log R^2, p} (z_0) \geqslant \sigma_{2 n} e^{nt} B_{k, \tau, p} (z_0),
\end{equation}
\tag{3.2}
$$
for any $p > 0$, the first equality holds because $\log |z - z_0 | < \log R^2$ for any $z, z_0 \in D$.
Now, for any $f \in \mathcal{O} (D)$ with $f (z_0) = 1$ satisfying since $\psi \geqslant 0$, we have
$$
\begin{equation*}
\int_D |f|^2 e^{- k \varphi - p \psi_{\tau}} \leqslant \int_D |f|^2 e^{- k \varphi} < \infty.
\end{equation*}
\notag
$$
It then follows that
$$
\begin{equation*}
\sigma^{- 1}_{2 n} e^{-nt} \frac{1}{B_{k, \tau, p}} \leqslant \sigma^{-1}_{2 n} e^{-nt} \int_D |f|^2 e^{- k \varphi - p \psi_{\tau}}.
\end{equation*}
\notag
$$
Therefore, by (3.2) and the above inequality,
$$
\begin{equation*}
\begin{aligned} \, \sigma^{- 1}_{2 n} R^{- 2 n} \frac{1}{B_{k, \log R^2}}&\leqslant \sigma^{- 1}_{2 n} e^{- nt} \int_{D (\log |z - z_0 |^2 < t)} |f|^2 e^{- k \varphi} \\ &\qquad+\sigma^{- 1}_{2 n} e^{- nt} \int_{D (\log |z - z_0 |^2 \geqslant t)} |f|^2 e^{- k \varphi - p \psi}. \end{aligned}
\end{equation*}
\notag
$$
Now making $p \to \infty$, we have
$$
\begin{equation*}
\sigma^{- 1}_{2 n} R^{- 2 n} \frac{1}{B_{\log R^2}} \leqslant \sigma^{- 1}_{2 n} e^{- nt} \int_{D (\log |z - z_0|^2 < t)} |f|^2 e^{- k \varphi},
\end{equation*}
\notag
$$
and further, by letting $t \to - \infty$, since $z_0$ is a Lebesgue point of $e^{- k\varphi}$ and since $\varphi (z_0) > - \infty$, we have
$$
\begin{equation}
\sigma^{- 1}_{2 n} R^{- 2 n} \frac{1}{B_{k, \log R^2}} \leqslant e^{- k \varphi (z_0)}.
\end{equation}
\tag{3.3}
$$
By our assumption that $B_{k, \log R^2} (z_0) = B_{k, \log R^2, p} (z_0) > 0$ and by Proposition 2.13, there exists $F \in \mathcal{O} (D)$ with $F (z_0) = 1$ satisfying
$$
\begin{equation*}
\sigma^{- 1}_{2 n} R^{- 2 n} \int_D |F|^2 e^{- k \varphi} = \sigma^{-1}_{2 n} R^{- 2 n} \frac{1}{B_{k, \log R^2}},
\end{equation*}
\notag
$$
it then follows from (3.3) that
$$
\begin{equation*}
\sigma^{- 1}_{2 n} R^{- 2 n} \int_D |F|^2 e^{- k \varphi} \leqslant e^{-k \varphi (z_0)}.
\end{equation*}
\notag
$$
By Theorem 2.7 and Remark 2.8, there exists a sequence of plurisubharmonic functions $\{\varphi_m \} \subset \operatorname{PSH} (D)$ converging to $\varphi$ decreasingly on any Lebesgue point $z$ of $e^{- k\varphi}$ with $B_{k,{\tau},p}(z)>0$, $\varphi (z) > - \infty$ for any $k\in\mathbb{N}^+, {\tau}\in H$ and $p>0$. Since $\{\varphi_m \} \subset \operatorname{PSH} (D)$ does not converge to $- \infty$ uniformly on $D$, there exists a plurisubharmonic function $\widetilde{\varphi}$ on $D$ such that
$$
\begin{equation*}
\lim_{m \to \infty} \varphi_m (z) = \widetilde{\varphi} (z).
\end{equation*}
\notag
$$
By Lemma 2.17, almost every $z \in D$ is a Lebesgue point of $e^{- k \varphi}$ for any $k \in \mathbb{N}^+$. Since $\bigcup_k \{z \in D \colon B_{k,{\tau},p} (z) = 0 \}$ and $\{ z \in D \colon \varphi (z) = - \infty \}$ are subsets of measure zero, it follows that, for almost every $z \in D$, On the other hand, on $D$ we have
$$
\begin{equation}
\limsup_{w \to z,\, w \in D \setminus E} \varphi (w) = \varphi (z)
\end{equation}
\tag{3.4}
$$
for any subset $E$ of measure zero. For any $z \in D$, we may choose $E$ in (3.4) to be the union of the set of non Lebesgue points of $e^{- k \varphi}$, $k = 1, 2, \dots $, $\bigcup_k \{ z \in D\colon B_{k, \log R^2} (z) = 0 \}$ and $\{ z \in D \colon \varphi (z) = - \infty \}$, then therefore, $\varphi$ is a plurisubharmonic function on $D$. $\Box$ The next result establishes implication $(1)\,{\Rightarrow}\,(4)$ in Theorem 1.2. Proposition 3.3. With the same assumptions and notation in Theorem 1.2, if $\varphi$ is plurisubharmonic, then, for any domain $H \subset \mathbb{C}^m_{\tau}$ and any locally bounded $\psi \in \operatorname{PSH} (H \times D)$, the direct image bundle $F$ of the trivial line bundle $(\underline{\mathbb{C}}, e^{- k \pi^{\ast} \varphi - \psi}) \to H \times D$ is semi-positive in the sense of Nakano. Proof. By Theorem 1.3 in [5], $F$ is semi-positive in the sense of Griffiths. We may assume that $H$ is pseudoconvex. By Definition 2.10, it suffices to show that, for any $f \in C^{\infty}_{\mathrm{c}} (H, \bigwedge^{m, 1} T^{\ast}_H \otimes F)$ with $\overline{\partial}_{\tau} f = 0$ and for any plurisubharmonic function $\rho$ on $H$, there exists $u \in L^2 (H, \bigwedge^{m, 0} T^{\ast}_H \otimes F)$ satisfying $\overline{\partial} u = f$ and
Let $f \in C^{\infty}_{\mathrm{c}} (H, \bigwedge^{m, 1} T^{\ast}_H \otimes F)$ with $\overline{\partial}_{\tau} f = 0$. We can regard $f$ as $f (\tau, z)$ which is smooth and compact supported in $\tau$ and holomorphic in $z$, therefore, $\overline{\partial} f = \overline{\partial}_{\tau} f+\overline{\partial}_z f = 0$. We write $f = \sum_{j = 1}^m f_j (\tau, z)\, d \tau \wedge d \overline{\tau}_j$ with $f_j(\tau, z) \in F_{\tau}$ for $\tau \in H$ and $d \tau$ denotes $d \tau_1\, \wedge d \tau_2 \wedge \dots \wedge d \tau_m$. Since $\overline{\partial} f = 0$, it follows from Hörmander’s $L^2$-existence theorem (Theorem 2.1) that there exists $u \in L^2 (H \times D)$ such that $\overline{\partial}_{\tau}u=f$ and
$$
\begin{equation}
\int_{H \times D} | u |^2 e^{- k \varphi - \psi - \rho} \leqslant \int_{H \times D} | f |^2_{i\, \partial\, \overline{\partial} \rho} e^{- k \varphi - \psi - \rho}.
\end{equation}
\tag{3.5}
$$
It follows from $\overline{\partial} u = f$ that $\partial u/\partial \overline{z}_j = 0$ for $j = 1, 2, \dots, n$, that is $\overline{\partial}_{\tau}u=f$. Therefore, we can regard $u$ as a section of $F$ which satisfies the integral condition.
By the Fubini theorem, we may rewrite (3.5) as
$$
\begin{equation*}
\begin{aligned} \, \int_H | u |^2_{\tau} e^{- \rho} &=\int_{H \times D} | u |^2 e^{- k \varphi - \psi - \rho} \leqslant \int_{H \times D} | f |^2_{i \,\partial\, \overline{\partial} \rho} e^{- k \varphi - \psi - \rho} \\ &= \int_H \langle [i\, \partial\,\overline{\partial}\rho\otimes \operatorname{Id}_{F},\Lambda_{\omega}] ^{-1} f, f \rangle^2_{\tau} e^{- \rho}, \end{aligned}
\end{equation*}
\notag
$$
the last equality follows from Lemma 2.11 and Remark 2.12. Thus, by Definition 2.10, $F$ is semi-positive in the sense of Nakano. $\Box$ Remark 3.4. Proposition 3.3 is closely related to the question raised by Lempert in [26] which asked whether a $C^2$ hermitian metric whose curvature dominates $0$ is positive in the sense of Nakano. Actually, results in [5] lead to an affirmative answer to this question, see [27], [28]. With Propositions 3.2 and 3.3, one reaches Theorem 1.2 immediately. We continue the proof of Theorem 1.2. $(4)\Rightarrow(3)$ follows from Definition 2.10. It follows from the assumptions and Remark 3.1 that fiberwised Bergman kernel $B_{k,\tau} $ does not vanish for any $k\in\mathbb{N}^+$ and $\tau\in H$. $(3)\Rightarrow (2)$ follows from that $B_{k,{\tau}}$ can be regarded as a non-trivial holomorphic section of the dual bundle. Indeed, for any $f \in \mathcal{O} (H \times D)$ such that $f (\,{\cdot}\,, z)$ is square integrable with respect to $k \varphi+\psi$,
$$
\begin{equation*}
f (\tau, w) = \int_D f (\tau,{\cdot}\,) B_{k,{\tau}} (\,{\cdot}\,, w) e^{- k \varphi - \psi},
\end{equation*}
\notag
$$
for any $w \in D$. It again follows from Definition 2.10 that $B_{k,{\tau}}(z)$ is log-plurisubharmonic. $\Box$ Remark 3.5. With the same assumptions, it can be seen from the proof of Theorem 1.2 that it only needs $ B_{k,\tau} (z)$, a specific family of elements in the dual bundle, to satisfy the log-plurisubharmonic property and non-triviality to obtain Nakano positivity. Moreover, it follows that if one has just $\log B_{k,\tau} (z)\not\equiv -\infty$ (for any $k\in\mathbb{N}^+$ and $\tau\in H$) to be plurisubharmonic, then $\log | \xi |^2_\tau$ is plurisubharmonic for every $\xi$ in the dual bundle. The method adopted here with a suitable modification can also be used to obtain an analogue of Theorem 1.2 for the direct image of a hermitian holomorphic vector bundle. More precisely, we have the following result. Theorem 3.6. Let $D \subset \mathbb{C}^n_z$ be a bounded domain and $(E, h) \to D$ be a holomorphic vector bundle with a hermitian metric $h$ whose minimal eigenvalue is locally lower bounded. We assume that, for any open set $U\subset D$ and any local holomorphic section $\xi \in H^0(U, E^{\ast})$, for every $z_0 \in U$, where $E$ is any set of measure zero. Assume that for any domain $H \subset \mathbb{C}^m_{\tau}$ and any locally bounded $\psi \in \operatorname{PSH} (H \times D)$, the direct image $F$ (over $H$) of $(\pi^{\ast} E^{\otimes k}, \pi^{\ast} h^k e^{- \psi}) \to H \times D$ is semi-positive in the sense of Griffiths for any positive integer $k$. Assume further that (1) for any constant local section $a$ of $E$, $| a^{\otimes k}|^2_{h^k}$ is locally integrable for every $k \in \mathbb{N}^+$; (2) for almost every $z \in D$, $k \in \mathbb{N}^+$, $\tau \in H$ and $a \in E_z$ with $|a^{\otimes k} |^2_{h^k e^{- \psi_{\tau}}} < \infty$ at $z$, there exists a holomorphic section $v_{k, \tau}$ of $E^{\otimes k}$ such that $v_{k, \tau} (z) = a^{\otimes k}$ and Then $(E, h)$ is semi-positive in the sense of Griffiths. Proof. Since $D \subset \mathbb{C}^n$ is a bounded domain, we assume that the diameter of $D$ is $R$, therefore, for any $z, z_0 \in D$, one has $\log |z - z_0 |^2 \leqslant \log R^2$. Again, when there is not any confusion, we will write $h$ instead of $\pi^{\ast} h$. Denote by $\psi (\tau, z) = \max \{\log |z - z_0 |^2 - \operatorname{Re} \tau,\, 0\}$ and $H = \{\tau \in \mathbb{C}\colon \operatorname{Re} \tau < \log R^2 \}$, the half plane of $\mathbb{C}$. Then $\psi$ is a plurisubharmonic function defined on $H \times D$ and depends only on $t = \operatorname{Re} \tau$. For a given positive integer $k$ and for any $p > 0$, As vector spaces, all $F_{k, \tau, p}$ are locally equal for any $p$. Note that and $\|u\|^2_{k, \log R^2} = \|u\|^2_{k, \log R^2, p}$ for any $p > 0$, where $D (\log |z - z_0 |^2 < t) = \{z \in D\colon \log |z - z_0 |^2 < t\}$ and similarly $\{z \in D \colon \log |z - z_0 |^2 \geqslant t\}$ will be denoted by $D (\log |z - z_0 |^2 \geqslant t)$, etc.
Without loss of generality, we assume that $E$ is trivial on $D$ since it only needs the local property of $E$ in the last step of our proof. By our assumption, any constant section $a\in H^0 (D, E)$, $| a^{\otimes k}|^2_{h^k}$ is locally integrable on $D$, and by Lemma 2.17, almost every $z\in D$ is a Lebesgue point for $|a^{\otimes k}|^2_{h^k}$ for every $k\in\mathbb{N}^+$. Without loss of generality, we may assume $|a^{\otimes k}|^2_{h^k e^{-\psi_{\tau}"}}<\infty$ at $z$. By our second condition, there exists a section $v_{k,\tau, p}\in L^2(D, E^{\otimes k}, h^{\otimes k} e^{- p \psi_\tau})$ such that $v_{k, \tau, p} (z_0) = a^{\otimes k}$ and $v_{k, \tau, p}$ has the minimal $L^2$-norm, in particular, $v_{k, \log R^2}$ is the minimal $L^2$-extension on $D$ such that $v_{k,\log R^2} (z_0) = a^{\otimes k}$ and $\|v_{k, \log R^2}\|^2_{h^k }<\infty$. Up to a set of measure zero, we may also assume that $z_0$ is a Lebesgue point of $\| v_{k, \log R^2} \|^2_{h^k }$. It follows that
$$
\begin{equation}
\|v_{k,\tau, p} \|^2_{k,\tau, p} = \int_D |v_{k, \tau, p} |^2_{h^{ k}} e^{- p\psi_\tau} = \sup \frac{| \langle \xi, v_{k, \tau, p} \rangle |^2}{\| \xi \|^2_{k, \tau, p}},
\end{equation}
\tag{3.6}
$$
where the supreme is taken over elements in $F^{\ast}_{k,\tau, p}$ that vanish on $I = \{s \in F_{k,\tau, p}\colon s (z_0) = 0\}$. We can take $\xi = \xi_g$ of the form where $g$ is a smooth section of $E^{\otimes k}$ with compact support and $(\xi_g, u)$ represents the standard (global) Euclidean inner product, that is, $g \cdot \overline{u}^{\,T}$. It is obvious that such $\xi_g$ forms a dense subspace of the subspace formed by all elements in $F^{\ast}_{k,\tau, p}$ that vanish on $I = \{s \in F_{k,\tau, p} \colon s (z_0) = 0\}$.
By our hypothesis, $(F, \|\,{\cdot}\,\|_{k, \tau, p})$ is semi-positive in the sense of Griffiths, it follows that $\log \| \xi_g \|^2_{k, \tau, p}$ is a plurisubharmonic function. Since $\log \| \xi_g \|^2_{k, \tau, p}$ depends only on $t = \operatorname{Re} \tau$, it follows that this is a convex function. Since the minimal eigenvalue of $h$ is locally lower bounded, there exist $C_1 > 0$ such that, for any $u \in F_{k, \tau, p}$, one has on $D (\log |z - z_0 |^2 \leqslant t)$, where again $|u|^2$ denotes the standard Euclidean norm of $u$ on $E$. It then follows form the sub-mean inequality that
$$
\begin{equation*}
|u (z_0) |^2 \leqslant \sigma_{2 n}^{- 1} e^{- nt} \int_{D (\log |z - z_0 |^2 < t)} |u|^2 \leqslant C_1 \sigma_{2 n}^{- 1} e^{- nt} \int_{D (\log |z - z_0 |^2 < t)} |u|^2_{h^k},
\end{equation*}
\notag
$$
which means that $\| \xi_g \|^2_{k, \tau, p} \sigma_{2 n} e^{nt}$ is bounded above as $t$ small enough and $p$ large enough. Since $\log \| \xi_g \|^2_{k, \tau, p}$ is convex, we have is an increasing function.
By (3.6), we have, for any $t < \log R^2$ and any $p > 0$,
$$
\begin{equation*}
\sigma_{2 n}^{- 1} R^{- 2 n} \|v_{k, \log R^2} \|^2_{k, \log R^2} = \sup \frac{| \langle \xi_g, v_{k, \log R^2} \rangle |^2}{\sigma_{2 n} R^{2 n} \| \xi_g \|^2_{k, \log R^2}} \leqslant \sup \frac{| \langle \xi_g, v_{k, \tau, p} \rangle|^2}{\sigma_{2 n} e^{nt} \| \xi_g \|^2_{k, \tau, p}}.
\end{equation*}
\notag
$$
Another appeal to (3.6) shows that
$$
\begin{equation*}
\begin{aligned} \, \sup \frac{| \langle \xi_g, v_{k, \tau, p} \rangle |^2}{\sigma_{2 n} e^{nt}\| \xi_g \|^2_{k, \tau, p}} &= \sigma_{2 n}^{- 1} e^{- nt} \int_D |v_{k,\tau, p} |^2_{h^k} e^{- p \psi_\tau} \leqslant \sigma_{2 n}^{- 1} e^{- nt} \int_D |v_{k, \log R^2}|^2_{h^k} e^{- p \psi_\tau} \\ &=\sigma_{2 n}^{- 1} e^{- nt} \int_{D (\log |z - z_0 |^2 < t)} |v_{k, \log R^2} |^2_{h^k} \\ &\qquad +\sigma_{2 n}^{- 1} e^{- nt} \int_{D (\log |z - z_0 |^2 \geqslant t)} |v_{k, \log R^2} |^2_{h^k} e^{- p \psi_\tau} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\begin{aligned} \, \sigma_{2 n}^{- 1} R^{- 2 n} \|v_{k, \log R^2} \|^2_{k, \log R^2} &\leqslant \sigma_{2 n}^{- 1} e^{- nt} \int_{D (\log |z - z_0 |^2 < t)} |v_{k, \log R^2}|^2_{h^k} \nonumber \\ &\qquad+\sigma_{2 n}^{- 1} e^{- nt} \int_{D (\log |z - z_0 |^2 \geqslant t)} |v_{k, \log R^2} |^2_{h^k} e^{- p \psi_\tau}. \end{aligned}
\end{equation}
\tag{3.7}
$$
Letting $p \to \infty$, we find that the second term of the right-hand side of (3.7) tends to zero. Since $z_0$ is a Lebesgue point of $|v_{k, \log R^2} |^2_{h^k}$, we have
$$
\begin{equation*}
\lim_{t \to - \infty} \sigma_{2 n}^{- 1} e^{- nt} \int_{D (\log |z - z_0 |^2 < t)} |v_{k, \log R^2} |^2_{h^k} = |a^{\otimes k} |^2_{h^k}|_{z_0}.
\end{equation*}
\notag
$$
Thus, by letting $t \to - \infty$ and using (3.7), we get
$$
\begin{equation*}
\sigma_{2 n}^{- 1} R^{- 2 n} \|v_{\log R^2} \|^2_{\log R^2} \leqslant \liminf_{t \to - \infty} \sigma_{2 n}^{- 1} e^{-nt} \int_{D (\log |z - z_0 |^2 < t)} |v_{k, \log R^2} |^2_{h^k} = |a^{\otimes k} |^2_{h^k} |_{z_0}.
\end{equation*}
\notag
$$
Now let $\widetilde{\xi}$ be a local holomorphic section of $E^{\ast}$ on a Stein neighbourhood $U$ of $z_0$. We may assume that at $z_0$, $|a|_h = 1$ and $\langle \widetilde{\xi} (z_0), a\rangle_h = | \widetilde{\xi} (z_0) |_h$. We regard $\widetilde{\xi}^{\otimes k}$ as a local holomorphic section of $(E^{\ast})^{\otimes k}$. It then follows that
$$
\begin{equation*}
| \widetilde{\xi}^{\otimes k} |_{h^k} = | \xi |^k_h,\qquad | \widetilde{\xi}^{\otimes k} (z_0) |_{h^k} = \langle \widetilde{\xi}^{\otimes k} (z_0), a^{\otimes k} \rangle
\end{equation*}
\notag
$$
and by definition,
$$
\begin{equation}
| \widetilde{\xi} (z) |^k_h \geqslant \frac{| \langle \widetilde{\xi}^{\otimes k} (z), v_{k, \log R^2} \rangle |}{|v_{k, \log R^2} |_{h^k}},
\end{equation}
\tag{3.8}
$$
where $z \in U$, $v_{k, \log R^2}$ is the minimal $L^2$ holomorphic section of $E^{\otimes k}$ such that $v_{k, \log R^2} (z_0) = a^{\otimes k}$. We rewrite (3.8) as
$$
\begin{equation}
e^{- k \log | \widetilde{\xi} (z) |_h} \leqslant e^{- \log | \langle \widetilde{\xi}^{\otimes k} (z), v_{k, \log R^2} \rangle |} |v_{k, \log R^2} |_{h^k}.
\end{equation}
\tag{3.9}
$$
Since $\widetilde{\xi}^{\otimes k}$ and $v_{k, \log R^2}$ are local holomorphic sections of $(E^{\ast})^{\otimes k}$ and $E^{\otimes k}$, respectively, $\langle\widetilde{\xi}^{\otimes k} (z), v_{k, \log R^2} \rangle$ is a holomorphic function on $U$. By the Ohsawa–Takegoshi extension theorem, there is a holomorphic section $H$ on $U$ such that $H (z_0) = 1$ and
$$
\begin{equation*}
\int_U |H|^2 e^{- 2 \log | \langle \widetilde{\xi}^{\otimes k} (z), v_{k, \log R^2} \rangle |} \leqslant Ce^{- 2 \log | \langle \widetilde{\xi}^{\otimes k} (z_0), a^{\otimes k} \rangle |} = Ce^{- 2 \log | \widetilde{\xi}^k (z_0) |_{h^k}}.
\end{equation*}
\notag
$$
It then follows from (3.9) that
$$
\begin{equation*}
\begin{aligned} \, \int_U |H| e^{- k \log | \widetilde{\xi} (z) |_h} &\leqslant \int_U |H| e^{- \log |\langle \widetilde{\xi}^{\otimes k} (z), v_{k, \log R^2} \rangle |} |v_{k, \log R^2}|_{h^k} \\ &\leqslant \biggl[ \int_U |H|^2 e^{- 2 \log | \langle \widetilde{\xi}^{\otimes k}(z), v_{k, \log R^2} \rangle |} \int_U |v_{k, \log R^2} |^2_{h^k} \biggr]^{1/2} \\ &\leqslant \bigl[ Ce^{- 2 \log | \widetilde{\xi}^k (z_0) |_{h^k}} \sigma_{2 n}^{-1} R^{- 2 n} \bigr]^{1/2} = \sqrt{C \sigma_{2 n}^{- 1} R^{- 2 n}}\, e^{- k \log |\widetilde{\xi} (z_0)|}. \end{aligned}
\end{equation*}
\notag
$$
By Theorem 2.7 and Remark 2.8, there exists a sequence of plurisubharmonic functions $\{\varphi_m \} \subset \operatorname{PSH} (U)$ converging to $\log | \widetilde{\xi} |_h$ decreasingly on the set of Lebesgue point of $| a^{\otimes k} |_{h^k}$ up to a set of measure zero, $k = 1, 2, 3, \dots$, with $| a^{\otimes k}|_{h^k} < \infty$ and the requirement of our second condition. Since $\{\varphi_m \} \subset \operatorname{PSH} (U)$ does not converge to $- \infty$ uniformly on $U$, there exists a plurisubharmonic function $\widetilde{\varphi}$ on $U$ such that
$$
\begin{equation*}
\lim_{m \to \infty} \varphi_m (z) = \widetilde{\varphi} (z).
\end{equation*}
\notag
$$
By Lemma 2.17, almost every $z \in U$ is a Lebesgue point of $| a^{\otimes k}|_{h^k}$ (up to a set of measure zero) for any $k \in \mathbb{N}^+$. Since the set of points which do not satisfy our second condition and the set of points at which $|a|_h = \infty$ are of measure zero, it follows that, for almost every $z \in U$,
$$
\begin{equation}
\log | \widetilde{\xi} |_h = \widetilde{\varphi} (z).
\end{equation}
\tag{3.10}
$$
On the other hand, we have on $U$,
$$
\begin{equation}
\limsup_{z \to z_0,\, z \in U \setminus E} \log | \widetilde{\xi} (z) |_h = \log |\widetilde{\xi} (z_0)|_h
\end{equation}
\tag{3.11}
$$
for any subset $E$ of measure zero. For any $z \in U$, we may choose $E$ in (3.11) to be the union of the set of non Lebesgue points of $|a^{\otimes k}|_{h^k}, k = 1, 2, \dots$, and $\{ z \in U \colon |a|_{h} = - \infty \}$, then therefore, $\log | \widetilde{\xi} |_h$ is a plurisubharmonic function on $U$. $\Box$ § 4. Proof of Theorem 1.5In this section, we prove Theorem 1.5 with the method which is similar to that of Theorem 1.2. Proof of Theorem 1.5. We suppose the diameter of $D$ is $R$, then, for any $z, w \in D$, we have $\log | z - w |^2 < \log R^2$. By our hypothesis, is plurisubharmonic for any $k > 0$ and any plurisubharmonic function $\psi(\tau, z)$ on $H_{\tau} \times D_z$. All the integrals are taken with respect to $d V_z$ except written explicitly. Since $e^{-k \varphi}$ is integrable for $k\in\mathbb{N}^+$, almost every $z\in D$ is a Lebesgue point of $e^{-k \varphi}$, $k=1,2,\dots$ . Since $\varphi\in L^1_{\mathrm{loc}}(D)$, for almost every $z\in D$, $\varphi(z)>-\infty$.
In particular, we choose that $H = \{ \tau \in \mathbb{C} \colon \operatorname{Re} \tau < \log R^2 \}$ is the half plane of $\mathbb{C}$ and $\psi (\tau, z) = \max \{ \log | z - z_0 |^2 - \operatorname{Re} \tau,\, 0 \}$, where $z_0$ is a Lebesgue point of $e^{-k\varphi}$, $k=1,2,\dots$, with $\varphi(z)>-\infty$. Then $\widetilde{\psi}$ is a plurisubharmonic function defined on $H \times D$ and depends only on $t = \operatorname{Re} \tau$. Note that
$$
\begin{equation*}
\widetilde{\psi} (\tau) = - \log \int_D e^{- k \varphi (z) - p \psi(\tau, z)}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, &\log \int_D e^{- k \varphi (z) - p \psi (\tau, z)} \\ &\qquad= \log \biggl(\int_{D (\log | z - z_0 |^2 < t)} e^{- k \varphi(z)} + \int_{D (\log | z - z_0 |^2 \geqslant t)} e^{- k\varphi (z) - p \psi (\tau, z)}\biggr) \\ &\qquad >\log \int_{D (\log | z - z_0 |^2 < t)} e^{- k \varphi(z)}, \end{aligned}
\end{equation*}
\notag
$$
where again $D (\log | z - z_0 |^2 < t)$ denotes $\{ z \in D \colon \log | z - z_0 |^2 < t \}$ and similarly, for $D (\log | z - z_0 |^2 \geqslant t)$. Since
$$
\begin{equation*}
\limsup_{z\to z_0,\, z\in D\setminus E}\varphi(z)=\varphi(z_0),
\end{equation*}
\notag
$$
for any set $E$ of measure zero, it follows that
$$
\begin{equation*}
\liminf_{z\to z_0,\, z\in D}e^{-\varphi(z)}=e^{-\varphi(z_0)}.
\end{equation*}
\notag
$$
That is to say, for any $\varepsilon > 0$, there exists $\delta \in\mathbb{R}$ such that on $D (\log | z - z_0 |^2 < \delta)$,
$$
\begin{equation*}
e^{- k \varphi (z)}>e^{- k \varphi (z_0)} - \varepsilon,
\end{equation*}
\notag
$$
and thus if $t<\delta$,
$$
\begin{equation*}
\int_{D (\log | z - z_0 |^2 < t)} e^{- k \varphi}>(e^{- k \varphi (z_0)} - \varepsilon) \sigma_{_{2 n}} e^{nt},
\end{equation*}
\notag
$$
that is,
$$
\begin{equation*}
\frac{1}{\sigma_{_{2 n}} e^{nt}} \int_{D (\log | z - z_0 |^2 < t)} e^{- k \varphi}> e^{- k \varphi (z_0)} - \varepsilon,
\end{equation*}
\notag
$$
where $\sigma_{2 n}$ is the volume of $2 n$-dimensional unit ball. Thus for any $p$, when $t < 0$ small enough, we have
$$
\begin{equation}
\begin{aligned} \, &\widetilde{\psi} (\tau)+\log \sigma_{2 n}+n t = - \log \frac{1}{\sigma_{2 n} e^{n t}} \int_D e^{- k \varphi (z) - p\psi (\tau, z)} \nonumber \\ &\qquad\leqslant -\log \frac{1}{\sigma_{2 n} e^{n t}} \int_{D (\log | z - z_0 |^2 < t)} e^{- k \varphi (z)} \leqslant -\log ( e^{- k \varphi (z_0)} - \varepsilon). \end{aligned}
\end{equation}
\tag{4.1}
$$
On the other hand, since $\widetilde{\psi} (\tau)+\log \sigma_{2 n}+n t$ depends only on $t = \operatorname{Re} \tau$, $\widetilde{\psi} (\tau)+\log \sigma_{2 n}+n t$ is a convex function. It then follows from (4.1) that is an increasing function. Now, for any $t < \log R^2$,
$$
\begin{equation}
\widetilde{\psi} (\log R^2)+\log \sigma_{2 n}+n \log R^2 \geqslant \widetilde{\psi} (t)+\log \sigma_{2 n}+n t.
\end{equation}
\tag{4.2}
$$
Since $\psi \geqslant 0$, it follows from (4.2) that
$$
\begin{equation*}
\begin{aligned} \, &\frac{1}{\sigma_{2 n} R^{2 n}} \int_D e^{- k \varphi(z)} \leqslant \frac{1}{\sigma_{2 n} e^{n t}} \int_D e^{- k \varphi(z) - p \psi (\tau, z)} \\ &\qquad =\frac{1}{\sigma_{2 n} e^{n t}} \int_{D (\log | z - z_0 |^2 < t)} e^{- k \varphi (z)}+\frac{1}{\sigma_{2 n} e^{n t}} \int_{D (\log | z - z_0 |^2 \geqslant t)} e^{- k \varphi (z) - p \psi (\tau, z)}. \end{aligned}
\end{equation*}
\notag
$$
Therefore, let $p \to \infty$, we have
$$
\begin{equation*}
\frac{1}{\sigma_{2 n} R^{2 n}} \int_D e^{- k \varphi (z)} \leqslant \frac{1}{\sigma_{2 n} e^{n t}} \int_{D (\log | z - z_0 |^2 < t)} e^{- k \varphi (z)},
\end{equation*}
\notag
$$
and now, making $t \to - \infty$, since $z_0$ is a Lebesgue point of $e^{-\varphi}$, we have
$$
\begin{equation*}
\frac{1}{\sigma_{2 n} R^{2 n}} \int_D e^{- k \varphi (z)} \leqslant e^{- k \varphi (z_0)}.
\end{equation*}
\notag
$$
By Theorem 2.7 and Remark 2.8, there exists a sequence of plurisubharmonic functions $\{\varphi_m\}\subset \operatorname{PSH}(D)$ converging to $\varphi$ decreasingly on any Lebesgue point of $e^{-k\varphi}$, $k=1,2,\dots$, with $\varphi(z)>-\infty$. Since $\{\varphi_m\}$ does not converge to $-\infty$ uniformly on $D$, there exists a plurisubharmonic function $\widetilde{\varphi}$ on $D$ such that
$$
\begin{equation*}
\lim_{m\to \infty}\varphi_m(z)=\widetilde{\varphi}(z).
\end{equation*}
\notag
$$
By Proposition 2.16, almost every $z\in D$ is a Lebesgue point of $e^{-k\varphi}$, $k=1,2,\dots$, and moreover, almost every point of $D$ satisfies $\varphi(z)>-\infty$. It then follows that, for almost every $z\in D$, On the other hand, we have on $D$,
$$
\begin{equation}
\limsup_{w\to z,\, w\in D\setminus E}\varphi(w)=\varphi(z)
\end{equation}
\tag{4.3}
$$
for any subset $E$ of measure zero. For any $z\in D$, we may choose $E$ in (4.3) to be the union of the non Lebesgue points of $e^{-k\varphi}$, $k=1,2,\dots$, and $\{z\in D\colon \varphi(z)=-\infty\}$, then therefore, $\varphi$ is a plurisubharmonic function on $D$. Remark 4.1. As we have mentioned, the conditions on group action invariance for $D$ and $\varphi$ can be omitted.
Citation in format AMSBIB
\Bibitem{LiZho23} |
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