# SPIN-PARITY EFFECT IN VIOLATION OF BELL’S INEQUALITIES FOR ENTANGLED STATES
OF PARALLEL POLARIZATION

HAIFENG ZHANG, JIANHUA WANG, ZHIGANG SONG  Institute of Theoretical Physics
and Department of Physics, Shanxi University, Taiyuan, Shanxi 030006, China
J. -Q. LIANG  Institute of Theoretical Physics and Department of Physics,
Shanxi U niversity, Taiyuan, Shanxi 030006, China
<EMAIL_ADDRESS>L. -F. WEI  State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China  Quantum Optoelectronics Laboratory, School of Physics and Technology, Southwest Jiaotong University, Chengdu 610031, China

( 15 September 2016; 1 December 2016)

###### Abstract

Bell inequalities (BIs) derived in terms of quantum probability statistics are
extended to general bipartite-entangled states of arbitrary spins with
parallel polarization. The original formula of Bell for the two-spin singlet
is slightly modified in the parallel configuration, while, the inequality
formulated by Clauser-Horne-Shimony-Holt remains not changed. The violation of
BIs indeed resulted from the quantum non-local correlation for spin-$1/2$
case. However, the inequalities are always satisfied for the spin-$1$
entangled states regardless of parallel or antiparallel polarizations of two
spins. The spin parity effect originally demonstrated with the antiparallel
spin-polarizations (Mod. Phys. Lett. B28, 145004) still exists for the
parallel case. The quantum non-locality does not lead to the violation for
integer spins due to the cancellation of non-local interference effects by the
quantum statistical-average. Again the violation of BIs seems a result of the
measurement induced nontrivial Berry-phase for half-integer spins.

Bell’s inequality; entanglement; Berry phase.

###### pacs:

03.65.Ud; 03.65.Vf; 03.67.Bg; 42.50.Xa

##  Part I 1\. Introduction

Non-locality being the most peculiar characteristic of quantum mechanics seems
inconsistency with the classical field-theory based on the relativistic
causality. Quantum entanglement as one of the most striking feature of quantum
mechanics has become a resource for quantum computing and quantum information
with technological breakthroughs in these areas.2 ; 3 ; 4  Bell’s theorem
known as Bell’s inequality (BI) proved that the existence of entangled quantum
states has no classical counterpart and therefore provides a possibility of
quantitative test for non-local correlations. BI was originally derived under
classical statistics with the assumption of ”locality”, which means that
physical systems cannot be instantly affected by distant objects in a space-
like distance. Various extensions of the original BI have investigated 26 ; 5
and attracted considerable attentions both theoretically and experimentally.2
; 3 ; 11 ; 12 ; 13 ; 14 ; 16  The violation of BIs is predicted in some
particular entangled-states with overwhelming experimental evidence,5 ; 6 ; 7
; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14  which invalidates local realistic
interpretations of quantum mechanics. The profound application of non-locality
is mainly in the quantum information with a space–time separation not
achievable for the classical systems. The violation of BIs constitutes an
ability to faithfully produce, control and read out entangled states of qubit-
pairs. The non-locality now has become a powerful resource3 ; 16  of quantum
information science.17 ; 18 ; 19 ; 20 ; 21 ; 22 ; 23 ; 24 ; 25  Nevertheless,
there are still many open problems in relation with both bipartite and
multipartite entanglements. Although the experimental evidence seems strongly
support the non-local nature the underlying physical-principle is obscure15
and many aspects in relation with the initial debate remain to be fully
understood. The non-determinism and non-locality have been bearing
continuously theoretical scrutiny ever since the birth of quantum mechanics.

To understand underlying physical principle of the violation of BIs we in a
previous paper1  adopted the density operators of entangled states to evaluate
the measurement-outcome-correlations in terms of quantum probability-
statistics. With the separation of density operators into local and non-local
parts the non-local outcome-correlations are obtained explicitly. The local
part of density operator describes the classical probability-statistics in the
absence of quantum interference between two components of the entangled state.
The BI is verified from the local part of density operator27  alone with the
assumption of measurement-outcome-independence. On the other hand the
violation of BIs indeed resulted from non-local correlations of entangled
states. A spin parity effect in the violation of BIs was predicted1  that BIs
are violated by the half-integer spin entangled-states but not the integer
spins. Moreover the violation in entangled spin-states is seen to be an effect
of Berry phase (BP) induced by relative-reversal measurements of two spins. We
established for the first time a relation between two kinds of non-localities,
namely, the violation of BI violation and the so-called dynamic non-locality
regarded as the geometric phase interference of quantum states.1  The spin-
parity phenomenon was originally demonstrated in terms of a spin singlet
following the model of Bell. It is a natural question that whether or not the
spin-parity effect exists only for these particular states with antiparallel
spin-polarizations. We in the present paper consider bipartite-entangled state
with parallel spin-polarization, which leads to the modification of original
form of BI. However the inequality derived by Clauser-Horne-Shimony-Holt
(CHSH) remains not changed. The spin-parity phenomenon for the violation of
BIs seems independent of the particular form of entangled states.

##  Part II 2\. Bell’s inequalities for spin-1/2 and spin-1 entangled states
of parallel spin-polarization and their violations

##  Part III 2.1 Spin-1/2 state

The violation of BIs was theoretically verified long ago for spin-$1/2$
bipartite entanglement with arbitrary superposition of opposite spin-
polarization states33 ; 34  in stead of the spin singlet in the original
formulation of Bell. It was revisited in a recent paper1  based on quantum
probability statistics, which has advantage that both the BI and its violation
can be formulated in a unified manner with the help of state density-
operator.1  We consider in the present paper a two-spin entangled state with
parallel polarization such that

$|\psi\rangle=c_{+}|+,+\rangle+c_{-}|-,-\rangle$ (1)

where $c_{\pm}$ are two arbitrary complex coefficients with the normalization
condition $|$$c_{+}|^{2}+$$|$$c_{-}|^{2}=1$ and $|\pm\rangle$ denote the usual
spin-$1/2$ eigenstates ($\widehat{\sigma}_{z}|\pm\rangle=\pm|\pm\rangle$).
Without losing generality, the two arbitrary coefficients $c_{\pm}$ can be
parameterized as

$c_{+}=e^{i\eta}\cos\xi,\qquad c_{-}=e^{-i\eta}\sin\xi,$

with $\xi$ and $\eta$ being two real parameters. The density-operator
$\overset{\wedge}{\rho}$ of entangled state Eq.(1) can be split into two parts

$\overset{\wedge}{\rho}=\overset{\wedge}{\rho}_{lc}+\overset{\wedge}{\rho}_{nlc}$

in order to see the non-local quantum correlation explicitly. Here, the first
part is a density operator of the complete mixed-state

$\overset{\wedge}{\rho}_{lc}=\cos^{2}\xi|+,+\rangle\langle+,+|+\sin^{2}\xi|-,-\rangle\langle-,-|,$

which describes two particles obeying the local or classical statistics in the
absence of quantum interference at all. While, the quantum interference term

$\overset{\wedge}{\rho}_{nlc}=e^{2i\eta}\sin\xi\cos\xi|+,+\rangle\langle-,-|+e^{-2i\eta}\sin\xi\cos\xi|-,-\rangle\langle+,+|$

denotes the non-local correlation, which remains even if the two particles are
separated in a space-like interval. Following Bell, two spins are measured
independently along arbitrary directions respectively, say ${\mathbf{a}}$ and
${\mathbf{b}}$. According to the quantum measurement principle, measuring
outcomes fall into the eigenvalues of projection spin-operators
$\hat{\sigma}\cdot{\mathbf{a}}$ and $\hat{\sigma}\cdot{\mathbf{b}}$, i.e.
$\hat{\sigma}\cdot\mathbf{a|\pm a}\rangle=\pm\mathbf{|\pm a}\rangle$, and
$\hat{\sigma}\cdot\mathbf{b|\pm b}\rangle=\pm\mathbf{|\pm b}\rangle$ . Two
orthogonal eigenstates of a projection spin-operator
$\hat{\sigma}\cdot\mathbf{r}$ can be found explicitly as

$\mathbf{|+r}\rangle\mathbf{=}\cos(\frac{\theta_{r}}{2})|+\rangle+\sin(\frac{\theta_{r}}{2})e^{i\phi_{r}}|-\rangle$

and

$\mathbf{|-r}\rangle\mathbf{=}\sin(\frac{\theta_{r}}{2})|+\rangle-\cos(\frac{\theta_{r}}{2})e^{i\phi_{r}}|-\rangle$

which are called the spin coherent states of north- and south- pole gauges.30
Here,
$\mathbf{r=(}\sin\theta_{r}\cos\phi_{r},\sin\theta_{r}\sin\phi_{r}$,$\cos\theta_{r})$
with $\mathbf{r}=\mathbf{a},\mathbf{b}$ is an unit vector parameterized by the
polar and azimuthal angles ($\theta_{r}$ and $\phi_{r}$) in the coordinate
frame with $z$-axis along the direction of the initial spin-polarization.
Outcome-independent base vectors for two-particle measurements are seen to be
the direct product of eigenstates for operators $\hat{\sigma}\cdot\mathbf{a}$
and $\hat{\sigma}\cdot\mathbf{b}.$ We can arbitrarily label these four base
vectors as1

$|1\rangle\\!\\!=\\!\\!\mathbf{|\\!\\!+\\!a,\\!\\!+\\!b}\rangle,\\!|2\rangle\\!\\!=\\!\\!\mathbf{|\\!\\!+\\!a,\\!\\!-\\!b}\rangle,\\!|3\rangle\\!\\!=\\!\\!\mathbf{|\\!\\!-\\!a,\\!+\\!b}\rangle,|4\rangle\\!\\!=\\!\\!\mathbf{|\\!\\!-\\!a,\\!\\!-\\!b}\rangle.$
(2)

Then the correlation probability for independent measurements of two spins is
the quantum statistical average of the correlation operator

$\hat{\Omega}(ab)=(\hat{\sigma}\cdot\mathbf{a)}(\hat{\sigma}\cdot\mathbf{b)}$

in the state $\overset{\wedge}{\rho}$

$P(ab)=Tr[\hat{\Omega}(ab)(\hat{\rho}_{lc}+\hat{\rho}_{nlc})]=P_{lc}(ab)+P_{nlc}(ab),$
(3)

Notice that the non-vanishing matrix elements of correlation operator in the
outcome-independent base vectors Eq.(2) are obviously

$\Omega_{11}(ab)=\Omega_{44}(ab)=1$

and

$\Omega_{22}(ab)=\Omega_{33}(ab)=-1,$

the correlation probability is found as

$P(ab)=\rho_{11}+\rho_{44}-\rho_{22}-\rho_{33}.$

The density-matrix elements denoted by $\rho_{ij}=\langle
i|\overset{\wedge}{\rho}|j\rangle$, ($i,j=1,2,3,4$) can be split into two
parts

$\rho_{ii}=\rho_{ii}^{lc}+\rho_{ii}^{nlc},$

with local elements given by

$\rho_{11}^{lc}=\cos^{2}(\xi)K_{a}^{2}K_{b}^{2}+\sin^{2}(\xi)\Gamma_{a}^{2}\Gamma_{b}^{2}$
$\rho_{22}^{lc}=\cos^{2}(\xi)K_{a}^{2}\Gamma_{b}^{2}+\sin^{2}(\xi)\Gamma_{a}^{2}K_{b}^{2}$
$\rho_{33}^{lc}=\cos^{2}(\xi)\Gamma_{a}^{2}K_{b}^{2}+\sin^{2}(\xi)K_{a}^{2}\Gamma_{b}^{2}$
$\rho_{44}^{lc}=\cos^{2}(\xi)\Gamma_{a}^{2}\Gamma_{b}^{2}+\sin^{2}(\xi)K_{a}^{2}K_{b}^{2}.$

Where $K_{r}^{m}=\cos^{m}(\theta_{r}/2)$ and
$\Gamma_{r}^{m}=\sin^{m}(\theta_{r}/2)$ with $r=a,b,c,d$ denoting measurement
directions and $m$ the integer power-index throughout the paper. The non-local
elements $\rho_{11}^{nlc}$, $\rho_{44}^{nlc}$ have equal value but opposite
signs with respect to $\rho_{22}^{nlc}$, $\rho_{33}^{nlc}$

$\displaystyle\rho_{11}^{nlc}$ $\displaystyle=$
$\displaystyle\rho_{44}^{nlc}=-\rho_{22}^{nlc}=-\rho_{33}^{nlc}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sin\xi\cos\xi\sin\theta_{a}\sin\theta_{b}\cos\left(\phi_{a}+\phi_{b}-2\eta\right).$

This property plays a crucial role, which we will see, in the violation of
BIs. The measurement outcome correlation has the same form

$P_{lc}(ab)=Tr[\hat{\Omega}(ab)\hat{\rho}_{lc}]=\cos\theta_{a}\cos\theta_{b},$
(5)

but a positive sign different from that for the spin singlet with opposite
spin-polarizations1  where the correlation formula has a negative sign seen
from Eq.(4) of Ref.(1 ). This simple sign-difference leads to a necessary
modification of the original formula of BI. In other words, the specific form
of BI is state dependent. From the local correlation Eq.(5) it is easy to find
the modified BI being

$|P_{lc}(ab)-P_{lc}(ac)|\leq 1-P_{lc}(bc),$ (6)

different from the original1  BI by a sign change in front of the local
correlation $P_{lc}(bc)$ on the greater side of inequality. As a matter of
fact, substitution of the correlation Eq.(5) into the less side of BI Eq.(6)
yields

$|P_{lc}(ab)-P_{lc}(ac)|\leq|\cos\theta_{b}-\cos\theta_{c}|.$

Therefor it is straightforward to prove the modified BI in the parallel spin-
polarization

$|P_{lc}(ab)-P_{lc}(ac)|+P_{lc}(bc)\leq|\cos\theta_{b}-\cos\theta_{c}|+\cos\theta_{b}\cos\theta_{c}\leq
1.$

The modified BI can be also verified by means of classical statistics
following Bell as shown in Appendix. The non-local correlation

$P_{nlc}(ab)=2\sin\xi\cos\xi\sin\theta_{a}\sin\theta_{b}\cos(\phi_{a}-\phi_{b}-2\eta),$

which comes from the quantum interference of coherent superposition of two-
particle states, is responsible for the violation of BI. To see the violation
of BI explicitly we assume that the parameters of superposition coefficients
are $\xi=\pi/4$ and $\eta=n\pi$ with $n$ being zero or integer. The total
measurement correlation including the non-local parts becomes the well known
quantum correlation-probability being a scalar product of the two unit-vectors
$\mathbf{a}$ and $\mathbf{b}$

$P(ab)=\mathbf{a}\cdot\mathbf{b},$ (7)

which has a sign difference compared with the opposite spin-polarization.1
The violation of BI has been investigated extensively1  in terms of the
quantum correlation-probability.The CHSH inequality remains not changed, since
a common sign-difference of the measurement outcome correlation Eq.(5) for any
two directions does not affect the absolute value

$P_{\mathrm{CHSH}}^{lc}=|P_{lc}(ab)+P_{lc}(ac)+P_{lc}(db)-P_{lc}(dc)|.$

We still have

$P_{CHSH}^{lc}\leq 2.$

With the quantum correlation-probability Eq.(7) it is also obviously to have
the well known formula

$P_{\mathrm{CHSH}}=|P(ab)+P(ac)+P(db)-P(dc)|\leq 2\sqrt{2}.$

##  Part IV 2.2 Spin-1 state

For the spin-$1$ entangled state of parallel spin-polarizations

$|\psi\rangle=c_{+}|+1,+1\rangle+c_{-}|-1,-1\rangle,$

the spin-coherent states of spin-$1$ projection operator
$\hat{s}\cdot\mathbf{a}$ are found as1

$|+\mathbf{a}\rangle_{1}=K_{a}^{2}|+1\rangle+\frac{1}{\sqrt{2}}e^{i\phi_{a}}\sin\theta_{a}|0\rangle+\Gamma_{a}^{2}e^{i2\phi_{a}}|-1\rangle$

and

$|-\mathbf{a}\rangle_{1}=\Gamma_{a}^{2}|+1\rangle-\frac{1}{\sqrt{2}}e^{i\phi_{a}}\sin\theta_{a}|0\rangle+K_{a}^{2}e^{i2\phi_{a}}|-1\rangle,$

where $\hat{s}_{z}|\pm 1\rangle=\pm|\pm 1\rangle$. The measurement-correlation
probability of two spin-$1$ particles initially prepared in the entangled
state with parallel spin-polarization can be obtained by the local matrix
elements

$\displaystyle\rho_{(1)11}^{lc}$ $\displaystyle=$
$\displaystyle\cos^{2}(\xi)K_{a}^{4}K_{b}^{4}+\sin^{2}(\xi)\Gamma_{a}^{4}\Gamma_{b}^{4},$
$\displaystyle\rho_{(1)44}^{lc}$ $\displaystyle=$
$\displaystyle\cos^{2}(\xi)\Gamma_{a}^{4}\Gamma_{b}^{4}+\sin^{2}(\xi)K_{a}^{4}K_{b}^{4},$
$\displaystyle\rho_{(1)22}^{lc}$ $\displaystyle=$
$\displaystyle\cos^{2}(\xi)K_{a}^{4}\Gamma_{b}^{4}+\sin^{2}(\xi)\Gamma_{a}^{4}K_{b}^{4},$
$\displaystyle\rho_{(1)33}^{lc}$ $\displaystyle=$
$\displaystyle\cos^{2}(\xi)\Gamma_{a}^{4}K_{b}^{4}+\sin^{2}(\xi)K_{a}^{4}\Gamma_{b}^{4}.$

However the four non-local matrix elements are all equal

$\rho_{(1)22}^{nlc}=\rho_{(1)33}^{nlc}=\rho_{(1)11}^{nlc}=\rho_{(1)44}^{nlc}=\frac{1}{8}\cos\xi\sin\xi\sin^{2}\theta_{a}\sin^{2}\theta_{b}\cos
2(\phi_{a}-\phi_{b}-\eta),$ (8)

the same as in the case of antiparallel spin-polarizations.1  It was explained
in our previous paper1  that the minus sign difference between
$\rho_{11}^{nlc}$,$\rho_{44}^{nlc}$ and $\rho_{22}^{nlc}$,$\rho_{33}^{nlc}$ in
Eq.(4) for the spin-$1/2$ is actually a nontrivial BP resulted from the
relative reversal of two-spin measurements
($\rho_{22}^{nlc},\rho_{33}^{nlc}$). In the spin-$1$ case the BP factor is
only a trivial one $e^{i2\pi}=1$. Therefore, the contributions of non-local
interference between the same and opposite spin-polarization measurements
cancel each other. The total outcome correlation is originated from the local
density operator $\hat{\rho}_{(1)}^{lc}$ only, i.e.,

$P_{(1)}(ab)=P_{(1)}^{lc}(ab)=\cos\theta_{a}\cos\theta_{b}.$

There is no room for the violation of BIs in agreement with the previous
observation for the spin-$1$ entangled state with antiparallel spin
polarizations.1 __ It is interesting to see a fact that the original form of
BI is valid only for entangled states of antiparallel spin-polarizations.
However our observation, that BIs are violated by the spin-$1/2$ but not the
spin-$1$ entangled states, is true regardless of antiparallel or parallel
polarizations. Now we extend our theorem to arbitrarily high spins.

##  Part V 3\. Spin-parity phenomenon

For two spin-$s$ particles the entangled macroscopic quantum-state (MQS) with
parallel spin-polarization is defined as

$|\psi\rangle=c_{+}|+s,+s\rangle+c_{-}|-s,-s\rangle.$ (9)

The density operator of it can also be separated into the local part

$\hat{\rho}_{(s)}^{lc}=\cos^{2}\xi|+s,+s\rangle\langle+s,+s|+\sin^{2}\xi|-s,-s\rangle\langle-s,-s|,$

obeying the local or classical theory and the non-local part

$\displaystyle\hat{\rho}_{(s)}^{{}_{nlc}}$ $\displaystyle=$ $\displaystyle
e^{2i\eta}\cos\xi\sin\xi|+s,+s\rangle\langle-s,-s|$
$\displaystyle+e^{-2i\eta}\cos\xi\sin\xi|-s,-s\rangle\langle+s,+s|,$

respectively. Here, the extreme state $|\pm s\rangle$ ( $\hat{s}_{z}|\pm
s\rangle=\pm s|\pm s\rangle$ ) is called the MQS wherein the minimum
uncertainty relation
$|\langle\widehat{s}_{z}\rangle|=2\langle(\Delta\widehat{s}_{x})^{2}\rangle^{1/2}\langle(\Delta\widehat{s}_{y})^{2}\rangle^{1/2}$
is satisfied. So that the state $|\psi\rangle$ defined in Eq.(9) is called the
Bell cat, which is actually entangled Schrődinger cat-states for both ”dead”
and both ”life” cats. We assume that the measurements are restricted on MQS,
namely the spin coherent states $\mathbf{|\pm a}\rangle_{s}$ with
$\hat{s}\cdot\mathbf{a|\pm a}\rangle_{s}=\pm s\mathbf{|\pm a}\rangle_{s}$.
These spin coherent states can be generated from the extreme states $|\pm
s\rangle$ such that

$\mathbf{|\pm a}\rangle_{s}=\hat{R}|\pm s\rangle$

with the generation operator

$\hat{R}=e^{i\theta_{a}\mathbf{m}\cdot\hat{s}}.$

The unit-vector $\mathbf{m}$ in the $x-y$ plane is perpendicular to the plane
expanded by $z$-axis and the unit vector $\mathbf{a}$.

The explicit forms of spin coherent-states in the representation of Dicke
states are given by30 ; 31

$|+\mathbf{a}\rangle_{s}=\sum_{m=-s}^{s}\binom{2s}{s+m}\\!\\!^{\frac{1}{2}}K_{a}^{s+m}\Gamma_{a}^{s-m}e^{i(s-m)\phi_{a}}|m\rangle,$
$|-\mathbf{a}\rangle_{s}=\sum_{m=-s}^{s}\binom{2s}{s+m}^{\frac{1}{2}}K_{a}^{s-m}\Gamma_{a}^{s+m}e^{i(s-m)(\phi_{a}+\pi)}|m\rangle.$

In the outcome-independent base vector of two-particle measurements
(corresponding to Eq. (2), however, with the spin-$1/2$ replaced by $s$, )
density-matrix elements for the Bell cat-state are given by:

$\displaystyle\rho_{11}^{lc}$ $\displaystyle=$
$\displaystyle\cos^{2}(\xi)K_{a}^{4s}K_{b}^{4s}+\sin^{2}(\xi)\Gamma_{a}^{4s}\Gamma_{b}^{4s},$
$\displaystyle\rho_{22}^{lc}$ $\displaystyle=$
$\displaystyle\cos^{2}(\xi)K_{a}^{4s}\Gamma_{b}^{4s}+\sin^{2}(\xi)\Gamma_{a}^{4s}K_{b}^{4s},$
$\displaystyle\rho_{33}^{lc}$ $\displaystyle=$
$\displaystyle\cos^{2}(\xi)\Gamma_{a}^{4s}K_{b}^{4s}+\sin^{2}(\xi)K_{a}^{4s}\Gamma_{b}^{4s},$
$\displaystyle\rho_{44}^{lc}$ $\displaystyle=$
$\displaystyle\cos^{2}(\xi)\Gamma_{a}^{4s}\Gamma_{b}^{4s}+\sin^{2}(\xi)K_{a}^{4s}K_{b}^{4s},$
(10)

for the local part. With the above density-matrix elements Eq.(10) the
normalized (the correlation per spin value) local correlations become

$P_{(s)}^{lc}(ab)=(K_{a}^{4s}-\Gamma_{a}^{4s})(K_{b}^{4s}-\Gamma_{b}^{4s}),$
(11)

which is also different from the opposite spin-polarization1  by a negative
sign compared with the Eq.(7) in Ref.(1 ). Using the corresponding local
correlations Eq.(11) and notice $K_{\alpha}^{4s}-\Gamma_{\alpha}^{4s}\leq 1$
for $\alpha=a,b,c,d$, it is obviously that CHSH inequality remains the same

$\left|P_{(s)}^{lc}(ab)+P_{(s)}^{lc}(ac)+P_{(s)}^{lc}(db)-P_{(s)}^{lc}(dc)\right|\leq
2.$

While the BI is also modified as

$1-P_{(s)}^{lc}(bc)\geq|P_{(s)}^{lc}(ab)-P_{(s)}^{lc}(ac)|.$

The non-local density-matrix elements are evaluated by

$\rho_{11}^{nlc}=\rho_{44}^{nlc}=2\sin\xi\cos(\xi)K_{a}^{2s}\Gamma_{a}^{2s}K_{b}^{2s}\Gamma_{b}^{2s}\cos\left[2s\left(\phi_{a}+\phi_{b}-2\eta\right)\right],$

for the measurements along the same spin-polarization direction. The density-
matrix elements for the measurements on opposite directions

$\rho_{(s)22}^{nlc}=\rho_{(s)33}^{nlc}=(-1)^{2s}\rho_{(s)11}^{nlc},$ (12)

possess an additional geometric phase factor $e^{i2s\pi}=(-1)^{2s}$ compared
with $\rho_{(s)11}^{nlc}=\rho_{(s)44}^{nlc}$. It was well explained in the
previous paper1  that the geometric phase or BP resulted from the relative
reversal of spin-polarization measurements in the following matrix-element
evaluation

$\displaystyle\rho_{{}_{(s)}22}^{nlc}$ $\displaystyle=$ $\displaystyle
e^{2i\eta}\sin\xi\cos(\xi)\langle+s|+\mathbf{a}\rangle\langle+\mathbf{a}|+s\rangle\langle-s|-\mathbf{b}\rangle\langle-\mathbf{b}|-s\rangle$
(13)
$\displaystyle+e^{-2i\eta}\sin\xi\cos(\xi)\langle-s|+\mathbf{a}\rangle\langle+\mathbf{a}|-s\rangle\langle+s|-\mathbf{b}\rangle\langle-\mathbf{b}|+s\rangle.$

Remarkably we come to the same conclusion as in the case of antiparallel spin-
polarizations1  that the non-local outcome correlation vanishes
$P_{(s)}^{nlc}(a,b)=0$ in the integer-spin Bell cat-state, since the BP factor
is trivial and thus the four elements become equal
($\rho_{(s)11}^{nlc}=\rho_{(s)22}^{nlc}$) seen from Eq.(12). However, in the
half-integer spin case an additional minus sign of the BP factor leads to
$\rho_{(s)11}^{nlc}=-\rho_{(s)22}^{nlc}$, and the non-local part of
correlation probability $P_{(s)}^{nlc}(ab)$ does not vanish any more. The spin
parity phenomenon discovered in the previous paper1  still exists for the Bell
cat-states with parallel spin-polarization (both ”dead” and both ”life” cats)
:  the BI can be violated for the Bell cat-states of half-integer spins but
not for the states of integer spins.

##  Part VI 4\. Conclusions and Discussions

The proposed formulation of quantum probability-statistics with state-density
operator has advantage to separate the local and non-local measurement
correlations. The non-local part, which comes from the quantum interference
between two superposed state-components, can be evaluated independently to see
why and how the violation of BIs takes place. For the bipartite-entanglement
states of parallel spin-polarization the local measurement-correlations have
only a sign difference with respect to the opposite spin-polarizations
demonstrated in Ref.(1 ). The original BI is slightly modified due to the sign
difference, however, CHSH inequality is not changed in the entangled state of
parallel spin-polarization. The previous observation of spin-parity
phenomenon1  still holds, that the BIs are indeed violated for entangled MQS
of half-integer spins but not the integer-spins. The violation for the half-
integer spins can be understood as the effect of geometric phase induced by
the relative reversal of spin measurements. We establish a relation between
two-type of non-localities, namely, the violation of BI and the dynamic non-
locality resulted from the geometrical phase.

Although the specific form of original BI depends on the initial state the
spin-parity effect seems state independent. Our generic arguments of the spin-
parity effect particularly for the nonviolation of BIs in the spin-$1$
entangled states can be tested by utilizing the orbital angular-momenta
entanglement.28 ; 29 ; 32

##  Part VII Acknowledgements

This work was supported in part by the National Natural Science Foundation of
China, under Grants No. 11275118, U1330201.

##  Part VIII Appendix

Proof of the modified BI $|P_{lc}(ab)-P_{lc}(ac)|\leq 1-P_{lc}(bc)$ for the
entangled state with parallel spin-polarization in terms of classical
statistics following Bell.

The correlation of product expectation-values for measuring two spins
respectively along unit-vector directions $\mathbf{a},$ and $\mathbf{b}$ is
evaluated as

$P\left(a,b\right)=\int
d\lambda\rho\left(\lambda\right)A\left(\mathbf{a},\lambda\right)B\left(\mathbf{b},\lambda\right),$

where $\rho\left(\lambda\right)$ is the probability distribution and $\lambda$
is a hidden variable. $A\left(\mathbf{a},\lambda\right)=\pm 1$ and
$B\left(\mathbf{b},\lambda\right)=\pm 1$ denote the outcome expectation-values
of measurements for two spins. The measuring outcome expectation-values of two
spins are equal $A(\mathbf{a}$,$\lambda)=B\left(\mathbf{b},\lambda\right)$ for
the entangled state Eq.(1) with parallel spin-polarization. Hence

$P\left(a,b\right)=\int
d\lambda\rho\left(\lambda\right)A\left(\mathbf{a},\lambda\right)A\left(\mathbf{b},\lambda\right),$

and

$p\left(a,b\right)-P\left(a,c\right)=\int
d\lambda\rho\left(\lambda\right)A\left(\mathbf{a},\lambda\right)A\left(\mathbf{b},\lambda\right)\left[1-A\left(\mathbf{b},\lambda\right)A\left(\mathbf{c},\lambda\right)\right].$

It is obviously that

$\displaystyle\left|P\left(a,b\right)-P\left(a,c\right)\right|$
$\displaystyle\leqslant$ $\displaystyle\int
d\lambda\rho\left(\lambda\right)\left[1-A\left(\mathbf{b},\lambda\right)A\left(\mathbf{c},\lambda\right)\right]$
$\displaystyle=$ $\displaystyle 1-P\left(b,c\right)$

since $\rho$ is a normalized probability distribution $\int
d\lambda\rho\left(\lambda\right)=1$.

##  Part IX References

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# On the Ohsawa-Takegoshi $L^{2}$ extension theorem and removable
singularities of plurisubharmonic functions

Xieping Wang  CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of
Mathematical Sciences, University of Science and Technology of China, Hefei
230026, Anhui, People’s Republic of China<EMAIL_ADDRESS>To Li and
Rui’an

###### Abstract.

The celebrated Ohsawa-Takegoshi extension theorem for $L^{2}$ holomorphic
functions on bounded pseudoconvex domains in $\mathbb{C}^{n}$ is an important
and powerful tool in several complex variables and complex geometry. Ohsawa
conjectured in 1995 that the same theorem holds for more general bounded
complete Kähler domains in $\mathbb{C}^{n}$. Recently, Chen-Wu-Wang confirmed
this conjecture in a special case. In this paper we extend their result to the
case of holomorphic sections of twisted canonical bundles over relatively
compact complete Kähler domains in Stein manifolds. As an application we prove
a Hartogs type extension theorem for plurisubharmonic functions across a
compact complete pluripolar set, which is complementary to a classical theorem
of Shiffman and can be thought of as an analogue of the Skoda-El Mir extension
theorem for plurisubharmonic functions.

###### Key words and phrases:

$L^{2}$ extension, $\bar{\partial}$-equation, complete Kähler manifolds,
removable singularities, plurisubharmonic functions, complete pluripolar sets

###### 2020 Mathematics Subject Classification:

Primary 32D15, 32D20, 32U30, 32U05; Secondary 32W05, 32Q15

The author was partially supported by NSFC grant 12001513, NSF of Anhui
Province grant 2008085QA18 and the Fundamental Research Funds for the Central
Universities.

##  1\. Introduction

One of the most important and powerful tools in several complex variables and
complex geometry is the celebrated Ohsawa-Takegoshi $L^{2}$ extension theorem,
which can be stated in an equivalent way as follows:

######  Theorem A (see [OT87]).

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^{n}$ and $H$ be a
complex subspace of $\mathbb{C}^{n}$ intersecting $\Omega$. Then for every
plurisubharmonic function $\varphi$ on $\Omega$ and every holomorphic function
$f$ on $\Omega\cap H$ such that

$\int_{\Omega\cap H}|f|^{2}e^{-\varphi}dV_{H}<\infty,$

there exists a holomorphic function $F$ on $\Omega$ such that
$\left.F\right|_{\Omega\cap H}=f$ and

$\int_{\Omega}|F|^{2}e^{-\varphi}dV\leq C\int_{\Omega\cap
H}|f|^{2}e^{-\varphi}dV_{H},$

where $C>0$ is a constant depending only on the diameter of $\Omega$ and the
codimension of $H$, and $dV$ (resp. $dV_{H}$) denotes the Lebesgue measure on
$\mathbb{C}^{n}$ (resp. $H$).

Since the work of Ohsawa-Takegoshi [OT87], there has been a great deal of work
in the literature on the above theorem and its various generalizations as well
as applications. We refer the reader to [Ber10, Bło14, Dem12b, MV15, Ohs18,
Ohs20, Pau10, Siu02] and the references therein for a comprehensive exposition
of this subject and to [Bło13, GZ15a, GZ15b, ZZ18] for recent significant
developments. In particular, Guan, Zhou and Zhu [GZ15a, ZZ18] established an
optimal constant version of the Ohsawa-Takegoshi $L^{2}$ extension theorem for
holomorphic sections of twisted canonical bundles over weakly pseudoconvex
Kähler manifolds. After this, there is also some interesting work closely
related to the Ohsawa-Takegoshi $L^{2}$ extension theorem; see [ZZ20, DNWZ23,
CP20, CC22] and others.

Despite the above achievements, a number of questions about the $L^{2}$
extension of holomorphic functions remain to be answered, including the
following longstanding one posed by Ohsawa himself:

######  Question B (see [Ohs95]).

111Although formulated slightly differently, the question here is actually
equivalent to Ohsawa’s original question, as can be easily seen by induction
on $n$.

Does Theorem A still hold for more general bounded complete Kähler domains in
$\mathbb{C}^{n}$ ?

For the background and motivation for asking this question, we refer the
reader to Ohsawa’s original paper [Ohs95]. At present, it seems difficult to
answer the question in full generality; to the author’s knowledge, the only
known result is the one recently obtained by Chen-Wu-Wang [CWW15], who gave an
affirmative answer in the special case when $H$ is $0$-dimensional, i.e., a
single point. This result has found some interesting applications; see [CWW15]
for details.

The purpose of this paper is to extend the above result of Chen-Wu-Wang to the
case of holomorphic sections of twisted canonical bundles over relatively
compact complete Kähler domains in Stein manifolds, and to give some new
applications. The first main result reads as follows.

######  Theorem 1.1.

Let $X$ be an $n$-dimensional Stein manifold with a continuous volume form
$dV_{X}$ and $\Omega$ be a relatively compact complete Kähler domain in $X$.
Suppose $L$ is a holomorphic line bundle over $\Omega$ and $h$ is a singular
Hermitian metric on $L$ with semi-positive curvature current. Then for every
$x_{0}\in\Omega$ and $s_{x_{0}}\in\Lambda^{n}T^{\ast}_{x_{0}}\Omega\otimes
L_{x_{0}}$, there is an $L$-valued holomorphic $n$-form $s$ on $\Omega$ such
that $s(x_{0})=s_{x_{0}}$ and

$\int_{\Omega}\sqrt{-1}^{\,n^{2}}\\{s,s\\}_{h}\leq
C\frac{\sqrt{-1}^{\,n^{2}}\\{s_{x_{0}},s_{x_{0}}\\}_{h}}{dV_{X}(x_{0})}$

provided the right-hand side is finite, where $C>0$ is a constant depending
only on $n$, $\Omega$ and $dV_{X}$.

Here $\\{\,\,,\,\\}_{h}$ denotes the sesquilinear product combining the wedge
product of scalar valued forms with the Hermitian metric $h$ on $L$ (see
Section 4 below for a precise definition).

We now indicate the idea of the proof of Theorem 1.1. Following the strategy
of Ohsawa [Ohs95] and Chen-Wu-Wang [CWW15], we reduce Theorem 1.1 to an
Ohsawa-Takegoshi type $L^{2}$ existence theorem for the
$\bar{\partial}$-operator on Hermitian holomorphic line bundles, subject to
certain curvature constraints, over complete Kähler manifolds (see Theorem 3.2
below). Since the Hermitian metric on the line bundle under consideration is
allowed to be singular, the proof of Theorem 3.2 exploits in an essential way
Demailly’s approximation theory [Dem82, Dem92, Dem94], in which there are
mainly two approximation theorems for singular Hermitian metrics: one is a
local approximation by Hermitian metrics that are smooth on the complement of
suitably chosen complex subvarieties in the manifold, with arbitrarily small
loss of positivity in curvature; the other is a global approximation by
Hermitian metrics that are smooth on the entire manifold, but with less
controllable loss of positivity in curvature. The first approximation theorem
has been used by Zhou-Zhu [ZZ18] (see also [ZZ20]) to establish an optimal
constant version of the Ohsawa-Takegoshi-Manivel $L^{2}$ extension theorem on
weakly pseudoconvex Kähler manifolds. In the complete Kähler manifold case,
however, the absence of weak pseudoconvexity makes the first approximation
theorem inapplicable. We therefore appeal to the second one, which combined
with the a priori inequality for the twisted $\bar{\partial}$-operator due to
Ohsawa-Takegoshi [OT87] allows us to solve the $\bar{\partial}$-equation with
two correcting terms by an induction argument, thus proving Theorem 3.2. Such
an idea is inspired by the work of Demailly [Dem82].

As an application of Theorem 1.1 we obtain the following

######  Theorem 1.2.

Let $\Omega$ be a domain in a Stein manifold of dimension $\geq 2$ and $K$ be
a compact complete pluripolar subset of $\Omega$. Then every plurisubharmonic
function on $\Omega\\!\setminus\\!K$ admits a unique plurisubharmonic
extension to $\Omega$.

A weaker version of this theorem, with some additional assumptions, was proved
by the author in his previous work [Wan22b] using a different approach.

Recently, Chen [Che17] discovered that the well-known Hartogs extension
theorem for holomorphic functions is also valid for pluriharmonic functions
(see also [Wan22a]). But it has long been known that this is not the case for
plurisubharmonic functions (see, e.g., [BT88]). In such a situation, it is
meaningful to prove a slightly different type of Hartogs’ extension theorem
for plurisubharmonic functions, such as Theorem 1.2. Our motivation for such a
result also stems from a renewed interest in recent years in the original
Hartogs extension theorem; for details see the most recent work [BDP22, Vij22]
and the references therein.

Let us now highlight the main feature of Theorem 1.2. To put the theorem in a
historical context, we recall that there are two landmark results in the
extension theory of closed positive currents, namely the Harvey extension
theorem [Har74] and the Skoda-El Mir extension theorem [Sko82, ElM84], which
are complementary to each other and either of which unifies and generalizes
many important results previously proved by Bishop, Remmert-Stein, Shiffman,
Siu and others; for more information see the original papers just mentioned
and also [HP75, Har77, Sib85]. Corresponding to Harvey’s extension theorem
there is also an extension theorem for plurisubharmonic functions due to
Shiffman [Shi72], which states that every plurisubharmonic function on an
$n$-dimensional complex manifold extends plurisubharmonically across a closed
set of Hausdorff $(2n-2)$-measure zero. Theorem 1.2 can be thought of as, in
more or less, an analogue of the Skoda-El Mir extension theorem for
plurisubharmonic functions, which seemed to be missing until this paper, at
least as far as the author found.

To say more about Theorem 1.2, we now recall a standard fact in pluripotential
theory that pluripolar sets in an $n$-dimensional complex manifold have
Hausdorff dimension at most $2n-2$ and mention the existence of compact
complete pluripolar sets with positive (finite or infinite) Hausdorff
$(2n-2)$-measure in $\mathbb{C}^{n}$; see Subsection 2.1 for precise
references. Therefore, Theorem 1.2 is complementary to Shiffman’s theorem as
stated above. Also, it is worth noting that Theorem 1.2 is global in nature,
while Shiffman’s theorem (and many other related results in the literature;
see [Siu74, HP75, CWW15], etc.) is local in nature.

The proof of Theorem 1.2 is a mixture of the $L^{2}$ theory for the
$\bar{\partial}$-operator (via Theorem 1.1) and some basic geometric theory.
Note that in most previous work on the extension of closed positive currents
(particularly, complex varieties) and plurisubharmonic functions, the $L^{2}$
method and the geometric measure theory method have often been used
separately, notably by Siu [Siu74] and Shiffman [Shi68], Harvey [Har74],
respectively. The use of Theorem 1.1 in the proof of Theorem 1.2 is inspired
by the work of Chen-Wu-Wang [CWW15] and Demailly [Dem92].

We also pose the following question concerning Theorem 1.2:

######  Question 1.3.

Does Theorem 1.2 still hold when $K$ is assumed to be only a compact
pluripolar subset of $\Omega$ ?

This question seems to have an affirmative answer, but the author has no idea
of proof at the time of the writing. We also refer the interested reader to
[CWW15, Conjecture 4.1] for a different but closely related problem.

The paper is organized as follows. In Section 2 we review some necessary
notions and results that will be used in this paper. We then prove in Section
3 an Ohsawa-Takegoshi type $L^{2}$ existence theorem for the
$\bar{\partial}$-operator on complete Kähler manifolds, which is in turn used
in Section 4 to prove Theorem 1.1. Finally, the proof of Theorem 1.2 is
carried out in Section 5.

Acknowledgements. The author would like to thank Professor Bo-Yong Chen for
bringing to the author’s attention the question of Ohsawa described in the
introduction to this paper. The author would also like to thank Professor
Xiangyu Zhou, his mentor when he was a postdoctoral fellow at the Institute of
Mathematics, AMSS, Chinese Academy of Sciences from 2017 to 2019, for his
constant support and encouragement over the years.

##  2\. Preliminaries

In this section we fix some necessary notation and collect some fundamental
results from pluripotential theory and the $L^{2}$ theory of the
$\bar{\partial}$-operator, which will be used in subsequent sections.

###  2.1. Complete pluripolar sets and their defining functions

We begin by recalling the definition of complete pluripolar sets. Let $X$ be a
complex manifold and ${\rm Psh}(X)$ denote the set of all plurisubharmonic
(psh for short) functions on $X$.

######  Definition 2.1.

A set $E\subset X$ is called complete pluripolar if for every point $x\in E$
there exists a neighborhood $U$ of $x$ and a function $\varphi\in{\rm Psh}(U)$
such that

$E\cap U=\varphi^{-1}(-\infty).$

The set of all complex subvarieties of $X$ forms a particularly important
class of (closed) complete pluripolar sets, but complete pluripolar sets are
much more general: for instance, the Cartesian product of finitely many
(possibly distinct) Cantor type sets of logarithmic capacity zero in the
complex plane is a compact complete pluripolar set in the corresponding
complex Euclidean space (see, e.g., [Ran95]), but obviously not complex-
analytic. One may also consult [ElM84, LMP92, Edl04, DM14] and the references
therein for many other nontrivial and very interesting examples of compact
complete pluripolar sets in $\mathbb{C}^{n}$, in particular those of minimal
Hausdorff codimension (i.e., of Hausdorff codimension two).

On Stein manifolds, Colţoiu [Col90] proved the following important result
concerning the existence of a global defining function for a closed complete
pluripolar set.

######  Theorem 2.2 (see [Col90]).

Let $X$ be a Stein manifold and $E\subset X$ be a closed complete pluripolar
set. Then there is a function $\rho\in{\rm Psh}(X)\cap
C^{\infty}(X\\!\setminus\\!E)$ such that $\rho^{-1}(-\infty)=E$ and
$\sqrt{-1}\partial\bar{\partial}\rho>0$ on $X\\!\setminus\\!E$.

As we shall see later, this result plays a fundamental role in the proof of
Theorem 1.2.

###  2.2. Twisted Bochner-Kodaira-Nakano inequality

Let $(X,\omega)$ be an $n$-dimensional Kähler manifold and $(L,h)$ be a
Hermitian holomorphic line bundle over $X$. Given integers $0\leq p,q\leq n$,
we let $L^{2}(X,\,\Lambda^{p,\,q}T^{\ast}X\otimes L;\,\omega,h)$ denote the
space of $L$-valued $(p,q)$-forms $u$ with measurable complex coefficients on
$X$, satisfying

$\|u\|_{\omega,\,h}:=\bigg{(}\int_{X}|u|^{2}_{\omega,\,h}\,dV_{\omega}\bigg{)}^{1/2}<\infty,$

where $|\,\ |^{2}_{\omega,\,h}$ is the pointwise norm on
$\Lambda^{p,\,q}T^{\ast}X\otimes L$ induced by $\omega$ and $h$, and
$dV_{\omega}=\omega^{n}/n!$ is the volume form on $X$ induced by $\omega$. We
also denote by $\langle\,\,,\,\rangle_{\omega,\,h}$ (resp.
$\langle\\!\langle\,\,,\,\rangle\\!\rangle_{\omega,\,h}$) the pointwise (resp.
global) inner product corresponding to $|\,\ |^{2}_{\omega,\,h}$ (resp. $\|\,\
\|^{2}_{\omega,\,h}$). Obviously
$\langle\\!\langle\,\,,\,\rangle\\!\rangle_{\omega,\,h}$ makes
$L^{2}(X,\,\Lambda^{p,\,q}T^{\ast}X\otimes L;\,\omega,h)$ a Hilbert space,
which contains as a dense subspace the space
$\mathcal{D}(X,\,\Lambda^{p,\,q}T^{\ast}X\otimes L)$ of $L$-valued
$C^{\infty}$ $(p,q)$-forms with compact support in $X$. For our later
purposes, we also need to consider the maximal weak extension of the
$\bar{\partial}$-operator to $L^{2}(X,\,\Lambda^{p,\,q}T^{\ast}X\otimes
L;\,\omega,h)$, which we still denote by $\bar{\partial}$. Then it is a
closed, densely defined operator, and so is its Hilbert adjoint
${\bar{\partial}}^{\ast}_{\omega,\,h}$ with respect to
$\langle\\!\langle\,\,,\,\rangle\\!\rangle_{\omega,\,h}$.

Sometimes we also need to consider $L$-valued $(p,q)$-forms with local square-
integrability on $X$, instead of (global) square-integrability. The space
consisting of all such forms is independent of the choice of the $C^{\infty}$
metrics $\omega$ and $h$, and we denote it by $L^{2}_{\rm
loc}(X,\,\Lambda^{p,\,q}T^{\ast}X\otimes L)$.

We now record the following fundamental a priori inequality due to Ohsawa-
Takegoshi [OT87] (see also [Ohs95, Dem00, Dem12b]).

######  Lemma 2.3.

Let $(X,\omega)$ be an $n$-dimensional Kähler manifold and $(L,h)$ be a
Hermitian holomorphic line bundle over $X$. Given two $C^{\infty}$ functions
$\eta$, $\lambda>0$ on $X$, we then have

(2.1)
$\begin{split}\big{\|}&\sqrt{\eta+\lambda}\,\bar{\partial}^{\ast}_{\omega,\,h}u\big{\|}^{2}_{\omega,\,h}+\big{\|}\sqrt{\eta}\bar{\partial}u\big{\|}^{2}_{\omega,\,h}\\\
&\;\geq\big{\langle}\negmedspace\big{\langle}\big{[}\eta\sqrt{-1}\Theta_{h}(L)-\sqrt{-1}\partial\bar{\partial}\eta-\lambda^{-1}\sqrt{-1}\partial\eta\wedge\bar{\partial}\eta,\,\Lambda_{\omega}\big{]}u,\,u\big{\rangle}\negmedspace\big{\rangle}_{\omega,\,h}\end{split}$

for all $u\in\mathcal{D}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes L)$ and $q\geq
1$, where $\Theta_{h}(L)$ denotes the Chern curvature tensor of $(L,h)$ and
$\Lambda_{\omega}$ is the contraction operator with respect to $\omega$.

Inequality (2.1) also applies to the case where $(L,h)$ is a Hermitian
holomorphic vector bundle over $X$. Since we only use the case of line bundles
in this paper, we restrict ourselves to this particular case.

##  3\. An Ohsawa-Takegoshi type $L^{2}$ existence theorem

In this section we prove an Ohsawa-Takegoshi type $L^{2}$ existence theorem
for the $\bar{\partial}$-operator on complete Kähler manifolds, which contains
as a special case Demailly’s generalization of the celebrated $L^{2}$
existence theorem due to Hörmander and Andreotti-Vesentini (cf. [Dem82,
Théorème 5.1] and also Remark 3.3 (iii)).

To begin with, we recall the notion of singular Hermitian metrics.

######  Definition 3.1.

Let $L$ be a holomorphic line bundle over a complex manifold $X$. A singular
Hermitian metric $h$ on $L$ is a measurable Hermitian metric that can be
expressed on $U$ as $e^{-\varphi_{U}}$, where $U\subset X$ is an open set over
which $L$ is trivial, and $\varphi_{U}$ is an upper semi-continuous function
on $U$ and lies in $L^{1}_{{\rm loc}}(U)$.

It is easy to see that in terms of a given smooth Hermitian metric $h_{0}$,
every singular Hermitian metric $h$ on $L$ can be written exactly in the form
$h=h_{0}e^{-\varphi}$, where $\varphi\in L^{1}_{{\rm loc}}(X)$ is upper semi-
continuous on $X$; and the curvature current

$\sqrt{-1}\Theta_{h}(L):=-\sqrt{-1}\partial\bar{\partial}\log h$

of $(L,h)$ is semi-positive exactly when the weight $\varphi=-\log(h/h_{0})$
of $h$ (with respect to $h_{0}$) satisfies

$\sqrt{-1}\Theta_{h_{0}}(L)+\sqrt{-1}\partial\bar{\partial}\varphi\geq
0\quad{\rm on}\,\,X$

in the sense of currents.

Given a singular Hermitian metric $h$ on $L$, we can also define
$L^{2}(X,\,\Lambda^{p,\,q}T^{\ast}X\otimes L;\,\omega,h)$ and
$\langle\\!\langle\,\,,\,\rangle\\!\rangle_{\omega,\,h}$ in a similar way as
in Subsection 2.2. We now state and prove the following

######  Theorem 3.2.

Let $X$ be an $n$-dimensional complete Kähler manifold with a (not necessarily
complete) Kähler metric $\omega$ and $L$ be a holomorphic line bundle over $X$
with a singular Hermitian metric $h$. Suppose there are bounded $C^{\infty}$
functions $\eta$, $\lambda>0$ on $X$ such that

(3.1)
$\eta\sqrt{-1}\Theta_{h}(L)-\sqrt{-1}\partial\bar{\partial}\eta-\lambda^{-1}\sqrt{-1}\partial\eta\wedge\bar{\partial}\eta\geq\Theta$

in the sense of currents, where $\Theta$ is a continuous positive $(1,1)$-form
on $X$. Then for every integer $q\geq 1$ and $\bar{\partial}$-closed $v\in
L^{2}_{\rm loc}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes L)$ such that

$\int_{X}\big{\langle}[\Theta,\Lambda_{\omega}]^{-1}v,\,v\big{\rangle}_{\omega,\,h}\,dV_{\omega}<\infty,$

there exists $u\in L^{2}(X,\,\Lambda^{n,\,q-1}T^{\ast}X\otimes L;\,\omega,h)$
such that $\bar{\partial}u=v$ and

(3.2)
$\int_{X}(\eta+\lambda)^{-1}|u|^{2}_{\omega,\,h}\,dV_{\omega}\leq\int_{X}\big{\langle}[\Theta,\Lambda_{\omega}]^{-1}v,\,v\big{\rangle}_{\omega,\,h}\,dV_{\omega}.$

###### Proof.

First of all, a standard monotonicity argument as in [Dem82] allows us to
reduce the theorem to the case when $\omega$ is complete, which we shall
henceforward assume.

We next make use of Demailly’s approximation theorem [Dem82, Théorème 9.1],
according to which there is an increasing sequence
$\\{h_{j}\\}_{j\in\mathbb{N}}$ of $C^{\infty}$ Hermitian metrics on $L$ such
that

  1. (i)

$h_{j}\nearrow h$ on $X$ as $j\to\infty$, and

  2. (ii)

the curvature of each $h_{j}$ satisfies

(3.3)
$\eta\sqrt{-1}\Theta_{h_{j}}(L)-\sqrt{-1}\partial\bar{\partial}\eta-\lambda^{-1}\sqrt{-1}\partial\eta\wedge\bar{\partial}\eta\geq\Theta-\delta_{j}\omega,$

where $\\{\delta_{j}\\}_{j\in\mathbb{N}}$ is a decreasing sequence of
continuous functions on $X$ that converges to $0$ almost everywhere on $X$ as
$j\to\infty$.

Since $(X,\omega)$ is complete, we can easily construct a $C^{\infty}$
exhaustion function $\rho$ for $X$ with $|d\rho|_{\omega}\leq 1$ on $X$.
Choose also a $C^{\infty}$ cut-off function $0\leq\chi\leq 1$ on $\mathbb{R}$
satisfying $\chi|_{(-\infty,\,1)}=1$, $\chi|_{(2,\,\infty)}=0$ and
$|\chi^{\prime}|\leq 2$ on $\mathbb{R}$. Define

$\chi_{j}:=\chi(\rho/j),\quad j\in\mathbb{N}.$

We now proceed to establish some necessary estimates. For notational
simplicity, we set

$L^{2}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes
L)=L^{2}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes L;\,\omega,h),$
$L^{2}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes
L)_{k}=L^{2}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes L;\,\omega,h_{k})$

and

$\langle\\!\langle\,\,,\,\rangle\\!\rangle=\langle\\!\langle\,\,,\,\rangle\\!\rangle_{\omega,\,h},\quad\langle\\!\langle\,\,,\,\rangle\\!\rangle_{k}=\langle\\!\langle\,\,,\,\rangle\\!\rangle_{\omega,\,h_{k}}$

as well as
${\bar{\partial}}^{\ast}_{k}={\bar{\partial}}^{\ast}_{\omega,\,h_{k}}$ for
$k\in\mathbb{N}$. The same simplification applies to other notation, and we
may even omit subscripts when no confusion arises. Given $v$ as in the theorem
we have

(3.4)
$\displaystyle\begin{split}\big{|}\langle\\!\langle\chi_{j}v,w\rangle\\!\rangle_{k}\big{|}^{2}&\leq\big{\langle}\negmedspace\big{\langle}[\Theta,\Lambda]^{-1}v,\,v\big{\rangle}\negmedspace\big{\rangle}_{k}\,\big{\langle}\negmedspace\big{\langle}[\Theta,\Lambda](\chi_{j}w),\,\chi_{j}w\big{\rangle}\negmedspace\big{\rangle}_{k}\\\
&\leq\big{\langle}\negmedspace\big{\langle}[\Theta,\Lambda]^{-1}v,\,v\big{\rangle}\negmedspace\big{\rangle}\big{\langle}\negmedspace\big{\langle}[\Theta,\Lambda](\chi_{j}w),\,\chi_{j}w\big{\rangle}\negmedspace\big{\rangle}_{k}\\\
\end{split}$

for all $w\in{\rm Dom}\,\bar{\partial}\cap{\rm
Dom}\,{\bar{\partial}}^{\ast}_{k}$, in view of the Cauchy-Schwarz inequality
and the fact that $h_{k}\leq h$. Since $\chi_{j}w\in{\rm
Dom}\,\bar{\partial}\cap{\rm Dom}\,{\bar{\partial}}^{\ast}_{k}$ as easily
checked by definition, Ohsawa-Takegoshi’s a priori inequality (2.1) together
with (3.3) and Andreotti-Vesentini’s density theorem (see, e.g., [Dem12b,
Chap. 5]) implies

(3.5)
$\displaystyle\begin{split}\big{\|}\sqrt{\eta+\lambda}\,\bar{\partial}^{\ast}_{k}(\chi_{j}w)\big{\|}^{2}_{k}+\big{\|}\sqrt{\eta}\,\bar{\partial}(\chi_{j}w)\big{\|}^{2}_{k}&\geq\big{\langle}\negmedspace\big{\langle}[\Theta-\delta_{k}\omega,\,\Lambda](\chi_{j}w),\,\chi_{j}w\big{\rangle}\negmedspace\big{\rangle}_{k}\\\
&=\big{\langle}\negmedspace\big{\langle}[\Theta,\,\Lambda](\chi_{j}w),\,\chi_{j}w\big{\rangle}\negmedspace\big{\rangle}_{k}-q\big{\|}\chi_{j}\delta_{k}^{1/2}w\big{\|}^{2}_{k},\end{split}$

where the last equality follows from the fact that $[\omega,\Lambda]=q\,{\rm
Id}$ on $\Lambda^{n,\,q}T^{\ast}X$. Observe also that

$\bar{\partial}(\chi_{j}w)=\chi_{j}\bar{\partial}w+\bar{\partial}\chi_{j}\wedge
w$

and

$\bar{\partial}^{\ast}_{k}(\chi_{j}w)=\chi_{j}\bar{\partial}^{\ast}_{k}w-({\rm
grad}\,\chi_{j})^{0,\,1}\lrcorner\,w,$

where $\lrcorner$ denotes the contraction operator and $({\rm
grad}\,\chi_{j})^{0,\,1}$ is the $(0,1)$-part of the gradient of $\chi_{j}$
under the metric corresponding to $\omega$. Substituting this into (3.5) and
applying the Cauchy-Schwarz inequality, we get

(3.6)
$\displaystyle\begin{split}\big{\langle}\negmedspace\big{\langle}[\Theta,\,\Lambda](\chi_{j}w),\,\chi_{j}w\big{\rangle}\negmedspace\big{\rangle}_{k}&\leq
q\big{\|}\chi_{j}\delta_{k}^{1/2}w\big{\|}^{2}_{k}+\big{(}1+\frac{1}{j}\big{)}\Big{(}\big{\|}\sqrt{\eta+\lambda}\,\bar{\partial}^{\ast}_{k}w\big{\|}^{2}_{k}+\big{\|}\sqrt{\eta}\,\bar{\partial}w\big{\|}^{2}_{k}\Big{)}\\\
&\quad\;+(1+j)\Big{(}\big{\|}\sqrt{\eta+\lambda}\,({\rm
grad}\,\chi_{j})^{0,\,1}\lrcorner\,w\big{\|}^{2}_{k}+\big{\|}\sqrt{\eta}\,\bar{\partial}\chi_{j}\wedge
w\big{\|}^{2}_{k}\Big{)}\\\
&\leq\big{(}1+\frac{1}{j}\big{)}\Big{(}\big{\|}\sqrt{\eta+\lambda}\,\bar{\partial}^{\ast}_{k}w\big{\|}^{2}_{k}+\big{\|}\sqrt{\eta}\,\bar{\partial}w\big{\|}^{2}_{k}+\frac{C}{j}\|w\|^{2}_{k}\Big{)}\\\
&\quad\;+q\big{\|}\chi_{j}\delta_{k}^{1/2}w\big{\|}^{2}_{k}\end{split}$

with $C=2\sup_{X}(2\eta+\lambda)$, where in the last inequality we have used
the facts that

$\big{|}\bar{\partial}\chi_{j}\wedge
w\big{|}_{k}\leq|\bar{\partial}\chi_{j}||w|_{k}\leq\frac{\sqrt{2}}{j}|w|_{k}$

and

$\big{|}({\rm
grad}\,\chi_{j})^{0,\,1}\lrcorner\,w\big{|}_{k}\leq|\bar{\partial}\chi_{j}||w|_{k}\leq\frac{\sqrt{2}}{j}|w|_{k}.$

Now combining (3.4) with (3.6) yields

(3.7) $\big{|}\langle\\!\langle\chi_{j}v,w\rangle\\!\rangle_{k}\big{|}^{2}\leq
C_{j}\Big{(}\big{\|}\sqrt{\eta+\lambda}\,\bar{\partial}^{\ast}_{k}w\big{\|}^{2}_{k}+\big{\|}\sqrt{\eta}\,\bar{\partial}w\big{\|}^{2}_{k}+q\big{\|}\chi_{j}\delta_{k}^{1/2}w\big{\|}^{2}_{k}+\frac{C}{j}\|w\|^{2}_{k}\Big{)}$

for all $w\in{\rm Dom}\,\bar{\partial}\cap{\rm
Dom}\,{\bar{\partial}}^{\ast}_{k}$, with

$C_{j}:=\big{(}1+\frac{1}{j}\big{)}\big{\langle}\negmedspace\big{\langle}[\Theta,\Lambda]^{-1}v,\,v\big{\rangle}\negmedspace\big{\rangle}.$

This estimate enables us to argue by induction on $n-q$, as we shall now
explain. The idea is due to Demailly [Dem82], as mentioned in the
Introduction.

When $q=n$ we have

$\big{|}\langle\\!\langle\chi_{j}v,w\rangle\\!\rangle_{k}\big{|}^{2}\leq
C_{j}\Big{(}\big{\|}\sqrt{\eta+\lambda}\,\bar{\partial}^{\ast}_{k}w\big{\|}^{2}_{k}+q\big{\|}\chi_{j}\delta_{k}^{1/2}w\big{\|}^{2}_{k}+\frac{C}{j}\|w\|^{2}_{k}\Big{)}$

for all $w\in{\rm Dom}\,{\bar{\partial}}^{\ast}_{k}$. This shows that the
anti-linear functional
$w\mapsto\langle\\!\langle\chi_{j}v,w\rangle\\!\rangle_{k}$ is continuous on
the Hilbert space

$\Big{(}{\rm
Dom}\,{\bar{\partial}}^{\ast}_{k},\,\big{\|}\sqrt{\eta+\lambda}\,\bar{\partial}^{\ast}_{k}\cdot\big{\|}^{2}_{k}+q\big{\|}\chi_{j}\delta_{k}^{1/2}\cdot\big{\|}^{2}_{k}+\frac{C}{j}\|\cdot\|^{2}_{k}\Big{)}$

with norm at most $\sqrt{C_{j}}$. Thus by the Riesz representation theorem,
there exist

$u_{jk}\in L^{2}(X,\,\Lambda^{n,\,n-1}T^{\ast}X\otimes L)_{k}$

and

$v_{jk},\,w_{jk}\in L^{2}(X,\,\Lambda^{n,\,n}T^{\ast}X\otimes L)_{k}$

such that

(3.8)
$\big{\|}(\eta+\lambda)^{-\frac{1}{2}}u_{jk}\big{\|}^{2}_{k}+\|v_{jk}\|^{2}_{k}+\|w_{jk}\|^{2}_{k}\leq
C_{j}$

and

$\langle\\!\langle\chi_{j}v,w\rangle\\!\rangle_{k}=\langle\\!\langle
u_{jk},{\bar{\partial}}^{\ast}_{k}w\rangle\\!\rangle_{k}+\sqrt{q}\,\big{\langle}\negmedspace\big{\langle}v_{jk},\,\chi_{j}\delta_{k}^{1/2}w\big{\rangle}\negmedspace\big{\rangle}_{k}+\sqrt{C/j}\,\langle\\!\langle
w_{jk},w\rangle\\!\rangle_{k}$

for all $w\in{\rm Dom}\,{\bar{\partial}}^{\ast}_{k}$. This means that
$u_{jk}\in{\rm Dom}\,{\bar{\partial}}^{\ast\ast}_{k}={\rm
Dom}\,\bar{\partial}$ and

$\chi_{j}v=\bar{\partial}u_{jk}+\sqrt{q}\,\chi_{j}\delta_{k}^{1/2}v_{jk}+\sqrt{C/j}\,w_{jk}$

for all $j,\,k\in\mathbb{N}$. Moreover, since $\\{h_{k}\\}_{k\in\mathbb{N}}$
is increasing, (3.8) and the Banach-Alaoglu theorem imply that
$\\{u_{jk}\\}_{k\in\mathbb{N}}$ (resp. $\\{w_{jk}\\}_{k\in\mathbb{N}}$) has a
subsequence that converges weakly to some $u_{j}$ (resp. $w_{j}$) in
$L^{2}_{\rm loc}(X,\,\Lambda^{n,\,n-1}T^{\ast}X\otimes L)$ (resp. $L^{2}_{\rm
loc}(X,\,\Lambda^{n,\,n}T^{\ast}X\otimes L)$). Also since
$\\{\delta_{k}\\}_{k\in\mathbb{N}}$ decreases to $0$ almost everywhere on $X$,
it follows from (3.8), the Cauchy-Schwarz inequality and the dominated
convergence theorem that
$\big{\\{}\chi_{j}\delta_{k}^{1/2}v_{jk}\big{\\}}_{k\in\mathbb{N}}$ converges
to $0$ in $L^{1}_{\rm loc}(X,\,\Lambda^{n,\,n}T^{\ast}X\otimes L)$.
Consequently

$\chi_{j}v=\bar{\partial}u_{j}+\sqrt{C/j}\,w_{j}$

and

$\displaystyle\begin{split}\int_{\Omega}\big{(}(\eta+\lambda)^{-1}|u_{j}|^{2}_{l}+|w_{j}|^{2}_{l}\big{)}\,dV&\leq\limsup\limits_{k\to\infty}\int_{\Omega}\big{(}(\eta+\lambda)^{-1}|u_{jk}|^{2}_{l}+|w_{jk}|^{2}_{l}\big{)}\,dV\\\
&\leq\limsup\limits_{k\to\infty}\big{(}\big{\|}(\eta+\lambda)^{-\frac{1}{2}}u_{jk}\big{\|}^{2}_{k}+\|w_{jk}\|^{2}_{k}\big{)}\\\
&\leq C_{j}\end{split}$

for all $j,\,l\in\mathbb{N}$ and open sets $\Omega\subset\subset X$. Letting
first $l\to\infty$ and then $\Omega\to X$ we obtain

$\big{\|}(\eta+\lambda)^{-\frac{1}{2}}u_{j}\big{\|}^{2}+\|w_{j}\|^{2}\leq
C_{j}$

for all $j\in\mathbb{N}$, in view of Levi’s theorem. Taking a weak limit point
$u$ of $\\{u_{j}\\}_{j\in\mathbb{N}}$ in $L^{2}_{\rm
loc}(X,\,\Lambda^{n,\,n-1}T^{\ast}X\otimes L)$, we then conclude immediately
that $\bar{\partial}u=v$ and

$\big{\|}(\eta+\lambda)^{-\frac{1}{2}}u\big{\|}^{2}\leq\lim\limits_{j\to\infty}C_{j}=\big{\langle}\negmedspace\big{\langle}[\Theta,\Lambda]^{-1}v,\,v\big{\rangle}\negmedspace\big{\rangle}.$

This completes the proof of the case $q=n$.

Now assume that $1\leq q\leq n-1$ and the conclusion of the theorem is true
for all $\bar{\partial}$-closed $(n,q+1)$-forms with the prescribed
integrability. We proceed to show that the conclusion is true for all
$(n,q)$-forms with the same property. Let $v$ be such a form. Then by the
induction hypothesis, for every $j\in\mathbb{N}$ there exists $v_{j}\in
L^{2}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes L)$ such that

$\bar{\partial}v_{j}=\bar{\partial}(\chi_{j}v)=\bar{\partial}\chi_{j}\wedge v$

and

(3.9)
$\displaystyle\begin{split}\big{\|}(\eta+\lambda)^{-\frac{1}{2}}v_{j}\big{\|}^{2}&\leq\big{\langle}\negmedspace\big{\langle}[\Theta,\Lambda]^{-1}(\bar{\partial}\chi_{j}\wedge
v),\,\bar{\partial}\chi_{j}\wedge v\big{\rangle}\negmedspace\big{\rangle}\\\
&\leq\sup_{X}|\bar{\partial}\chi_{j}|^{2}\,\big{\langle}\negmedspace\big{\langle}[\Theta,\Lambda]^{-1}v,\,v\big{\rangle}\negmedspace\big{\rangle}\\\
&\leq\frac{2}{j^{2}}\big{\langle}\negmedspace\big{\langle}[\Theta,\Lambda]^{-1}v,\,v\big{\rangle}\negmedspace\big{\rangle}.\end{split}$

Also for every $w\in{\rm Dom}\,{\bar{\partial}}^{\ast}_{k}$ we have the
orthogonal decomposition $w=w_{1}+w_{2}$, where

$w_{1}\in{\rm Ker}\,\bar{\partial}\quad{\rm and}\quad w_{2}\in({\rm
Ker}\,\bar{\partial})^{\perp}\subset{\rm Ker}\,{\bar{\partial}}^{\ast}_{k}.$

Since

$v_{j}\in L^{2}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes L)\subset
L^{2}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes L)_{k}$

and $\chi_{j}v-v_{j}\in{\rm Ker}\,\bar{\partial}$, it then follows that

$\langle\\!\langle\chi_{j}v-v_{j},w\rangle\\!\rangle_{k}=\langle\\!\langle\chi_{j}v-v_{j},w_{1}\rangle\\!\rangle_{k},$

whence

$\langle\\!\langle\chi_{j}v,w\rangle\\!\rangle_{k}=\langle\\!\langle
v_{j},w\rangle\\!\rangle_{k}+\langle\\!\langle\chi_{j}v-v_{j},w_{1}\rangle\\!\rangle_{k}=\langle\\!\langle\chi_{j}v,w_{1}\rangle\\!\rangle_{k}+\langle\\!\langle
v_{j},w_{2}\rangle\\!\rangle_{k}.$

Applying the Cauchy-Schwarz inequality to this and taking into account (3.9),
we get

$\displaystyle\begin{split}\big{|}\langle\\!\langle\chi_{j}v,w\rangle\\!\rangle_{k}\big{|}^{2}&\leq\big{(}1+\frac{1}{j}\big{)}\big{|}\langle\\!\langle\chi_{j}v,w_{1}\rangle\\!\rangle_{k}\big{|}^{2}+(1+j)\|v_{j}\|^{2}\|w_{2}\|^{2}_{k}\\\
&\leq\big{(}1+\frac{1}{j}\big{)}\Big{(}\big{|}\langle\\!\langle\chi_{j}v,w_{1}\rangle\\!\rangle_{k}\big{|}^{2}+\frac{2}{j}\sup_{X}(\eta+\lambda)\big{\langle}\negmedspace\big{\langle}[\Theta,\Lambda]^{-1}v,\,v\big{\rangle}\negmedspace\big{\rangle}\|w\|^{2}_{k}\Big{)}.\end{split}$

On the other hand, since $w_{1}\in{\rm Ker}\,\bar{\partial}\cap{\rm
Dom}\,{\bar{\partial}}^{\ast}_{k}$ and $w_{2}\in{\rm
Ker}\,{\bar{\partial}}^{\ast}_{k}$, it follows from (3.7) that

$\displaystyle\begin{split}\big{|}\langle\\!\langle\chi_{j}v,w_{1}\rangle\\!\rangle_{k}\big{|}^{2}&\leq
C_{j}\Big{(}\big{\|}\sqrt{\eta+\lambda}\,\bar{\partial}^{\ast}_{k}w_{1}\big{\|}^{2}_{k}+q\big{\|}\chi_{j}\delta_{k}^{1/2}w_{1}\big{\|}^{2}_{k}+\frac{C}{j}\|w_{1}\|^{2}_{k}\Big{)}\\\
&\leq
C_{j}\Big{(}\big{\|}\sqrt{\eta+\lambda}\,\bar{\partial}^{\ast}_{k}w\big{\|}^{2}_{k}+q\big{\|}\chi_{j}\delta_{k}^{1/2}w_{1}\big{\|}^{2}_{k}+\frac{C}{j}\|w\|^{2}_{k}\Big{)}\end{split}$

Combining this with the preceding inequalities we arrive at

$\big{|}\langle\\!\langle\chi_{j}v,w\rangle\\!\rangle_{k}\big{|}^{2}\leq\big{(}1+\frac{1}{j}\big{)}C_{j}\Big{(}\big{\|}\sqrt{\eta+\lambda}\,\bar{\partial}^{\ast}_{k}w\big{\|}^{2}_{k}+q\big{\|}\chi_{j}\delta_{k}^{1/2}P_{k}w\big{\|}^{2}_{k}+\frac{2C}{j}\|w\|^{2}_{k}\Big{)}$

for all $w\in{\rm Dom}\,{\bar{\partial}}^{\ast}_{k}$, where $P_{k}$ denotes
the orthogonal projection from $L^{2}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes
L)_{k}$ onto ${\rm Ker}\,\bar{\partial}$. Then the Riesz representation
theorem applies and implies that there exist

$u_{jk}\in L^{2}(X,\,\Lambda^{n,\,q-1}T^{\ast}X\otimes L)_{k}$

and

$v_{jk},\,w_{jk}\in L^{2}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes L)_{k}$

such that

(3.10)
$\big{\|}(\eta+\lambda)^{-\frac{1}{2}}u_{jk}\big{\|}^{2}_{k}+\|v_{jk}\|^{2}_{k}+\|w_{jk}\|^{2}_{k}\leq\big{(}1+\frac{1}{j}\big{)}C_{j}$

and

$\langle\\!\langle\chi_{j}v,w\rangle\\!\rangle_{k}=\langle\\!\langle
u_{jk},{\bar{\partial}}^{\ast}_{k}w\rangle\\!\rangle_{k}+\sqrt{q}\,\big{\langle}\negmedspace\big{\langle}v_{jk},\,\chi_{j}\delta_{k}^{1/2}P_{k}w\big{\rangle}\negmedspace\big{\rangle}_{k}+\sqrt{2C/j}\,\langle\\!\langle
w_{jk},w\rangle\\!\rangle_{k}$

for all $w\in{\rm Dom}\,{\bar{\partial}}^{\ast}_{k}$. It follows that

$\chi_{j}v=\bar{\partial}u_{jk}+\sqrt{q}P_{k}(\chi_{j}\delta_{k}^{1/2}v_{jk})+\sqrt{2C/j}\,w_{jk}$

for all $j,\,k\in\mathbb{N}$. Now what remains is essentially to show that for
every $j\in\mathbb{N}$,

$f_{jk}:=P_{k}(\chi_{j}\delta_{k}^{1/2}v_{jk})\to 0\quad{\rm
weakly\,\,in}\quad L^{2}_{\rm loc}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes L)$

as $k\to\infty$. Once this is proved, an analogous argument as before leads to
the existence of a solution $u$ to equation $\bar{\partial}u=v$ together with
the estimate

$\big{\|}(\eta+\lambda)^{-\frac{1}{2}}u\big{\|}^{2}\leq\big{\langle}\negmedspace\big{\langle}[\Theta,\Lambda]^{-1}v,\,v\big{\rangle}\negmedspace\big{\rangle},$

thus completing the induction procedure (and hence the proof).

Observe first that by (3.10),

(3.11)
$\|f_{jk}\|_{k}\leq\big{\|}\chi_{j}\delta_{k}^{1/2}v_{jk}\big{\|}_{k}\leq\sup_{X}(\chi_{j}\delta_{1}^{1/2})\,\|v_{jk}\|_{k}\leq{\rm
const}_{j}$

for all $k\in\mathbb{N}$, where ${\rm const}_{j}$ denotes a constant depending
only on $j$. This observation is crucial in the following argument and will be
used repeatedly. For the time being, combined with the Banach-Alaoglu theorem,
it implies that the set of weak limit points of
$\\{f_{jk}\\}_{k\in\mathbb{N}}$ is non-empty. Thus what we need to show is
equivalent to saying that $0$ is the only element of the set. To establish the
validity of the latter, we may assume without loss of generality that the
sequence $\\{f_{jk}\\}_{k\in\mathbb{N}}$ itself is weakly convergent in
$L^{2}_{\rm loc}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes L)$, with limit $f_{j}$.
Let $H_{k}$ (resp. $H$) denote the only positive definite Hermitian operator
on $(L,h_{1})$ that satisfies $|H_{k}\zeta|_{1}=|\zeta|_{k}$ (resp.
$|H\zeta|_{1}=|\zeta|$) for all $\zeta\in L$. Then

${\rm Id}_{L}\leq H_{k}\leq H_{k+1}$

for all $k\in\mathbb{N}$, and $\\{H_{k}\\}_{k\in\mathbb{N}}$ converges to $H$
almost everywhere on $X$. Extending $H_{k}$ and $H$ to
$\Lambda^{n,\,q}T^{\ast}X\otimes L$ in a natural way, we claim that

$H_{k}f_{jk}\to Hf_{j}\quad{\rm weakly\,\,in}\quad L^{2}_{\rm
loc}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes L)$

as $k\to\infty$. To show this, it suffices to check that $Hf_{j}$ is the only
weak limit point of $\\{H_{k}f_{jk}\\}_{k\in\mathbb{N}}$. Again by (3.11) and
the Banach-Alaoglu theorem, we may assume that the sequence
$\\{H_{k}f_{jk}\\}_{k\in\mathbb{N}}$ itself is weakly convergent in
$L^{2}_{\rm loc}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes L)$, with limit $g_{j}$.
Observe that for every compactly supported $\varphi\in
L^{2}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes L)_{1}$,

$\big{|}\big{\langle}\negmedspace\big{\langle}H_{k}f_{jk},\,(H_{k}^{-1}-H^{-1})\varphi\big{\rangle}\negmedspace\big{\rangle}_{1}\big{|}\leq\|f_{jk}\|_{k}\big{\|}(H_{k}^{-1}-H^{-1})\varphi\big{\|}_{1}\leq{\rm
const}_{j}\big{\|}(H_{k}^{-1}-H^{-1})\varphi\big{\|}_{1}\to 0$

as $k\to\infty$, in view of (3.11), the Cauchy-Schwarz inequality and the
dominated convergence theorem. Then

$\langle\\!\langle
f_{j},\varphi\rangle\\!\rangle_{1}=\lim\limits_{k\to\infty}\langle\\!\langle
H_{k}f_{jk},\,H_{k}^{-1}\varphi\rangle\\!\rangle_{1}=\lim\limits_{k\to\infty}\langle\\!\langle
H_{k}f_{jk},\,H^{-1}\varphi\rangle\\!\rangle_{1}=\langle\\!\langle
g_{j},H^{-1}\varphi\rangle\\!\rangle_{1},$

hence $g_{j}=Hf_{j}$ and the claim follows. Moreover by (3.11) and the
definition of $H_{k}$, $Hf_{j}\in L^{2}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes
L)_{1}$. Now using (3.11) again and a standard $\varepsilon/3$-argument, we
further conclude that

$H_{k}f_{jk}\to Hf_{j}\quad{\rm weakly\,\,in}\quad
L^{2}(X,\,\Lambda^{n,\,q}T^{\ast}X\otimes L)_{1}$

as $k\to\infty$. In particular

$\|f_{j}\|^{2}=\|Hf_{j}\|^{2}_{1}=\lim\limits_{k\to\infty}\langle\\!\langle
H_{k}f_{jk},Hf_{j}\rangle\\!\rangle_{1}=\lim\limits_{k\to\infty}\langle\\!\langle
H_{k}f_{jk},H_{k}f_{j}\rangle\\!\rangle_{1},$

where the last equality follows from the fact that

$\big{|}\big{\langle}\negmedspace\big{\langle}H_{k}f_{jk},\,(H_{k}-H)f_{j}\big{\rangle}\negmedspace\big{\rangle}_{1}\big{|}\leq\|f_{jk}\|_{k}\big{\|}(H_{k}-H)f_{j}\big{\|}_{1}\leq{\rm
const}_{j}\big{\|}(H_{k}-H)f_{j}\big{\|}_{1}\to 0$

as $k\to\infty$. Equivalently,

$\|f_{j}\|^{2}=\lim\limits_{k\to\infty}\langle\\!\langle
f_{jk},f_{j}\rangle\\!\rangle_{k}.$

On the other hand, since $\bar{\partial}f_{jk}=0$ for all $k\in\mathbb{N}$, it
follows immediately that $\bar{\partial}f_{j}=0$ as well, hence

$\langle\\!\langle
f_{jk},f_{j}\rangle\\!\rangle_{k}=\big{\langle}\negmedspace\big{\langle}\chi_{j}\delta_{k}^{1/2}v_{jk},\,f_{j}\big{\rangle}\negmedspace\big{\rangle}_{k}.$

Moreover

$\big{|}\langle\\!\langle\chi_{j}\delta_{k}^{1/2}v_{jk},\,f_{j}\rangle\\!\rangle_{k}\big{|}\leq\|v_{jk}\|_{k}\big{\|}\chi_{j}f_{j}\delta_{k}^{1/2}\big{\|}_{k}\leq{\rm
const}_{j}\big{\|}\chi_{j}f_{j}\delta_{k}^{1/2}\big{\|}\to 0$

as $k\to\infty$, in view of (3.10) and the dominated convergence theorem.
Consequently $\|f_{j}\|=0$, i.e., $f_{j}=0$ almost everywhere on $X$. This
establishes the desired weak convergence of $\\{f_{jk}\\}_{k\in\mathbb{N}}$
and hence the theorem. ∎

######  Remark 3.3.

Several remarks are in order:

  1. (i)

When $q=1$, the estimate in (3.2) is independent of the choice of the Kähler
metric $\Omega$. In fact, Demailly [Dem82] and Ohsawa [Ohs95] have explained
in detail why this is the case for the right-hand side of (3.2); as for the
left-hand side, things are relatively simple, since it is easy to see that
$|\,\ |^{2}_{\omega,\,h}\,dV_{\omega}$ can be expressed in an intrinsic way in
terms of $\\{\,\,,\,\\}_{h}$ (to be recalled in Section 4):

(3.12) $|u|^{2}_{\omega,\,h}\,dV_{\omega}=\sqrt{-1}^{\,n^{2}}\\{u,u\\}_{h}.$

Moreover in the case when $q=1$, if the form $v$ in Theorem 3.2 is smooth then
every solution to equation $\bar{\partial}u=v$ is smooth as well. This follows
immediately from the ellipticity of $\bar{\partial}$ as an operator acting on
$(n,0)$-forms.

  2. (ii)

In [Dem00] (see also [Ohs95, Dem12b]), Demailly proved using Ohsawa-
Takegoshi’s a priori inequality (2.1) an abstract $L^{2}$ existence theorem
for the $\bar{\partial}$-operator under the assumption that the metric $h$ on
$L$ is smooth and $v$ satisfies the additional $L^{2}$ integrability

$\int_{X}|v|^{2}_{\omega,\,h}\,dV_{\omega}<\infty.$

Compared with this result, the point of Theorem 3.2 is that it allows the
metric $h$ to be singular. This is crucial in proving Theorem 1.1 (and hence
Theorem 1.2).

  3. (iii)

As the preceding proof indicates, the curvature condition (3.1) can be
weakened to an analogous one as in [Dem82, Théorème 5.1] and the bundle $L$
can also be allowed to have higher rank. We restrict ourselves to the present
case in order to make the argument as less technical as possible. As we shall
see in the next section, Theorem 3.2 is sufficient for the purpose of the
present paper.

  4. (iv)

In the special case when $X$ is a bounded complete Kähler domain in
$\mathbb{C}^{n}$ and

$(L,h)=(X\times\mathbb{C},\,e^{-\varphi})$

with $\varphi\in{\rm Psh}(X)$, Theorem 3.2 was first proved by Chen-Wu-
Wang222They only considered the case when $v$ is smooth and of type $(n,1)$,
but their method actually works equally well for not necessarily smooth
$(n,q)$-forms $v$, at least under the additional assumption that
$\int_{X}|v|^{2}_{\omega}e^{-\varphi}<\infty$. in [CWW15] using a different
method, namely first solving the twisted Laplace equation with Dirichlet
boundary condition on a sequence of smooth subdomains increasing to $X$ and
then using these solutions to approximate the needed solution of the twisted
$\bar{\partial}$-equation on $X$ with $L^{2}$ estimate. Although quite
interesting, this approach does not seem to work in the manifold case, since
as far as the author finds, it requires that the strict positivity of the
curvature term on the left-hand side of (3.1) be preserved during the
regularization process of the singular Hermitian metric $h$, but the loss of
positivity (i.e., the error term $\delta_{j}\omega$ in (3.3)) coming from any
regularization sequence $\\{h_{j}\\}_{j\in\mathbb{N}}$ is inevitable and
difficult to control even locally.

##  4\. Ohsawa-Takegoshi $L^{2}$ extension theorems: the complete Kähler case

Let $\Omega$ be a domain in a complex manifold. Denote by
$C^{\infty}(\Omega,\,\Lambda^{p,\,q}T^{\ast}\Omega\otimes L)$ the set of all
$L$-valued $C^{\infty}$ $(p,q)$-forms on $\Omega$. Given a Hermitian
holomorphic line bundle $(L,h)$ over $\Omega$, there is a canonical
sesquilinear map

$\\{\,\,,\,\\}_{h}\\!:C^{\infty}(\Omega,\,\Lambda^{p,\,q}T^{\ast}\Omega\otimes
L)\times C^{\infty}(\Omega,\,\Lambda^{r,\,s}T^{\ast}\Omega\otimes L)\to
C^{\infty}(\Omega,\,\Lambda^{p+s,\,q+r}T^{\ast}\Omega)$

which combines the wedge product of scalar valued forms with the Hermitian
metric $h$ on $L$. More explicitly, for any given $u\in
C^{\infty}(\Omega,\,\Lambda^{p,\,q}T^{\ast}\Omega\otimes L)$ and $v\in
C^{\infty}(\Omega,\,\Lambda^{r,\,s}T^{\ast}\Omega\otimes L)$, if we write

$\left.u\right|_{U}=f\otimes\zeta\quad{\rm
and}\quad\left.v\right|_{U}=g\otimes\zeta$

where $U$ is an open subset of $\Omega$ and $\zeta$ is a frame of $L$ over
$U$, then

$\left.\\{u,v\\}_{h}\right|_{U}=f\wedge\bar{g}\,|\zeta|^{2}_{h}.$

Clearly the definition of $\\{\,\,,\,\\}_{h}$ is pointwise and hence makes
sense for $u\in\Lambda^{p,\,q}T^{\ast}_{x_{0}}\Omega\otimes L_{x_{0}}$ and
$v\in\Lambda^{r,\,s}T^{\ast}_{x_{0}}\Omega\otimes L_{x_{0}}$ with
$x_{0}\in\Omega$, even when $h$ is merely a singular Hermitian metric on $L$.

We now come to prove Theorem 1.1, whose statement we recall for ease of
reference.

######  Theorem 4.1.

Let $X$ be an $n$-dimensional Stein manifold with a continuous volume form
$dV_{X}$ and $\Omega$ be a relatively compact complete Kähler domain in $X$.
Suppose $L$ is a holomorphic line bundle over $\Omega$ and $h$ is a singular
Hermitian metric on $L$ with semi-positive curvature current. Then for every
$x_{0}\in\Omega$ and $s_{x_{0}}\in\Lambda^{n}T^{\ast}_{x_{0}}\Omega\otimes
L_{x_{0}}$, there is an $L$-valued holomorphic $n$-form $s$ on $\Omega$ such
that $s(x_{0})=s_{x_{0}}$ and

$\int_{\Omega}\sqrt{-1}^{\,n^{2}}\\{s,s\\}_{h}\leq{\rm
const}_{n,\,\Omega,\,dV_{X}}\frac{\sqrt{-1}^{\,n^{2}}\\{s_{x_{0}},s_{x_{0}}\\}_{h}}{dV_{X}(x_{0})}$

provided the right-hand side is finite.

Here and henceforward we use ${\rm const}_{n,\,\Omega,\,dV_{X}}$ to denote a
generic constant which depends only on $n$, $\Omega$ and $dV_{X}$, and may be
different in different appearances. We shall also use similar notation with
obvious meaning in the context.

Since $\overline{\Omega}$ is compact in $X$, we can associate to every point
$x\in\Omega$ a local holomorphic coordinate
$z_{x}=(z_{x}^{1},\ldots,z_{x}^{n})$ of $X$ defined on a neighborhood of the
closure of $B_{x}:=\\{|z_{x}|<1\\}$ such that $z_{x}(x)=0$ and

(4.1) $\frac{\sqrt{-1}^{\,n^{2}}dz_{x}\wedge d\bar{z}_{x}}{dV_{X}}\geq{\rm
const}_{n,\,\Omega,\,dV_{X}}\quad{\rm on}\,\,B_{x},$

where $dz_{x}:=dz_{x}^{1}\wedge\cdots\wedge dz_{x}^{n}$. Set

$\varepsilon B_{x}=\\{|z_{x}|<\varepsilon\\}$

for $x\in\Omega$ and $\varepsilon\in(0,1)$. We claim the following

######  Lemma 4.2.

Given $x_{0}\in\Omega$, $f_{x_{0}}\in\Lambda^{n}T^{\ast}_{x_{0}}\Omega$ and
$\varepsilon\in(0,1)$, $\varphi\in{\rm Psh}(\varepsilon B_{x_{0}})$, there is
a holomorphic $n$-form $f_{\varepsilon}$ on $\varepsilon B_{x_{0}}$ such that
$f_{\varepsilon}(x_{0})=f_{x_{0}}$ and

$\varepsilon^{-2n}\int_{\varepsilon
B_{x_{0}}}\sqrt{-1}^{\,n^{2}}f_{\varepsilon}\wedge\bar{f}_{\varepsilon}\,e^{-\varphi}\leq{\rm
const}_{n,\,\Omega,\,dV_{X}}\frac{\sqrt{-1}^{\,n^{2}}f_{x_{0}}\wedge\bar{f}_{x_{0}}}{dV_{X}(x_{0})}e^{-\varphi(x_{0})}.$

###### Proof.

Write $f_{x_{0}}=c\,dz_{x_{0}}$ with $c\in\mathbb{C}$. As in [CWW15], an
application of the Ohsawa-Takegoshi extension theorem shows that for every
$\varepsilon\in(0,1)$ there is a holomorphic $n$-form $f_{\varepsilon}$ on
$\varepsilon B_{x_{0}}$ such that $f_{\varepsilon}(x_{0})=f_{x_{0}}$ and

$\varepsilon^{-2n}\int_{\varepsilon
B_{x_{0}}}\sqrt{-1}^{\,n^{2}}f_{\varepsilon}\wedge\bar{f}_{\varepsilon}\,e^{-\varphi}\leq{\rm
const}_{n}|c|^{2}e^{-\varphi(x_{0})}.$

Combining this with inequality (4.1) yields the desired result immediately. ∎

We now proceed to prove Theorem 4.1. With Theorem 3.2 in hand, we do this by
adapting the argument in the proof of [CWW15, Theorem 1.3].

Without loss of generality we may assume that $dV_{X}=\omega^{n}/n!$, $\omega$
being a Kähler form on $X$. Fix an arbitrary point $x_{0}\in\Omega$. Let us
write $(B,z)=(B_{x_{0}},z_{x_{0}})$ for the sake of simplicity, and pick a
function $\rho\in C^{\infty}(X,\,\mathbb{R})$ which is positive, strictly log-
psh on $X\\!\setminus\\!\\{x_{0}\\}$ and satisfies $\rho<e^{-1}$ on
$\overline{\Omega}$ and $\rho=|z|^{2}$ on a small neighborhood of $x_{0}$.
Similarly as in [Ohs95, Che11, CWW15] we set

$\psi_{\varepsilon}=\log(\rho+\varepsilon^{2})\quad{\rm
and}\quad\eta_{\varepsilon}=-\psi_{\varepsilon}+\log(-\psi_{\varepsilon})$

for $0<\varepsilon\leq\varepsilon_{0}$, where $\varepsilon_{0}>0$ is chosen so
small that $\psi_{\varepsilon_{0}}<-1$ on $\overline{\Omega}$ and the given
Hermitian holomorphic line bundle $(L,h)$ is trivial over $\varepsilon_{0}B$.
A straightforward calculation shows

$-\partial\eta_{\varepsilon}=(1-\psi_{\varepsilon}^{-1})\partial\psi_{\varepsilon}\quad{\rm
and}\quad-\partial\bar{\partial}\eta_{\varepsilon}=(1-\psi_{\varepsilon}^{-1})\partial\bar{\partial}\psi_{\varepsilon}+\psi_{\varepsilon}^{-2}\partial\psi_{\varepsilon}\wedge\bar{\partial}\psi_{\varepsilon},$

whence

$-\sqrt{-1}\partial\bar{\partial}\eta_{\varepsilon}-\lambda_{\varepsilon}^{-1}\sqrt{-1}\partial\eta_{\varepsilon}\wedge\bar{\partial}\eta_{\varepsilon}\geq\sqrt{-1}\partial\bar{\partial}\psi_{\varepsilon}\quad{\rm
on}\ \,\Omega,$

where $\lambda_{\varepsilon}=(1-\psi_{\varepsilon})^{2}$.

Now let $s_{x_{0}}\in\Lambda^{n}T^{\ast}_{x_{0}}\Omega\otimes L_{x_{0}}$ be
given such that

$\frac{\sqrt{-1}^{\,n^{2}}\\{s_{x_{0}},s_{x_{0}}\\}_{h}}{\omega^{n}(x_{0})}<\infty.$

Take a holomorphic frame $\zeta$ of $L$ over $\varepsilon_{0}B$ and write
$s_{x_{0}}$ as $s_{x_{0}}=f_{x_{0}}\otimes\zeta_{x_{0}}$. Let
$\\{f_{\varepsilon}\\}_{0<\varepsilon\leq\varepsilon_{0}}$ be a family of
holomorphic $n$-forms as in Lemma 4.2, corresponding to the weight function

$\varphi=-\log|\zeta|^{2}_{h},$

which is plurisubharmonic on $\varepsilon_{0}B$ because of the semi-positivity
of $\sqrt{-1}\Theta_{h}(L)$. Consider

$\Theta_{\varepsilon}=\sqrt{-1}\partial\bar{\partial}(\rho+\psi_{\varepsilon})\quad{\rm
and}\quad
v_{\varepsilon}=\bar{\partial}\big{(}\chi(|z|^{2}/\varepsilon^{2})f_{\varepsilon}\big{)}\otimes\zeta,$

where $\chi$ is a $C^{\infty}$ function on $\mathbb{R}$ satisfying
$\chi|_{(-\infty,\,1/4)}=1$, $\chi|_{(3/4,\,\infty)}=0$ and
$|\chi^{\prime}|\leq 4$. Observe that

$\Theta_{\varepsilon}\geq\sqrt{-1}\partial\bar{\partial}\psi_{\varepsilon}=\sqrt{-1}\partial\bar{\partial}\log(|z|^{2}+\varepsilon^{2})\geq\frac{\varepsilon^{2}}{(|z|^{2}+\varepsilon^{2})^{2}}\sqrt{-1}\partial\bar{\partial}|z|^{2}$

on a neighborhood of $x_{0}$. Using this and [Dem82, Lemme 3.2] (see also the
paragraph following Proposition 1.5 in [Ohs95]), we can estimate

$\displaystyle\begin{split}\big{\langle}\negmedspace\big{\langle}[\Theta&{}_{\varepsilon},\Lambda_{\omega}]^{-1}v_{\varepsilon},\,v_{\varepsilon}\big{\rangle}\negmedspace\big{\rangle}_{\omega,\,he^{-(\rho+n\log\rho)}}\\\
&=\int_{{\rm
supp}\,v_{\varepsilon}}\big{\langle}[\Theta_{\varepsilon},\Lambda_{\sqrt{-1}\partial\bar{\partial}|z|^{2}}]^{-1}v_{\varepsilon},\,v_{\varepsilon}\big{\rangle}_{\sqrt{-1}\partial\bar{\partial}|z|^{2},\,h}\,e^{-(\rho+n\log\rho)}\frac{(\sqrt{-1}\partial\bar{\partial}|z|^{2})^{n}}{n!}\\\
&\leq\int_{{\rm
supp}\,v_{\varepsilon}}\frac{(|z|^{2}+\varepsilon^{2})^{2}}{\varepsilon^{2}}|v_{\varepsilon}|^{2}_{\sqrt{-1}\partial\bar{\partial}|z|^{2},\,h}\,e^{-(\rho+n\log\rho)}\frac{(\sqrt{-1}\partial\bar{\partial}|z|^{2})^{n}}{n!}\\\
&\leq\|\chi^{\prime}\|^{2}_{L^{\infty}(\mathbb{R})}\int_{\frac{\sqrt{3}\varepsilon}{2}B\setminus\frac{\varepsilon}{2}B}\frac{|z|^{2}}{\varepsilon^{4}}\frac{(|z|^{2}+\varepsilon^{2})^{2}}{\varepsilon^{2}}\sqrt{-1}^{\,n^{2}}f_{\varepsilon}\wedge\bar{f}_{\varepsilon}\,|z|^{-2n}e^{-\varphi-|z|^{2}}\\\
&\leq{\rm const}_{n}\,\varepsilon^{-2n}\int_{\varepsilon
B}\sqrt{-1}^{\,n^{2}}f_{\varepsilon}\wedge\bar{f}_{\varepsilon}\,e^{-\varphi}\\\
&\leq{\rm
const}_{n,\,\Omega,\,\omega}\frac{\sqrt{-1}^{\,n^{2}}f_{x_{0}}\wedge\bar{f}_{x_{0}}}{\omega^{n}(x_{0})}e^{-\varphi(x_{0})}\\\
&={\rm
const}_{n,\,\Omega,\,\omega}\frac{\sqrt{-1}^{\,n^{2}}\\{s_{x_{0}},s_{x_{0}}\\}_{h}}{\omega^{n}(x_{0})}\end{split}$

for all sufficiently small $\varepsilon>0$. In this way,
$(\Omega,\omega,L,he^{-(\rho+n\log\rho)},\eta_{\varepsilon},\lambda_{\varepsilon},\Theta_{\varepsilon},v_{\varepsilon})$
fits into the framework of Theorem 3.2. Therefore, there is a solution
$u_{\varepsilon}$ to equation $\bar{\partial}u=v_{\varepsilon}$ with the
estimate

(4.2)
$\int_{\Omega}\frac{\sqrt{-1}^{\,n^{2}}\\{u_{\varepsilon},u_{\varepsilon}\\}_{h}}{\eta_{\varepsilon}+\lambda_{\varepsilon}}e^{-(\rho+n\log\rho)}\leq{\rm
const}_{n,\,\Omega,\,\omega}\frac{\sqrt{-1}^{\,n^{2}}\\{s_{x_{0}},s_{x_{0}}\\}_{h}}{\omega^{n}(x_{0})}.$

Here we have used the identity in (3.12). Define

$s_{\varepsilon}:=\chi(|z|^{2}/\varepsilon^{2})f_{\varepsilon}\otimes\zeta-
u_{\varepsilon}.$

Then $s_{\varepsilon}$ is an $L$-valued holomorphic $n$-form on $\Omega$.
Moreover, since $u_{\varepsilon}$ is holomorphic on a small neighborhood of
$x_{0}$ and $e^{-n\log\rho}=|z|^{-2n}$ is not integrable near $x_{0}$, it
follows from the finiteness of the left-hand side of (4.2) that
$u_{\varepsilon}(x_{0})=0$ and hence

$s_{\varepsilon}(x_{0})=f_{x_{0}}\otimes\zeta_{x_{0}}=s_{x_{0}}.$

Also recalling that $\rho<e^{-1}$ on $\overline{\Omega}$ and taking into
account the trivial fact that

$\eta_{\varepsilon}+\lambda_{\varepsilon}\leq 6\,\psi_{\varepsilon}^{2}\leq
6\log^{2}\\!\rho,$

we conclude that

$(\eta_{\varepsilon}+\lambda_{\varepsilon})^{-1}e^{-(\rho+n\log\rho)}\geq\frac{{\rm
const}}{\rho^{n}\log^{2}\\!\rho}\geq{\rm const}_{n}.$

Hence (4.2) implies

$\int_{\Omega}\sqrt{-1}^{\,n^{2}}\\{u_{\varepsilon},u_{\varepsilon}\\}_{h}\leq{\rm
const}_{n,\,\Omega,\,\omega}\frac{\sqrt{-1}^{\,n^{2}}\\{s_{x_{0}},s_{x_{0}}\\}_{h}}{\omega^{n}(x_{0})}.$

Now combining this with the choice of $f_{\varepsilon}$ and applying the
Cauchy-Schwarz inequality, we end up with

$\int_{\Omega}\sqrt{-1}^{\,n^{2}}\\{s_{\varepsilon},s_{\varepsilon}\\}_{h}\leq{\rm
const}_{n,\,\Omega,\,\omega}\frac{\sqrt{-1}^{\,n^{2}}\\{s_{x_{0}},s_{x_{0}}\\}_{h}}{\omega^{n}(x_{0})}.$

Consequently $s=s_{\varepsilon}$ is a holomorphic section of
$\Lambda^{n}T^{\ast}\Omega\otimes L$ as desired in the theorem, provided
$\varepsilon>0$ is sufficiently small. This completes the proof of Theorem
4.1.

Applying Theorem 4.1 to the trivial line bundle $L=\Omega\times\mathbb{C}$
equipped with the metric $e^{-\varphi}$, $\varphi\in{\rm Psh}(\Omega)$, we get
immediately the following

######  Theorem 4.3.

Let $X$ be an $n$-dimensional Stein manifold with a continuous volume form
$dV_{X}$ and $\Omega$ be a relatively compact complete Kähler domain in $X$.
Then for every $x_{0}\in\Omega$,
$f_{x_{0}}\in\Lambda^{n}T^{\ast}_{x_{0}}\Omega$ and $\varphi\in{\rm
Psh}(\Omega)$, there exists a holomorphic $n$-form $f$ on $\Omega$ such that
$f(x_{0})=f_{x_{0}}$ and

(4.3) $\int_{\Omega}\sqrt{-1}^{\,n^{2}}f\wedge\bar{f}\,e^{-\varphi}\leq{\rm
const}_{n,\,\Omega,\,dV_{X}}\frac{\sqrt{-1}^{\,n^{2}}f_{x_{0}}\wedge\bar{f}_{x_{0}}}{dV_{X}(x_{0})}e^{-\varphi(x_{0})}.$

######  Remark 4.4.

Note that the conclusion of Theorem 4.3 holds trivially when the right-hand
side of (4.3) is infinite: in this case every holomorphic $n$-form $f$ on
$\Omega$ that takes value $f_{x_{0}}$ at $x_{0}$ satisfies the desired
estimate (4.3), and such an $f$ always exists (even on $X$) since $X$ is
Stein. Incidentally, as Theorem 4.3 (and also Theorem 4.1)) itself implies, in
order for its conclusion to hold it is sufficient to assume that $dV_{X}$ is
defined on a neighborhood of $\overline{\Omega}$.

##  5\. Proof of Theorem 1.2

The key ingredients of the proof are Theorem 4.3 and some basic geometric
measure theory. As mentioned in the Introduction, the use of Theorem 4.3 in
the proof is inspired by the work of Chen-Wu-Wang [CWW15] and Demailly
[Dem92]. We first prove the following result:

######  Theorem 5.1.

Let $X$ be a Stein manifold and $E\subset X$ be a closed complete pluripolar
set. Then $X\\!\setminus\\!E$ carries a complete Kähler metric.

The very important special case of $E\subset X$ being a complex subvariety is
well-known and is due to Grauert [Gra56] (see also Demailly [Dem82] for
related results). Note that even in this special case, $X\\!\setminus\\!E$
cannot be Stein unless $E$ is empty or purely one-codimensional, in view of
the second Riemann extension theorem.

###### Proof.

The result is an easy consequence of Theorem 2.2. Let $\varphi$ be a
$C^{\infty}$ strictly psh exhaustion function for $X$ and let
$\psi\\!:X\\!\setminus\\!E\to\mathbb{R}$ be a $C^{\infty}$ function such that

$\psi=-\log(-\rho)\quad\mbox{on}\;\;U\\!\setminus\\!E,$

where $\rho$ is a psh function as in Theorem 2.2 with $\sup_{X}\rho>-1$ and
$U:=\\{\rho<-1\\}$. Then one can construct a $C^{\infty}$ convex, rapidly
increasing function $\chi$ on $\mathbb{R}$ such that

$\omega:=\sqrt{-1}\partial\bar{\partial}(\chi\circ\varphi)+\sqrt{-1}\partial\bar{\partial}\psi\geq\omega_{0}\quad\mbox{on}\;\;X\\!\setminus\\!E$

for some complete Kähler metric $\omega_{0}$ on $X$.

We claim that $\omega$ is complete on $X\\!\setminus\\!E$. For this, we may
assume without loss of generality that $X$ itself is connected (and so is
$X\\!\setminus\\!E$). Suppose $\\{x_{j}\\}_{j\in\mathbb{N}}$ is a bounded
sequence in the metric space $(X\\!\setminus\\!E,\,\omega)$. Then there is a
sequence of smooth curves $\\{\gamma_{j}\\}_{j\in\mathbb{N}}\subset
C^{\infty}([0,1],\,X\\!\setminus\\!E)$ with uniformly bounded lengths with
respect to $\omega$, joining every $x_{j}$ to a (fixed) reference point in
$X\\!\setminus\\!\overline{U}$. Since $\omega\geq\omega_{0}$ on
$X\\!\setminus\\!E$ and $\omega_{0}$ is complete on $X$, we may assume that
the sequence $\\{x_{j}\\}_{j\in\mathbb{N}}$ itself converges in $X$ by passing
to a subsequence if necessary. What now remains is to show that the limit of
$\\{x_{j}\\}_{j\in\mathbb{N}}$ lies outside $E$. Suppose the contrary and set

$t_{j}:=\inf\\!\big{\\{}t\in[0,1]\\!:\gamma_{j}([t,1])\subset U\big{\\}},\quad
j\in\mathbb{N}.$

Clearly $0<t_{j}<1$ and $\gamma_{j}(t_{j})\in\partial U=\\{\rho=-1\\}$ for all
sufficiently large $j$. Observe also that

$\omega\geq\sqrt{-1}\partial\bar{\partial}\big{(}-\log(-\rho)\big{)}\geq\sqrt{-1}\partial\log(-\rho)\wedge\bar{\partial}\log(-\rho)\quad\mbox{on}\;\;U\\!\setminus\\!E.$

It then follows that

$\begin{split}\sqrt{2}\,{\rm
length}_{\omega}(\gamma_{j})&\geq\int_{t_{j}}^{1}\big{|}\big{(}d\log(-\rho)\big{)}(\gamma^{\prime}(t))\big{|}dt\geq\int_{t_{j}}^{1}\big{(}d\log(-\rho)\big{)}(\gamma^{\prime}(t))dt\\\
&=\log(-\rho(x_{j}))\to\infty\quad{\rm as}\;\;j\to\infty,\end{split}$

contradicting the boundedness of $\\{{\rm
length}_{\omega}(\gamma_{j})\\}_{j\in\mathbb{N}}$. Therefore the limit of
$\\{x_{j}\\}_{j\in\mathbb{N}}$ lies outside $E$, and hence $\omega$ is
complete on $X\\!\setminus\\!E$. ∎

######  Remark 5.2.

If we denote by $d_{\omega}$ and $d_{\omega_{0}}$ the distance functions on
$X\\!\setminus\\!E$ and $X$ associated to $\omega$ and $\omega_{0}$,
respectively, and by $\chi_{U\setminus E}$ the characteristic function of
$U\\!\setminus\\!E$, then the preceding argument actually shows the estimate

$d_{\omega}(x,x_{0})\geq\max\Big{\\{}d_{\omega_{0}}(x,x_{0}),\,\chi_{U\setminus
E}\log(-\rho(x))/\sqrt{2}\Big{\\}},\quad x\in X\\!\setminus\\!E,\,x_{0}\in
X\\!\setminus\\!\overline{U}.$

This, of course, implies the completeness of $\omega$.

We are now ready to prove Theorem 1.2.

###### Proof of Theorem 1.2.

The uniqueness part of the theorem is clear, since two psh functions on
$\Omega$ which coincide almost everywhere are actually equal everywhere. So it
suffices to prove the existence part.

Let us first reduce the problem to the case when $\Omega\supset K$ is a
relatively compact Stein domain in the ambient manifold, which we denote by
$X$. To this end, let $\rho$ be a psh function on $X$, continuous outside $K$
and satisfying $\rho^{-1}(-\infty)=K$ (see Theorem 2.2). Choose an open set
$U\subset X$ such that $K\subset U\subset\subset X$, and set

$\widetilde{\rho}:=\left\\{\begin{array}[]{lll}\\!\\!\max\big{\\{}\rho,\,\inf\limits_{\partial
U}\rho\big{\\}}\quad\mbox{on}\;\;X\\!\setminus\\!\overline{U};\\\
\\!\\!\qquad\quad\rho\quad\quad\quad\;\;\,\mbox{on}\;\;\overline{U}.\end{array}\right.$

Then $\widetilde{\rho}$ is a psh function on $X$ with
$\widetilde{\rho}^{\,-1}(-\infty)=K$. Replacing $\Omega$ by any connected
component of $\\{\widetilde{\rho}<\inf_{\partial U}\rho\\}$, we may assume in
what follows that $\Omega$ is a relatively compact Stein domain in $X$.

Now $\Omega\\!\setminus\\!K$ is a relatively compact complete Kähler domain in
$X$, in view of Theorem 5.1. Write $n={\rm dim}_{\mathbb{C}}X$ and fix a
continuous volume form $dV_{X}$ on $X$. Given $\varphi\in{\rm
Psh}(\Omega\\!\setminus\\!K)$, we can therefore invoke Theorem 4.3 to assign
to every point $x\in\Omega\\!\setminus\\!K$ a holomorphic $n$-form $f_{x}$ on
$\Omega\\!\setminus\\!K$ with the property that

(5.1)
$\frac{\sqrt{-1}^{\,n^{2}}f_{x}\wedge\bar{f}_{x}}{dV_{X}}(x)=e^{\varphi(x)}\quad{\rm
and}\quad\int_{\Omega\setminus
K}\sqrt{-1}^{\,n^{2}}f_{x}\wedge\bar{f}_{x}\,e^{-\varphi}\leq{\rm const},$

where the constant involved is independent of $x\in\Omega\\!\setminus\\!K$. By
the Hartogs extension theorem for holomorphic forms (see, e.g., [Ser55, p. 22,
Corollaire du Théorème 3]), each such $f_{x}$ extends holomorphically to
$\Omega$. With a slight abuse of notation, we still denote the extension by
$f_{x}$.

We now come to establish the psh extendibility of $\varphi$ across $K$. By a
classical result of Lelong, it suffices to show that every point of $K$ admits
a small neighborhood on which $\varphi$ is bounded above. Fix an arbitrary
point $x_{0}\in K$. Because of the local nature of the problem, we may assume
without loss of generality that $\Omega$ is a bounded pseudoconvex domain in
$\mathbb{C}^{n}$ containing $x_{0}$. Let $z=(z_{1},\ldots,z_{n})$ be the
coordinate of $\Omega$. Identifying $x$ with its coordinate $z=z(x)$ and
$f_{x}$ with $f_{x}/dz_{1}\wedge\cdots\wedge dz_{n}$, and shrinking $\Omega$
if necessary, we conclude from (5.1) that

(5.2) $e^{\varphi(z)}\leq{\rm const}\,|f_{z}(z)|^{2}\quad{\rm
and}\quad\int_{\Omega\setminus K}|f_{z}|^{2}e^{-\varphi}\leq{\rm const},$

where the integral involved is taken against the Lebsesgue measure on
$\mathbb{C}^{n}$. To proceed we recall that being a polar subset of
$\mathbb{C}^{n}\cong\mathbb{R}^{2n}$, $K$ has Hausdorff dimension at most
$2n-2$ (see, e.g., [AG01, Theorem 5.9.6]). This allows us to make use of a
result of Shiffman (see [Har77, Lemma 2.3] or [Dem12a, Chapter 3, Lemma
4.7])), according to which we can find by suitably selecting affine linear
coordinates for $\mathbb{C}^{n}$ a polydisc
$\Delta^{\prime}\times\Delta^{\prime\prime}\subset\mathbb{C}^{n-1}\times\mathbb{C}$
centered at $z_{0}=z(x_{0})=:(z^{\prime}_{0},z^{\prime\prime}_{0})$ such that

$(\Delta^{\prime}\times\partial\Delta^{\prime\prime})\cap K=\emptyset.$

By shrinking $\Delta^{\prime}$ if necessary, we further arrive at

$\big{(}\Delta^{\prime}\times(\Delta^{\prime\prime}\\!\setminus\\!(1-\varepsilon)\overline{\Delta^{\prime\prime}})\big{)}\cap
K=\emptyset$

for some sufficiently small $\varepsilon>0$. The remaining argument is the
same as in the last paragraph of the proof of [CWW15, Theorem 1.2]. Choose
$R,\,r>0$ such that $1-\varepsilon<r<R<1$ and ball
$B\subset\subset\Delta^{\prime}$ centered at $z^{\prime}_{0}$. Then the Cauchy
estimate for holomorphic functions and inequalities in (5.2) imply

$\begin{split}e^{\varphi(z)}&\leq{\rm const}\,|f_{z}(z)|^{2}\leq{\rm
const}_{B,\,R,\,r,\,\varepsilon}\int_{B\times(R\Delta^{\prime\prime}\setminus
r\Delta^{\prime\prime})}|f_{z}|^{2}\\\ &\leq{\rm
const}_{B,\,R,\,r,\,\varepsilon}\sup_{B\times(R\Delta^{\prime\prime}\setminus
r\Delta^{\prime\prime})}e^{\varphi}\int_{B\times(R\Delta^{\prime\prime}\setminus
r\Delta^{\prime\prime})}|f_{z}|^{2}e^{-\varphi}\\\ &\leq
C\sup_{B\times(R\Delta^{\prime\prime}\setminus
r\Delta^{\prime\prime})}e^{\varphi}\end{split}$

for all
$z\in\big{(}(1-\varepsilon)(B\times\Delta^{\prime\prime})\big{)}\setminus K$,
where $C>0$ is a constant independent of $z$. Consequently, $\varphi$ is
bounded above on $(1-\varepsilon)(B\times\Delta^{\prime\prime})\ni z_{0}$
outside $K$. This completes the proof. ∎

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