# Singular Nakano positivity
of direct image sheaves of adjoint bundles

Takahiro INAYAMA  Department of Mathematics, Faculty of Science and
Technology, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510,
Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Shin-ichi
MATSUMURA  Mathematical Institute, Tohoku University, 6-3, Aramaki Aza-Aoba,
Aoba-ku, Sendai 980-8578, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Yuta WATANABE  Department of Mathematics, Faculty of
Science and Engineering, Chuo University. 1-13-27 Kasuga, Bunkyo-ku, Tokyo
112-8551, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: , version 0.01)

###### Abstract.

In this paper, we consider a proper Kähler fibration $f\colon X\to Y$ and a
singular Hermitian line bundle $(L,h)$ on $X$ with semi-positive curvature. We
prove that the direct image sheaf
$f_{*}(\mathcal{O}_{X}(K_{X/Y}+L)\otimes\mathcal{I}(h))$, equipped with the
Narasimhan-Simha metric, is singular Nakano semi-positive in the sense that
the $\overline{\partial}$-equation can be solved with optimal
$L^{2}$-estimate. Our proof does not rely on the theory of Griffiths
positivity for the direct image sheaf.

###### Key words and phrases:

Singular Hermitian metrics, Vector bundles, Nakano positivity, Direct image
sheaves, $\overline{\partial}$-equations, optimal $L^{2}$-estimates.

###### 2020 Mathematics Subject Classification:

Primary 32U05, Secondary 32A70, 32L20.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Proof of the main result

##  1\. Introduction

The theory of positivity of direct image sheaves of adjoint bundles originates
from Griffiths’ work [Gri70], which is based on the theory of variation of
Hodge structures. This theory has been further developed in subsequent works
[Fuj78, Kaw81, Vie83] and other subsequent contributions from the viewpoint of
algebraic geometry. On the other hand, the analytic theory of the positivity
of direct image sheaves was initiated by the theory of variation of Bergman
kernels by [MY04] and further advanced in [Ber09]. This analytic approach has
been extended by [HPS18, PT18] to establish the theory of singular Griffiths
positivity of direct image sheaves as a more applicable framework. The main
result of this paper (see Theorem 1.1 below) establishes singular Nakano
positivity of direct image sheaves in the same framework as in [HPS18, PT18],
and further generalizes the previous works [Wat23, Theorem 1.2] and [DNWZ23,
Theorem 1.5] on singular Nakano positivity. Here, we adopt the definition of
singular Nakano positivity based on [DNWZ23, Ina22], which uses solutions of
the $\overline{\partial}$-equation with optimal $L^{2}$-estimates of Hörmander
type (see Definition 2.1 for the precise definition).

######  Theorem 1.1.

Let $f\colon X\to Y$ be a proper Kähler fibration between complex manifolds
$X$ and $Y$ $($i.e., a proper surjective morphism with connected fibers such
that the inverse image $f^{-1}(U_{Y})$ is a Kähler manifold for a sufficiently
small open set $U_{Y}\subset Y$$)$. Let $(L,h)$ be a singular Hermitian line
bundle on $X$ such that $\sqrt{-1}\Theta_{h}\geq f^{*}\theta$ holds on $Y$ in
the sense of currents, where $\theta$ is a continuous $(1,1)$-form on $Y$.
Then, we have the following$:$

$(1)$ The induced Narasimhan-Simha metric $H$ on the direct image sheaf
$f_{*}(\mathcal{O}_{X}(K_{X/Y}+L)\otimes\mathcal{I}(h))$ satisfies that

$\sqrt{-1}\Theta_{H}\geq^{L^{2}}_{\mathrm{loc.}\mathrm{Nak.}}\theta\otimes\mathrm{Id}\text{
on }Y\text{ in the sense of Definition }\ref{def-dbarNak-sheaf}.$

$(2)$ If we further assume that $X$ is Kähler, then the Narasimhan-Simha
metric $H$ satisfies that

$\sqrt{-1}\Theta_{H}\geq^{L^{2}}_{\mathrm{glo.}\mathrm{Nak.}}\theta\otimes\mathrm{Id}\text{
on }Y\text{ in the sense of Definition }\ref{def-dbarNak-sheaf}.$

Here $K_{X/Y}$ is the relative canonical bundle and $\mathcal{I}(h)$ is the
multiplier ideal sheaf associated with $h$.

To clarify our contribution, we review and compare the previous works to
Theorem 1.1. Consider the setting of Theorem 1.1 and suppose that $\theta=0$.
When $h$ is a smooth Hermitian metric and $f\colon X\to Y$ is a smooth
fibration (i.e., a holomorphic submersion), the breakthrough work [Ber09]
proved that the Narasimhan-Simha metric $H$ is a smooth Hermitian metric on
$f_{*}(\mathcal{O}_{X}(K_{X/Y}+L))$ whose Chern curvature is Nakano positive.
This result has been generalized to the “non-smooth” setting by [HPS18, PT18].
More specifically, building upon the work [PT18], the work [HPS18] proved that
$H$ is a (possibly) singular Hermitian metric on
$f_{*}(\mathcal{O}_{X}(K_{X/Y}+L)\otimes\mathcal{I}(h))$ satisfying Griffiths
positivity. This is an elegant application of the Ohsawa-Takegoshi
$L^{2}$-extension theorem with optimal $L^{2}$-estimate. For smooth Hermitian
metrics, Nakano positivity is a stronger notion than Griffiths positivity.
Therefore, the study of Nakano positivity in the “non-smooth” setting is a
natural and important problem. In tackling this problem, we face the
difficulty that the Chern curvature cannot be defined for singular Hermitian
metrics (see [Rau15]), and the correct definition of Nakano positivity is not
immediately obvious. The work [DNWZ23] showed that Nakano positivity of smooth
Hermitian metrics can be characterized by the $\overline{\partial}$-equation
with optimal $L^{2}$-estimate. In this paper, by adopting the definition based
on this characterization, we prove the main result, generalizing not only
[Ber09] to the “non-smooth” setting but also [HPS18, PT18] to singular Nakano
positivity. We emphasize that our proof does not rely on the Griffiths
positivity.

A vanishing theorem of Kodaira type can be expected for singular Nakano
positive vector bundles, which is an advantage of considering Nakano
positivity compared to Griffiths positivity. In this context, as an
application of Theorem 1.1, we prove a vanishing theorem of Kollár-Ohsawa type
(see Corollary 1.2 and cf. [Mat16, Ohs84, Kol86a, Kol86b]). Corollary 1.2
follows directly from Theorem 1.1 and [Ina22, Theorem 1.5] when $Y$ is
projective and $f_{*}(\mathcal{O}_{X}(K_{X}+L)\otimes\mathcal{I}(h))$ is a
locally free sheaf. However, in the general case, we utilize the proof of
Theorem 1.1 to solve the $\overline{\partial}$-equation with optimal
$L^{2}$-estimates, thereby deducing the vanishing theorem. In this regard,
Corollary 1.2 can be viewed as a quantitative version of the vanishing theorem
of Kollár-Ohsawa type.

######  Corollary 1.2 (cf. [Mat16, Ohs84, Kol86a, Kol86b]).

Consider the same situation as in Theorem 1.1. Assume that $X$ is a weakly
pseudo-convex Kähler manifold and $\sqrt{-1}\Theta_{h}\geq f^{*}\omega_{Y}$
holds on $Y$ for some Kähler form $\omega_{Y}$ on $Y$.

Then, we have the following vanishing theorem$:$

$H^{q}(Y,f_{*}(\mathcal{O}_{X}(K_{X}+L)\otimes\mathcal{I}(h)))=0\quad\text{
for all $q>0$.}$

### Organization of the paper

The paper is organized as follows: In Section 2, we review the definition of
singular Nakano positivity and prove the technical lemmas necessary for the
proof of the main result. Section 3 is divided into two parts: In the first
half, we solve the $\bar{\partial}$-equation with optimal $L^{2}$-estimates in
an appropriate setting. In the second half, we derive Theorem 1.1 and
Corollary 1.2.

### Acknowledgment

The authors would like to thank Professor Masataka Iwai and Professor Hisashi
Kasuya, the organizers of the “Workshop on Complex Geometry in Osaka,” where
their collaboration began. T. I. is supported by Grant-in-Aid for Early-Career
Scientists $\sharp$23K12978 from Japan Society for the Promotion of Science
(JSPS). S. M. is supported by Grant-in-Aid for Scientific Research (B)
$\sharp$21H00976 from JSPS.

##  2\. Preliminaries

We begin by reviewing the definition of singular Nakano positivity. The direct
image sheaf treated in this paper is a torsion-free sheaf, but not necessarily
locally free. Therefore, for our purpose, we require Definition 2.2.

######  Definition 2.1 (Singular Nakano positivity for vector bundles,
[DNWZ23, Ina22]).

Let $E$ be a vector bundle on a complex manifold $X$ and $\theta$ be a
continuous $(1,1)$-form on $X$. Consider a singular Hermitian metric $h$ on
$E$ on $X$ with $h^{*}$ being upper semi-continuous (i.e., the function
$|u|^{2}_{h^{*}}$ is upper semi-continuous for any local section $u$ of
$E^{*}$).

(1) We define the global $\theta$-Nakano positivity in the sense of
$L^{2}$-estimates, denoted by

$\sqrt{-1}\Theta_{h}\geq^{L^{2}}_{\mathrm{glo.}\mathrm{Nak.}}\theta\otimes\mathrm{Id}_{E}\text{
on }X,$

as follows: For any data consisting of:

  * •

a Stein coordinate $\Omega\subset X$ admitting a trivialization
$E|_{\Omega}\cong\Omega\times\mathbb{C}^{r}$;

  * •

a Kähler form $\omega_{\Omega}$ on $\Omega$;

  * •

a smooth function $\psi$ on $\Omega$ such that
$\theta+\sqrt{-1}\partial\overline{\partial}\psi>0$;

  * •

a positive integer $q$ with $1\leq q\leq n$;

  * •

a $\overline{\partial}$-closed $g\in
L^{2}_{n,q}(\Omega,E;\omega_{\Omega},he^{-\psi})$ satisfying

$\int_{\Omega}\langle
B^{-1}_{\omega_{\Omega},\psi,\theta}g,g\rangle_{\omega_{\Omega},h}e^{-\psi}\,dV_{\omega_{\Omega}}<\infty,$

there exists $u\in L^{2}_{n,q-1}(\Omega,E;\omega_{\Omega},he^{-\psi})$ such
that

$\overline{\partial}u=g\text{ and
}\int_{\Omega}|u|^{2}_{\omega_{\Omega},h}e^{-\psi}\,dV_{\omega_{\Omega}}\leq\int_{\Omega}\langle
B^{-1}_{\omega_{\Omega},\psi,\theta}g,g\rangle_{\omega_{\Omega},h}e^{-\psi}\,dV_{\omega_{\Omega}},$

where
$B_{\omega_{\Omega},\psi,\theta}=[(\sqrt{-1}\partial\overline{\partial}\psi+\theta)\otimes\mathrm{Id}_{E},\Lambda_{\omega_{\Omega}}]$.

(2) We define the local $\theta$-Nakano positivity in the sense of
$L^{2}$-estimates, denoted by

$\sqrt{-1}\Theta_{h}\geq^{L^{2}}_{\mathrm{loc.}\mathrm{Nak.}}\theta\otimes\mathrm{Id}_{E}\text{
on }X,$

as follows: For any point $x\in X$, there exists an open neighborhood $U$ of
$x$ such that
$\sqrt{-1}\Theta_{h}\geq^{L^{2}}_{\mathrm{glo.}\mathrm{Nak.}}\theta\otimes\mathrm{Id}_{E}\text{
on }U.$

######  Definition 2.2 (Singular Nakano positivity for torsion-free sheaves).

Let $\mathcal{E}$ be a torsion-free sheaf and $\theta$ be a continuous
$(1,1)$-form on a complex manifold $X$. Let $X_{\mathcal{E}}$ denote the
largest Zariski open set where $\mathcal{E}$ is locally free. Consider a
singular Hermitian metric $h$ on $\mathcal{E}$ (i.e., a singular Hermitian
metric on the vector bundle $\mathcal{E}|_{X_{\mathcal{E}}}$). Assume that the
function $|u|^{2}_{h^{*}}$ is upper semi-continuous for any local section $u$
of $(\mathcal{E}|_{X_{\mathcal{E}}})^{*}$. We say that the sheaf
$(\mathcal{E},h)$ is (global/local) $\theta$-Nakano positive in the sense of
$L^{2}$-estimates if $(\mathcal{E}|_{X_{\mathcal{E}}},h)$ satisfies this
property.

The following lemmas play an important role in the proof of our main result.
For the reader’s convenience, we provide the proofs and relevant references
below. Lemma 2.4 is directly derived from [GMY23, Lemma 9.10] by setting
$\eta=1$ and $g=k$ in [GMY23, Lemma 9.10], and then taking the limit as
$k\to\infty$. Lemma 2.5 is derived from the proof of [Dem-book, Chapter VIII,
Lemma (6.3)].

######  Lemma 2.3 ([GMY23, Lemma 9.10]).

Let $X$ be a complete Kähler manifold with a $($not necessarily complete$)$
Kähler form $\omega$, and $(Q,h)$ be a vector bundle $Q$ with a smooth
Hermitian metric $h$. Let $\eta>0$ and $g>0$ be smooth functions on $X$ such
that $(\eta+g)$ and $(\eta+g)^{-1}$ are bounded. Let $\lambda\geq 0$ be a
bounded continuous function on $X$ such that $(B+\lambda I)$ is semi-positive
definite everywhere on $\wedge^{n,q}T^{*}X\otimes Q$, where $n:=\dim X$ and
$B:=B^{n,q}_{h,\eta,g,\omega}=[\eta\sqrt{-1}\Theta_{Q,h}-\sqrt{-1}\partial\overline{\partial}\eta-\sqrt{-1}g\partial\eta\wedge\overline{\partial}\eta,\Lambda_{\omega}]$.

Then, for a given form $v\in L^{2}_{n,q}(X,Q;\omega,h)$ with

$\overline{\partial}v=0$ and $\int_{X}\langle(B+\lambda
I)^{-1}v,v\rangle_{\omega,h}\,dV_{\omega}<\infty$,

there exist an approximate solution $u\in L^{2}_{n,q-1}(X,Q;\omega,h)$ and a
correcting term $\tau\in L^{2}_{n,q}(X,Q;\omega,h)$ such that
$\overline{\partial}u+P_{h}(\sqrt{\lambda}\tau)=v$ and

$\int_{X}(\eta+g^{-1})^{-1}|u|^{2}_{\omega,h}\,dV_{\omega}+\int_{X}|\tau|^{2}_{\omega,h}\,dV_{\omega}\leq\int_{X}\langle(B+\lambda
I)^{-1}v,v\rangle_{\omega,h}\,dV_{\omega},$

where $P_{h}\colon
L^{2}_{n,q}(X,Q;\omega,h)\to\operatorname{Ker}\overline{\partial}$ is the
orthogonal projection.

######  Lemma 2.4 (cf. [ZZ18, Lemma 3.2]).

Let $X$ be a complete Kähler manifold with a $($not necessarily complete$)$
Kähler form $\omega$, and $(Q,h)$ be a vector bundle $Q$ with a smooth
Hermitian metric $h$. Let $\delta\geq 0$ be a semi-positive constant such that
$(B+\delta I)$ is semi-positive definite everywhere on
$\wedge^{n,q}T^{*}X\otimes Q$, where
$B:=B^{n,q}_{Q,h,\omega}=[\sqrt{-1}\Theta_{Q,h},\Lambda_{\omega}]$.

Then, for a given form $v\in L^{2}_{n,q}(X,Q;\omega,h)$ with

$\overline{\partial}v=0$ and $\int_{X}\langle(B+\delta
I)^{-1}v,v\rangle_{\omega,h}\,dV_{\omega}<\infty$,

there exist an approximate solution $u\in L^{2}_{n,q-1}(X,Q;\omega,h)$ and a
correcting term $\tau\in L^{2}_{n,q}(X,Q;\omega,h)$ such that
$\overline{\partial}u+\sqrt{\delta}\tau=v$ and

$\int_{X}|u|^{2}_{\omega,h}\,dV_{\omega}+\int_{X}|\tau|^{2}_{\omega,h}\,dV_{\omega}\leq\int_{X}\langle(B+\delta
I)^{-1}v,v\rangle_{\omega,h}\,dV_{\omega}.$

######  Lemma 2.5.

Let $X$ be a complex manifold of dimension $n$ and $\omega_{1},\omega_{2}$ be
Hermitian forms on $X$ with $\omega_{1}\geq\omega_{2}$. Let $(E,h)$ be a
vector bundle on $X$ with a smooth Hermitian metric $h$ and $\theta_{E}\in
C^{\infty}(X,\wedge^{2}T^{*}X\otimes\mathrm{Hom}(E,E))$ be a smooth operator
with $\theta_{E}^{*}=-\theta_{E}$. Assume that the operator
$B_{\theta_{E},\omega_{2}}:=[\sqrt{-1}\theta_{E},\Lambda_{\omega_{2}}]$ is
positive $($resp. semi-positive$)$ definite on $\wedge^{n,q}T^{*}X\otimes E$,
which we simply write as $B_{\theta_{E},\omega_{2}}>0$ $($resp. $\geq 0$$)$.
Then, for any $(n,q)$-form $u\in\wedge^{n,q}T^{*}X\otimes E$ with $q\geq 1$,
we have

$B_{\theta_{E},\omega_{1}}>0\,\,(\text{resp.}\,\geq 0)\text{ and }\langle
B_{\theta_{E},\omega_{1}}^{-1}u,u\rangle_{\omega_{1},h}dV_{\omega_{1}}\leq\langle
B_{\theta_{E},\omega_{2}}^{-1}u,u\rangle_{\omega_{2},h}dV_{\omega_{2}}.$

The following lemma is well-known and often applied in the case where $E$ is a
line bundle. In this paper, we need a generalization of this lemma to vector
bundles. While the proof is straightforward, we provide a proof for the
general case to ensure completeness.

######  Lemma 2.6.

Let $(E,h)$ be a singular Hermitian vector bundle on a complex manifold $X$.
Assume that the norm $|t|_{h^{*}}$ with respect to $h^{*}$ is locally bounded
for any smooth section $t$ of $E^{*}$, where $h$ is the dual metric on the
dual vector bundle $E^{*}$. Then, we have:

  * $(1)$

Let $s=\sum_{i=1}^{r}s_{i}e_{i}$ be a $($not necessarily holomorphic or
smooth$)$ section of $E$, where $\\{e_{i}\\}_{{i=1}}^{r}$ is a local frame of
$E$. For any $p$ with $1\leq p\leq\infty$, the following holds: If $|s|_{h}$
is locally $L^{p}$-integrable, then the absolute value $|s_{i}|$ of each
component $s_{i}$ is also locally $L^{p}$-integrable.

  * $(2)$

Let $\omega_{X}$ be a Hermitian form on $X$. Let $u$, $v$ be $E$-valued-
differential forms on $X$ such that $|u|_{h,\omega_{X}}$ is locally
$L^{1}$-integrable and $|v|_{h,\omega_{X}}$ is locally $L^{2}$-integrable. If
$\overline{\partial}u=v$ holds on a $($non-empty$)$ Zariski open set
$X_{0}\subset X$ in the sense of distributions, then $\overline{\partial}u=v$
also holds on $X$.

###### Proof.

Note that the singular Hermitian metric $h$ is not defined on a set of
Lebesgue measure zero, thus the lemma and its proof are valid almost
everywhere on $X$. The norm $|s|_{h}$ is characterized as follows:

$|s|_{h}=\sup\\{|\langle s,t\rangle_{\rm{pair}}|\,\mid\,t\text{ is a section
of $E^{*}$ with }|t|_{h^{*}}\leq 1\\},$

where $\langle\bullet,\bullet\rangle_{\rm{pair}}$ is the point-wise natural
pairing. By assumption, we can find a constant $\delta>0$ with $|\delta
e^{*}_{i}|\leq 1$, where $\\{e^{*}_{i}\\}_{i=1}^{r}$ is the dual frame of
$\\{e_{i}\\}_{{i=1}}^{r}$. Consequently, conclusion (1) follows from the
inequality:

$\delta|s_{i}|=|\langle s,\delta e^{*}_{i}\rangle_{\rm{pair}}|\leq|s|_{h}.$

Conclusion (2) is then derived from Conclusion (1) and [Dem82, Lemma 6.9]. ∎

##  3\. Proof of the main result

In this section, we first establish Theorem 3.2, solving the
$\overline{\partial}$-equation with optimal $L^{2}$-estimate under Setting
3.1. At the end of this section, as applications of Theorem 3.2, we prove
Theorem 1.1 and Corollary 1.2.

######  Setting 3.1.

Under the setting of Theorem 1.1, we further assume that $X$ is a Kähler
manifold and the direct image sheaf
$\mathcal{E}:=f_{*}(\mathcal{O}_{X}(K_{X/Y}+L)\otimes\mathcal{I}(h))$ is a
locally free on $Y$. The locally free sheaf $\mathcal{E}$ is identified with a
vector bundle on $Y$ and often written as $E$. Furthermore, we assume that

  * •

$Y$ is a weakly pseudo-convex Kähler manifold;

  * •

$\omega_{Y}$ is a Kähler form on $Y$;

  * •

$\psi$ is a smooth function on $Y$ such that
$\theta_{\psi}:=\theta+\sqrt{-1}\partial\overline{\partial}\psi>0$;

  * •

$q$ is a positive integer with $1\leq q\leq m:=\dim Y$;

  * •

$g\in L^{2}_{m,q}(Y,E;\omega_{Y},He^{-\psi})$ is an $E$-valued
$\overline{\partial}$-closed form with

$\int_{Y}\langle
B^{-1}_{\omega_{Y},\theta_{\psi}}\,g,g\rangle_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}<\infty,$

where
$B_{\omega_{Y},\theta_{\psi}}=[\theta_{\psi}\otimes\mathrm{Id}_{E},\Lambda_{\omega_{Y}}]$
and $H$ is the Narasimhan-Simha metric on $E$ (see [HPS18] or the proof of
Theorem 3.2 for the precise definition).

Note that different from Definition 2.1, we do not assume that $Y$ is Stein
nor that $Y$ admits a trivialization $E\cong Y\times\mathbb{C}^{r}$.

Theorem 3.2 is the main technical result of this paper and solves the
$\overline{\partial}$-equation with optimal $L^{2}$-estimate under Setting
3.1, from which Theorem 1.1 and Corollary 1.2 can be deduced. The assumptions
of Setting 3.1 appear somewhat technical, but they are needed to prove Theorem
1.1 and Corollary 1.2 in a unified way. The Narasimhan-Simha metric $H$ on $E$
is known to be Griffiths $\theta$-semi-positive (see [HPS18]), but we never
use this property in the proof of Theorem 3.2.

######  Theorem 3.2.

Under Setting 3.1, there exists $u\in
L^{2}_{m,q-1}(Y,E;\omega_{Y},He^{-\psi})$ such that

$\overline{\partial}u=g\quad\text{ and
}\quad\int_{Y}|u|^{2}_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}\leq\int_{Y}\langle
B^{-1}_{\omega_{Y},\theta_{\psi}}g,g\rangle_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}.$

###### Proof.

The proof can be divided into four steps.

######  Step 1 (Strategy of the proof).

The purpose of this step is to explain the notation and the strategy of the
proof.

Let $(t_{1},\ldots,t_{m})$ be a coordinate on an open subset $U\subset Y$.
Then, the $E$-valued $(m,q)$-form $g$ can be locally written as

$g=\sum_{|J|=q}g_{J}\,dt\wedge d\overline{t}_{J}\text{ and
}g_{J}=\sum_{i=1}^{r}g_{J,i}e_{i},$

where

  * $\bullet$

$J$ runs through the multi-indices of degree $q$;

  * $\bullet$

$dt:=\,dt_{1}\wedge\cdots\wedge\,dt_{m}$ and
$d\overline{t}_{J}:=d\overline{t}_{j_{1}}\wedge\cdots\wedge
d\overline{t}_{j_{q}}$ for $J=(j_{1},j_{2},\cdots,j_{q})$;

  * $\bullet$

$g_{J}$ is a section of $\mathcal{E}|_{U}$;

  * $\bullet$

$\\{e_{i}\\}_{i=1}^{r}$ is a frame of $\mathcal{E}|_{U}$ and
$r:=\operatorname{rank}\mathcal{E}$;

  * $\bullet$

$g_{J,i}=g_{J,i}(t)$ is a function defined on $U\subset Y$.

The section $g_{J}$ can be regarded as a section of $K_{X/Y}\otimes
L\otimes\mathcal{I}(h)$ on the inverse image $f^{-1}(U)\subset X$ by the
definition of direct image sheaves, and thus $g_{J}\otimes f^{*}dt$ can be
identified with a section of $K_{X}\otimes L\otimes\mathcal{I}(h)$ since
$f^{*}dt$ is the local frame of $f^{*}K_{Y}$. Under this identification, the
$L$-valued $(n+m,q)$-form $\widetilde{g}$ is defined as follows:

(3.1) $\displaystyle\widetilde{g}:=\sum_{|J|=q}\big{(}g_{J}\otimes
f^{*}dt\big{)}\wedge f^{*}d\overline{t}_{J}$

where $n$ is the fiber dimension of $f\colon X\to Y$. Although $\widetilde{g}$
is defined by a local description of $g$, we can easily check that
$\widetilde{g}$ is globally defined on $X$ as an $L$-valued $(n+m,q)$-form.

The strategy of the proof is as follows: In Step 2, we show that
$\widetilde{g}$ is $\overline{\partial}$-closed and $L^{2}$-integrable in a
suitable sense, by comparing $\widetilde{g}$ to $g$. In Step 3, we solve the
$\overline{\partial}$-equation
$\overline{\partial}\widetilde{u}=\widetilde{g}$ on $X$, where after we apply
Demailly’s approximation theorem to $h$ and the result for a “twisted”
$\overline{\partial}$-equation, we take the limit to obtain the desired
solution $\widetilde{u}$ focusing the positivity in horizontal direction. In
Step 4, we finally descend $\widetilde{u}$ to an $E$-valued $(m,q-1)$-form $u$
on $Y$ and confirm that $u$ satisfies the desired properties in Theorem 3.2,
where the uniform $L^{2}$-estimate obtained in Step 3 plays a crucial role.

######  Step 2 ($L^{2}$-integrability and $\overline{\partial}$-closedness of
$\tilde{g}$).

For a fixed Kähler form $\omega_{X}$ on $X$, we define the Kähler form
$\omega_{\varepsilon}$ on $X$ by

$\omega_{\varepsilon}:=f^{*}\omega_{Y}+\varepsilon\omega_{X}.$

The purpose of this step is to prove Claim 3.3. To this end, we investigate
the construction of $\widetilde{g}$ and the Narasimhan-Simha metric $H$ in
detail.

######  Claim 3.3.

Under the same situation as above, we have$:$

  * (1)

$\widetilde{g}$ is a $\overline{\partial}$-closed and
$L\otimes\mathcal{I}(h)$-valued $(n+m,q)$ on $X$;

  * (2)

For a positive integer $\varepsilon>0$, we have

$\displaystyle\int_{X}\langle
B^{-1}_{\omega_{\varepsilon},f^{*}\theta_{\psi}}\widetilde{g},\widetilde{g}\rangle_{\omega_{\varepsilon},h}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}\leq\int_{Y}\langle
B^{-1}_{\omega_{Y},\theta_{\psi}}g,g\rangle_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}<\infty,$

where
$B_{\omega_{\varepsilon},f^{*}\theta_{\psi}}:=[f^{*}\theta_{\psi}\otimes\mathrm{Id}_{L},\Lambda_{\omega_{\varepsilon}}]$.

###### Proof.

Following [HPS18, Section 22], we first recall the definition of Narasimhan-
Simha metrics. We take a Zariski open subset $Y_{0}\subset Y$ such that

  * $\bullet$

$f\colon X_{0}:=f^{-1}(Y_{0})\to Y_{0}$ is a smooth fibration over $Y_{0}$;

  * $\bullet$

The quotient sheaf $f_{*}(\mathcal{O}_{X}(K_{X/Y}+L))/\mathcal{E}$ is locally
free on $Y_{0}$;

  * $\bullet$

$\mathcal{E}$ has the base change property.

Then, for any $y\in Y_{0}$, we have the inclusions

$H^{0}(X_{y},\mathcal{O}_{X_{y}}(K_{X_{y}}+L)\otimes\mathcal{I}(h|_{X_{y}}))\subset
E_{y}\subset H^{0}(X_{y},\mathcal{O}_{X_{y}}(K_{X_{y}}+L))$

under the natural identification $K_{X/Y}|_{X_{y}}\cong K_{X_{y}}$.
Furthermore, the norm $|\alpha|_{H}$ of $\alpha\in E_{y}$ can be described by
the fiber integral:

$|\alpha|^{2}_{H}:=\int_{X_{y}}|\alpha|^{2}_{\omega_{X_{y}},h}\,dV_{\omega_{X_{y}}}$

where $X_{y}$ is the fiber at $y\in Y_{0}$ and
$\omega_{X_{y}}:={\omega_{X}}|_{X_{y}}$. By the property of $(n,0)$-forms, the
integrand of the right-hand side can be written as

$|\alpha|^{2}_{\omega_{X_{y}},h}\,dV_{\omega_{X_{y}}}=|\alpha^{\prime}|^{2}_{h}\,c_{n}dz\wedge
d\overline{z},$

where $\alpha=\alpha^{\prime}\,dz$, $c_{n}:=\sqrt{-1}^{n^{2}}$ and $z$ is a
local coordinate of the fiber $X_{y}$. In particular, the integrand does not
depend on the choice of Kähler forms on $X_{y}$.

(1) It is obvious that $\widetilde{g}$ is an $L\otimes\mathcal{I}(h)$-valued
$(n+m,q)$-form by construction (see (3.1)). We now show that
$\overline{\partial}\widetilde{g}=0$ on $X_{0}$. Since $f\colon X\to Y$ is a
smooth fibration at $y_{0}\in Y_{0}$, we can take a local coordinate
$(z_{1},\ldots,z_{n})$ of the smooth fiber $f^{-1}(y_{0})$ so that

$(z_{1},\cdots,z_{n},f^{*}t_{1},\cdots,f^{*}t_{m})=(z_{1},\cdots,z_{n},t_{1},\cdots,t_{m})$

gives a local coordinate of $X$. Throughout the proof, we denote $f^{*}t_{j}$
by $t_{j}$ when considering it as part of the coordinates. However, we retain
the notation $f^{*}t_{j}$ when emphasizing that it is the pullback of $t_{j}$.
The $L$-valued $(n+m,0)$-form $g_{J}\otimes f^{*}dt$ can be written as

$g_{J}\otimes f^{*}dt=\sum_{i=1}^{r}g_{J,i}\,e_{i}\otimes
f^{*}dt=G_{J}\,dz\wedge\,dt$

where $G_{J}=G_{J}(t,z)$ is a local section of $L$. Hence, we obtain

$\widetilde{g}=\sum_{|J|=q}G_{J}\,dz\wedge\,dt\wedge d\overline{t}_{J}.$

The function $g_{J,i}=g_{J,i}(t)$ does not depend on the fiber coordinate $z$
and $e_{i}$ is a holomorphic section, and thus we can deduce that
$\overline{\partial}_{z}G_{J}=0$ holds, which indicates that
$\overline{\partial}_{z}\widetilde{g}=0$. Furthermore, since we have

$\sum_{|J|=q}\sum_{i=1}^{r}\sum_{k=1}^{m}\frac{\partial
g_{J,i}}{\partial\overline{t}_{k}}e_{i}\otimes\,d\overline{t}_{k}\wedge
d\overline{t}_{J}=0$

by the assumption $\overline{\partial}g=0$, we can see that

$\displaystyle\overline{\partial}_{t}\widetilde{g}$
$\displaystyle=\sum_{|J|=q}\sum_{k=1}^{m}\frac{\partial
G_{J}(t,z)}{\partial\overline{t}_{k}}\,d\overline{t}_{k}\wedge
dz\wedge\,dt\wedge d\overline{t}_{J}$
$\displaystyle=\sum_{|J|=q}\sum_{k=1}^{m}\sum_{i=1}^{r}\frac{\partial
g_{J,i}}{\partial\overline{t}_{k}}\,e_{i}\otimes d\overline{t}_{k}\wedge
dz\wedge dt\wedge d\overline{t}_{J}=0.$

Hence, we obtain $\overline{\partial}\widetilde{g}=0$ on $X_{0}$. The property
of $\overline{\partial}\widetilde{g}=0$ can be extended to the ambient space
$X$ by the (local) $L^{2}$-integrability of $\widetilde{g}$ (see [Dem82, Lemma
6.9]). This finishes the proof of (1).

(2) It is sufficient to prove the desired inequality for the integrals after
we replace $f\colon X\to Y$ with $f\colon X_{0}\to Y_{0}$. Let
$\\{\lambda_{j}\\}_{j=1}^{m}$ be eigenvalues of $\theta_{\psi}$ with respect
to $\omega_{Y}$. We take a coordinate
$(t_{1},\ldots,t_{m},z_{1},\ldots,z_{n})$ around a fixed point $x_{0}\in X$
with $f(x_{0})=y_{0}$ such that

$\displaystyle\theta_{\psi}=\sqrt{-1}\sum^{m}_{j=1}\lambda_{j}\,dt_{j}\wedge
d\overline{t}_{j}\quad\text{ and
}\quad\omega_{Y}=\sqrt{-1}\sum^{m}_{j=1}\,dt_{j}\wedge
d\overline{t}_{j}\quad\text{ at }y_{0}.$

We first show that the integrand $I_{\varepsilon}$ of the left-hand side in
Claim 3.3 (2), defined by

$I_{\varepsilon}:=\langle
B^{-1}_{\omega_{\varepsilon},f^{*}\theta_{\psi}}\widetilde{g},\widetilde{g}\rangle_{\omega_{\varepsilon},h}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}},$

converges to $I$ monotonically from below, where

$I:=\sum_{|J|=q}\Bigl{(}f^{*}\sum_{j\in J}\lambda_{j}\Bigr{)}^{-1}\cdot
f^{*}|d\overline{t}_{J}|^{2}_{\omega_{Y}}\cdot|G_{J}|_{h}^{2}\cdot
e^{-f^{*}\psi}\,c_{n+m}dz\wedge\,dt\wedge d\overline{z}\wedge\,d\overline{t}.$

Note that $I$ is a priori defined by local coordinates at each point $y_{0}$,
but $I$ is independent of this choice of the coordinates and globally defined,
since $I$ is the limit of the globally defined $I_{\varepsilon}$. To this end,
we consider the Kähler form $\omega_{\mathrm{st}}$ locally defined around
$x_{0}\in X_{y_{0}}$:

$\omega_{\mathrm{st}}:=\sum_{i=1}^{m}\sqrt{-1}\,dt_{i}\wedge
d\overline{t}_{i}+\sum_{j=1}^{n}\sqrt{-1}\,dz_{j}\wedge d\overline{z}_{j}$

and take a positive constant $C>0$ so that
$(1/C)\omega_{\mathrm{st}}\leq\omega_{X}\leq C\omega_{\mathrm{st}}$. Lemma 2.5
enables us to compare the norms with respect to $\omega_{\varepsilon}$,
$f^{*}\omega_{Y}+C\varepsilon\omega_{\mathrm{st}}$, and
$f^{*}\omega_{Y}+(1/C)\varepsilon\omega_{\mathrm{st}}$. This implies that it
is sufficient to show that $I_{\varepsilon,r}$ converges to $I$, where $r>0$,
$\omega_{\varepsilon,r}:=f^{*}\omega_{Y}+\varepsilon r\omega_{\mathrm{st}}$
and

$I_{\varepsilon,r}:=\langle
B^{-1}_{\omega_{\varepsilon,r},f^{*}\theta_{\psi}}\widetilde{g},\widetilde{g}\rangle_{\omega_{\varepsilon,r},h}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon,r}}.$

A straightforward calculation yields

$B^{-1}_{\omega_{\varepsilon,r},f^{*}\theta_{\psi}}\widetilde{g}=\sum_{|J|=q}\Bigl{(}\sum_{j\in
J}\frac{f^{*}\lambda_{j}}{\sqrt{1+\varepsilon r}}\Bigr{)}^{-1}G_{J}\,dz\wedge
dt\wedge d\overline{t}_{J}\quad\text{ at }y_{0},$

and thus, we can see that

$\displaystyle I_{\varepsilon,r}=\sum_{|J|=q}\Bigl{(}\sum_{j\in
J}\frac{f^{*}\lambda_{j}}{\sqrt{1+\varepsilon
r}}\Bigr{)}^{-1}\cdot|f^{*}d\overline{t}_{J}|^{2}_{\omega_{\varepsilon,r}}\cdot|G_{J}|_{h}^{2}e^{-f^{*}\psi}\,c_{n+m}dz\wedge\,dt\wedge
d\overline{z}\wedge\,d\overline{t}\quad\text{ at }y_{0}.$

Since $|f^{*}d\overline{t}_{J}|^{2}_{\omega_{\varepsilon,r}}$ and
$f^{*}\lambda_{j}/\sqrt{1+\varepsilon r}$ converges to
$f^{*}|d\overline{t}_{J}|^{2}_{\omega_{Y}}$ and $f^{*}\lambda_{j}$
respectively. Hence, Lemma 2.5 shows that $I_{\varepsilon}$ converges to $I$
monotonically from below.

By the definition of the Narasimhan-Simha metric $H$ on $\mathcal{E}$, we can
see that

(3.2) $\displaystyle|g_{J}\,dt\wedge
d\overline{t}_{J}|^{2}_{\omega_{Y},H}=|\,d\overline{t}_{J}|^{2}_{\omega_{Y}}\cdot|\,dt|^{2}_{\omega_{Y}}\cdot\int_{X_{y}}|G_{J}|^{2}_{h}\,c_{n}dz\wedge
d\overline{z}.$

Then, due to the same calculation as above, we can see that

$B_{\omega_{Y},\theta_{\psi}}^{-1}\,g=\sum_{|J|=q}(\sum_{j\in
J}\lambda_{j})^{-1}g_{J}\,dt\wedge d\overline{t}_{J}\quad\text{ at }y_{0}.$

Together with (3.2), we obtain that

$\displaystyle\langle
B_{\omega_{Y},\theta_{\psi}}^{-1}\,g,g\rangle_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}$
$\displaystyle=$ $\displaystyle\Big{(}\sum_{|J|=q}(\sum_{j\in
J}\lambda_{j})^{-1}\cdot|\,d\overline{t}_{J}|^{2}_{\omega_{Y}}\cdot
e^{-\psi}\cdot\int_{X_{y}}|G_{J}|^{2}_{h}\,c_{n}dz\wedge
d\overline{z}\Big{)}\cdot c_{m}\,dt\wedge d\overline{t}$ $\displaystyle=$
$\displaystyle\int_{X_{y}}\Bigl{(}\sum_{|J|=q}(f^{*}\sum_{j\in
J}\lambda_{j})^{-1}\cdot
f^{*}|d\overline{t}_{J}|^{2}_{\omega_{Y}}\cdot|G_{J}|_{h}^{2}e^{-f^{*}\psi}\,c_{n}dz\wedge
d\overline{z}\Bigr{)}\cdot c_{m}dt\wedge d\overline{t}\quad\text{ at }y_{0}.$

Hence, Fubini’s theorem shows that

(3.3) $\displaystyle\int_{Y}\langle
B^{-1}_{\omega_{Y},\theta_{\psi}}g,g\rangle_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}$
$\displaystyle=$ $\displaystyle\int_{X}\Bigl{(}\sum_{|J|=q}(f^{*}\sum_{j\in
J}\lambda_{j})^{-1}\cdot
f^{*}|d\overline{t}_{J}|^{2}_{\omega_{Y}}\cdot|G_{J}|_{h}^{2}e^{-f^{*}\psi}\,c_{n}dz\wedge
d\overline{z}\Bigr{)}\cdot c_{m}dt\wedge d\overline{t}.$

The desired conclusion follows from Lebesgue’s monotone convergence theorem
since $I_{\varepsilon}$ converges to the integrand in the right-hand side
monotonically from below. ∎

######  Step 3 ($\overline{\partial}$-equations on $X$ with optimal
$L^{2}$-estimate).

The purpose of this step is to prove Claim 3.4. To this end, we apply
Demailly’s approximation theorem to $h$, and then take the limit of
appropriate solutions of “twisted” $\overline{\partial}$-equations.

######  Claim 3.4.

Under the same situation as above, there exists $\widetilde{u}\in
L^{2}_{n+m,q-1}(X,L;\omega_{X},he^{-f^{*}\psi})$ such that
$\overline{\partial}\widetilde{u}=\widetilde{g}$ on $X$ and

$\displaystyle\int_{X}|\widetilde{u}|^{2}_{\omega_{\varepsilon},h}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}\leq\int_{Y}\langle
B^{-1}_{\omega_{Y},\theta_{\psi}}g,g\rangle_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}$

for any $\varepsilon>0$.

###### Proof.

Let $\\{Y_{j}\\}_{j=1}^{\infty}$ be an open cover of $Y$ such that
$Y_{j}\Subset Y_{j+1}$ and $Y_{j}\Subset Y$ hold for any $j$. We may assume
that $Y_{j}$ is a sub-level set of an exhaustive psh function on $Y$ since $Y$
is weakly pseudo-convex. The inverse image $X_{j}=f^{-1}(Y_{j})$ also
satisfies that $X_{j}\Subset X_{j+1}$ and ${X}_{j}\Subset X$. By applying
Demailly’s approximation theorem [Dem92] to $h|_{X_{j}}$ and
$\omega_{\varepsilon}|_{X_{j}}$, we can take a family
$\\{h_{j,\varepsilon,\delta}\\}_{\delta>0}$ of singular Hermitian metrics on
$L|_{X_{j}}$ such that

  * $(1)$

$h_{j,\varepsilon,\delta}$ has analytic singularities along a Zariski closed
subset $Z_{j,\varepsilon,\delta}\subset X_{j}$;

  * $(2)$

$\\{h_{j,\varepsilon,\delta}\\}_{\delta>0}$ increasingly converges to
$h|_{X_{j}}$ as $\delta\searrow 0$;

  * $(3)$

$\sqrt{-1}\Theta_{h_{j,\varepsilon,\delta}}(L)\geq
f^{*}\theta-\delta\omega_{\varepsilon}$ holds on $X_{j}$.

Note that Demailly’s approximation theorem remains valid for the relatively
compact subset $X_{j}\Subset X$ (see [Mat22, Theorem 2.9] for the detailed
argument). We consider the curvature operator $B_{j,\varepsilon,\delta}$
defined by

$B_{j,\varepsilon,\delta}:=[\sqrt{-1}\Theta_{h_{j,\varepsilon,\delta}}(L)+f^{*}\sqrt{-1}\partial\overline{\partial}\psi,\Lambda_{\omega_{\varepsilon}}]\text{
on }X_{j}\setminus Z_{j,\varepsilon,\delta}.$

Then, we can easily check that

$\displaystyle
B_{j,\varepsilon,\delta}+2\delta[\omega_{\varepsilon},\Lambda_{\omega_{\varepsilon}}]>$
$\displaystyle
B_{j,\varepsilon,\delta}+\delta[\omega_{\varepsilon},\Lambda_{\omega_{\varepsilon}}]$
$\displaystyle\geq$
$\displaystyle[f^{*}\theta_{\psi}\otimes\mathrm{Id}_{L},\Lambda_{\omega_{\varepsilon}}]=B_{\omega_{\varepsilon},f^{*}\theta_{\psi}}\geq
0.$

Note that
$[\omega_{\varepsilon},\Lambda_{\omega_{\varepsilon}}]=q\cdot\mathrm{Id}_{L}$
holds on $\wedge^{n+m,q}T^{*}X\otimes L$. Thus, together with Claim 3.3, we
can deduce that

$\displaystyle\int_{X_{j}\setminus
Z_{j,\varepsilon,\delta}}\langle(B_{j,\varepsilon,\delta}+2q\delta\cdot\mathrm{Id}_{L})^{-1}\widetilde{g},\widetilde{g}\rangle_{\omega_{\varepsilon},h_{j,\varepsilon,\delta}}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}$
$\displaystyle\leq$ $\displaystyle\int_{X}\langle
B^{-1}_{\omega_{\varepsilon},f^{*}\theta_{\psi}}\widetilde{g},\widetilde{g}\rangle_{\omega_{\varepsilon},h}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}$
$\displaystyle\leq$ $\displaystyle\int_{Y}\langle
B^{-1}_{\omega_{Y},\theta_{\psi}}g,g\rangle_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}.$

Note that $X_{j}\setminus Z_{j,\varepsilon,\delta}$ is a complete Kähler
manifold by [Dem82, Theorem 1.5] since $X_{j}$ is a sub-level set of an
exhaustive psh function. By applying Lemma 2.4, we can find an approximate
solution $\widetilde{u}_{j,\varepsilon,\delta}$ and a correcting term
$\tau_{j,\varepsilon,\delta}$:

$\displaystyle\widetilde{u}_{j,\varepsilon,\delta}\in
L^{2}_{n+n,q-1}(X_{j}\setminus
Z_{j,\varepsilon,\delta},L;\omega_{\varepsilon},h_{j,\varepsilon,\delta}e^{-f^{*}\psi})\text{
and }$ $\displaystyle\tau_{j,\varepsilon,\delta}\in
L^{2}_{n+m,q-1}(X_{j}\setminus
Z_{j,\varepsilon,\delta},L;\omega_{\varepsilon},h_{j,\varepsilon,\delta}e^{-f^{*}\psi})$

satisfying that

$\displaystyle\overline{\partial}\widetilde{u}_{j,\varepsilon,\delta}+\sqrt{2q\delta}\,\tau_{j,\varepsilon,\delta}$
$\displaystyle=\widetilde{g}\,\,\text{on}\,\,X_{j}\setminus
Z_{j,\varepsilon,\delta}\quad\text{and}$ $\displaystyle\int_{X_{j}\setminus
Z_{j,\varepsilon,\delta}}|\widetilde{u}_{j,\varepsilon,\delta}|^{2}_{\omega_{\varepsilon},h_{j,\varepsilon,\delta}}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}$
$\displaystyle+\int_{X_{j}\setminus
Z_{j,\varepsilon,\delta}}|\tau_{j,\varepsilon,\delta}|^{2}_{\omega_{\varepsilon},h_{j,\varepsilon,\delta}}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}$
$\displaystyle\leq\int_{Y}\langle
B^{-1}_{\omega_{Y},\theta_{\psi}}g,g\rangle_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}.$

Note that the right-hand side does not depend on the data
$j,\varepsilon,\delta$. The first property of $\overline{\partial}$-equation
on $X_{j}\setminus Z_{j,\varepsilon,\delta}$ can be extended to $X_{j}$ by the
(local) $L^{2}$-integrability (see [Dem82, Lemma 6.9]). Hereafter, we regard
both $\widetilde{u}_{j,\varepsilon,\delta}$ and $\tau_{j,\varepsilon,\delta}$
as $L^{2}$-sections on the ambient space $X$ via the zero extensions. We will
take an appropriate weak limit to construct the desired solution
$\widetilde{u}$.

We first fix $j$, $\varepsilon$, and consider a weak limit as $\delta\to 0$.
The $L^{2}$-norms
$\|\widetilde{u}_{j,\varepsilon,\delta}\|_{\omega_{\varepsilon},h_{j,\varepsilon,\delta}}$
and
$\|\tau_{j,\varepsilon,\delta}\|_{\omega_{\varepsilon},h_{j,\varepsilon,\delta}}$
are uniformly bounded in $\delta$ and
$\\{h_{j,\varepsilon,\delta}\\}_{\delta>0}$ monotonically increases as
$\delta\searrow 0$. Thus, by the same argument as in [IM24, Proof of Theorem
1.2] (see also [GMY23, Mat18, Mat22]), we can take a subsequence of
$\\{\widetilde{u}_{j,\varepsilon,\delta}\\}_{\delta>0}$ (for which we use the
same notation for simplicity) and

$\widetilde{u}_{j,\varepsilon}\in
L^{2}_{n+m,q-1}(X_{j},L;\omega_{\varepsilon},he^{-f^{*}\psi})$

such that $\widetilde{u}_{j,\varepsilon,\delta}$ weakly converges to
$\widetilde{u}_{j,\varepsilon}$ on
$L^{2}_{n,q-1}(X,L;\omega_{\varepsilon},h_{\delta_{0}}e^{-f^{*}\psi})$ for a
positive number $\delta_{0}>0$. Note that this subsequence and its weak limit
is independent of $\delta_{0}>0$ by the diagonal argument. Furthermore, by the
lower semi-continuity of the weak limit and property (2) of
$\\{h_{j,\varepsilon,\delta}\\}_{\delta>0}$, we can see that for
$\delta_{0}>0$

$\displaystyle\int_{X_{j}}|\widetilde{u}_{j,\varepsilon}|^{2}_{\omega_{\varepsilon},h_{j,\varepsilon,\delta_{0}}}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}$
$\displaystyle\leq\liminf_{\delta\to
0}\int_{X_{j}}|\widetilde{u}_{j,\varepsilon,\delta}|^{2}_{\omega_{\varepsilon},h_{j,\varepsilon,\delta_{0}}}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}$
$\displaystyle\leq\liminf_{\delta\to
0}\int_{X_{j}}|\widetilde{u}_{j,\varepsilon,\delta}|^{2}_{\omega_{\varepsilon},h_{j,\varepsilon,\delta}}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}$
$\displaystyle\leq\int_{Y}\langle
B^{-1}_{\omega_{Y},\theta_{\psi}}g,g\rangle_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}.$

By applying Fatou’s lemma, we obtain that

$\displaystyle\int_{X_{j}}|\widetilde{u}_{j,\varepsilon}|^{2}_{\omega_{\varepsilon},h}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}$
$\displaystyle\leq\liminf_{\delta_{0}\to
0}\int_{X_{j}}|\widetilde{u}_{j,\varepsilon}|^{2}_{\omega_{\varepsilon},h_{j,\varepsilon,\delta_{0}}}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}$
$\displaystyle\leq\int_{Y}\langle
B^{-1}_{\omega_{Y},\theta_{\psi}}g,g\rangle_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}.$

On the other hand, in the same way applied to
$\\{\tau_{j,\varepsilon,\delta}\\}_{\delta}$, we may assume that
$\\{\tau_{j,\varepsilon,\delta}\\}_{\delta>0}$ weakly converges to some
$\tau_{j,\varepsilon}$. Then it follows that
$\sqrt{2q\delta}\,\tau_{j,\delta,\varepsilon}$ weakly converges to zero by
$\delta\to 0$. Consequently, by taking the weak limit as $\delta\to 0$, we
obtain that

$\displaystyle\overline{\partial}\widetilde{u}_{j,\varepsilon}$
$\displaystyle=\widetilde{g}\,\,\text{on}\,\,X_{j}\quad\text{and}$
$\displaystyle\int_{X_{j}}|\widetilde{u}_{j,\varepsilon}|^{2}_{\omega_{\varepsilon},h}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}$
$\displaystyle\leq\int_{Y}\langle
B^{-1}_{\omega_{Y},\theta_{\psi}}g,g\rangle_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}.$

We now fix $\varepsilon$, and consider a weak limit as $j\to\infty$. The
right-hand side does not depend on $j$ and the norm
$\int_{X_{j}}|\bullet|^{2}_{\omega_{\varepsilon},h}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}$
monotonically increases as $j\nearrow\infty$. Therefore, for a fixed positive
number $\varepsilon>0$, the same argument as above works for
$\\{\widetilde{u}_{j,\varepsilon}\\}_{j=1}^{\infty}$. Hence, the subsequence
$\\{\widetilde{u}_{j,\varepsilon}\\}_{j=1}^{\infty}$ (for which we use the
same notation) weakly converges to some $\widetilde{u}_{\varepsilon}\in
L^{2}_{n,q-1}(X,L;\omega_{\varepsilon},he^{-f^{*}\psi})$ and the weak limit
$\widetilde{u}_{\varepsilon}$ satisfies that

$\displaystyle\overline{\partial}\widetilde{u}_{\varepsilon}$
$\displaystyle=\widetilde{g}\,\,\text{on}\,\,X_{j}\quad\text{and}$
$\displaystyle\int_{X}|\widetilde{u}_{\varepsilon}|^{2}_{\omega_{\varepsilon},h}e^{-\psi}\,dV_{\omega_{\varepsilon}}$
$\displaystyle\leq\int_{Y}\langle
B^{-1}_{\omega_{Y},\theta_{\psi}}g,g\rangle_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}.$

By noting that that
$|w|^{2}_{\omega_{\varepsilon}}\,dV_{\omega_{\varepsilon}}$ monotonically
increases as $\varepsilon\searrow 0$ for any $(n+m,q)$-form $w$, we can apply
the same argument as above again, which yields the desired solution
$\widetilde{u}$. ∎

######  Step 4 (Descent of $\widetilde{u}$ to $Y$).

The purpose of this step is to show that $\widetilde{u}$ determines an
$E$-valued $(m,q-1)$ form $u$ on $Y$ with the desired properties in Theorem
3.2 to complete the proof. To this end, from the uniform $L^{2}$-estimate in
Claim 3.4, we deduce that $\widetilde{u}$ can be locally written

$\displaystyle\widetilde{u}=\sum_{|K|=q-1}U_{K}\,dz\wedge\,dt\wedge
d\overline{t}_{K}$

on $X_{0}=f^{-1}(Y_{0})$, where $U_{K}$ is a local section of $E$, in other
words, the term containing $d\overline{z}_{i}$ never appear in the local
expression. To derive a contradiction, we assume that $\widetilde{u}$ contains
the term $Udz\wedge\,dt\wedge d\overline{z}_{I}\wedge d\overline{t}_{L}$,
where $I\not=\emptyset$ and $L$ are multi-indices. We take the Kähler form
$\omega_{\mathrm{st}}$ locally defined on $X_{0}$ as in Step 2, and the Kähler
form $\omega_{\mathrm{st},Y}:=\sqrt{-1}\sum_{i=1}^{m}\,dt_{i}\wedge
d\overline{t}_{i}$ locally defined on $Y_{0}$. Fix a positive constant $C>0$
so that
$\omega_{\varepsilon}\leq\omega^{\prime}_{\varepsilon}:=C(f^{*}\omega_{\mathrm{st},Y}+\varepsilon\omega_{\mathrm{st}})$
holds. Then, by noting that $dt_{j}$ and $dz_{k}$ are orthogonal with respect
to the new metric $\omega^{\prime}_{\varepsilon}$, we can see that

$\displaystyle|\widetilde{u}|^{2}_{h,\omega_{\varepsilon}}\,dV_{\omega_{\varepsilon}}$
$\displaystyle\geq|\widetilde{u}|^{2}_{h,\omega^{\prime}_{\varepsilon}}\,dV_{\omega^{\prime}_{\varepsilon}}$
$\displaystyle\geq|U\,dz\wedge dt\wedge d\overline{z}_{I}\wedge
d\overline{t}_{L}|^{2}_{h,\omega^{\prime}_{\varepsilon}}\,dV_{\omega^{\prime}_{\varepsilon}}$
$\displaystyle=|U|_{h}^{2}|d\overline{z}_{I}|^{2}_{\omega^{\prime}_{\varepsilon}}|d\overline{t}_{L}|^{2}_{\omega^{\prime}_{\varepsilon}}|dz\wedge\,dt|^{2}_{\omega_{X}}\,dV_{\omega_{X}}$

on an open set in $X_{0}$ Furthermore, by the definition of
$\omega^{\prime}_{\varepsilon}$, we can see that
$|d\overline{z}_{I}|^{2}_{\omega^{\prime}_{\varepsilon}}=1/(C\varepsilon)^{|I|}$
and
$|d\overline{t}_{L}|^{2}_{\omega^{\prime}_{\varepsilon}}=1/(C(1+\varepsilon))^{|L|}$.
This indicate that the integral of
$|\widetilde{u}|^{2}_{h,\omega_{\varepsilon}}\,dV_{\omega_{\varepsilon}}$ is
not uniformly bounded in $\varepsilon$, which contradicts Claim 3.4.

By $\overline{\partial}\widetilde{u}=\widetilde{g}$ and the above local
expression, we obtain that

$\sum_{|K|=q-1}\big{(}\sum_{j=1}^{m}\frac{\partial
U_{K}}{\partial\overline{t}_{j}}\,d\overline{t}_{j}+\sum_{k=1}^{n}\frac{\partial
U_{K}}{\partial\overline{z}_{k}}\,d\overline{z}_{k}\big{)}\wedge
d\overline{t}_{K}=\sum_{|J|=q}G_{J}\,d\overline{t}_{J}.$

By considering the term containing $d\overline{z}_{k}$, we can see that
$U_{K}$ is holomorphic in the fiber coordinate $z_{i}$, which implies that
$u_{K}(t):=U_{K}(t,\bullet)\,dz\in H^{0}(X_{t},K_{X_{t}}\otimes L|_{X_{t}})$
Thus, we can conclude that $\widetilde{u}$ can be determined by the $E$-valued
$(m,q-1)$-form $u$ on $Y_{0}$ defined by

$u:=\sum_{|K|=q-1}u_{K}(t)\,dt\wedge d\overline{t}_{K}.$

Furthermore, by considering the term containing $d\overline{t}_{k}$, we can
see that $\overline{\partial}u=g$ holds on $Y_{0}$.

We now consider the $L^{2}$-norm of $u$. Note that the norm
$|d\overline{t}_{K}|^{2}_{\omega_{\varepsilon}}$ converges to
$f^{*}|d\overline{t}_{K}|^{2}_{\omega_{Y}}$ by the same calculation as in Step
2 and

$|\widetilde{u}|^{2}_{\omega_{\varepsilon},h}\,dV_{\omega_{\varepsilon}}\nearrow\sum_{|K|=q-1}f^{*}|d\overline{t}_{K}|^{2}_{\omega_{Y}}\cdot|U_{K}|^{2}_{h}\,c_{n+m}dt\wedge
dz\wedge d\bar{t}\wedge d\bar{z}$

as $\varepsilon\searrow 0$. Hence, by Fatou’s lemma and the argument in Step 2
based on Fubini’s theorem, we can see that

$\displaystyle\liminf_{\varepsilon\to
0}\int_{X_{0}}|\widetilde{u}|^{2}_{\omega_{\varepsilon},h}\cdot
e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}$ $\displaystyle\geq$
$\displaystyle\int_{X_{0}}\sum_{|K|=q-1}f^{*}|d\overline{t}_{K}|^{2}_{\omega_{Y}}\cdot|U_{K}|^{2}_{h}\cdot
e^{-f^{*}\psi}\,c_{n+m}dz\wedge dt\wedge d\overline{z}\wedge\,d\overline{t}$
$\displaystyle=$ $\displaystyle\int_{y\in
Y_{0}}\sum_{|K|=q-1}|d\overline{t}_{K}|^{2}_{\omega_{Y}}e^{-\psi}\Big{(}\int_{X_{y}}|U_{K}|^{2}_{h}\,c_{n}dz\wedge
d\overline{z}\Big{)}\,c_{m}dt\wedge d\overline{t}$ $\displaystyle=$
$\displaystyle\int_{Y_{0}}|u|^{2}_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}$

Hence, together with Claim 3.4, we obtain the desired $L^{2}$-estimate:

$\displaystyle\int_{Y}|u|^{2}_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}\leq\int_{Y}\langle
B^{-1}_{\omega_{Y},\theta_{\psi}}g,g\rangle_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}.$

[HPS18, Proposition 23.3] shows that the Narasimhan-Simha metric $H$ satisfies
the assumption in Lemma 2.6, (i.e., the norm $|t|_{H^{*}}$ is bounded for any
section of $E^{*}$). This can be proved without using the Griffiths
$\theta$-semi-positivity (see the proof of [HPS18, Proposition 23.3]). Thus,
even in this step, our proof never uses the Griffiths $\theta$-semi-positivity
of $H$ (see Remark 3.5). By Lemma 2.6, the $\overline{\partial}$-equation
$\overline{\partial}u=g$ on $Y_{0}$ can be extended on $Y$, completing the
proof.

∎

######  Remark 3.5.

As noted, the Griffiths $\theta$-semi-positivity of $H$ is not used in the
proof of Theorem 1.1. More precisely, the only result from [HPS18, PT18] that
we use is [HPS18, Proposition 23.3] in the final step. This result follows
from the Ohsawa-Takegoshi $L^{2}$-extension theorem with a “non-optimal”
$L^{2}$-estimate. Due to [BL16], the Griffiths semi-positive of $H$ is
equivalent to the Ohsawa-Takegoshi $L^{2}$-extension theorem with “optimal”
$L^{2}$-estimate. In this sense, our proof does not depend on the Griffiths
$\theta$-semi-positive essentially.

At the end of this section, as an application of Theorem 3.2 we give proofs
for Theorem 1.1 and Corollary 1.2.

###### Proof of Theorem 1.1.

We first remark that the Narasimhan-Simha metric $H$ is lower semi-continuous
by [HPS18]. We emphasize that in this paper, the Griffiths semi-positivity is
used only to check this lower semi-continuous. More precisely, we do not use
the Griffiths semi-positivity in checking that $H$ is lower semi-continuous on
$Y\setminus Z$, where $Z$ is a suitable Zariski closed subset (see [HPS18,
Proposition 22.5]), but we use the Griffiths semi-positivity in extending this
property to $Y$.

Let $Y_{\mathcal{E}}\subset Y$ be the maximal Zariski open subset where the
torsion-free sheaf
$\mathcal{E}:=f_{*}(\mathcal{O}_{X}(K_{X/Y}+L)\otimes\mathcal{I}(h))$ is
locally free. Take a Stein coordinate $\Omega\subset Y_{\mathcal{E}}$ equipped
with the data as in Definition 2.1. If the inverse image $f^{-1}(\Omega)$ is
Kähler, then the $\overline{\partial}$-equation with optimal $L^{2}$-estimate
in Definition 2.1 can be solved by Theorem 3.2. This Kähler condition is
satisfied if $\Omega$ is sufficiently small, which proves conclusion (1).
Also, this Kähler condition is satisfied if $X$ is Kähler, which proves
conclusion (2). ∎

###### Proof of Corollary 1.2.

Our strategy of the proof is based on the argument in [Ohs84, Mat16].

We first fix a Stein open cover $\mathcal{U}:=\\{U_{i}\\}_{i\in I}$ of $Y$ and
a partition $\\{\rho_{i}\\}_{i\in I}$ of unity associated with $\mathcal{U}$.
Then, we consider the diagram

$\textstyle{H^{q}(X,K_{X}\otimes
L\otimes\mathcal{I}(h))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\scriptstyle{\circlearrowleft}$$\textstyle{H^{q}(X_{\mathcal{E}},K_{X}\otimes
L\otimes\mathcal{I}(h))}$$\textstyle{\check{H}^{q}(\mathcal{U},K_{Y}\otimes\mathcal{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi_{Y}}$$\textstyle{H^{q}(Y_{\mathcal{E}},K_{Y}\otimes\mathcal{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\widetilde{\bullet}}$

where $\mathcal{E}:=f_{*}(\mathcal{O}_{X}(K_{X/Y}+L)\otimes\mathcal{I}(h))$
and the above morphisms are defined as follows: The morphism
$\widetilde{\bullet}$ is defined by the same operation $[\widetilde{g}]$ as in
Step 1 of the proof of Theorem 3.2 by regarding
$H^{q}(Y_{\mathcal{E}},K_{Y}\otimes\mathcal{E})$ as the
$\overline{\partial}$-cohomology group and $r$ is the restriction map from $X$
to $X_{\mathcal{E}}:=f^{-1}(Y_{\mathcal{E}})$, where $Y_{\mathcal{E}}\subset
Y$ is the maximal Zariski open where $\mathcal{E}$ is locally free. To explain
the definitions of $\varphi$ and $\varphi_{Y}$, we take a $\rm{\check{C}}$ech
cohomology class $[\\{\alpha_{i_{0}\cdots
i_{q}}\\}]\in\check{H}^{q}(\mathcal{U},K_{Y}\otimes\mathcal{E})$. Then, the
$\overline{\partial}$-cohomology class $\varphi_{Y}(\alpha)$ is defined by the
$\overline{\partial}$-closed $E$-valued $(m,q)$-form $\varphi_{Y}(\alpha)$
constructed by the standard method:

$\varphi_{Y}(\alpha):=\overline{\partial}\sum_{i_{q-1}}\rho_{i_{q-1}}\Big{(}\cdots\overline{\partial}\sum_{i_{1}}\rho_{i_{1}}\big{(}\overline{\partial}\sum_{i_{0}}\rho_{i_{0}}(\alpha_{i_{0}\cdots
i_{q}}|_{U_{i_{0}\cdots i_{q}}\cap Y_{\mathcal{E}}})\big{)}\Big{)},$

where ${U}_{i_{0}...i_{q}}:=U_{i_{0}}\cap\cdots\cap U_{i_{q}}$ is a Stein open
set. By the projection formula

$f_{*}\mathcal{O}_{X}(K_{X}\otimes L\otimes\mathcal{I}(h))\cong
K_{Y}\otimes\mathcal{E},$

we have the natural morphism

$H^{0}({U}_{i_{0}...i_{p}},f_{*}(K_{X}\otimes
L\otimes\mathcal{I}(h)))\xrightarrow{\quad\cong\quad}H^{0}(f^{-1}({U}_{i_{0}...i_{p}}),K_{X}\otimes
L\otimes\mathcal{I}(h)),$

which corresponds to $\widetilde{\bullet}$ of the case $q=0$. Hence, we can
regard $\alpha_{i_{0}\cdots i_{q}}$ as the section

$\widetilde{\alpha}_{i_{0}\cdots i_{q}}\in
H^{0}(f^{-1}({U}_{i_{0}...i_{p}}),K_{X}\otimes L\otimes\mathcal{I}(h)).$

Then $\varphi(\alpha)$ is defined by the same way as $\varphi_{Y}(\alpha)$
using $\\{\widetilde{\alpha}_{i_{0}\cdots i_{q}}\\}$ and
$\\{\widetilde{\rho}_{i}:=f^{*}\rho_{i}\\}_{i\in I}$. We can easily see that
the above diagram is commutative. Furthermore, the morphism $\varphi$ is
injective by the same argument as in [Mat16, Subsection 3.3] (and [Ohs84]).

We will apply the proof of Theorem 3.2 to the case where
$g:=\varphi_{Y}(\alpha)$, $\theta:=\omega_{Y}$, and $\psi=0$. Then, by
construction and noting that $\theta_{0}=\omega_{Y}$, we can see that

$\langle
B^{-1}_{\omega_{Y},\theta_{0}}\,g,g\rangle_{\omega_{Y},H}=\frac{1}{q}|g|^{2}_{\omega_{Y},H}$

is locally $L^{2}$-integrable on $Y$ (not only on $Y_{\mathcal{E}}$)

Since $Y$ is a weakly pseudo-convex manifold, we can find an exhaustive psh
$\psi$ function on $Y$ such that $\langle
B^{-1}_{\omega_{Y},\theta_{\psi}}\,g,g\rangle_{\omega_{Y},H}e^{-\psi}$ is
$L^{2}$-integrable on $Y$. Indeed, this can be achieved by replacing $\psi$
with $\chi\circ\psi$ and noting that $B^{-1}_{\omega_{Y},\theta_{\psi}}\leq
B^{-1}_{\omega_{Y},\theta_{0}}$, where $\chi\colon\mathbb{R}\to\mathbb{R}$ is
a rapidly increasing convex function. Thus, Claim 3.4 shows that there exists
$\widetilde{u}$ on $X_{\mathcal{E}}$ satisfying

$\displaystyle\overline{\partial}\widetilde{u}$
$\displaystyle=\widetilde{g}\,\,\text{on}\,\,X_{\mathcal{E}}\quad\text{and}$
$\displaystyle\int_{X_{\mathcal{E}}}|\widetilde{u}|^{2}_{\omega_{\varepsilon},h}e^{-f^{*}\psi}\,dV_{\omega_{\varepsilon}}$
$\displaystyle\leq\int_{Y_{\mathcal{E}}}\langle
B^{-1}_{\omega_{Y},\theta_{\psi}}g,g\rangle_{\omega_{Y},H}e^{-\psi}\,dV_{\omega_{Y}}.$

The smooth function $f^{*}\psi$ is defined on $X$ (not only on
$X_{\mathcal{E}}$). Thus, the $\overline{\partial}$-equation
$\overline{\partial}\widetilde{u}=\widetilde{g}$ can be extended to $X$, which
implies that $\varphi(\alpha)$ is $\overline{\partial}$-exact. This finishes
the proof since $\varphi$ is injective and $\widetilde{g}$ is a representation
of the $\overline{\partial}$-cohomology $\varphi(g)$. ∎

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