#  $\partial\bar{\partial}$-lemmas and a conjecture of O. Fujino

Junyan CAO, Mihai PAUN  Université Côte d’Azur, CNRS, LJAD, France
<EMAIL_ADDRESS>Universität Bayreuth, Mathematisches Institut, Lehrstuhl
Mathematik VIII, Universitätsstrasse 30, D-95447, Bayreuth, Germany
<EMAIL_ADDRESS>
##  1\. Introduction

Let $(X,\omega_{X})$ be a $n$-dimensional compact Kähler manifold, and let
$(F,h_{F})$ be a holomorphic line bundle on $X$ endowed with a Hermitian
metric $h_{F}$. If the curvature form $\displaystyle i\Theta(F,h_{F})\geqslant
0$ is semi-positive, then for every $(n,q)$-form $v$ with values in $F$ such
that

$\bar{\partial}v=0,\qquad\int_{X}\langle[i\Theta(F,h_{F}),\Lambda_{\omega_{X}}]^{-1}v,v\rangle
dV<\infty$

and whose $L^{2}$ norm is finite one can solve the equation

(1.0.1) $\bar{\partial}u=v,$

where $u$ is an $F$-valued $(n,q-1)$-form whose $L^{2}$–norm is finite.
Moreover, the solution $u$ of (1.0.1) which is orthogonal to
$\mathrm{Ker}(\bar{\partial})$ satisfies very precise $L^{2}$ estimates, cf.
[6].

Consider next an snc divisor $E$ on $X$, and the corresponding space of forms
with log poles along $E$. Hodge theory results in [8], [10] show that for any
logarithmic $(n,q)$-form $v$

$\bar{\partial}v=0,\qquad v=\partial w$

(where the first equality holds pointwise in the complement of the support of
$E$ and $w$ is a $(n,q-1)$ log form) there exists a form $u$ with log poles
along $E$ such that

(1.0.2) $\bar{\partial}u=v,$

holds on $X\setminus E$. This is the log-version of the familiar
$\partial\bar{\partial}$ lemma in Hodge theory.

We remark that one could try to obtain (1.0.2) as consequence of the
$L^{2}$-theory, i.e. by interpreting $v$ as $(n,q)$-form with values in
$F:=\mathcal{O}(E)$, the line bundle corresponding to $E$. However, the only
natural, semi-positively curved metric $h_{F}$ we have at hand is singular
along $E$, and the $L^{2}$ condition we have mentioned above means that we
impose $v$ to vanish on $E$ –in conclusion, we would get a much weaker result.

In this article we are aiming at the following general vanishing theorem,
which loosely speaking interpolates between Hodge and $L^{2}$-theory,
respectively.

######  Theorem 1.1.

Let $X$ be a $n$-dimensional compact Kähler manifold and let $E$ be a snc
divisor on $X$, $s_{E}$ be the canonical section of $E$. Let $(L,h_{L})$ be a
holomorphic line bundle on $X$ such that

$i\Theta_{h_{L}}(L)=\sum q_{i}[Y_{i}]+\theta_{L},$

where $q_{i}\in]0,1[\cap\mathbb{Q}$, $E+\sum Y_{i}$ is snc, and the form
$\theta_{L}$ is smooth, semi-positive. Let $\lambda$ be a
$\bar{\partial}$-closed smooth $(n,q)$-form with value in $L+E$. If there
exists $\beta_{1}$ and $\beta_{2}$, two $L$-valued $(n-1,q)$-form and
$(n-1,q-1)$-form with log poles along $E+\sum Y_{i}$ respectively, such that

(1.1.1)
$\frac{\lambda}{s_{E}}=D^{\prime}_{h_{L}}\beta_{1}+\theta_{L}\wedge\beta_{2}\qquad\text{
on }X\setminus(E+\sum Y_{i}),$

then the form $\lambda$ is $\bar{\partial}$-exact, i.e, the class
$[\lambda]=0\in H^{q}(X,K_{X}+L+E)$.

As consequence of the methods developed in this paper (especially Theorem 1.1)
and in [4] we obtain the following injectivity statement, conjectured by O.
Fujino long ago (cf. [12]).

######  Theorem 1.2.

[12, Conj 2.21] Let $X$ be a compact Kähler manifold and let $E=\sum E_{i}$ be
a simple normal crossing divisor on $X$. We consider a semi-positive line $F$
bundle on $X$ and a holomorphic section $s$ of $mF$ (for some
$m\in\mathbb{N}$). We assume moreover that the zero locus of $s$ contains no
lc centres of the lc pair $(X,E)$. Then the multiplication map induced by the
tensor product with $s$

(1.2.1) $s\otimes:\qquad H^{q}(X,K_{X}+E+F)\rightarrow
H^{q}(X,K_{X}+E+(m+1)F)$

is injective for every $q\in\mathbb{N}$.

We refer to [16, 5] for some special cases of the above result.

Given the results obtained in [15], it would be natural to adapt our method to
the case of a singular metric $h_{F}$ (with arbitrary singularities). More
precisely, we propose the following question.

######  Question 1.3.

Let $E=\sum E_{i}$ be a simple normal crossing divisor on a compact Kähler
manifold $X$ and let $F$ be a holomorphic line bundle on $X$, endowed with a
possible singular metric $h_{F}$ such that $\displaystyle
i\Theta_{h_{F}}(F)\geqslant 0$. We suppose that
$\mathcal{I}(h_{F})=\mathcal{I}(h_{F}\cdot
e^{-(1-\varepsilon)\log|s_{E}|^{2}})$ for every $\varepsilon>0$.

Let $s$ be a (holomorphic) section of $mF$ (for some $m\in\mathbb{N}$) such
that $\displaystyle|s|_{h^{m}_{F}}<+\infty$ point-wise and the zero locus of
$s$ contains no lc centres of the lc pair $(X,E)$. Then the multiplication map
induced by the tensor product with $s$

(1.3.1) $s\otimes:\qquad H^{q}\big{(}X,K_{X}\otimes E\otimes
F\otimes\mathcal{I}(h)\big{)}\rightarrow H^{q}\big{(}X,K_{X}\otimes
E\otimes(m+1)F\otimes\mathcal{I}(h^{m+1})\big{)}$

is injective for every $q\in\mathbb{N}$.

Theorem 1.1 can be seen as a generalisation of the familiar
$\partial\bar{\partial}$ lemma. Hence it should be no surprise that Hodge
theory plays a key role in its proof. The corresponding versions of Hodge
decomposition theorem needed here, for metrics with Poincaré or conic
singularities, respectively will be established in Section 2. We refer to [17]
and the references therein for statements with a similar flavour.

In this paper we introduce and study the notion of a $(p,q)$-current with
values in a singular Hermitian bundle $(L,h_{L})$, cf. Definition 2.30 in
Section 3. The terminology refers to the fact that under the hypothesis of
Theorem 1.1 we are forced to work with a Kähler metric with conic
singularities on $X$; moreover, the model for the type of currents we have to
manipulate here is provided by the RHS of (1.1.1).

The Green operator of such a current $T$ is defined by duality

(1.3.2)
$\int_{X}\mathcal{G}(T)\wedge\overline{\star\phi}:=\int_{X}T\wedge\overline{\star\mathcal{G}(\phi)},$

for any $L$-valued $(p,q)$-form $\phi$, smooth in conic sense –for this
important notion, as well for the definition of $\mathcal{G}(\phi)$ we refer
to Sections 2 and 3.

As consequence of our version of Hodge decomposition in Section hodge, we have
a decomposition of the current $T$ which parallels the case of differential
forms

(1.3.3) $T=\mathcal{H}(T)+\Delta^{\prime\prime}\big{(}\mathcal{G}(T)\big{)}.$

where $\mathcal{H}(T)$ is the projection of $T$ on the space of harmonic,
$L$-valued forms of type $(p,q)$. We call this equality _Kodaira-de Rham
decomposition of the current_ $T$, by analogy with the untwisted case treated
in [13]. Together with the properties of the current $\mathcal{G}(T)$ and an
important lemma from [14], this is a key technique in the proof of Theorem
1.1, and interesting in its own right.

Acknowledgments. During part of the preparation of this article we have
enjoyed the hospitality and excellent working conditions offered by the
_Freiburg Institut for Advanced Study_. At FRIAS we had the opportunity to
meet many visitors, but the math exchanges we have had with S. Kebekus and C.
Schnell have been exceptionally fruitful. We thank A. Höring, J. Lott, R.
Mazzeo and Y. Rubinstein for valuable discussions. JC thanks the Institut
Universitaire de France and the A.N.R JCJC project Karmapolis
(ANR-21-CE40-0010) for providing excellent working condition. Finally, our
work was completed during MP’s visit at _Center for Complex Geometry_ (in
Daejeon, South Korea). Many thanks to J.-M. Hwang for the invitation and
vibrant working atmosphere in this institute!

This paper is organised as follows.

###### Contents

  1. 1 Introduction
  2. 2 A few results from $L^{2}$ Hodge theory
    1. 2.1 Hodge decomposition for metrics with Poincaré singularities
    2. 2.2 Hodge decomposition for metrics with conic singularities, I
    3. 2.3 Hodge decomposition for metrics with conic singularities, II
      1. 2.3.1 Examples
  3. 3 De Rham-Kodaira decompostion for conic currents
    1. 3.1 Regularity of the Green kernel for certain currents
  4. 4 Proof of Theorem 1.1 and Theorem 1.2
    1. 4.1 A general vanishing theorem
    2. 4.2 Application: an injectivity theorem
  5. 5 Appendix
    1. 5.0.1 Taylor expansions

##  2\. A few results from $L^{2}$ Hodge theory

###  2.1. Hodge decomposition for metrics with Poincaré singularities

Let $X$ be a $n$-dimensional compact Kähler manifold, and let $(L,h_{L})$ be a
line bundle endowed with a (singular) metric $h_{L}=e^{-\varphi_{L}}$. The
terminology in this section is as follows.

  1. (a)

We say that the metric $h_{L}$ has log poles, if its local weights can be
written as follows

$\varphi_{L}\equiv\sum a_{i}\log|f_{i}|^{2}$

modulo a smooth function, where $a_{i}$ are rational numbers and $f_{i}$ are
holomorphic functions.

  2. (b)

If the metric $h_{L}$ has log poles, then we will consistently write

$\displaystyle i\Theta_{h_{L}}(L)=\sum a_{i}[Y_{i}]+\theta_{L}$

where $\theta_{L}$ is a smooth form.

We consider a modification $\pi:\widehat{X}\to X$ of $X$ such that the support
of the singularities of $\varphi_{L}\circ\pi$ is a simple normal crossing
divisor $E$. As usual, we can construct $\pi$ such that its restriction to
$\widehat{X}\setminus E$ is a biholomorphism . We write

(2.0.1)
$\varphi_{L}\circ\pi|_{\Omega}\equiv\sum_{\alpha=1}^{p}e_{\alpha}\log|z_{\alpha}|^{2}$

modulo a smooth function. Here $\Omega\subset\widehat{X}$ is a coordinate
subset, and $(z_{\alpha})_{\alpha=1,\dots,n}$ are coordinates such that
$E\cap\Omega=(z_{1}\dots z_{p}=0)$.

Let $\widehat{\omega}_{E}$ be a complete Kähler metric on
$\widehat{X}\setminus E$, with Poincaré singularities along $E$, and let

(2.0.2) $\omega_{E}:=\pi_{\star}(\widehat{\omega}_{E})$

be the direct image metric. We note that in this way $(X_{0},\omega_{E})$
becomes a complete Kähler manifold, where $X_{0}:=X\setminus(h_{L}=\infty)$.

The following statement is well-known, but we recall the precise version we
need here. We denote by $s$ the section of a line bundle for which the support
of its zero divisor coincides with the support of $(h_{L}=\infty)$.

######  Lemma 2.1.

There exist a family of smooth functions $(\mu_{\varepsilon})_{\varepsilon>0}$
with the following properties.

  1. (a)

For each $\varepsilon>0$, the function $\mu_{\varepsilon}$ has compact support
in $X_{0}$, and $0\leqslant\mu_{\varepsilon}\leqslant 1$.

  2. (b)

The sets $(\mu_{\varepsilon}=1)$ are providing an exhaustion of $X_{0}$.

  3. (c)

There exists a positive constant $C>0$ such that we have

$\displaystyle\sup_{X_{0}}\left(|\partial\mu_{\varepsilon}|_{\omega_{E}}^{2}+|\partial\bar{\partial}\mu_{\varepsilon}|^{2}_{\omega_{E}}\right)\leqslant
C.$

###### Proof.

It is enough to obtain the corresponding statement on $\widehat{X}$, so that
the divisor $E$ is snc. Then we take

(2.1.1)
$\mu_{\varepsilon}=\rho_{\varepsilon}\Big{(}\log\log\frac{1}{|\widehat{s}|^{2}}\Big{)}$

where $\rho_{\varepsilon}$ is equal to 1 on the interval
$[1,\varepsilon^{-1}]$ and is equal to zero on $[1+\varepsilon^{-1},\infty]$.
Also, we denote by $|\widehat{s}|$ the inverse image of the norm of the
section $s$ with respect to an arbitrary, smooth metric. Actually we can also
impose the condition that

(2.1.2) $\max_{j=1,\dots
N}\sup_{\varepsilon>0}\sup_{\mathbb{R}_{+}}|\rho^{(j)}_{\varepsilon}|<C_{N}<\infty,$

for any positive integer $N$. The properties (a)-(c) are then verified by a
simple computation. ∎

In this context we have the following statement, which is a slight
generalization of the usual result in $L^{2}$ Hodge theory.

######  Theorem 2.2.

Consider a line bundle $(L,h_{L})\to X$ endowed with a metric $h_{L}$ such
that the requirement (a) above is satisfied and assume that
$\theta_{L}\geqslant 0$ on $X$. Let $(X_{0},\omega_{E})$ be the corresponding
complete Kähler manifold cf. (2.0.2). Then the following assertions are true.

  1. (i)

We have the Hodge decomposition for $(n,q)$ forms, i.e.

$L^{2}_{n,q}(X_{0},L)={\mathcal{H}}_{n,q}(X_{0},L)\oplus\mathrm{Im}\,\bar{\partial}\oplus\mathrm{Im}\,\bar{\partial}^{\star},$

where ${\mathcal{H}}_{n,q}(X_{0},L)$ is the space of $L^{2}$
$\Delta^{{}^{\prime\prime}}$-harmonic $(n,q)$-forms with respect to
$(\omega_{E},h_{L})$.

  2. (ii)

Let $u$ be an $L$-valued $\bar{\partial}$-closed $L^{2}$-form of type $(n,q)$
on $X_{0}$, and assume that we have $u=D^{\prime}_{h_{L}}w$, where $w\in
L^{2}_{n-1,q}(X_{0},L)$. Then there exists $v\in L^{2}_{n,q-1}(X_{0},L)$ such
that $u=\bar{\partial}v$.

The proof of Theorem 2.2, which we give below, makes use of the following
statement which is the $\bar{\partial}$-version of the Poincaré inequality
established in [3].

######  Theorem 2.3.

Let $p\leqslant n$ be an integer. There exists a positive constant $C>0$ such
that the following inequality holds

(2.3.1)
$\int_{X_{0}}|u|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}\leqslant
C\int_{X_{0}}|\bar{\partial}u|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}$

for any $L$-valued form $u$ of type $(p,0)$ which belongs to the domain of
$\bar{\partial}$ and which is orthogonal to the space of $L^{2}$ holomorphic
$(p,0)$–forms. Note that here $\theta_{L}$ is arbitrary.

###### Proof of Theorem 2.3.

The first observation is that we can assume from the very beginning that the
polar set $h_{L}=\infty$ has snc support. Indeed, via the map $\pi$ the
hypothesis and the conclusion of our Poincaré inequality transform as follows.

$\bullet$ The form

(2.3.2) $\widehat{u}=\pi^{\star}u$

on $\widehat{X}$ is $\pi^{\star}L$-valued and $L^{2}$ with respect to
$\widehat{\omega}_{E}$ and $\pi^{\star}h_{L}$.

$\bullet$ The form $\widehat{u}$ is orthogonal on the space of
$L^{2}$–harmonic $(p,0)$ forms with values in $\pi^{\star}L$. Indeed, given
any $\pi^{\star}L$-valued form of $(p,0)$-type $\widehat{\gamma}$ on
$\widehat{X}\setminus E$ there exists a form $\gamma$ on $X\setminus Z$ such
that

(2.3.3) $\widehat{\gamma}=p^{\star}\gamma$

because $p:\widehat{X}\setminus E\to X\setminus Z$ is a biholomorphism. Also,
$p$ establish an isometry between $(\widehat{X}\setminus
E,\widehat{\omega}_{E})$ and $(X\setminus Z,\omega_{E})$ respectively. The
same is true for the pairs $(p^{\star}L,p^{\star}h_{L})$ and $(L,h_{L})$, so
our claim is clear.

This means that it is enough to assume that $\widehat{X}=X$, i.e. the metric
$\omega_{E}$ has Poincaré singularities along an snc divisor $E$ which
coincides with the support of the analytic set $h_{L}=\infty$.

The following statement represents an important step towards the proof of
Theorem 2.3.

######  Proposition 2.4.

Let $\Omega$ be some co-ordinate open set of $X$ and let $\tau$ be a
$L$-valued $(p,0)$-form with compact support in $\Omega$. Then we have

(2.4.1)
$\int_{\Omega}|\tau|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}\leqslant
C\int_{\Omega}|\bar{\partial}\tau|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}$

where $C$ is a constant.

The arguments in [3], Lemma 1.10 correspond to the case $p=0$ and
$\varphi_{L}=0$. For the general case we need the following auxiliary
statement.

######  Lemma 2.5.

Let $\delta$ be an arbitrary real number. Then there exists $x_{\delta}<1$
such that the following holds. For any smooth function $f$ defined on the
closure of the unit disk $\mathbb{D}$ in $\mathbb{C}$, such that there exists
$\varepsilon>0$ with the property that $f(z)=0$ on the set
$\\{|z|\leqslant\varepsilon\\}\cup\\{|z|\geqslant 1-x_{\delta}\\}$, we have

(2.5.1) $\int_{\mathbb{D}}|f|^{2}\frac{d\lambda(z)}{|z|^{2\delta}}\leqslant
C\int_{\mathbb{D}}\left|\frac{\partial
f}{\partial\overline{z}}\right|^{2}|z|^{2-2\delta}\log^{2}|z|d\lambda(z)$

as well as

(2.5.2)
$\int_{\mathbb{D}}|f|^{2}\frac{d\lambda(z)}{|z|^{2+2\delta}\log^{2}|z|}\leqslant
C\int_{\mathbb{D}}\left|\frac{\partial
f}{\partial\overline{z}}\right|^{2}\frac{d\lambda(z)}{|z|^{2\delta}}.$

###### Proof of Lemma 2.5.

Let $x_{\delta}<1$ be a positive real number whose expression will be
determined in what follows. Let $v:[0,1]\to\mathbb{C}$ be a smooth function
whose support is contained in the interval $[\varepsilon,x_{\delta}]$ for some
positive, small enough $\varepsilon$.

We claim that the following inequalities hold

(2.5.3) $\int_{0}^{1}|v|^{2}t^{k}dt\leqslant
C\int_{0}^{1}|v^{\prime}|^{2}t^{k+2}\log^{2}tdt$

as well as

(2.5.4) $\int_{0}^{1}|v|^{2}t^{k}\frac{dt}{\log^{2}t}\leqslant
C\int_{0}^{1}|v^{\prime}|^{2}t^{k+2}dt$

where $k$ is any element of the set $\delta+\mathbb{Z}$, and $C$ is a
numerical constant (in particular independent of $v,\varepsilon,k,...$).

We first check that (2.5.3) holds true: we have

(2.5.5) $-\int_{0}^{1}|v|^{2}t^{k+1}\log^{2}d\left(\frac{1}{\log
t}\right)=\int_{0}^{1}\frac{1}{\log
t}\frac{d}{dt}\left(v^{2}t^{k+1}\log^{2}t\right)dt$

and this equals

(2.5.6)
$2\int_{0}^{1}(\overline{v}v^{\prime}+v\overline{v^{\prime}})t^{k+1}\log
tdt+\int_{0}^{1}\frac{t^{k}v^{2}}{\log t}\left((k+1)\log^{2}t+2\log
t\right)dt.$

Now two things can happen: either $k+1=0$, and then the second term in (2.5.6)
is equal to twice the quantity we are interested in, or $k+1\neq 0$. Note that
in the latter case it is possible to fix $x_{\delta}$ uniformly with respect
to $k$ so that

(2.5.7) $|(k+1)\log^{2}t+2\log t|\geqslant 2|\log t|$

for any $t\in[0,x_{\delta}]$ precisely because $k\in\delta+\mathbb{Z}$.

As a consequence, we have

(2.5.8)
$\left|\int_{0}^{1}(\overline{v}v^{\prime}+v\overline{v^{\prime}})t^{k+1}\log
tdt\right|\geqslant\int_{0}^{1}t^{k}v^{2}dt.$

for any $v$ of compact support in $]0,x_{\delta}[$.

Moreover, we have

(2.5.9)
$\left|\int_{0}^{1}(\overline{v}v^{\prime}+v\overline{v^{\prime}})t^{k+1}\log
tdt\right|^{2}\leqslant\int_{0}^{1}|v^{\prime}|^{2}t^{k+2}\log^{2}tdt\int_{0}^{1}|v|^{2}t^{k}dt.$

by Cauchy-Schwarz, so the inequality (2.5.3) is settled.

The arguments for (2.5.4) are completely similar: we have

(2.5.10)
$\int_{0}^{1}|v|^{2}t^{k}\frac{dt}{\log^{2}t}=\int_{0}^{1}|v|^{2}\frac{t^{k+1}}{\log^{2}t}d\log
t$

which equals

(2.5.11) $-\int_{0}^{1}{\log
t}\frac{d}{dt}\left(|v|^{2}\frac{t^{k+1}}{\log^{2}t}\right)dt=\int_{0}^{1}\frac{|v|^{2}t^{k}}{\log^{2}t}\left(2+(k+1)\log\frac{1}{t}\right)dt-\int_{0}^{1}\frac{t^{k+1}}{\log
t}(v^{\prime}\overline{v}+v\overline{v^{\prime}})dt.$

and two things can happen. If $k+1\geqslant 0$, then we have $\displaystyle
2+(k+1)\log\frac{1}{t}\geqslant 2$ for any $t\in[0,1]$ and (2.5.4) follows.
For each element $k\in\delta+\mathbb{Z}$ such that $k+1<0$ we can assume that

(2.5.12) $(k+1)\log\frac{1}{t}+2\leqslant\frac{1}{2}$

for any $t\in[0,x_{\delta}]$ by imposing a supplementary (but uniform)
condition on $x_{\delta}$ if necessary, and we proceed exactly as we did for
(2.5.3).

Therefore, we obtain (2.5.3) and (2.5.4) for any $k\in\delta+\mathbb{Z}$.

Lemma 2.5 follows from this: we consider the Fourier series

(2.5.13) $f=\sum_{k\in\mathbb{Z}}a_{k}(t)e^{\sqrt{-1}k\theta}$

of $f$, and then we have

(2.5.14) $\frac{\partial
v}{\bar{\partial}z}=\sum_{k\in\mathbb{Z}}\left(a_{k}^{\prime}(t)-\frac{k}{t}a_{k}(t)\right)e^{\sqrt{-1}k\theta}.$

The identity

(2.5.15)
$t^{k}\frac{d}{dt}\left(\frac{a_{k}}{t^{k}}\right)=a_{k}^{\prime}(t)-\frac{k}{t}a_{k}(t)$

reduces the proof of our statement to inequalities of type (2.5.3) and (2.5.4)
which are already established. In conclusion, Lemma 2.5 is completely
established. ∎

We now return to the proof of Proposition 2.4.

###### Proof of Proposition 2.4.

We can suppose that the intersection $\Omega\cap E$ is of type $z_{1}\cdots
z_{r}=0$.

We first consider the case when $\tau$ is of compact support in
$\Omega\setminus E$. Then $\tau$ can be written as sum of forms of type

(2.5.16) $\tau_{I}:=f_{I}dz_{I}.$

After changing the orders of the index, there are two cases:
$I\cap\\{1,\dots,r\\}=\\{1,\dots,p\\}$ for some $p$ or
$I\cap\\{1,\dots,r\\}=\emptyset.$

In the first case we have

(2.5.17)
$\frac{|\tau_{I}|_{g}^{2}e^{-\varphi_{L}}dV_{g}}{d\lambda}=\frac{|f_{I}|^{2}e^{-\varphi_{L}}}{\prod_{\alpha=p+1}^{r}|z_{\alpha}|^{2}\log^{2}|z_{\alpha}|^{2}}$

where $\displaystyle
e^{-\varphi_{L}}=\frac{1}{\prod_{i=1}^{r}|z_{i}|^{2\delta_{i}}}$ and in the
second case this is the same with $p=0$. In (2.5.17) we denote by $g$ the
model Poincaré metric

$\sum_{i=1}^{r}\frac{\sqrt{-1}dz_{i}\wedge
d\overline{z}_{i}}{|z_{i}|^{2}\log^{2}|z_{i}|^{2}}+\sum_{i=r+1}^{n}{\sqrt{-1}dz_{i}\wedge
d\overline{z}_{i}}.$

Now for the $\bar{\partial}\tau_{I}$ we have

$\displaystyle\frac{|\bar{\partial}\tau_{I}|_{g}^{2}e^{-\varphi_{L}}dV_{g}}{d\lambda}=$
$\displaystyle\sum_{i=1}^{r}\left|\frac{\partial
f_{I}}{\partial\overline{z}_{i}}\right|^{2}\frac{|z_{i}|^{2}\log^{2}|z_{i}|^{2}e^{-\varphi_{L}}}{\prod_{\alpha=p+1}^{r}|z_{\alpha}|^{2}\log^{2}|z_{\alpha}|^{2}}$
(2.5.18) $\displaystyle+$ $\displaystyle\sum_{i=r+1}^{n}\left|\frac{\partial
f_{I}}{\partial\overline{z}_{i}}\right|^{2}\frac{e^{-\varphi_{L}}}{\prod_{\alpha=p+1}^{r}|z_{\alpha}|^{2}\log^{2}|z_{\alpha}|^{2}}$

and the observation is that for every $\xi\in\Omega$ there exist some index
$j\in\\{1,\dots,r\\}$ for which the support condition we require in Lemma 2.5
is satisfied for

(2.5.19) $f(z):=f_{I}(\xi_{1},\dots\xi_{j-1},z,\xi_{j+1},\dots\xi_{n}).$

We use (2.5.3) or (2.5.4) according to the case where $j\geqslant p$ in
(2.5.18) or not.

In general, if $\tau$ is of compact support in $\Omega$, we use the cut-off
function $\mu_{\varepsilon}$ in Lemma 2.1. We apply the above estimation for
$\mu_{\varepsilon}\tau$. Note that the constant $C$ in Lemma 2.5 is uniform
with respect to $\varepsilon$. Proposition 2.4 is therefore completely proved
by letting $\varepsilon\to 0$. ∎

Now we finish the proof of Theorem 2.3, which follows the arguments in [3,
Lemma 1.10]: if a positive constant as in (2.3.1) does not exists, then we
obtain a sequence $u_{j}$ of $L$-valued forms of type $(p,0)$ orthogonal to
the space of holomorphic forms such that

(2.5.20)
$\int_{X}|u_{i}|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}=1,\qquad\lim_{i}\int_{X}|\bar{\partial}u_{i}|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}=0$

It follows that the weak limit $u_{\infty}$ of $(u_{i})$ is holomorphic. On
the other hand, each $u_{i}$ is perpendicular to the space of holomorphic
forms, so it follows that $u_{\infty}$ is equal to zero. Moreover, by Bochner
formula combined with Proposition 2.4 we can assume that $u_{i}$ converges
weakly in the Sobolev space $W^{1}$, so strongly in $L^{2}$ to zero. In
particular we have

(2.5.21) $u_{i}|_{K}\to 0$

in $L^{2}$ for any compact subset $K\subset X\setminus E$.

The last step in the proof is to notice that the considerations above
contradict the fact that the $L^{2}$ norm of each $u_{i}$ is equal to one.
This is not quite immediate, but is precisely as the end of the proof of Lemma
1.10 in [3], so we will not reproduce it here. The idea is however very clear:
we decompose each $u_{j}=\chi u_{j}+(1-\chi)u_{j}$ where $\chi$ is a cutoff
function which is equal to 1 in $U$ and whose norm of the corresponding
gradient is small. Then the $L^{2}$ norm of $\chi u_{j}$ is small by (2.4.1)
and (2.5.20). The $L^{2}$ norm of $(1-\chi)u_{j}$ is equally small by
(2.5.21), and this is how we reach a contradiction. ∎

We have the following consequences of Proposition 2.4.

######  Corollary 2.6.

In the setting of Theorem 2.2, we can find a positive constant $C>0$ such that
the following holds true. Let $v$ be a $L$-valued form of type $(n,q)$ for
some $q\geqslant 1$. We assume that $v$ is $L^{2}$, and orthogonal to the
kernel of the operator $\bar{\partial}^{\star}$. Then we have

(2.6.1)
$\int_{X_{0}}|v|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}\leqslant
C\int_{X_{0}}|\bar{\partial}^{\star}v|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}.$

###### Proof.

To start with, we consider the Hodge star $u:=\star v$ (of type $(n-q,0)$ and
values in $L$) is orthogonal to the space $\mathrm{Ker}(D^{\prime}_{h_{L}})$.
This can be seen as follows: let $\phi$ be a $L^{2}$-form of type $(n-q,0)$
such that $D^{\prime}_{h_{L}}\phi=0$. Then we have

$\int_{X_{0}}\langle u,\phi\rangle
e^{-\varphi_{L}}dV_{\omega_{E}}=\int_{X_{0}}\langle v,\star\phi\rangle
e^{-\varphi_{L}}dV_{\omega_{E}}=0.$

since $\displaystyle\bar{\partial}^{\star}(\star\phi)=-\star
D^{\prime}_{h_{L}}(\phi)=0$.

We therefore have to show that there exists some constant $C>0$ such that the
inequality

(2.6.2)
$\int_{X_{0}}|u|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}\leqslant
C\int_{X_{0}}|D^{\prime}_{h_{L}}u|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}.$

holds for any $L$-valued form $u$ of type $(n-q,0)$ orthogonal to the kernel
of $D^{\prime}_{h_{L}}$.

As before, this is done contradiction: if we cannot find a constant $C$ as in
(2.6.2), then we get a sequence of forms $u_{i}$ such that

(2.6.3)
$\|u_{i}\|^{2}=1,\qquad\lim_{i\to\infty}\|D^{\prime}_{h_{L}}(u_{i})\|^{2}=0,$

where the norms in (2.6.3) are precisely the ones in (2.6.2). It follows that
any weak limit $u_{\infty}$ of the sequence $(u_{i})$ must be zero, since it
belongs to both $\mathrm{Ker}(D^{\prime}_{h_{L}})$ and to its orthogonal.
Therefore we get

(2.6.4) $u_{i}|_{K}\to 0$

in $L^{2}$ for any compact subset $K\subset X\setminus E$.

On the other hand, for each index $i$ we have $D^{\prime\star}(u_{i})=0$. So
Bochner formula reads

(2.6.5)
$\int_{X_{0}}|\bar{\partial}u_{i}|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}=\int_{X_{0}}|D^{\prime}_{h_{L}}u_{i}|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}+\int_{X_{0}}\langle[\theta_{L},\Lambda_{\omega_{E}}]u_{i},u_{i}\rangle
e^{-\varphi_{L}}dV_{\omega_{E}}$

and since -by the assumption $\theta_{L}\geqslant 0$, we have

(2.6.6)
$\int_{X_{0}}\langle[\theta_{L},\Lambda_{\omega_{E}}]u_{i},u_{i}\rangle
e^{-\varphi_{L}}dV_{\omega_{E}}\leqslant 0$

because of the type of $u_{i}$.

It follows that $\displaystyle\lim_{i}\|\bar{\partial}u_{i}\|^{2}=0$ and we
obtain a contradiction exactly as in end of the proof of Theorem 2.3. ∎

The following is the ”adjoint” version of the preceding corollary.

######  Corollary 2.7.

In the setting of Theorem 2.2, there exists a positive constant $C>0$ such
that the following holds true. Let $v$ be a $L$-valued form of type $(n,q)$
for some $q\geqslant 1$. We assume that $v$ is $L^{2}$, and orthogonal to the
kernel of the operator $\bar{\partial}$. Then we have

(2.7.1)
$\int_{X_{0}}|v|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}\leqslant
C\int_{X_{0}}|\bar{\partial}v|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}.$

###### Proof.

The arguments are completely identical: if $v$ is orthogonal on
$\mathrm{Ker}(\bar{\partial})$, then $u:=\star v$ is orthogonal to the kernel
of $D^{\prime\star}$. Thus the inequality to be established is

(2.7.2)
$\int_{X_{0}}|u|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}\leqslant
C\int_{X_{0}}|D^{\prime\star}u|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}.$

holds for any $L$-valued form $u$ of type $(n-q,0)$ orthogonal to the kernel
of $D^{\prime\star}$. This is (again) done by contradiction –note that in this
case $D^{\prime}_{h_{L}}u=0$– , and we will skip the details. ∎

###### Proof of Theorem 2.2.

This statement is almost contained in [6, chapter VIII, pages 367-370].
Indeed, in the context of complete manifolds one has the following
decomposition

(2.7.3)
$L^{2}_{n,q}(X_{0},L)={\mathcal{H}}_{n,q}(X_{0},L)\oplus\overline{\mathrm{Im}\,\bar{\partial}}\oplus\overline{\mathrm{Im}\,\bar{\partial}^{\star}}.$

We also know (see _loc. cit._) that the adjoints $\bar{\partial}^{\star}$ and
$D^{\prime\star}$ in the sense of von Neumann coincide with the formal
adjoints of $\bar{\partial}$ and $D^{\prime}$ respectively.

It remains to show that the ranges of the
$\bar{\partial},\bar{\partial}^{\star}$-operators are closed with respect to
the $L^{2}$ topology. In our set-up, this is a consequence of the particular
shape of the metric $\omega_{E}$ at infinity (i.e. near the support of
$\pi(E)$). The image of $\bar{\partial}^{\star}$ is closed thanks to Corollary
2.6 and the fact that the image of $\bar{\partial}$ is closed follows from
Corollary 2.7

The assertion $(ii)$ is an easy consequences of $(i)$, together with standard
considerations. ∎

######  Remark 2.8.

It would be very interesting to know if the images of the operators
$\bar{\partial}$ and $\bar{\partial}^{\star}$ are closed for $L$-valued forms
of arbitrary type, especially if the non-singular part $\theta_{L}$ of the
curvature of $(L,h_{L})$ has arbitrary sign. In our arguments, the requirement
$\theta_{L}\geqslant 0$ is imposed by the -extensive- use of Bochner formula.
We refer to [10] and [11] for results in this direction.

In the setting of the current section, the harmonic $(n,1)$-forms have the
following properties.

######  Lemma 2.9.

In the setting of Theorem 2.2, let $\alpha$ be a $L^{2}$
$\Delta^{\prime\prime}$-harmonic form of type $(n,1)$ with values in $L$. We
denote by $F:=\star\alpha$ its Hodge dual. Then the following hold.

  1. (1)

We have

$\bar{\partial}F=0\qquad\text{and }\qquad D^{\prime}_{h_{L}}F=0.$

  2. (2)

Let $\Omega\subset X$ be a coordinate open set, and let $f$ be any holomorphic
function defined on $\overline{\Omega}$. We assume that the support of the set
of zeroes of $f$ is contained in $(h_{L}=\infty)$. Then the form
$\displaystyle F\wedge\frac{df}{f}$ is non-singular on $X$ (but it is not
clear whether this form is $L^{2}$ with respect to $e^{-\varphi_{L}}$ or not).

  3. (3)

Let $\Omega\subset X$ be a coordinate open set, and let $g$ be any holomorphic
function defined on $\overline{\Omega}$. Then we have

$\int_{\Omega}|F\wedge dg|^{2}e^{-\varphi_{L}}<\infty.$

###### Proof.

Note that given any $\Delta^{\prime\prime}$-harmonic $(n,1)$-form $\alpha$, by
Bochner formula combined with the fact that the curvature of $(L,h_{L})$ is
semi-positive, we have

(2.9.1) $D^{\prime\star}\alpha=0\qquad\text{on }X\setminus(h_{L}=\infty).$

So if we write $\displaystyle\alpha=\omega_{E}\wedge F$, it follows that

(2.9.2) $F\in H^{0}\big{(}X,\Omega_{X}^{n-1}\otimes
L\big{)},\qquad\int_{X}|F|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}<\infty$

thanks to the property (2.9.1). Moreover, as
$\displaystyle\bar{\partial}^{*}\alpha=0$ it turns out that we have
$D^{\prime}_{h_{L}}F=0$ as well. We obtain thus $(1)$ and $(3)$. Note that
(2.9.2) is valid in a more general setting, cf. [DPS01], [19].

As for $(2)$, by our choice of the metric $\omega_{E}$, we show first that we
have

(2.9.3) $|df|^{2}_{\omega_{E}}\sim|f|^{2}\log^{2}|f|.$

Indeed, assume that via the bimeromorphism $\pi:\widehat{X}\rightarrow X$ the
function $f$ corresponds locally to the function
$\displaystyle\prod_{i=1}^{r}z_{i}^{p_{i}}$ (notations as in the beginning of
this section). Then the metric $\omega_{E}$ is quasi-isometric to the Poincaré
metric with singularities along $z_{1}\dots z_{r}=0$. The $\pi$-inverse image
of $\displaystyle\frac{df}{f}$ is equal to

$\sum_{i=1}^{r}p_{i}\frac{dz_{i}}{z_{i}}$

and thus we have

$\left|\frac{\partial f}{f}\right|^{2}_{\omega_{E}}\simeq\sum
p_{i}^{2}\log^{2}|z_{i}|^{2}$

and the RHS of this quantity is the same as $\log^{2}|f|^{2}$.

Therefore we infer

(2.9.4)
$\int_{\Omega}\left|F\wedge\frac{df}{f}\right|^{2}\frac{e^{-\varphi_{L}}}{\log^{2}|f|}\leqslant
C\int_{X}{|F|^{2}_{\omega_{E}}}{}e^{-\varphi_{L}}dV_{\omega_{E}}.$

Since the RHS of (2.9.4) is finite and $\varphi_{L}$ has non zero Lelong
number over any component of $Div(f)$, it follows that $F\wedge df$ vanishes
along $Div(f)$. ∎

Consider the equation

(2.9.5)
$\bar{\partial}u=D^{\prime}_{h_{L}}w+\Theta(L,h_{L})\wedge\tau\qquad\text{ on
}X\setminus\\{h_{L}=\infty\\}$

where $\tau,w$ and $D^{\prime}w$ are $L^{2}$, and
$D^{\prime}_{h_{L}}w+\Theta(L,h_{L})\wedge\tau$ is $\bar{\partial}$-closed on
$X\setminus\\{h_{L}=\infty\\}$. We show next that one can solve it in a fair
general context, and moreover obtain $L^{2}$ estimates for the solution with
minimal norm. We refer to [Wan17, Thm A.5] for a similar argument in the non-
singular case.

######  Theorem 2.10.

Let $(L,h_{L})$ be a holomorphic line bundle on $X$ with a possible singular
metric $h_{L}$ with analytic singularities along a subvarities $Z\subset X$
and whose curvature current is semi-positive. Consider a complete Kähler
metric $(X\setminus Z,\omega_{E})$. Let $w$ be an $L$-valued $(n-1,1)$–form on
$X\setminus Z$ such that $w$ and $D^{\prime}_{h_{L}}w$ are in $L^{2}$. Let
also $\tau$ be an $L$-valued $(n-1,0)$–form on $X\setminus Z$ such that $\tau$
and $\Theta(L,h_{L})\wedge\tau$ are $L^{2}$.

If $D^{\prime}_{h_{L}}w+\Theta(L,h_{L})\wedge\tau$ is $\bar{\partial}$-closed
on $X\setminus Z$, then there exists a solution $u$ of the equation (2.9.5)
such that

(2.10.1)
$\int_{X}|u|^{2}e^{-\varphi_{L}}\leqslant\int_{X}|w|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}-\int_{X\setminus
Z}\langle[\Theta(L,h_{L}),\Lambda_{\omega_{E}}]\tau,\tau\rangle_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}$

###### Proof.

Let $\xi$ be a smooth $(n,1)$-form of compact support in $X\setminus Z$. Let
$\xi=\xi_{1}+\xi_{2}$ be its decomposition according to
$\mathrm{Ker}(\bar{\partial})$ and its orthogonal.

By using the $L^{2}$-assumptions, we have

$\int_{X\setminus Z}\langle
D^{\prime}w+\Theta(L,h_{L})\wedge\tau,\xi\rangle_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}=\int_{X\setminus
Z}\langle
D^{\prime}w+\Theta(L,h_{L})\wedge\tau,\xi_{1}\rangle_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}.$

Then the semipositive curvature condition and the Bochner equality imply that

(2.10.2) $\left|\int_{X\setminus Z}\langle
D^{\prime}w,\xi_{1}\rangle_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}\right|^{2}$

is smaller than

(2.10.3)
$\int_{X}|w|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}\cdot\int_{X}|\bar{\partial}^{\star}\xi|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}.$

For the term containing the curvature of $(L,h_{L})$ we are using Cauchy
inequality and we infer that

(2.10.4) $\left|\int_{X\setminus
Z}\langle\Theta(L,h_{L})\wedge\tau,\xi_{1}\rangle_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}\right|^{2}$

is smaller than

(2.10.5)
$\int_{X}\langle-[\Theta(L,h_{L}),\Lambda_{\omega_{E}}]\tau,\tau\rangle_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}\cdot\int_{X}\langle[\Theta(L,h_{L}),\Lambda_{\omega_{E}}]\xi_{1},\xi_{1}\rangle_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}.$

We are using again Bochner equality and we infer that

(2.10.6) $\left|\int_{X\setminus
Z}\langle\Theta(L,h_{L})\wedge\tau,\xi_{1}\rangle_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}\right|^{2}$

is smaller than

(2.10.7)
$\int_{X}\langle-[\Theta(L,h_{L}),\Lambda_{\omega_{E}}]\tau,\tau\rangle_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}\cdot\int_{X}|\bar{\partial}^{\star}\xi|^{2}_{\omega_{E}}e^{-\varphi_{L}}dV_{\omega_{E}}.$

Then the theorem is proved by the standard $L^{2}$-estimate argument. ∎

###  2.2. Hodge decomposition for metrics with conic singularities, I

We discuss in this section the analog of Theorem 2.2 for metrics with conic
singularities. This will be important for the applications we are aiming at.
There are two possibilities to deal with this sort of problems: either we
approximate our metrics with complete ones (e.g. with Poincaré singularities),
or directly, by using local uniformisations. In this subsection we use the
former approach and in the next subsection we present the later.

Since most of the arguments in the proof of Theorem 2.2 will be used here, we
will skip a few details. In the last part of this section we offer a few
comments about the $L^{2}$ estimates in this class of singularities. The
setting of this subsection is as follows.

Setting: Let $X$ be a compact Kähler manifold, and let $\displaystyle Y=\sum
Y_{i}$ be a snc divisor. We denote by $\omega_{\mathcal{C}}$ a metric with
conic singularities along the $\mathbb{Q}$-divisor

(2.10.8) $\sum_{i\in I}\left(1-\frac{1}{m}\right)Y_{i}$

where $m$ is a positive integer. By this we mean that locally we have

(2.10.9) $\omega_{\mathcal{C}}\simeq\sum_{i=1}^{r}\frac{\sqrt{-1}dz_{i}\wedge
d\overline{z}_{i}}{|z_{i}|^{2-\frac{2}{m}}}+\sum_{i\geqslant
r+1}\sqrt{-1}dz_{i}\wedge d\overline{z}_{i}$

that is to say, $\omega_{\mathcal{C}}$ is quasi-isometric with the RHS of
(2.10.9), where $z_{1}\dots z_{r}=0$ is the local equation of the divisor $Y$.

Let $(L,h_{L})$ be a line bundle endowed with a (singular) metric $h_{L}$ such
that the following holds.

  1. (i)

The singularities of the metric $h_{L}$ are contained in the support of $Y$,
i.e. $\displaystyle\varphi_{L}\simeq\sum q_{i}\log|z_{i}|^{2}$ modulo a smooth
function, where $q_{i}\in\mathbb{Q}$ and as above, $z_{i}$ represent the local
equations of the components of $Y$. Note that the $q_{i}$ are not necessarily
positive.

  2. (ii)

For each $q_{i}\in\mathbb{Q}\setminus\mathbb{Z}$, we assume that
$\displaystyle\left\lfloor q_{i}-\frac{1}{m}\right\rfloor=\lfloor
q_{i}\rfloor$ holds true and that $mq_{i}\in\mathbb{Z}$.

Note that the requirement $(i)$ implies that the curvature current of
$(L,h_{L})$ can be written as

$i\Theta(L,h_{L})=\sum q_{i}[Y_{i}]+\theta_{L}$

where $\theta_{L}$ is a smooth form on $X$.

Our next result states as follows.

######  Theorem 2.11.

Let $(L,h_{L})\to X$ be a line bundle endowed with a metric $h_{L}$ such that
the requirements (i)-(ii) above are satisfied; in addition, we assume
$i\Theta(L,h_{L})\geqslant 0$. Let $\omega_{\mathcal{C}}$ be a Kähler metric
with conic singularities as above.

  1. (i)

The following equality holds

$L^{2}_{n,q}(X_{0},L)={\mathcal{H}}_{n,q}(X_{0},L)\oplus\mathrm{Im}\,\bar{\partial}\oplus\mathrm{Im}\,\bar{\partial}^{\star},$

for any $q\geqslant 0$, where $X_{0}:=X\setminus Y$.

  2. (ii)

For any harmonic form $\xi\in{\mathcal{H}}_{n,q}(X_{0},L)$, we have

$\bar{\partial}\xi=\bar{\partial}^{\star}\xi=(D^{\prime})^{\star}\xi=0.$

In particular, $\star\xi$ is a $L^{2}$-holomorpic form on $X$.

The proof of Theorem 2.11 is based on the corresponding version for metrics
with Poincaré singularities, together with the following result. A different
argument for a more general version of this statement will be given in the
next subsection. We have decided to include this first proof here because the
statement 2.12 below could be useful in other contexts.

Let

(2.11.1)
$\omega_{Y,\varepsilon}:=\omega_{\mathcal{C}}+\varepsilon\omega_{\mathcal{P}}$

be a sequence approximating $\omega_{\mathcal{C}}$, where
$\omega_{\mathcal{P}}$ is a metric with Poincaré singularities on $Y$. We have
the following statement.

######  Theorem 2.12.

There exists a positive constant $C>0$ (independent of $\varepsilon$) such
that the following inequality holds

(2.12.1)
$\int_{X_{0}}|u|^{2}_{\omega_{Y,\varepsilon}}e^{-\varphi_{L}}dV_{\omega_{Y,\varepsilon}}\leqslant
C\int_{X_{0}}|\bar{\partial}u|^{2}_{\omega_{Y,\varepsilon}}e^{-\varphi_{L}}dV_{\omega_{Y,\varepsilon}}$

for any $L$-valued form $u$ of type $(p,0)$ which belongs to the domain of
$\bar{\partial}$ and which is orthogonal to the space of $L^{2}$ harmonic
$(p,0)$–forms (with respect the metric $\omega_{Y,\varepsilon}$).

We first prove Theorem 2.12, and then derive Theorem 2.11 as consequence.

###### Proof of Theorem 2.12.

The case $\varepsilon=0$ is quite easy and even if the general case does not
follows from it, we give here a direct proof.

We argue by contradiction: if (2.12.1) does not holds, then there exists a
sequence $(u_{k})$ of $(p,0)$-forms orthogonal to
$\mathrm{Ker}(\bar{\partial})$ such that

(2.12.2)
$\int_{X_{0}}|u_{k}|^{2}_{\omega_{\mathcal{C}}}e^{-\varphi_{L}}dV_{\omega_{\mathcal{C}}}=1,\qquad\lim_{k\to\infty}\int_{X_{0}}|\bar{\partial}u_{k}|^{2}_{\omega_{\mathcal{C}}}e^{-\varphi_{L}}dV_{\omega_{\mathcal{C}}}=0.$

It follows that the weak limit $u_{0}$ of $(u_{k})_{k\geqslant 1}$ has the
following properties

(2.12.3)
$\int_{X_{0}}|u_{0}|^{2}_{\omega_{\mathcal{C}}}e^{-\varphi_{L}}dV_{\omega_{\mathcal{C}}}\leqslant
1,\qquad\bar{\partial}u_{0}=0.$

Since on the other hand $u_{0}$ is orthogonal to
$\mathrm{Ker}(\bar{\partial})$, it follows that $u_{0}=0$. We show next that
this contradicts the first equality of (2.12.2) for $k$ large enough.

Let $\Omega$ be a coordinate ball whose origin belongs to $Y$. The restriction
$\displaystyle\bar{\partial}u_{k}|_{\Omega\setminus Y}$ is a closed
$(p,1)$-form, but we can also see it as $(n,1)$-form with values in the vector
bundle $\displaystyle\Lambda^{n-p}T_{X}\otimes L|_{\Omega}$. We equip the
tangent bundle $\displaystyle E:=T_{X}|_{\Omega}$ with the standard metric
with conic singularities, and $L$ with $e^{-\phi_{L}}$ where

(2.12.4) $\phi_{L}=\varphi_{L}+\|z\|^{2}+C\sum|z_{i}|^{\frac{2}{m_{i}}}.$

In (2.12.4) we denote by $\displaystyle(z_{i})_{i=1,\dots,n}$ a coordinate
system adapted to $Y$. Then the curvature of $E$ is greater than
$\omega_{\mathcal{C}}\otimes\mathrm{Id}_{E}$ and by the $L^{2}$ estimates
applied on $\Omega\setminus Y$ we can solve the equation

(2.12.5)
$\bar{\partial}v_{\Omega,k}=\bar{\partial}u_{k},\qquad\int_{\Omega}|v_{\Omega,k}|^{2}e^{-\varphi_{L}}\leqslant
C\int_{X_{0}}|\bar{\partial}u_{k}|^{2}_{\omega_{\mathcal{C}}}e^{-\varphi_{L}}dV_{\omega_{\mathcal{C}}}.$

Here we are using the fact that local term added to the global metric of $L$
in (2.12.4) is two-sided bounded.

We cover $X$ with a finite subset of coordinate subsets $\Omega$, and we
construct the forms $v_{\Omega,k}$ as above (if the set $\Omega$ do not
intersects the support of the divisor $Y$, then we just use the flat metric).
If all the coefficients $q_{i}$ in (i) are positive it follows that the
holomorphic forms $\displaystyle(u_{k}-v_{\Omega,k})$ are converging to zero
weakly, hence strongly as well. Moreover, the $L^{2}$ norm of each
$v_{\Omega,k}$ tends to zero as $k\to\infty$, so for $k$ large enough the
first equality in (2.12.2) cannot hold. The contradiction we obtained ends the
proof of the case $\varepsilon=0$.

Now assume that we only have one negative coefficient $q_{1}<0$, then we write
$q_{1}=\lfloor q_{1}\rfloor+\\{q_{1}\\}$ and replace this coefficient with its
fractionally part $\\{q_{1}\\}$ in the expression of the metric. Then can
argue as above, by considering $\displaystyle z_{1}^{-\lfloor
q_{1}\rfloor}(u_{k}-v_{\Omega,k})$.

The general case (i.e. for an arbitrary $\varepsilon$) is quite similar,
except that the we cannot rely on the ”correction” forms $v_{\Omega,k}$ for
$\Omega\cap Y\neq\emptyset$. Instead we are using the following two facts.

The first fact is a version of Proposition 2.4: let $U$ be a finite union of
coordinate subsets of $X$ containing $Y$. There exists a positive constant
$C>0$ (independent of $\varepsilon$) such that for any $L$-valued $(p,0)$ form
$u$ with support in $U$ we have the inequality

(2.12.6)
$\frac{1}{C}\int_{U}|u|^{2}_{\omega_{Y,\varepsilon}}e^{-\varphi_{L}}dV_{\omega_{Y,\varepsilon}}\leqslant\int_{U}|\bar{\partial}u|^{2}_{\omega_{Y,\varepsilon}}e^{-\varphi_{L}}dV_{\omega_{Y,\varepsilon}}+\int_{U\setminus\frac{1}{2}U}|u|^{2}_{\omega_{Y,\varepsilon}}e^{-\varphi_{L}}dV_{\omega_{Y,\varepsilon}}.$

We will not prove (2.12.6) here, because it follows directly from the
arguments invoked for Proposition 2.4.

The second fact is that the space of $L^{2}$ holomorphic $p$-forms is
independent of $\varepsilon\geqslant 0$. Let $\Phi$ be a holomorphic $p$–form
on $X_{0}$ with values in $L$. If $\Phi$ is $L^{2}$ with respect to
$(\omega_{Y,\varepsilon},h_{L})$, it is also $L^{2}$ integrable with respect
to $(\omega_{Y},h_{L})$ because of the inequality

(2.12.7)
$\|\Phi\|_{\omega_{Y,\varepsilon}}^{2}\geqslant\|\Phi\|_{\omega_{Y}}^{2}.$

We next assume that $\Phi$ is $L^{2}$ with respect to $(\omega_{Y},h_{L})$. In
local coordinates this writes as

(2.12.8) $\int_{\Omega}\sum|\Phi_{I}(z)|^{2}\prod_{j\not\in
I}\lambda_{j}(z)e^{-\varphi_{L}(z)}d\lambda(z)<\infty$

where $\lambda_{j}$ are the eigenvalues of $\omega_{\mathcal{C}}$ with respect
to the Euclidean metric on $\big{(}\Omega,(z_{1},\dots,z_{n})\big{)}$. We take
the coordinates $z$ so that they are adapted to $(X,Y)$, i.e. $\Omega\cap
Y=z_{1}\dots z_{r}=0$. It follows that the holomorphic functions $\Phi_{I}$
are divisible by

(2.12.9) $\prod_{i\leqslant r,i\in I}z_{i}^{\lfloor
q_{i}\rfloor}\prod_{i\leqslant r,i\not\in I}z_{i}^{\lfloor
1+q_{i}-\frac{1}{m}\rfloor}$

cf. notations at the beginning. Recall that we have chosen $m$ so that

(2.12.10) $\left\lfloor q_{i}-\frac{1}{m}\right\rfloor=\left\lfloor
q_{i}\right\rfloor,$

for each non-integer coefficient $q_{i}$ which appears in the expression of
the metric $h_{L}$.

So summing up we see that by the careful choice of the singularities of the
metric $\omega_{\mathcal{C}}$ cf. $(ii)$, for any holomorphic form $\Phi$
satisfying $\|\Phi\|_{\omega_{\mathcal{C}}}^{2}<\infty$, we have

(2.12.11)
$\|\Phi\|_{\omega_{Y,\varepsilon}}^{2}<\infty\qquad\text{and}\qquad\lim_{\varepsilon\rightarrow
0}\|\Phi\|_{\omega_{Y,\varepsilon}}^{2}=\|\Phi\|_{\omega_{\mathcal{C}}}^{2}.$

The rest of the proof is absolutely similar (i.e. by contradiction) to the
case of Theorem 2.3: we suppose by contradiction that there exists a sequence
$u_{\varepsilon}$ such that $\|u_{\varepsilon}\|_{\omega_{Y,\varepsilon}}=1$,
$u_{\varepsilon}$ is orthogonal to the space of $L^{2}$-holomorphic forms with
respect to $\omega_{Y,\varepsilon}$ and $\lim_{\varepsilon\rightarrow
0}\|\bar{\partial}u_{\varepsilon}\|_{\omega_{Y,\varepsilon}}=0$. By passing to
some subsequence, $u_{\varepsilon}$ converges weakly to some $(p,0)$-form $u$
on $X_{0}$. Then $\|u\|_{\omega_{\mathcal{C}}}\leqslant 1$ and
$\bar{\partial}u=0$. Therefore $u$ is a holomorphic $L^{2}$-form on $X$. Since
$u_{\varepsilon}$ is orthogonal to the holomorphic forms, we have
$\int_{X}\langle u,u_{\varepsilon}\rangle_{\omega_{Y,\varepsilon}}=0$.
Together with (2.12.11), we obtain $u=0$. Then we obtain a contradiction by
the same arguments as in Theorem 2.3 combined with (2.12.6). ∎

We prove next Theorem 2.11.

###### Proof.

Let $\xi$ be any $L^{2}$-form of type $(n,q)$ with values in $(L,h_{L})$ (with
respect to the metric $\omega_{\mathcal{C}}$). Then we also have
$\displaystyle\|\xi\|_{\omega_{Y,\varepsilon}}<\infty$ and thus we write

(2.12.12)
$\xi=\xi_{\varepsilon}+\bar{\partial}(\eta_{\varepsilon})+\bar{\partial}^{\star}(\tau_{\varepsilon}),$

where $\xi_{\varepsilon}$ is harmonic, $\eta_{\varepsilon}$ is orthogonal to
$\mathrm{Ker}(\bar{\partial})$ and $\tau_{\varepsilon}$ orthogonal on
$\mathrm{Ker}(\bar{\partial}^{\star})$. We have

(2.12.13)
$\|\xi_{\varepsilon}\|^{2}_{\omega_{Y,\varepsilon}}\leqslant\|\xi\|^{2}_{\omega_{Y,\varepsilon}}\leqslant\|\xi\|^{2}_{\omega_{Y}}$

and therefore the limit of $(\xi_{\varepsilon})_{\varepsilon>0}$ will be
harmonic with respect to $\omega_{\mathcal{C}},h_{L}$. Moreover, thanks to
Theorem 2.12, we have

(2.12.14) $\|\eta_{\varepsilon}\|^{2}_{\omega_{Y,\varepsilon}}\leqslant
C\|\bar{\partial}\eta_{\varepsilon}\|^{2}_{\omega_{Y,\varepsilon}}\leqslant
C\|\xi\|^{2}_{\omega_{Y}}$

and thus we can extract a limit of $(\eta_{\varepsilon})_{\varepsilon>0}$ on
compacts of $X_{0}$.

In order to deal with the family $(\tau_{\varepsilon})_{\varepsilon>0}$,
thanks to the uniform estimate of Theorem 2.12, the proof of Corollary 2.6
applies. We get thus a constant $C>0$ so that

(2.12.15) $\|\tau_{\varepsilon}\|^{2}_{\omega_{Y,\varepsilon}}\leqslant
C\|\bar{\partial}^{\star}(\tau_{\varepsilon})\|^{2}_{\omega_{Y,\varepsilon}}$

and the RHS of (2.12.15) is bounded uniformly with respect to $\varepsilon$.
We can therefore take the limit $\varepsilon\to 0$ and we get $\xi_{0}$
harmonic, $\eta_{0}\in\mathrm{Dom}(\bar{\partial})$ and
$\tau_{0}\in\mathrm{Dom}(\bar{\partial}^{\star})$ such that

(2.12.16)
$\xi=\xi_{0}+\bar{\partial}\eta_{0}+\bar{\partial}^{\star}\\!(\tau_{0}).$

To prove that $\mathcal{H}_{n,q}(X_{0},L),\mathrm{Im}\,\bar{\partial}$ and
$\mathrm{Im}\,\bar{\partial}^{*}$ are orthogonal with each other, we use the
partition of unity and the local covering. For example, let
$\xi_{0}\in\mathcal{H}_{n,q}(X_{0},L)$ and $\eta_{0}\in L^{2}$. Let
$\theta_{i}$ be a partition of unity. We have

$\int_{X_{0}}\langle\xi_{0},\bar{\partial}\eta_{0}\rangle=\sum_{i}\int_{X_{0}}\langle\xi_{0},\bar{\partial}(\theta_{i}\eta_{0})\rangle.$

Let $\pi_{i}:V_{i}\rightarrow U_{i}$ be a local ramified cover such that
$\pi_{i}^{*}\omega_{\mathcal{C}}$ is smooth and $Supp\theta_{i}\Subset U_{i}$.
Then

$\int_{X_{0}}\langle\xi_{0},\bar{\partial}(\theta_{i}\eta_{0})\rangle=\int_{U_{i}}\langle\xi_{0},\bar{\partial}(\theta_{i}\eta_{0})\rangle=\frac{1}{r}\int_{V_{i}}\langle\pi_{i}^{*}\xi_{0},\bar{\partial}(\pi_{i}^{*}(\theta_{i}\eta_{0}))\rangle_{\pi_{i}^{*}\omega_{\mathcal{C}}}=0,$

where the last equality comes from the fact that
$\pi_{i}^{*}\omega_{\mathcal{C}}$ is smooth on $V_{i}$ and
$Supp\theta_{i}\Subset V_{i}$.

To conclude the proof, it remains to check ((ii)). Let
$\xi\in\mathcal{H}_{n,q}(X_{0},L)$ (with respect to the conic metric). Then
$\xi$ is still $L^{2}$ with respect to $\omega_{Y,\varepsilon}$ for every
$\varepsilon$. We consider the Hodge decomposition of $\xi$ with respect to
$\omega_{Y,\varepsilon}$:

$\xi=\xi_{\varepsilon}+\bar{\partial}\eta_{\varepsilon}+\bar{\partial}^{\star_{\varepsilon}}\tau_{\varepsilon}.$

Then $\xi_{\varepsilon}$ is harmonic with respect to the complete metric
$\omega_{Y,\varepsilon}$. In particular, by using Bochner, we know that

(2.12.17)
$\bar{\partial}\xi_{\varepsilon}=\bar{\partial}^{\star_{\varepsilon}}\xi_{\varepsilon}=(D^{\prime})^{\star_{\varepsilon}}\xi_{\varepsilon}=0.$

By passing to some subsequence, the above argument implies that
$\xi_{\varepsilon},\eta_{\varepsilon}$ and $\tau_{\varepsilon}$ converge to
some $\xi_{0},\eta_{0}$ and $\tau_{0}$ which are $L^{2}$ with respect to
$\omega_{\mathcal{C}}$ and we have

$\xi=\xi_{0}+\bar{\partial}\eta_{0}+\bar{\partial}^{\star}\tau_{0}.$

As we proved that $\mathcal{H}_{n,q}(X_{0},L),\mathrm{Im}\,\bar{\partial}$ and
$\mathrm{Im}\,\bar{\partial}^{*}$ are orthogonal with each other, we obtain
that $\xi=\xi_{0}$. Together with (2.12.17), ((ii)) is proved. ∎

######  Remark 2.13.

The decomposition

$L^{2}_{n,q}=\mathrm{Ker}\Delta^{\prime\prime}\oplus\mathrm{Im}\,\Delta^{\prime\prime}$

with respect to the Poincaré type/conic metrics, still holds, as we will see
next. Given our previous results, all we need to prove is the image of
$\Delta^{\prime\prime}$ is closed.

We prove it for the Poincaré case, and the conic case is similar. If (by
contradiction) this is not true, then there exists a sequence of smooth
$f_{i}$ of compact support such that

(2.13.1)
$\|f_{i}\|^{2}_{L^{2}}=1,\qquad\|\Delta^{\prime\prime}f_{i}\|^{2}_{L^{2}}\rightarrow
0.$

Moreover, each of the forms $f_{i}$ is orthogonal to the space of harmonic
forms, i.e. to $\mathrm{Ker}(\bar{\partial})$ and
$\mathrm{Ker}(\bar{\partial}^{\star})$.

As consequence of (2.13.1) we have

(2.13.2) $\langle f_{i},\Delta^{\prime\prime}f_{i}\rangle\rightarrow 0$

as $i\to\infty$. On the other hand, we have $\langle
f_{i},\Delta^{\prime\prime}f_{i}\rangle=\|\bar{\partial}f_{i}\|^{2}+\|\bar{\partial}^{*}f_{i}\|^{2}$,
and this leads to a contradiction as follows. We can write

(2.13.3) $f_{i}=\bar{\partial}\alpha_{i}+\bar{\partial}^{\star}\beta_{i},$

according to Theorem 2.2. Moreover, it follows from the proof of Theorem 2.2
that

(2.13.4) $\|f_{i}-\bar{\partial}\alpha_{i}\|^{2}\leqslant
C\|\bar{\partial}f_{i}\|^{2},\qquad\|f_{i}-\bar{\partial}^{\star}\beta_{i}\|^{2}\leqslant
C\|\bar{\partial}^{\star}f_{i}\|^{2}$

since $f_{i}-\bar{\partial}\alpha_{i}$ is orthogonal to
$\mathrm{Ker}(\bar{\partial})$ and $f_{i}-\bar{\partial}^{\star}\beta_{i}$ is
orthogonal to $\mathrm{Ker}(\bar{\partial}^{\star})$. Adding the two
inequalities shows that

(2.13.5) $\|f_{i}\|^{2}\leqslant
C(\|\bar{\partial}f_{i}\|^{2}+\|\bar{\partial}^{\star}f_{i}\|^{2}),$

which is impossible as $i\to\infty$.

###  2.3. Hodge decomposition for metrics with conic singularities, II

The setting here will be the same as in the previous subsection 2.2. We will
use systematically the ramified covers corresponding to the orbifold structure

(2.13.6) $\Delta:=\sum_{i\in I}\left(1-\frac{1}{m}\right)Y_{i}$

in order to establish a generalization of Theorem 2.11.

Let $\displaystyle(V_{i},z_{i})_{i\in I}$ be a finite cover of $X$ with
coordinate charts such that

(2.13.7) $z_{i}^{1}\dots z_{i}^{r}=0$

is the local equation of the divisor $Y$ when restricted to the set $V_{i}$.
We then consider the local ramified maps

(2.13.8) $\pi_{i}:U_{i}\to
V_{i},\qquad\pi_{i}(w_{i}^{1},\dots,w_{i}^{n}):=\left((w_{i}^{1})^{m},\dots,(w_{i}^{r})^{m},w_{i}^{r+1},\dots
w_{i}^{n}\right)$

which define the orbifold structure corresponding to $(X,\Delta)$.

We first define the analog of smooth forms in our geometric context.

######  Definition 2.14.

Let $\phi$ be a $\mathcal{C}^{\infty}$ form of $(p,q)$–type with values in $L$
defined on $X_{0}:=X\setminus Y$. We say that $\phi$ is smooth in conic sense
if the quotient of the local inverse images

(2.14.1) $\phi_{i}:=\frac{1}{w_{i}^{qm}}\pi_{i}^{\star}(\phi|_{V_{i}})$

admit a $\mathcal{C}^{\infty}$-extension to $U_{i}$. In order to simplify the
writing, in the equality above we are using the notation
$w_{i}^{qm}:=(w_{i}^{1})^{q_{1}m}\cdots(w_{i}^{r})^{q_{r}m}$.

In the absence of the twisting bundle $L$, this corresponds to the usual
definition of “orbifold differential forms”. Here in some sense the idea is
the same, except that we take into account the singularities of the metric
$h_{L}$ of $L$. In what follows we will establish a few simple properties.

######  Proposition 2.15.

Let $\phi$ be a smooth $(n,q)$-form on $X$ with value in $L$ in the standard
sense, and we suppose that $q_{i}<1$ for every $i$. Then $\phi$ is smooth in
conic sense.

###### Proof.

The matter is immediate, via a local calculation. This is based on the fact
that $\displaystyle\frac{d(z^{m})}{z^{qm}}$ is smooth (recall that $qm$ is an
integer). ∎

######  Remark 2.16.

Let $\phi$ be a smooth $L$-valued $(p,q)$-form on $X$. Even if $q_{i}\in]0,1[$
for every $i$, it is possible that $\phi$ is not smooth in conic sense. But we
will show later that it defines a current in conic sense.

The following statement establishes a correspondence between the intrinsic
differential operators associated to the data $(X,\omega_{\mathcal{C}})$ and
$(L,h_{L})$ and the local ones.

######  Proposition 2.17.

Let $\phi$ be a $L$-valued $(p,q)$-form, smooth in orbifold sense. Then its
”natural” derivatives $D^{\prime}_{h_{L}},D_{h_{L}}^{\star},\cdots$ are also
smooth in the orbifold sense. Moreover, we have:

  1. (1)

$\sup_{X\setminus Y}|\phi|_{h_{L},\omega_{\mathcal{C}}}<\infty$, i.e. forms
which are smooth in conic sense are bounded.

  2. (2)

The following equalities hold true

$\pi_{i}^{\star}(D^{\prime}\phi)=w_{i}^{qm}D^{\prime}\phi_{i};\qquad\pi_{i}^{\star}(\bar{\partial}\phi)=w_{i}^{qm}\bar{\partial}\phi_{i},$

as well as

$\pi_{i}^{\star}({D^{\prime}}^{\star}\phi)=w_{i}^{qm}{D^{\prime}}^{\star}\phi_{i};\qquad\pi_{i}^{\star}(\bar{\partial}^{\star}\phi)=w_{i}^{qm}\bar{\partial}^{\star}\phi_{i}$

where the $D^{\prime}$ on the LHS is $\displaystyle D^{\prime}_{h_{L}}$ and
the rest of the notations are
$D^{\prime}\psi:=\partial\psi-\partial(\varphi_{L,0}\circ\pi_{i})\wedge\psi$
and $\varphi_{L}=\sum q_{i}\log|z_{i}|^{2}+\varphi_{L,0}$.

  3. (3)

Let $\Delta^{\prime\prime}:=[\bar{\partial},\bar{\partial}^{\star}]$ be the
Laplace with respect to $(\omega_{\mathcal{C}},h_{L})$. Then we have

$\pi_{i}^{\star}\Delta^{\prime\prime}\phi=w_{i}^{qm}\cdot\Delta^{\prime\prime}_{sm}\phi_{i},$

where $\Delta^{\prime\prime}_{sm}$ is the Laplace for the local, non-singular
setting $(\pi_{i}^{\star}\omega_{\mathcal{C}},\varphi_{L,0}\circ\pi_{i})$

###### Proof.

The first inequality is clear; for the rest we will only discuss the first
equality of (2), since the verification of all the others is completely
analogue. Also, we will drop the index $i$ in order to simplify the notations.
By definition, locally on $V$ we have

(2.17.1)
$D^{\prime}_{h_{L}}\phi=\partial\phi-\Big{(}\partial\varphi_{L,0}+\sum
q_{\alpha}\frac{dz_{\alpha}}{z_{\alpha}}\Big{)}\wedge\phi$

and its pull-back to $U$ via the map $\pi$ is equal to

(2.17.2)
$\pi^{\star}(D^{\prime}_{h_{L}}\phi)=\partial\pi^{\star}\phi-\Big{(}\partial\varphi_{L,0}\circ\pi+m\sum
q_{\alpha}\frac{dw_{\alpha}}{w_{\alpha}}\Big{)}\wedge\pi^{\star}\phi.$

The equality

(2.17.3)
$\frac{1}{w^{mq}}\pi^{\star}(D^{\prime}_{h_{L}}\phi)=\partial\frac{\pi^{\star}\phi}{w^{mq}}-\partial(\varphi_{L,0}\circ\pi)\wedge\frac{\pi^{\star}\phi}{w^{mq}}$

follows immediately and the RHS equals precisely $\displaystyle
D^{\prime}\Big{(}\frac{1}{w^{qm}}\pi^{\star}\phi\Big{)}$. ∎

######  Example 2.18.

The terminology we are using could be sometimes misleading. Let $L$ be the
trivial bundle over $X=\mathbb{D}$, endowed with the metric
$\varphi_{L}=-\frac{1}{2}\ln|z|$. Then the form $\displaystyle\frac{dz}{z}$
defined over the pointed disk is smooth in the sense of our Definition 2.14.
Moreover, its $\bar{\partial}$ is also smooth in conic sense: the requirement
is that $\displaystyle
w\bar{\partial}\Big{(}\frac{dw}{w}|_{\mathbb{D}^{\star}}\Big{)}$ extends over
the disk and this is obviously the case.

A first consequence of the statement 2.17 is that forms which are smooth in
conic sense behave well by integration by parts.

######  Corollary 2.19.

Let $\alpha$ and $\beta$ be an $L$-valued $(p,q)$-form and an
$L^{\star}$-valued $(n-p-1,n-q)$ form, respectively, both smooth in conic
sense. Then the usual integration by parts formula holds true

(2.19.1)
$\int_{X}D^{\prime}_{h_{L}}\alpha\wedge\beta=(-1)^{p+q+1}\int_{X}\alpha\wedge
D^{\prime}_{h_{L}^{\star}}\beta.$

###### Proof.

Let $(\mu_{\varepsilon})_{\varepsilon>0}$ be the family of truncation
functions corresponding to $Y$. We obviously have

(2.19.2)
$\int_{X}D^{\prime}_{h_{L}}(\mu_{\varepsilon}\alpha)\wedge\beta=(-1)^{p+q+1}\int_{X}\mu_{\varepsilon}\alpha\wedge
D^{\prime}_{h_{L}^{\star}}\beta$

for each positive $\varepsilon$. The integrals
$\displaystyle\int_{X}\mu_{\varepsilon}\alpha\wedge
D^{\prime}_{h_{L}^{\star}}\beta$ and
$\displaystyle\int_{X}\mu_{\varepsilon}D^{\prime}_{h_{L}}\alpha\wedge\beta$
are converging to the expected value by Lebesgue theorem, since

(2.19.3) $\sup_{X\setminus
Y}(|D^{\prime}_{h_{L}}\alpha|+|\alpha|+|D^{\prime}_{h_{L}^{\star}}\beta|+|\beta|)<\infty$

(cf. Proposition 2.17 above). The remaining term is bounded by

(2.19.4) $\int_{X}|\partial\mu_{\varepsilon}|dV_{\mathcal{C}}$

up to a uniform constant –because of (2.19.3)– and this converges to zero as
$\varepsilon\to 0$. ∎

The following corollary is very useful.

######  Corollary 2.20.

Let $\phi$ be a $L$-valued $L^{2}$-form on $X$ such that
$\Delta^{\prime\prime}(\phi)=\psi$ holds in the sense of currents on $X$ for
some form $\psi$ smooth in conic sense. Then $\phi$ is smooth in conic sense.

###### Proof.

Let $V\subset X$ be a coordinate open subset, such that we have a local
ramified cover $\pi:U\rightarrow V$ given by the orbifold structure. By
hypothesis we have

(2.20.1)
$\int_{V}\phi\wedge\overline{\Delta^{\prime\prime}\tau}e^{-\varphi_{L}}=\int_{V}\psi\wedge\overline{\tau}e^{-\varphi_{L}}$

for any $L$-valued form $\tau$ with compact support in $V$, smooth in orbifold
sense. The Laplace operator in (2.20.1) is induced by $\omega_{\mathcal{C}}$
and $h_{L}$.

We first prove that

(2.20.2)
$\int_{U}\frac{\pi^{\star}(\phi)}{w^{mq}}\wedge\overline{\Delta^{\prime\prime}_{sm}(\rho)}=\int_{U}\frac{\pi^{\star}(\psi)}{w^{mq}}\wedge\overline{\rho}$

for any smooth form $\rho$ of compact support in $U$.

In fact, the $m$-ramified cover $\pi:U\rightarrow V$ induces the Galois group
$\gamma_{1},\cdots,\gamma_{m}$ which are the automorphisms
$\gamma_{i}:U\rightarrow U$ invariant over $V$. Since $\pi^{*}\psi$ is
$\gamma_{i}$-invariant, we have

$\int_{U}\frac{\pi^{\star}(\psi)}{w^{mq}}\wedge\overline{\rho}=\frac{1}{m}\sum_{i}\int_{U}\pi^{\star}(\psi)\wedge\gamma_{i}^{*}\frac{\overline{\rho}}{w^{mq}}=\frac{1}{m}\sum_{i}\int_{U}\pi^{\star}(\psi)\wedge\gamma_{i}^{*}(\overline{w^{mq}\cdot\rho})\cdot
e^{-\pi^{*}\varphi_{L}}$
$=\frac{1}{m}\int_{U}\pi^{\star}(\psi)\wedge\overline{G}\cdot
e^{-\pi^{*}\varphi_{L}},$

where $G:=\sum_{i}\gamma_{i}^{*}(w^{mq}\rho)$ is $\pi$-invariant and the
quotient $\frac{G}{w^{mq}}$ is smooth by construction. By definition, $G$ is
the pull back of some form $\tau$ on $V$, smooth in conic sense. Therefore we
have

(2.20.3)
$\int_{U}\frac{\pi^{\star}(\psi)}{w^{mq}}\wedge\overline{\rho}=\int_{V}\psi\wedge\overline{\tau}\cdot
e^{-\varphi_{L}}.$

For the term
$\int_{U}\frac{\pi^{\star}(\phi)}{w^{mq}}\wedge\overline{\Delta^{\prime\prime}_{sm}(\rho)}$,
we do the same argument:

$\int_{U}\frac{\pi^{\star}(\phi)}{w^{mq}}\wedge\overline{\Delta^{\prime\prime}_{sm}(\rho)}=\frac{1}{m}\sum_{i}\int_{U}\pi^{\star}(\phi)\wedge\gamma_{i}^{*}\frac{\overline{\Delta^{\prime\prime}_{sm}(\rho)}}{w^{mq}}=\frac{1}{m}\sum_{i}\int_{U}\pi^{\star}(\phi)\wedge\gamma_{i}^{*}(\overline{w^{mq}\cdot\Delta^{\prime\prime}_{sm}(\rho)})\cdot
e^{-\pi^{*}\varphi_{L}}$
$=\frac{1}{m}\int_{U}\pi^{\star}(\phi)\wedge\overline{H}\cdot
e^{-\pi^{*}\varphi_{L}},$

where $H:=\sum_{i}\gamma_{i}^{*}(w^{mq}\Delta^{\prime\prime}_{sm}(\rho))$ is
$\pi$-invariant and the quotient $\frac{H}{w^{mq}}$ is smooth. Then there
exists a form $\theta$ on $V$ smooth in conic sense such that
$H=\pi^{*}\theta$. Then

(2.20.4)
$\int_{U}\frac{\pi^{\star}(\phi)}{w^{mq}}\wedge\overline{\Delta^{\prime\prime}_{sm}(\rho)}=\int_{V}\phi\wedge\overline{\theta}\cdot
e^{-\varphi_{L}}.$

Since $\Delta^{{}^{\prime\prime}}_{sm}$ commute with $\gamma_{i}$, we obtain
$H=w^{qm}\Delta^{{}^{\prime\prime}}_{sm}(\frac{G}{w^{qm}})$. Together with
Proposition 2.17, we know that $\theta=\Delta^{{}^{\prime\prime}}\tau$ on $V$.
Combining this with (2.20.3), (2.20.4) and (2.20.1), we obtain finally

$\int_{U}\frac{\pi^{\star}(\phi)}{w^{mq}}\wedge\overline{\Delta^{\prime\prime}_{sm}(\rho)}=\int_{U}\frac{\pi^{\star}(\psi)}{w^{mq}}\wedge\overline{\rho}.$

Therefore, the equality

(2.20.5)
$\Delta^{\prime\prime}_{sm}(\widehat{\phi})=\frac{\pi^{\star}(\psi)}{w^{m}q}$

holds in the sense of distributions on $U$, where
$\displaystyle\widehat{\phi}:=\frac{\pi^{\star}(\phi)}{w^{mq}}$ is moreover
$L^{2}$. The corollary follows. ∎

Another consequence is the following version of Bochner formula.

######  Proposition 2.21.

Let $\phi$ be a $L$-valued $(p,q)$-form, smooth in orbifold sense. Then the
equality

(2.21.1)
$\Delta^{\prime\prime}\phi=\Delta^{\prime}\phi+[\theta_{L},\omega_{\mathcal{C}}]\phi$

holds pointwise on $X\setminus Y$. Moreover the following Bochner formula
holds.

$\displaystyle\int_{X}|\bar{\partial}\phi|^{2}e^{-\varphi_{L}}dV_{\mathcal{C}}+\int_{X}|\bar{\partial}^{\star}\phi|^{2}e^{-\varphi_{L}}dV_{\mathcal{C}}=$
$\displaystyle\int_{X}|D^{\prime}\phi|^{2}e^{-\varphi_{L}}dV_{\mathcal{C}}+\int_{X}|D^{\prime\star}\phi|^{2}e^{-\varphi_{L}}dV_{\mathcal{C}}$
(2.21.2) $\displaystyle+$
$\displaystyle\int_{X}\langle[\theta_{L},\omega_{\mathcal{C}}]\phi,\phi\rangle
e^{-\varphi_{L}}dV_{\mathcal{C}}$

###### Proof.

On every local ramified cover $\pi_{i}:U_{i}\rightarrow V_{i}$, thanks to
Proposition 2.17, we have

$\pi_{i}^{\star}(\Delta^{\prime\prime}\phi)=w_{i}^{qm}\cdot\Delta^{\prime\prime}_{sm}\phi_{i}.$

Using the same type of arguments we infer that we have

$\pi_{i}^{\star}(\Delta^{\prime}\phi)=w_{i}^{qm}\cdot\Delta^{\prime}_{sm}\phi_{i}.$

Together with the usual Bochner equality
$\displaystyle\Delta^{\prime\prime}_{sm}=\Delta^{\prime}_{sm}+[i\Theta_{\pi^{\star}\varphi_{0}},\pi_{i}^{\star}\omega_{\mathcal{C}}]$,
we obtain (2.21.1).

In order to obtain the integral identity, we first see that the formula holds
if we replace $\phi$ with $\mu_{\varepsilon}\phi$, where the cut-off function
$\mu_{\varepsilon}$ is defined in Lemma 2.1. Since the error terms e.g. of
type

(2.21.4)
$\int_{X}\langle\bar{\partial}\mu_{\varepsilon}\wedge\phi,\bar{\partial}\phi\rangle
e^{-\varphi_{L}}dV_{\mathcal{C}}$

tend to zero, we are done. Note that here we are using the fact that the
function $|\phi||\bar{\partial}\phi|e^{-\varphi_{L}}$ is bounded (and not only
$L^{1}$). ∎

Yet another corollary states as follows.

######  Corollary 2.22.

Let $\tau$ be a smooth $(n,q)$-form with values in $(L,h_{L})$ and support in
$X\setminus Y$. We consider the (usual) orthogonal decomposition

(2.22.1) $\tau=\tau_{1}+\tau_{2}$

where $\tau_{1}\in\mathrm{Ker}\bar{\partial}$ and
$\tau_{2}\in(\mathrm{Ker}\bar{\partial})^{\perp}$.

  * •

We have

$\bar{\partial}\tau_{1}=0,\qquad\bar{\partial}^{\star}\tau_{1}=0$

when restricted to $U$, the complement of the support of $\tau$. In other
words, $\tau_{1}|_{U}$ is harmonic, hence smooth in conic sense.

  * •

As a consequence of the first bullet, Bochner formula holds true for the
projection $\tau_{1}$ of $\tau$ on the kernel of $\bar{\partial}$. The is true
more generally for forms $\rho$ such that $\displaystyle\sup_{X\setminus
Y}|\rho|_{h_{L},\omega_{\mathcal{C}}}<\infty$.

######  Corollary 2.23.

If $i\Theta_{h_{L}}\geqslant 0$ (resp. $i\Theta_{h_{L}}\leqslant 0$), and
$\phi$ is a $L$-valued $\Delta^{\prime\prime}$-harmonic $(n,q)$-form (resp.
$(0,q)$-form) smooth in conic sense. Then
$D_{h_{L}}\phi=D_{h_{L}}^{\star}\phi=0$ and $\theta_{L}\wedge\phi=0$.

###### Proof.

All we have to do is to combine the previous results: harmonic forms are
smooth in conic sense, hence Bochner formula applies and the corollary
follows. ∎

Like in the smooth case, we can define the Hodge operators $\star$ and
$\sharp=\overline{\star}$ in our setting, namely give a $L$-value $(p,q)$-form
$s$, there exists a unique $L^{*}$-value $(n-p,n-q)$-form, denoted by $\sharp
t$, such that for every $L$-valued $(p,q)$-form $s$, we have

$\langle s,t\rangle_{h_{L}}dV_{\omega_{\mathcal{C}}}=s\wedge\sharp t,$

where $s\wedge\sharp t$ is calculated by using the natural pairing $L\otimes
L^{\star}\rightarrow\mathbb{C}$. We have the following statements, whose
proofs are left to the interested readers.

######  Proposition 2.24.

Let $t$ be a $L$-valued $(p,q)$-form, smooth in the conic sense. Then $\sharp
t$ is a $L^{\star}$-valued $(n-p,n-q)$-form, smooth in the conic sense (with
respect to $(\omega_{\mathcal{C}},h_{L}^{\star})$).

######  Proposition 2.25.

Let $t$ be a $\Delta^{\prime\prime}$-harmonic form with value in $L$. Then
$\sharp t$ is $\Delta^{\prime\prime}$-harmonic form with value in $L^{*}$

The usual Gårding and Sobolev inequalities together with Rellich embedding
theorem still hold in conic setting. They are established by their usual
versions via the local uniformisations of $(X,\Delta)$. We will state and
comment Gårding inequality, but prior to doing that, we introduce the analogue
of Sobolev spaces in our context.

######  Definition 2.26.

We denote by $W^{k}_{p,q}(X,L)$ the space of $L^{2}$ forms $u$ of type $(p,q)$
with values in $L$, such that its pull back
$\displaystyle\frac{\pi_{i}^{\star}u}{w_{i}^{qm}}$ via the ramified coverings
belongs to the usual Sobolev space $W^{k}(U_{i})$. The Sobolev norm of
$W^{k}_{p,q}(X,L)$ is obtained by a partition of unit and the usual Sobolev
norm of the local forms $\displaystyle\frac{\pi_{i}^{\star}u}{w_{i}^{qm}}$.

Given a form $u\in W^{k}_{p,q}(X,L)$, we can approximated it by smooth forms
$u_{\varepsilon}$ in the sense of conic. In fact, by using the partition of
unity, it is sufficient to consider the case when $u$ is supported in some
open set $V_{i}$. Let $z$ be the coordinate on $U_{i}$. Let
$\rho_{\varepsilon}(\|w\|^{2})$ be the standard convolution function. Let $G$
be the Galois group of the covering $\pi:U_{i}\rightarrow V_{i}$ and let
$\gamma\in G$. Then $\rho_{\varepsilon}(|w|^{2})$ is $G$-invariant. Assume for
simplicity that $p=q=0$; we then consider the function

$v_{\varepsilon}(w):=\frac{(w^{-qm}\cdot\pi^{\star}u)\star\rho_{\varepsilon}}{w^{-qm}}.$

Note that $\gamma(w)^{-qm}=c_{\gamma}\cdot w^{-qm}$ for some constant
$c_{\gamma}$ independent of $w$. Then

$v_{\varepsilon}(\gamma(w))=\frac{\int_{U_{i}}(\gamma(w)-\tau)^{-qm}\cdot\pi^{\star}u(\gamma(w)-\tau)\cdot\rho_{\varepsilon}(\|\tau\|^{2})dV_{\tau}}{\gamma(w)^{-qm}}$
$=\frac{c_{\gamma}\int_{U_{i}}(w-\gamma^{-1}(\tau))^{-qm}\cdot\pi^{\star}u(w-\gamma^{-1}(\tau))\cdot\rho_{\varepsilon}(\|\tau\|^{2})dV_{\tau}}{c_{\gamma}\cdot
w^{-qm}}$
$=\frac{c_{\gamma}\int_{U_{i}}(w-\tau)^{-qm}\cdot\pi^{\star}u(w-\tau)\cdot\rho_{\varepsilon}(\|\gamma(\tau)\|^{2})dV_{\gamma(\tau)}}{c_{\gamma}\cdot
w^{-qm}}=v_{\varepsilon}(w),$

where we use the fact that $\pi^{*}u$, $\rho_{\varepsilon}(\|w\|^{2})$ and
$dV_{w}$ are $G$-invariant. Therefore $v_{\varepsilon}$ is $G$-invariant and
we can find some form $u_{\varepsilon}$ on $U_{i}$ such that
$v_{\varepsilon}=\pi^{*}u_{\varepsilon}$. Then $u_{\varepsilon}$ is smooth in
conic sense and $\|u_{\varepsilon}-u\|_{W^{k}_{p,q}}\rightarrow 0$.

The general case of an arbitrary $(p,q)$–form is similar, obtained by
regularising locally its coefficients.

We have the following version of Gårding theorem.

######  Theorem 2.27.

Let $\Delta^{\prime\prime}:=[\bar{\partial}^{\star},\bar{\partial}]$ be the
Laplace operator induced by $(h_{L},\omega_{\mathcal{C}})$. Then for any $u\in
L^{2}_{p,q}(X,L)$, if $\Delta^{\prime\prime}u\in W^{k}_{p,q}(X,L)$, then $u\in
W^{k+2}_{p,q}(X,L)$ and

$\|u\|_{k+2}\leqslant C_{k}(\|\Delta^{\prime\prime}u\|_{k}+\|u\|_{0}).$

###### Proof.

We first observe that this is a direct consequence of the usual Gårding
inequality if the form $u$ has compact support contained in one of the sets
$V_{i}$. Moreover, the “dictionary” established in Proposition 2.17, shows
that one could define the space $\big{(}W^{k}_{p,q}(X,L),\|\cdot\|_{k}\big{)}$
directly on $X$, by using the covariant derivative on $L$-valued forms induced
by $h_{L}$ and $\omega_{\mathcal{C}}$. The general case is obtained by a
partition of unit argument. ∎

As a consequence, the Hodge decomposition still holds in the conic setting, as
we see e.g. from the proof presented in [6], Chapter 6.

######  Theorem 2.28.

Let $(L,h_{L})\to X$ be a line bundle endowed with a metric $h_{L}$ such that
the requirements (i)-(ii) at the beginning of this section are satisfied. Let
$\omega_{\mathcal{C}}$ be a Kähler metric with conic singularities as in
(2.10.8). Then we have the following Hodge decomposition

(2.28.1)
$L^{2}_{p,q}(X,L)=\mathrm{Ker}\Delta^{\prime\prime}_{h_{L}}\oplus\mathrm{Im}\,\bar{\partial}\oplus\mathrm{Im}\,\bar{\partial}^{\star}$

and

(2.28.2)
$\mathcal{C}^{\infty}_{p,q}\big{(}X,(L,h_{L})\big{)}=\mathrm{Ker}\Delta^{\prime\prime}_{h_{L}}\oplus\mathrm{Im}\,\Delta^{\prime\prime}_{h_{L}},$

where $\displaystyle\mathcal{C}^{\infty}_{p,q}\big{(}X,(L,h_{L})\big{)}$ and
$L^{2}_{p,q}(X,L)$ are the spaces of $L$-valued $(p,q)$-forms smooth in conic
sense and $L^{2}$, respectively.

We will not give a formal argument for this statement, because it follows as
in the non-singular case (by using the ellipticity of the Laplace operator
combined with the three big theorems mentioned above). Similarly, existence of
a positive constant $C>0$ such that the familiar elliptic estimate

(2.28.3) $\int_{X}|u|^{2}e^{-\varphi_{L}}dV_{\mathcal{C}}\leqslant
C\int_{X}|\Delta^{\prime\prime}_{h_{L}}u|^{2}e^{-\varphi_{L}}dV_{\mathcal{C}}$

holds true for all $L^{2}$-forms orthogonal to
$\mathrm{Ker}\Delta^{\prime\prime}_{h_{L}}$ follows as consequence of the
corresponding statement in non-singular setting.

To end this section, we collect a few results concerning the $L^{2}$ theory
for the data $(L,h_{L})$ and $(X,\omega_{\mathcal{C}})$. The first and
foremost of them is the following statement, due to J.-P. Demailly.

######  Theorem 2.29.

[6] Let $v\in L^{2}_{n,q}(X,L)$ be an $L$-valued $(n,q)$-form such that
$\bar{\partial}v=0$. We assume that $i\Theta(L,h_{L})\geqslant 0$ and that the
coefficients $q_{i}\in]0,1[$. Moreover, suppose that the integrability
condition

$\int_{X}\langle[\theta_{L},\Lambda_{\omega_{\mathcal{C}}}]^{-1}v,v\rangle
e^{-\varphi_{L}}dV_{\mathcal{C}}<\infty$

holds true. Then there exists $u\in L^{2}_{n,q-1}(X,L)$ such that

(2.29.1)
$\bar{\partial}u=v,\qquad\int_{X}|u|^{2}e^{-\varphi_{L}}dV_{\mathcal{C}}\leqslant\int_{X}\langle[\theta_{L},\Lambda_{\omega_{\mathcal{C}}}]^{-1}v,v\rangle
e^{-\varphi_{L}}dV_{\mathcal{C}}.$

This is done by an approximation process (by reducing to the complete case)
and it holds in much more general setting, see [7].

In our specific situation, an important observation is that is would be
sufficient to solve the equation

(2.29.2) $\bar{\partial}u=v$

in the sense of distributions on $X\setminus Y$, provided that $u$ is in
$L^{2}$. Again, this can be seen as in the usual, non-singular setting.
Indeed, let $\phi$ be any smooth $(0,n-q)$ form with values in the dual bundle
$L^{\star}$ of the bundle $L$. We have

(2.29.3)
$\int_{X}\bar{\partial}u\wedge(\mu_{\varepsilon}\phi)=\int_{X}v\wedge(\mu_{\varepsilon}\phi)$

so in order to show that (2.29.2) holds globally on $X$ it would be enough to
prove that we have

(2.29.4)
$\lim_{\varepsilon}\int_{X}u\wedge\phi\wedge\bar{\partial}\mu_{\varepsilon}=0.$

This is however clear, by Cauchy-Schwarz: we have
$\displaystyle\sup_{X\setminus
Y}|\phi|^{2}_{\omega_{\mathcal{C}}}e^{\varphi_{L}}<\infty$ since the
coefficients $q_{i}$ corresponding to the singularities of $h_{L}$ are
positive and $\phi$ is smooth on $X$. Thus (2.29.4) boils down to showing that
the limit of
$\displaystyle\int_{X}|\bar{\partial}\mu_{\varepsilon}|^{2}_{\omega_{\mathcal{C}}}dV_{\mathcal{C}}$
is equal to zero. But this is clear by a quick computation that we skip.

We introduce the following notion, which -so to speak- represents the
formalisation of the examples that we discuss afterwards.

######  Definition 2.30.

A $(p,q)$-current $T$ of with values in $(L,h_{L})$ is simply a $L$-valued
current such that there exist a constant $C>0$ and a positive integer $d$ such
that the inequality

(2.30.1)
$\left|\int_{X}T\wedge\overline{\star\phi}e^{-\varphi_{L}}\right|^{2}\leqslant
C\sum_{s=0}^{d}\sup_{X\setminus
Y}|\nabla^{s}\phi|^{2}_{h_{L},\omega_{\mathcal{C}}}$

holds for all $(p,q)$-forms $\phi$, with values in $L$ which moreover are
smooth in conic sense. Here $h_{0}$ is a non-singular metric on $L$.

######  Remark 2.31.

Instead of the expression (2.30.1) we could have as well said that $T$ acts on
$L^{\star}$-valued $(n-p,n-q)$ forms smooth in conic sense.

######  Remark 2.32.

A current $T$ as in the Definition 2.30 is “more singular” than a $L$-valued
current, since it only acts on forms which are smooth in conic sense.

The condition (2.30.1) is equivalent to the following.

######  Proposition 2.33.

A $(p,q)$-current $T$ with values in $(L,h_{L})$ is given by a collection of
invariant currents $T_{i}$ on $U_{i}$ (where $i\in I$) such that for each
compact $K\subset U_{i}$ there exists a constant $C_{K}>0$ and a number
$d\in\mathbb{N}$ such that following holds

(2.33.1)
$\left|\int_{U_{i}}\frac{1}{w_{i}^{qm}}T_{i}\wedge\overline{\rho}_{i}\right|^{2}\leqslant
C_{K}\sum_{s=0}^{d}\sup_{V_{i}\setminus Y}|\nabla^{s}\rho_{i}|^{2}$

for all $(n-q,n-p)$-forms $\rho$ which are $L$-valued, with compact support in
$V_{i}$ and smooth in conic sense, where
$\displaystyle\rho_{i}:=\frac{1}{w_{i}^{mq}}\pi_{i}^{\star}(\rho)$.

###### Proof.

This is due to the fact that we have

(2.33.2) $\sum_{s=0}^{d}\sup_{V_{i}\setminus
Y}|\nabla^{s}_{h_{L}}\rho|^{2}\simeq\sum_{s=0}^{d}\sup_{U_{i}}|\nabla^{s}\rho_{i}|^{2},$

and one can see by the same arguments as in Proposition 2.17. ∎

######  Remark 2.34.

We have the following alternative description of $T$ in terms of the local
currents $(T_{i})$. Let $\phi$ be an $L$-valued form, smooth in conic sense
and let $T$ be a current as in Definition 2.30. Then we have

(2.34.1) $\int_{X}T\wedge\overline{\phi}e^{-\varphi_{L}}=\sum_{i\in
I}\frac{1}{\delta_{i}}\int_{U_{i}}\frac{\theta_{i}\cdot(T_{i}\wedge\overline{\phi}_{i})}{w_{i}^{qm}}$

where we recall that
$\displaystyle\phi_{i}=\frac{1}{w_{i}^{mq}}\pi_{i}^{\star}(\phi)$ and
$(\theta_{i})$ is a partition of unit corresponding to $(V_{i})_{i\in I}$. The
integer $\delta_{i}$ is the degree of the map $\pi_{i}$. Note that
$\displaystyle\frac{T_{i}\wedge\overline{\phi}_{i}}{w_{i}^{qm}}$ is considered
as a $(n,n)$-current on $U_{i}$ by using the smooth metric $h_{0}$ on
$L_{0}:=L-\sum q_{i}Y_{i}$, where $h_{0}:=h_{L}e^{\sum
q_{i}\log|s_{Y_{i}}|^{2}}$.

####  2.3.1. Examples

We are discussing next a few examples.

$\bullet$ Let $\Theta$ be a smooth $L$-valued $(p,q)$-form on $X$. In general,
$\Theta$ is not smooth in the conic sense –the case $p=n$ and $q_{i}<1$ is
exceptional. However, $\Theta$ induces a current in conic sense (via its
principal value) as follows:

(2.34.2) $\int_{X}\Theta\wedge\overline{\rho}:=\lim_{\varepsilon\rightarrow
0}\sum_{i}\frac{1}{\delta_{i}}\int_{U_{i}}\mu_{\varepsilon}\frac{\theta_{i}}{w_{i}^{qm}}\pi_{i}^{\star}\Theta\wedge\overline{\rho}_{i},$

where $\theta_{i}$ is a partition of unity.

$\bullet$ More generally, let $E=\sum E_{i}$ be a divisor on $X$ such that
$E+Y$ is snc (so in particular $E$ and $Y$ do not have any common components).
Let $\Theta$ be an $L$-valued $(p,q)$-form on $X$ with log-poles along $E+Y$
and smooth on $X\setminus(E+Y)$. $\Theta$ induces naturally an
$(L,h_{L})$-valued current as follows:

(2.34.3) $\int_{X}\Theta\wedge\overline{\rho}:=\lim_{\varepsilon\rightarrow
0}\sum_{i}\frac{1}{\delta_{i}}\int_{U_{i}}\mu_{\varepsilon}\frac{\theta_{i}}{w_{i}^{qm}}\pi_{i}^{\star}\Theta\wedge\overline{\rho}_{i}.$

We do not detail here the verification of the fact that the equalities
(2.34.2) and (2.34.2) define currents in conic setting but rather refer to the
Appendix where this is extensively discussed.

$\bullet$ Let $E=\sum E_{i}$ and $W$ be a divisor on $X$ and an additional
hypersurface, respectively such that $E+W$ is snc. If $\alpha$ is an
$L$-valued $(n-1,0)$-form with log-poles along $E|_{W}$ on $W$, then the
corresponding current is defined by the formula

(2.34.4)
$\int_{X}T\wedge\overline{\phi}:=\sum_{i}\frac{1}{\delta_{i}}\int_{W_{i}}\theta_{i}\frac{1}{w_{i}^{qm}}\pi_{i}^{\star}\alpha\wedge\overline{\phi}_{i}$

where $W_{i}:=\pi_{i}^{-1}(W\cap V_{i})$. Because of the type of $\alpha$ it
is immediate to verify that (2.30.1) holds true (with $d=0$).

$\bullet$ Let $\lambda$ be a smooth, $(n,1)$ form with values in the bundle
$L+kE$. A more general version of the current introduced at first bullet above
is given as follows

(2.34.5)
$\int_{U_{i}}\frac{1}{w_{i}^{qm}}T_{i}\wedge\overline{\phi}:=\lim_{\varepsilon\to
0}\frac{1}{d_{i}}\int_{U_{i}}\frac{1}{w_{i}^{qm}}\pi_{i}^{\star}\Big{(}\theta_{\varepsilon}\frac{\lambda}{s_{E}^{k}}\Big{|}_{V_{i}}\Big{)}\wedge\overline{\phi}$

where $\theta_{i}$ is a sequence of cutoff functions with compact support in
$X\setminus E$ and converging to the identity function of this open subset of
$X$. By using the argument involving the Taylor expansion of
$\theta_{\varepsilon}$ we infer that the global object $T$ defined by (2.34.5)
has the following property

(2.34.6)
$\left|\int_{X}T\wedge\overline{\phi}e^{-\varphi_{L}}\right|^{2}\leqslant
C\sum_{i\in I}\sum_{s=0}^{k}\sup_{U_{i}}|\nabla^{s}\phi_{i}|^{2}.$

where $\phi$ is an $L$-valued form of type $(n-1,0)$, smooth in conic sense
and $\phi_{i}$ are defined in $\eqref{deck4}$. Again, the verification of
(2.34.6) is detailed at the end of the Appendix.

The next statement can be seen as ”mise en bouche” for the type of arguments
in the Appendix.

######  Lemma 2.35.

Let $\phi$ be a negative, $\mathcal{C}^{\infty}$ real function defined on the
unit disk in $\mathbb{C}$. We define

${\mathcal{D}}_{\varepsilon}:=\\{z\in\mathbb{D}:\varepsilon<|z|e^{-\phi(z)}<2\varepsilon\\},\qquad{\mathcal{D}}_{\varepsilon,r}:=\\{z\in\mathbb{D}:\varepsilon<|z|e^{-\phi(z)}<r\\}.$

The following hold true.

  1. (a)

For any smooth function $f$ and any $\delta<1$ we have
$\displaystyle\lim_{\varepsilon\to
0}\int_{{\mathcal{D}}_{\varepsilon}}f(z)\frac{d\lambda}{z|z|^{2\delta}}=0$.

  2. (b)

For any smooth function $f$, any $\delta<1$ and any fixed $r>0$ the set of
complex numbers
$\displaystyle\left(\int_{{\mathcal{D}}_{\varepsilon,r}}f(z)\frac{d\lambda}{z|z|^{2\delta}}\right)_{\varepsilon>0}$
is bounded.

###### Proof.

We discuss the point (a). A first remark is that it is enough to show that the
limit is question is zero for constant functions $f$ (as we see by writing
$f(z)=f(0)+zf_{1}(z)+\overline{z}f_{\overline{1}}(z)$ and observing that for
the second and third term things are immediate).

We take then $f=1$; since $\varphi$ is negative, we can write

(2.35.1)
$\int_{{\mathcal{D}}_{\varepsilon}}\frac{d\lambda}{z|z|^{2\delta}}=\int_{\varepsilon<|z|<2\varepsilon}\frac{d\lambda}{z|z|^{2\delta}}-\int_{2\varepsilon
e^{\phi(z)}<|z|<2\varepsilon}\frac{d\lambda}{z|z|^{2\delta}}+\int_{\varepsilon
e^{\phi(z)}<|z|<\varepsilon}\frac{d\lambda}{z|z|^{2\delta}}$

and the first term of the RHS is equal to zero. We can assume that $1\leqslant
2\delta<2$ and it is enough to show that we have

(2.35.2) $\lim_{\varepsilon\to
0}\int_{\varepsilon(1-C\varepsilon)<|z|<\varepsilon}\frac{d\lambda}{|z|^{1+2\delta}}=0,$

where $C$ is a positive constant depending on the function $\varphi$. This in
turn is obvious.

We leave the rest of the proof of the lemma to the interested readers.

∎

As application, we obtain the following result. Let $\beta$ be an $L$-valued
form of type $(n-1,1)$ and logarithmic poles along $E+Y$. Then we have

(2.35.3) $\lim_{\varepsilon\to
0}\int_{X}\theta_{\varepsilon}D^{\prime}\beta\wedge\overline{\gamma}e^{-\varphi_{L}}=0$

for any holomorphic $L^{2}$ form $\gamma$, where $\theta_{\varepsilon}$ is the
standard cut-off functions.

##  3\. De Rham-Kodaira decompostion for conic currents

In this subsection, we would like to show that the results of de Rham-Kodaira
in [9] concerning Hodge decomposition for currents on compact manifolds have a
complete analogue in our setting, i.e. for $(X,\omega_{\mathcal{C}})$ and
$(L,h_{L})$ in the setting of Subsection 2.2.

By Theorem 2.28, for any $L$-valued $(p,q)$-form $\phi$, smooth in conic
sense, we have

(3.0.1) $\phi=\mathcal{H}(\phi)+\Delta^{\prime\prime}\psi$

where $\mathcal{H}(\phi)$ is the harmonic part of $\phi$ and $\psi$ is an
$L$-valued $(p,q)$-form. If we add to $\psi$ a harmonic form, the equality
(3.0.1) still holds. Thus, as in the standard case we introduce the Green
operator.

######  Definition 3.1.

We denote by $\mathcal{G}(\phi)$ the unique $L$-valued $(p,q)$-form which is
orthogonal to the harmonic space and such that

(3.1.1)
$\phi=\mathcal{H}(\phi)+\Delta^{\prime\prime}\big{(}\mathcal{G}(\phi)\big{)}$

holds true.

######  Remark 3.2.

Since harmonic forms are in particular smooth in conic sense, we infer that
$\mathcal{G}(\varphi)$ has this property as well.

It is easy to see that $\mathcal{G}$ commutes with $\bar{\partial}$. As a
consequence, if $\bar{\partial}\phi=0$, then
$\bar{\partial}\mathcal{G}(\phi)=0$ and so the equality (3.0.1) becomes

(3.2.1) $\phi=\mathcal{H}(\phi)+\bar{\partial}S$

where $S:=\bar{\partial}^{\star}\mathcal{G}(\phi)$. The same is true for forms
with value in the dual $L^{\star}$ of $L$.

Let $T$ be an $(L,h_{L})$-valued $(p,q)$-current. Precisely as in the
untwisted case [9] (i.e. without any ”$L$”), we can define the _harmonic part_
of $T$ as

(3.2.2) $\mathcal{H}(T):=\sum_{i}\langle T,\xi_{i}\rangle\xi_{i}$

where $(\xi_{i})_{i}$ is a basis of $L^{2}$ harmonic $L$-valued forms of type
$(p,q)$ and where the notation in (3.2.2) is

(3.2.3) $\langle T,\xi\rangle:=\int_{X}T\wedge\overline{\gamma_{\xi}},$

and $\gamma_{\xi}:=\star\xi$ and so $\overline{\gamma_{\xi}}=\sharp\xi$
becomes an $(n-p,n-q)$ form with values in $L^{\star}$.

Let $T$ be a $(p,q)$-current with values in $(L,h_{L})$, cf. Definition 2.30.
We define next by duality the Green operator on $T$ by the formula

(3.2.4)
$\int_{X}\mathcal{G}(T)\wedge\overline{\star\phi}:=\int_{X}T\wedge\overline{\star\mathcal{G}(\phi)},$

for any $L$-valued $(p,q)$-form $\phi$, smooth in conic sense. This induces a
decomposition of the current $T$ as in the usual case

(3.2.5) $T=\mathcal{H}(T)+\Delta^{\prime\prime}\big{(}\mathcal{G}(T)\big{)}.$

######  Definition 3.3.

The equality (3.2.5) is called de Rham-Kodaira decomposition of the current
$T$.

######  Remark 3.4.

If moreover $T$ is closed, then we can write

(3.4.1) $T=\mathcal{H}(T)+\bar{\partial}T_{1}$

where $\displaystyle
T_{1}=\bar{\partial}^{\star}\big{(}\mathcal{G}(T)\big{)}$. This can be seen as
follows: thanks to ((ii)), we know that $\mathcal{H}(T)$ is
$\bar{\partial}$-closed. Then we have
$\Delta^{\prime\prime}\bar{\partial}\big{(}\mathcal{G}(T)\big{)}=0$. It
follows that $\bar{\partial}\big{(}\mathcal{G}(T)\big{)}$ is smooth in
orbifold sense and then we get
$\bar{\partial}^{\star}\bar{\partial}\big{(}\mathcal{G}(T)\big{)}=0$ by the
usual integration by parts.

We have the following compatibility relation between the notions we have
introduced above.

######  Lemma 3.5.

Let $s$ be a $L$-valued $(p,q)$-current and $t$ be a $L^{\star}$-valued
$(n-p,n-q)$-form smooth in conic sense. We have

(3.5.1) $\int_{X}\mathcal{G}(s)\wedge t=\int_{X}s\wedge{\mathcal{G}(t)}.$

###### Proof.

Note first that, since $\mathcal{G}(s)$ is orthogonal to the harmonic space,
by Proposition (2.25), we have

$\int_{X}\mathcal{G}(s)\wedge\mathcal{H}(t)=0.$

Then we have

(3.5.2) $\int_{X}\mathcal{G}(s)\wedge
t=\int_{X}\mathcal{G}(s)\wedge\Delta^{{}^{\prime\prime}}(\mathcal{G}{t})=\int_{X}\Delta^{{}^{\prime\prime}}(\mathcal{G}(s))\wedge\mathcal{G}(t)=\int_{X}s\wedge{\mathcal{G}(t)}.$

∎

###  3.1. Regularity of the Green kernel for certain currents

Coming back to the case of a current induced by a form $\alpha$ with log-
poles, the Green current $\mathcal{G}(\alpha)$ has a few interesting
properties that we collect in our next statement.

######  Proposition 3.6.

Let $E=\sum E_{i}$ be divisor on $X$ such that $E+Y$ is snc. Let $\alpha$ be a
smooth $L$-valued $(n,q)$-form on $X\setminus(E\cup Y)$, with log pole along
$E+Y$. The following statements hold true.

  1. (1)

The current $\mathcal{G}(\alpha)$ is a smooth $L$-valued $(n,q)$–form on
$X\setminus(E\cup Y)$ which moreover is $L^{2}$ on $X$.

  2. (2)

For each index $i$ the local form $\displaystyle v_{i}:=\frac{1}{\prod
w_{i}^{q_{i}m}}\pi_{i}^{\star}\left(\mathcal{G}(\alpha)\right)$ is a solution
of an equation of the following type

$\Delta^{\prime\prime}_{sm}(v_{i})=u_{i}$

where $u_{i}$ is a $(n,q)$-form with logarithmic poles along $E+Y$.

  3. (3)

For every $\delta>0$ form $v_{i}$ belongs to the Sobolev space
$W^{2,2-\delta}$ (i.e. it has two derivatives in $L^{2-\delta}$).

  4. (4)

The form $\tau_{i}:=\partial^{\star}v_{i}$ is $L^{2}$ and moreover “smooth”
with respect to the variables $w_{j}$ for each $j=r+1,\dots,n$, in the sense
that it admits a Taylor expansion

$\tau_{i}(w)=\sum_{|I|+|J|\leqslant
N-1}w^{\prime\alpha}\overline{w}^{\prime\beta}\tau_{I\overline{J}}(w^{\prime\prime})+\sum_{|I|+|J|=N}w^{\prime\alpha}\overline{w}^{\prime\beta}\tau_{I\overline{J}}(w)$

where

$w^{\prime}=(w_{r+1},\dots,w_{n}),\qquad w^{\prime\prime}=(w_{1},\dots,w_{r})$

and $\tau_{I\overline{J}}$ are $L^{2}$.

###### Proof.

In order to prove the point (1), our first remark is that

(3.6.1) $\int_{X}|\mathcal{G}(\Phi)|^{2}e^{-\varphi}\leqslant
C\int_{X}|\Phi-\mathcal{H}(\Phi)|^{2}e^{-\varphi}$

by Poincaré inequality. The inequality (3.6.1) is valid for $(n,q)$-forms and
it implies that

(3.6.2) $\int_{X}|\mathcal{G}(\Phi)|^{2}e^{-\varphi}\leqslant
C\int_{X}|\Phi|^{2}e^{-\varphi}$

because the $L^{2}$ norm of $\mathcal{H}(\Phi)$ is smaller than the $L^{2}$
norm of $\Phi$. The smoothness of $\mathcal{G}(\alpha)$ in the complement of
the support of the divisor $E+Y$ is standard. Then the Sobolev norm of order
two of $\mathcal{G}(\Phi)$ is bounded by the RHS of (3.6.2).

By the definition of the current $\mathcal{G}(\alpha)$ as in (3.2.4) it
follows that we have

(3.6.3)
$\left|\langle\mathcal{G}(\alpha),\overline{\Phi}\rangle\right|\leqslant\|\alpha\|_{L^{2-\delta}}\|\mathcal{G}(\Phi)\|_{L^{\frac{2-\delta}{1-\delta}}}.$

Now the $L^{\frac{2-\delta}{1-\delta}}$-norm of $\mathcal{G}(\Phi)$ is
certainly controlled by the Sobolev norm above (at least when $\delta$ is
small enough) so in conclusion we get

(3.6.4)
$\left|\langle\mathcal{G}(\alpha),\overline{\Phi}\rangle\right|^{2}\leqslant
C_{\delta}\|\Phi\|^{2}_{L^{2}}$

which shows that $\mathcal{G}(\alpha)$ is in $L^{2}$.

The first part of the point (2) (i.e. the equation satisfied by the form
$v_{i}$) was already discussed in Proposition 2.17. The fact that the RHS is a
form with log-poles is immediate and we skip the details.

The points (3) and (4) are a bit more interesting and we will provide a
complete treatment.

To start with, the arguments provided so far allow us to assume that the form
$v_{i}$ in ((2)) is in $L^{2}$. Together with the Gårding inequality
established in [2], Theorem 7.1 we obtain

(3.6.5) $\|v_{i}\|_{W^{2,2-\delta}}\leqslant
C_{p}(\|v_{i}\|_{L^{2-\delta}}+\|u_{i}\|_{L^{2-\delta}})$

for any $0<\delta<2$. The point $(3)$ is thus proved.

To prove the point $(4)$, let $\Xi$ be any local vector field with constant
coefficients of the following shape

(3.6.6) $\Xi=\sum_{i=r+1}^{n}a_{i}\frac{\partial}{\partial w_{i}}.$

The equation ((2)) implies that we have

(3.6.7) $\Delta^{\prime\prime}_{sm}(\Xi\cdot v_{i})=\Xi\cdot
u_{i}+\mathcal{L}(v_{i})$

where $\mathcal{L}$ is a second order operator. In (3.6.7) we denote by
$\Xi\cdot v_{i}$ the form whose coefficients are the derivative of those of
$v_{i}$ in the direction of $\Xi$.

By Sobolev embedding theorem we infer that $\Xi(v)$ is in $L^{2}$ (provided
that $\delta\ll 1$). It follows from (3.6.7) that $\Xi(v)$ has two derivatives
in $L^{p}$. We can iterate this method, and infer that $\Xi^{m}(v)\in L^{2}$
for any $m$.

Finally, the usual argument –by using a Fourier expansion– implies that
$v_{i}$ admits a Taylor expansion as required in (4). We discuss here only a
particular case, for notational reasons. Let $f\in L^{2}(\mathbb{D})$ be a
square integrable function defined on the unit disk in $\mathbb{R}^{2}$, and
such that $\displaystyle\frac{\partial^{m}}{\partial x^{m}}(f)$ is in $L^{2}$
for any positive integer $m$.

We consider the convolution $f_{\varepsilon}$ of $f$ with the usual
regularisation kernel and as usual we can assume that this functions are
1-periodic. We then write

(3.6.8)
$f_{\varepsilon}(x,y)=\sum_{k\in\mathbb{Z}}f_{\varepsilon,k}(y)e^{\sqrt{-1}kx}$

and by hypothesis, together with the $L^{2}$ convergence of convolutions and
their derivatives we infer in particular that for any $m\in\mathbb{Z}_{+}$ we
have

(3.6.9)
$\sum_{k\in\mathbb{Z}}k^{2m}\int_{|y|<1}|f_{\varepsilon,k}(y)|^{2}d\lambda(y)<C_{m}$

where $C_{m}$ is a constant independent of $\varepsilon$.

A first consequence of (3.6.9) is the existence of $C>0$ such that

(3.6.10) $\int_{|y|<1}|f_{\varepsilon}(0,y)|^{2}d\lambda(y)<C$

which can be seen by observing that

$\displaystyle\left|\sum_{k=-N}^{N}f_{\varepsilon,k}(y)\right|^{2}\leqslant
2\sum_{k=-N}^{N}k^{2}|f_{\varepsilon,k}(y)|^{2}\sum_{k=1}^{N}\frac{1}{k^{2}}$

and integrating.

We next write $\displaystyle
f_{\varepsilon}(x,y)=f_{\varepsilon}(0,y)+x\int_{0}^{1}\partial_{1}f_{\varepsilon}(tx,y)dt$.
Remark that we already know that the $L^{2}$ norm of the first term of the RHS
is uniformly bounded, hence it is weakly convergent towards a function in
$L^{2}$. We will analyse next the other function; notice that

(3.6.11)
$\int_{\mathbb{D}}\left|\int_{0}^{1}\partial_{1}f_{\varepsilon}(tx,y)dt\right|^{2}d\lambda\leqslant\int_{0}^{1}dt\int_{\mathbb{D}}\left|\partial_{1}f_{\varepsilon}(tx,y)\right|^{2}d\lambda$

which equals

(3.6.12)
$\int_{\mathbb{D}}\log\frac{1}{|x|}\left|\partial_{1}f_{\varepsilon}(x,y)\right|^{2}d\lambda.$

By Hölder inequality the expression (3.6.12) is smaller than

(3.6.13)
$C_{\delta}\int_{|y|<1}dy\left(\int_{|x|<1}\left|\partial_{1}f_{\varepsilon}(x,y)\right|^{2+2\delta}dx\right)^{\frac{1}{1+\delta}},$

and on the other hand, for each fixed $y$ we have

(3.6.14)
$\frac{1}{C}\left(\int_{|x|<1}\left|\partial_{1}f_{\varepsilon}(x,y)\right|^{2+2\delta}dx\right)^{\frac{1}{1+\delta}}\leqslant\int_{|x|<1}\left|\partial_{1}f_{\varepsilon}(x,y)\right|^{2}dx+\int_{|x|<1}\left|\partial_{11}f_{\varepsilon}(x,y)\right|^{2}dx$

by Sobolev inequality. The integration with respect to $y$ of (3.6.14) shows
that the LHS of (3.6.11) is uniformly bounded.

The arguments for the general case are completely similar –modulo the fact
that the number of indexes is greater– and we will not provide any further
detail. ∎

Let $E=\sum E_{i}$ be divisor on $X$ such that $E+Y$ is snc. Consider next a
current of the following type

(3.6.15) $T:=\frac{\lambda}{s_{E}}-\sum_{i}T_{\gamma_{i}},$

where $\lambda$ is a $L+E$-valued form, smooth in conic sense, $\gamma_{i}$ is
a $L$-valued form on $E_{i}$, smooth in conic sense, and $T_{\gamma_{i}}$ is
the current on $X$ defined by $\gamma$, cf. Subsection 2.3.1. We establish
next some of the properties of $\mathcal{G}(T)$. We will namely discuss the
following result.

######  Proposition 3.7.

Let $T$ be the current defined in (3.6.15). The associated current
$\mathcal{G}(T)$ is smooth in the complement of the divisor $E+F$ and it is
moreover $L^{2}$, i.e.

(3.7.1)
$\int_{X}|\mathcal{G}(T)|_{\omega_{\mathcal{C}}}^{2}e^{-\varphi}dV_{\omega_{\mathcal{C}}}<\infty.$

###### Proof.

We first remark that $T$ is more singular than e.g. a current induced by a
form with log poles, so it is natural that the regularity properties of
$\mathcal{G}(T)$ are considerably weaker than those established in Proposition
3.6.

By definition we have

(3.7.2)
$\langle\mathcal{G}(T),\Phi\rangle=\int_{X}\frac{\lambda}{s_{E}}\wedge\overline{\mathcal{G}(\Phi)}e^{-\varphi_{L}}+\sum_{i}\int_{E_{i}}\gamma_{i}\wedge\overline{\mathcal{G}(\Phi)}e^{-\varphi_{L}},$

and the absolute value of the first part of the sum is bounded by
$\displaystyle\|\Phi\|_{L^{2}}$ up to a constant: this is a consequence of the
proof of the first point of Proposition 3.6.

We treat now the second part of the RHS of (3.7.2). Let $\Gamma_{i}$ be a
$L^{2}$ and _smooth_ extension of the $L$-valued form $\gamma_{i}$. This is
constructed e.g. by a partition of unit –we notice the $L^{2}$ condition is
fulfilled precisely because $E+Y$ is a snc divisor. For each $i$ we can write

(3.7.3)
$\int_{E_{i}}\gamma_{i}\wedge\overline{\mathcal{G}(\Phi)}e^{-\varphi_{L}}=\int_{X}\left(\theta_{i}+dd^{c}\log|s_{i}|^{2}\right)\wedge\Gamma_{i}\wedge\overline{\mathcal{G}(\Phi)}e^{-\varphi_{L}}.$

where $\theta_{i}$ is the curvature of the bundle associated to the divisor
$E_{i}$ endowed with a non-singular metric.

The first term of the RHS of (3.7.3) is bounded by the $L^{2}$ norm of $\Phi$,
given that $\theta_{i}\wedge\Gamma_{i}$ is $L^{2}$ with respect to $h_{L}$.

For the second term of the RHS of (3.7.3), we are using Stokes and we have to
estimate

(3.7.4) $\int_{X}\frac{\partial
s_{i}}{s_{i}}\wedge\bar{\partial}\Gamma_{i}\wedge\overline{\mathcal{G}(\Phi)}e^{-\varphi_{L}},\qquad\int_{X}\frac{\partial
s_{i}}{s_{i}}\wedge\Gamma_{i}\wedge\overline{D^{\prime}\mathcal{G}(\Phi)}e^{-\varphi_{L}}.$

To this end we simply use the Gårding inequality (after composing with the
local ramification maps) and infer that

(3.7.5) $\|\mathcal{G}(\Phi)\|^{2}_{W^{2,2}}\leqslant
C\left(\|\Phi\|^{2}_{L^{2}}+\|\mathcal{G}(\Phi)\|^{2}_{L^{2}}\right).$

By Sobolev inequality the $L^{2+\delta}$-norm of $\mathcal{G}(\Phi)$ and
$D^{\prime}\mathcal{G}(\Phi)$ are bounded by the left hand side of (3.7.5)
provided that $\delta\ll 1$. This ends the proof of our proposition. ∎

##  4\. Proof of Theorem 1.1 and Theorem 1.2

###  4.1. A general vanishing theorem

The plan for this section is to prove Theorem 1.1 and Theorem 1.2. We prove
the Theorem 1.1 in this subsection. As an application, we will prove Theorem
1.2 in the next subsection. We would like to prove:

######  Theorem 4.1.

[= Theorem 1.1] Let $X$ be a $n$-dimensional compact Kähler manifold and let
$E$ be a snc divisor on $X$, $s_{E}$ be the canonical section of $E$. Let
$(L,h_{L})$ be a holomorphic line bundle on $X$ such that

$i\Theta_{h_{L}}(L)=\sum q_{i}[Y_{i}]+\theta_{L},$

where $q_{i}\in]0,1[\cap\mathbb{Q}$, $E+\sum Y_{i}$ is snc, and the form
$\theta_{L}$ is smooth, semi-positive.

Let $\lambda$ be a $\bar{\partial}$-closed smooth $(n,q)$-form with value in
$L+E$. If there exists $\beta_{1}$ and $\beta_{2}$, two $L$-valued
$(n-1,q)$-form and $(n-1,q-1)$-form with log poles along $E+\sum Y_{i}$
respectively, such that

(4.1.1)
$\frac{\lambda}{s_{E}}=D^{\prime}_{h_{L}}\beta_{1}+\theta_{L}\wedge\beta_{2}\qquad\text{
on }X\setminus(E+\sum Y_{i}),$

then the form $\lambda$ is $\bar{\partial}$-exact, i.e, the class
$[\lambda]=0\in H^{q}(X,K_{X}+L+E)$.

Remark that the case $E=0$ is quite simple: it follows from Theorem 2.2 and
standard argument.

###### Proof of Theorem 4.1.

We consider a test form $\displaystyle\varphi\in
C^{\infty}_{(0,n-q)}\big{(}X,-L-E\big{)}$ with compact support in $X\setminus
Y$.

Then we have the orthogonal decomposition

(4.1.2)
$\varphi=\varphi_{1}+\varphi_{2},\qquad\varphi_{1}\in\mathrm{Ker}\,\bar{\partial},\quad\varphi_{2}\in(\mathrm{Ker}\,\bar{\partial})^{\perp}$

where the reference metric on $X$ is $\omega_{\mathcal{C}}$. The bundle $-L$
is endowed with the dual metric $h_{L}^{*}$ and $-E$ with a smooth, arbitrary
metric.

Our proof relies on the following two statements:

(4.1.3)
$\int_{X}s_{E}(D^{\prime}_{h_{L}}\beta_{1}+\theta_{L}\wedge\beta_{2})\wedge\varphi_{1}=0$

and moreover there exists a constant $C>0$ such that

(4.1.4) $\int_{X}|\varphi_{2}|^{2}dV_{\mathcal{C}}\leqslant
C\int_{X}|\bar{\partial}\varphi_{2}|^{2}dV_{\mathcal{C}}=C\int_{X}|\bar{\partial}\varphi|^{2}dV_{\mathcal{C}}.$

To prove (4.1.3), note first that, thanks to Corollary LABEL:Kompsup, the form
$\varphi_{1}$ is smooth in conic sense. Then $s_{E}\cdot\varphi_{1}$ has the
same property. Moreover, $\bar{\partial}(s_{E}\varphi_{1})=0$ and
$s_{E}\varphi_{1}$ vanishes along $E$. Then we obtain (4.1.3) as consequence
of Lemma 4.3 below. For (4.1.4), we simply use Theorem 2.28 which implies that
the image of $\bar{\partial}$ operator is closed (regardless to the fact that
the curvature of $L^{\star}$ is negative).

We consider next the linear application

$\bar{\partial}\varphi\rightarrow\int_{X}s_{E}(D^{\prime}_{h_{L}}\beta_{1}+\theta_{L}\wedge\beta_{2})\wedge\varphi.$

It follows from (4.1.3) and (4.1.4) that it is bounded. By standard functional
analysis there exists a form $f\in L^{2}_{(n,q-1)}(X,L+E)$ such that

$\int_{X}s_{E}(D^{\prime}_{h_{L}}\beta_{1}+\theta_{L}\wedge\beta_{2})\wedge\varphi=\int_{X}\bar{\partial}\varphi\wedge
f.$

Therefore we infer that the equality

(4.1.5)
$\lambda=s_{E}(D^{\prime}_{h_{L}}\beta_{1}+\theta_{L}\wedge\beta_{2})=\bar{\partial}f\qquad\text{
on }X\setminus Y.$

Since the LHS in (4.1.5) is smooth, it is $L^{2}$. Moreover, $f$ is also
$L^{2}$. By using the remark after Theorem 2.29, we know that (4.1.5) holds on
the total space $X$. Theorem 4.1 is thus proved modulo the lemma 4.3 below. ∎

The following key lemma 4.3 is a generalization of [14, Lemma .2.3] in our
setting. Prior to stating it, we introduce a further notation. Let $\gamma$ be
an $L$-valued form of $(p,q)$-type which moreover has log-poles along $E$ and
it is $\bar{\partial}$-closed when restricted to the complement of the support
of $E$. Locally on a coordinate subset $\Omega\subset X$ such that
$E_{1}\cap\Omega=(z_{1}=0)$ we can write

(4.1.6) $\gamma|_{\Omega}=\frac{dz_{1}}{z_{1}}\wedge\gamma_{1}+\gamma_{2}$

where $\gamma_{1}$ and $\gamma_{2}$ are smooth along $E_{1}\cap\Omega$. We
define

(4.1.7) $\mathrm{Res}_{E_{1}}(\gamma)|_{\Omega}:=\gamma_{1}|_{\Omega}$

and it is not difficult to see that the following hold:

$\bullet$ The forms $\mathrm{Res}_{E_{1}}(\gamma)|_{\Omega}$, even if given
locally as in (4.1.7), defines a global $(p-1,q)$-form on $E_{1}$ with values
in $\displaystyle L|_{E_{1}}$ and log-poles on
$\displaystyle(E-E_{1})|_{E_{1}}$. We will denote it with
$\mathrm{Res}_{E_{1}}(\gamma)$

$\bullet$ We have

(4.1.8) $\mathrm{Res}_{E_{1}}(\gamma)=-\chi_{E_{1}}\bar{\partial}\gamma$

i.e. residue along $E_{1}$ can be obtained by multiplying the current
$\bar{\partial}\gamma$ with the characteristic function of $E_{1}$.

$\bullet$ The equality $\bar{\partial}\mathrm{Res}_{E_{1}}(\gamma)=0$ holds in
the complement of the support of $\displaystyle(E-E_{1})|_{E_{1}}$.

All the properties above are completely standard. In the specific context of
Theorem 1.1, we consider the current

(4.1.9)
$\widehat{\Theta}:=D^{\prime}_{h_{L}}\beta_{1}+\theta_{L}\wedge\beta_{2}$

and its residues along the components of $E$. As a preparation for the proof
of this result, we discuss next a few properties of $\widehat{\Theta}$ that we
will next use.

$\bullet$ The first fact to remark is that we have the equality

(4.1.10)
$\frac{\lambda}{s_{E}}=D^{\prime}_{h_{L}}\beta_{1}+\theta_{L}\wedge\beta_{2}$

in the sense of currents on the total space $X$. This is due to the fact that
the current $D^{\prime}_{h_{L}}\beta_{1}$ has no divisorial part supported in
$E+Y$. Note that this is absolutely not the case in general if we replace
$D^{\prime}_{h_{L}}$ with $\bar{\partial}$.

$\bullet$ Another important observation is that the residue of
$\widehat{\Theta}$ along any component of $E$ has the same “shape”. i.e. it
belongs to the image of $D^{\prime}_{h_{L}}+\theta_{L}\wedge$. This is a
simple calculation that we skip.

$\bullet$ The residue of $\widehat{\Theta}$ on the components of $Y$ is equal
to zero. This claim may look strange/suspicious, since the forms $\beta_{i}$
could have log-poles on $Y$, so the residues of each form
$D^{\prime}_{h_{L}}\beta_{1}$ and $\theta_{L}\wedge\beta_{2}$ could be non-
zero on some of the components of $Y$. However, the equality (4.1.10)
clarifies the matter.

We have the next auxiliary result which will be very useful in the proof of
Lemma 4.3 below.

######  Lemma 4.2.

Let $\gamma$ be a $\Delta^{{}^{\prime\prime}}_{\omega_{\mathcal{C}}}$-harmonic
$(0,n-q)$-form with values in the dual bundle $L^{\star}$ of $L$. Then the
equality

(4.2.1)
$\int_{X}(D^{\prime}_{h_{L}}\beta_{1}+\theta_{L}\wedge\beta_{2})\wedge\gamma=0$

holds (here as everywhere, the integral is computed in the complement of
$E+Y$as a matter of fact, the form in (4.2.1) is absolutely integrable).

###### Proof.

Since $\gamma$ is harmonic, it is in particular smooth in conic sense and
moreover, as explained above, the equality $\displaystyle
D^{\prime}_{h_{L}}\beta_{1}+\theta_{L}\wedge\beta_{2}=\frac{\lambda}{s_{E}}$
holds in the sense of currents. It follows that the form under the integral
(4.2.1) is absolutely integrable: this is seen as follows. Let $\pi:U\to V$ be
a ramified cover adapted to our setting and coordinates on $V$ such that

$Y\cap V=(z_{1}\dots z_{r}=0),\qquad E\cap V=(z_{r+1}\dots z_{r+s}=0).$

We have

$\frac{1}{w^{q}m}\pi^{\star}\Big{(}\frac{\lambda}{s_{E}}\Big{|}_{V}\Big{)}\wedge\tau=\frac{\mathcal{O}(1)}{\prod
w_{j}^{q_{j}m+1-m}\prod w_{i}}d\lambda(w)$

for any smooth form $\tau$ of type $(0,n-1)$, so our claim follows.

Thus, it would be enough to show that we have

(4.2.2) $\lim_{\varepsilon\to
0}\int_{X}\mu_{\varepsilon}(D^{\prime}_{h_{L}}\beta_{1}+\theta_{L}\wedge\beta_{2})\wedge\gamma=0$

where $(\mu_{\varepsilon})$ is the usual family of truncation functions. Now
$\gamma$ is harmonic, so we have

(4.2.3) $D^{\prime}\gamma=0,\qquad\theta_{L}\wedge\gamma=0.$

Therefore, it all boils down to the next equality

(4.2.4) $\lim_{\varepsilon\to
0}\int_{X}\partial\mu_{\varepsilon}\wedge\beta_{1}\wedge\gamma=0$

which in turn is clear by a local calculation (using the fact that $\gamma$ is
smooth in conic sense).

For example, say that $n=2$, so that $\beta_{1}$ is a $(1,1)$-form and
$Y=(z_{1}=0),E=(z_{2}=0)$. Then the expression (4.2.4) is locally equal to

(4.2.5)
$\int_{V}\frac{\rho_{\varepsilon}^{\prime}(z)}{\log(|z_{1}|^{2}e^{-\phi})}\frac{dz_{1}}{z_{1}}\wedge\frac{dz_{2}}{z_{2}}\wedge(ad\overline{z}_{1}+bd\overline{z}_{2})\wedge\gamma|_{V}$

and by the local transformation $(w_{1},w_{2})\to(w_{1}^{m},w_{2})$ this
expression above becomes

(4.2.6)
$\int_{U}\frac{\rho_{\varepsilon}^{\prime}(w)}{\log(|w_{1}|^{2}e^{-\phi})}\frac{dw_{1}}{w_{1}}\wedge\frac{dw_{2}}{w_{2}}\wedge(a_{1}d\overline{w}_{1}+b_{1}d\overline{z}_{2})\wedge\frac{\tau}{w_{1}^{mq}}$

where $a_{1},b_{1}$ are smooth functions with compact support on $U$ and
$\tau$ is a smooth $(0,1)$ form. This is a consequence of the fact that
harmonic forms are smooth in conic sense. Arguments as in Lemma 2.35 and the
examples above do apply, so we get

(4.2.7) $\lim_{\varepsilon\to
0}\int_{U}\rho_{\varepsilon}^{\prime}(w)\frac{f(w)}{w_{1}^{1+mq}w_{2}}\frac{d\lambda}{\log(|w_{1}|^{2}e^{-\phi})}=0.$

Actually we refer to the Appendix for a complete proof of (4.2.7). ∎

Before stating our main lemma, we introduce some notations. We suppose that
the snc divisor $E$ is written as $E=\sum_{1\leqslant i\leqslant r}E_{i}$. Let
$I_{d}\subset\\{1,\cdots,r\\}$ with $|I_{d}|=d$. Let $\alpha$ be a
$L^{*}$-valued smooth form (in the conic sense with respect to
$h_{L}^{\star},\omega_{\mathcal{C}}$) defined on some space which contains
$E_{I_{d}}$. We define

$L_{I_{d}}(\alpha):=\bar{\partial}^{\star_{I_{d}}}\mathcal{G}_{I_{d}}(\alpha|_{E_{I_{d}}}),$

where $\mathcal{G}_{I_{d}}$ is the Green operator on $E_{I_{d}}$ and
$\bar{\partial}^{\star_{I_{d}}}$ is the $\bar{\partial}^{\star}$-operator on
$E_{I_{d}}$ with respect to $h_{L}^{\star},\omega_{\mathcal{C}}$. If $d=0$,
$L_{I_{0}}\varphi:=\bar{\partial}^{\star}\mathcal{G}(\varphi)$ on $X$. For
every sequence $I_{k}\subset I_{k-1}\subset\cdots\subset I_{0}$ fixed, the
residue $\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta})$ is defined by the
following way:

(4.2.8)
$\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}):=\mathrm{Res}_{E_{I_{k}}}\Big{(}\mathrm{Res}_{E_{I_{k-1}}}\big{(}\cdots\mathrm{Res}_{E_{I_{1}}}(\widehat{\Theta})\big{)}\Big{)}.$

Now we can state the key lemma:

######  Lemma 4.3.

In the setting of Theorem 4.1, set
$\widehat{\Theta}:=D^{\prime}_{h_{L}}\beta_{1}+\theta_{L}\wedge\beta_{2}$. Let
$\varphi$ be a $\bar{\partial}$-closed $(0,n-q)$-form on $X$ with values in
the dual of $L$, which moreover is smooth in conic sense. We assume that
$\varphi$ vanishes along $E$. Then for every $k\in\mathbb{N}^{*}$, we have

(4.3.1) $\sum_{I_{k}\subset I_{k-1}\subset\cdots\subset
I_{0}}\int_{E_{I_{k}}}\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta})\wedge
L_{I_{k-1}}\circ L_{I_{k-2}}\cdots\circ L_{I_{0}}\varphi=0,$

and

(4.3.2) $\int_{X}\widehat{\Theta}\wedge\varphi=0.$

###### Proof.

We prove it by induction on $k$ from large to small. For $k=n+1$, it is
automatically true since $E_{I_{n+1}}$ is empty. We suppose that the lemma
holds for $k+1$. Now we prove the lemma for $k$.

First of all, we have

(4.3.3)
$\int_{E_{I_{k}}}\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta})\wedge\gamma=0$

for any harmonic $\gamma\in C^{\infty}_{(0,n-k-1)}(E_{I_{k}},L^{\star})$.
Indeed, we know that the residue $\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta})$
obtained by iterating the construction we have recalled before the lemma 4.2
can be written as

$\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta})=D^{\prime}(\beta_{k})+\theta_{L}\wedge\alpha_{k}$

for some forms $\beta_{k},\alpha_{k}$ with log poles. Then Lemma 4.2 implies
(4.3.3).

Then the deRham-Kodaira decomposition implies that the equality

$\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta})=\bar{\partial}\bar{\partial}^{\star_{I_{k}}}\mathcal{G}_{I_{k}}(\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}))+\bar{\partial}^{\star_{I_{k}}}\bar{\partial}\mathcal{G}_{I_{k}}(\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}))$

holds. As a consequence, we obtain

$\sum_{I_{k}\subset I_{k-1}\subset\cdots\subset
I_{0}}\int_{E_{I_{k}}}\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta})\wedge
L_{I_{k-1}}\circ L_{I_{k-2}}\cdots\circ L_{I_{0}}\varphi=A+B,$

where

$A:=\sum_{I_{k}\subset I_{k-1}\subset\cdots\subset
I_{0}}\int_{E_{I_{k}}}\bar{\partial}^{\star_{I_{k}}}\bar{\partial}\mathcal{G}_{I_{k}}(\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}))\wedge
L_{I_{k-1}}\circ\cdots\circ L_{I_{0}}\varphi$

and

$B:=\sum_{I_{k}\subset I_{k-1}\subset\cdots\subset
I_{0}}\int_{E_{I_{k}}}\bar{\partial}\bar{\partial}^{\star_{I_{k}}}\mathcal{G}_{I_{k}}(\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}))\wedge
L_{I_{k-1}}\circ\cdots\circ L_{I_{0}}\varphi.$

We would like to prove that $A=B=0$.

We first estimate $A$. Since $\mathcal{G}$ commutes with $\bar{\partial}$,
together with the relation (4.1.8), we have

$A=\sum_{I_{k}\subset I_{k-1}\subset\cdots\subset
I_{0}}\int_{E_{I_{k}}}\bar{\partial}^{\star_{I_{k}}}\mathcal{G}_{I_{k}}\bar{\partial}(\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}))\wedge
L_{I_{k-1}}\circ\cdots\circ L_{I_{0}}\varphi$ $=-\sum_{I_{k+1}\subset
I_{k}\subset\cdots\subset
I_{0}}\int_{E_{I_{k+1}}}\mathrm{Res}_{E_{I_{k+1}}}(\widehat{\Theta})\wedge
L_{I_{k}}\circ L_{I_{k-1}}\circ\cdots\circ L_{I_{0}}\varphi=0.$

where the last equality is a consequence of the induction hypothesis.

Next, we estimate the quantity $B$. We have

$B=\sum_{I_{k}\subset I_{k-1}\subset\cdots\subset
I_{0}}\int_{E_{I_{k}}}\bar{\partial}\bar{\partial}^{\star_{I_{k}}}\mathcal{G}_{I_{k}}(\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}))\wedge
L_{I_{k-1}}\circ\cdots\circ L_{I_{0}}\varphi$ $=\sum_{I_{k}\subset
I_{k-1}\subset\cdots\subset
I_{0}}\int_{E_{I_{k}}}\bar{\partial}^{\star_{I_{k}}}\mathcal{G}_{I_{k}}(\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}))\wedge\bar{\partial}(L_{I_{k-1}}\circ\cdots\circ
L_{I_{0}}\varphi).$

By applying the deRham-Kodaira decomposition (3.2.5) to
$(L_{I_{k-2}}\circ\cdots\circ L_{I_{0}}\varphi)|_{E_{I_{k-1}}}$, we obtain

$\bar{\partial}(L_{I_{k-1}}\circ\cdots\circ
L_{I_{0}}\varphi)=\gamma+(L_{I_{k-2}}\circ\cdots\circ
L_{I_{0}}\varphi)|_{E_{I_{k-1}}}+L_{I_{k-1}}\circ\bar{\partial}(L_{I_{k-2}}\cdots\circ
L_{I_{0}}\varphi),$

where $\gamma$ is a $\Delta^{\prime\prime}_{E_{I_{k-1}}}$-harmonic
$(0,n-k+1)$-form on $E_{I_{k-1}}$ with values in the restriction of the dual
of $L$.

We next estimate the integral involving the first two terms.

$\bullet$ The term involving $\gamma$. Since $\gamma$ is a harmonic
$(0,n-k+1)$-form on $E_{I_{k-1}}$ with values in $L^{\star}$, it follows that
$D^{\prime}\gamma=0$ and $\displaystyle\theta_{L}|_{E_{I_{k}}}\wedge\gamma=0$
on $E_{I_{k-1}}$. These crucial equalities obviously are still verified by
restriction one level deeper, to $E_{I_{k}}$. We infer that the restriction

$\gamma|_{E_{I_{k}}}$

is still harmonic and thus

$\int_{E_{I_{k}}}\bar{\partial}^{\star_{I_{k}}}\mathcal{G}_{I_{k}}(\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}))\wedge\gamma=0.$

$\bullet$ The term involving $L_{I_{k-2}}\circ\cdots\circ L_{I_{0}}\varphi$.
The sum

$\sum_{I_{k}\subset I_{k-1}\subset\cdots\subset
I_{0}}\int_{E_{I_{k}}}\bar{\partial}^{\star_{I_{k}}}\mathcal{G}_{I_{k}}(\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}))\wedge(L_{I_{k-2}}\circ\cdots\circ
L_{I_{0}}\varphi)$

equals zero as soon as $k-1\geqslant 1$: for every $I_{k}\subset I_{k-2}$
fixed, $I_{k-2}\setminus I_{k}$ contains only two elements. By construction of
$\mathrm{Res}_{I_{k}}$ (4.2.8), the alternate sum for $I_{k-1}$ satisfying
$I_{k}\subset I_{k-1}\subset I_{k-2}$ equals to $0$.

It follows that if $k-1\geqslant 1$, we obtain

$B=\sum_{I_{k}\subset I_{k-1}\subset\cdots\subset
I_{0}}\int_{E_{I_{k}}}\bar{\partial}^{\star_{I_{k}}}\mathcal{G}_{I_{k}}(\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}))\wedge
L_{I_{k-1}}\circ\bar{\partial}(L_{I_{k-2}}\circ\cdots\circ L_{I_{0}}\varphi).$

Now we can repeat the same argument for
$\bar{\partial}(L_{I_{k-2}}\circ\cdots\circ L_{I_{0}}\varphi)$ as long as
$k\geqslant 3$. The only difference is to treat the integral involving the
harmonic forms $\gamma$ on $E_{I_{k-2}}$: we use the fact that
$\mathcal{G}_{I_{k-1}}(\gamma|_{E_{I_{k-2}}})=0$. We finally get

$B=\sum_{I_{k}\subset I_{k-1}\subset\cdots\subset
I_{0}}\int_{E_{I_{k}}}\bar{\partial}^{\star_{I_{k}}}\mathcal{G}_{I_{k}}(\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}))\wedge
L_{I_{k-1}}\circ\bar{\partial}(L_{I_{k-2}}\circ\cdots\circ L_{I_{0}}\varphi)$
$\cdots=\sum_{I_{k}\subset I_{k-1}\subset\cdots\subset
I_{0}}\int_{E_{I_{k}}}\bar{\partial}^{\star_{I_{k}}}\mathcal{G}_{I_{k}}(\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}))\wedge
L_{I_{k-1}}\circ L_{I_{k-2}}\circ\cdots\circ
L_{I_{1}}\circ\bar{\partial}(L_{I_{0}}(\varphi)).$

Concerning the term
$\bar{\partial}(L_{I_{0}}(\varphi))=\bar{\partial}\bar{\partial}^{\star}\mathcal{G}(\varphi)$,
we consider the Hodge decomposition

$\bar{\partial}\bar{\partial}^{\star}\mathcal{G}(\varphi)=\gamma+\varphi-\bar{\partial}^{\star}\mathcal{G}(\bar{\partial}\varphi)=\gamma+\varphi,$

where $\gamma$ is harmonic. Since $\varphi=0$ on $E$, we have
$L_{I_{1}}(\varphi)=0$.

We obtain

$\sum_{I_{k}\subset I_{k-1}\subset\cdots\subset
I_{0}}\int_{E_{I_{k}}}\bar{\partial}^{\star_{I_{k}}}\mathcal{G}_{I_{k}}(\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}))\wedge
L_{I_{k-1}}\circ
L_{I_{k-2}}\circ\cdots\circ\bar{\partial}(L_{I_{0}}(\varphi))$
$=\sum_{I_{k}\subset I_{k-1}\subset\cdots\subset
I_{0}}\int_{E_{I_{k}}}\bar{\partial}^{\star_{I_{k}}}\mathcal{G}_{I_{k}}(\mathrm{Res}_{E_{I_{k}}}(\widehat{\Theta}))\wedge
L_{I_{k-1}}\circ\cdots\circ L_{I_{1}}(\gamma+\varphi)=0.$

This completes the proof of Lemma 4.3. ∎

######  Remark 4.4.

It is important and challenging to refine further Theorem 4.1 and solve the
equation $\lambda=\bar{\partial}f$ _with estimates_. A quick inspection of our
arguments shows that one ”only” needs to understand on what the constant $C$
in (4.1.4) depends. This might not be the easiest thing to do, given the proof
(by contradiction) for the existence of this constant.

###  4.2. Application: an injectivity theorem

As a corollary of Theorem 1.1, we obtain the following result.

######  Theorem 4.5.

Let $E=\sum E_{i}$ be a simple normal crossing divisor on a compact Kähler
manifold $X$ and $F$ be a holomorphic line bundle on $X$ with a smooth metric
$h_{F}$ such that $i\Theta_{h_{F}}(F)\geqslant 0$. Let
$m\in\mathbb{N}^{\star}$ and consider a section $s\in H^{0}(X,mF)$ such that
the zero locus $s$ contains no lc centers of the lc pair $(X,E)$. Then the
multiplication map induced by the tensor product with $s$

(4.5.1) $s\otimes:\qquad H^{q}(X,K_{X}+E+F)\rightarrow
H^{q}(X,K_{X}+E+(m+1)F)$

is injective for every $q\in\mathbb{N}$.

###### Proof.

Let $s_{E}$ be the canonical section of $E$. After some desingularisation, by
the assumption on the zero locus of $s$, we can suppose that

$\mathrm{Div}(s)=\sum a_{i}F_{i},$

and $\sum F_{i}+E$ is snc. Indeed, consider a birational map
$\pi:\widehat{X}\to X$ such that the inverse image of $E+\mathrm{Div}(s)$ is
snc. Given a $(n,q)$-form $\lambda$ with values in $E+F$, the $\pi$ inverse
image of $\displaystyle\frac{\lambda}{s_{E}}$ is a $(n,q)$-form with log-poles
along the support of the inverse image of $E$, i.e. we have

(4.5.2)
$\pi^{\star}\left(\frac{\lambda}{s_{E}}\right)=\frac{\widehat{\lambda}}{s_{\widehat{E}}}$

where $\widehat{\lambda}$ is a $(n,q)$ form on $\widehat{X}$ with values in
$\widehat{E}+\pi^{\star}(F)$. Another way to express this is the familiar
equality

(4.5.3) $G+\pi^{\star}(K_{X}+E)=K_{\widehat{X}}+\widehat{E}$

where $G$ is effective and the components of $\widehat{E}$ are projecting onto
the log-canonical centers of $(X,E)$ (since we are only blowing up
submanifolds which are transverse to $E$). In particular, the components of
the support of $\widehat{E}$ is disjoint from the inverse image of
$\mathrm{Div}(s)$.

Now if we are able to show that $\widehat{\lambda}$ is $\bar{\partial}$-exact,
then we write

(4.5.4)
$\pi^{\star}\left({\lambda}\right)=\frac{\widehat{\lambda}}{s_{\widehat{E}}}\otimes\pi^{\star}(s_{E}).$

The RHS is holomorphic, since $\widehat{E}$ is reduced and its support is
contained in the inverse image of $E$. It follows that the inverse image of
$\lambda$ is zero in cohomology and then the same is true for $\lambda$.

We suppose for simplicity that $\mathrm{Div}(s)=aF_{1}+bF_{2}$ (the general
case follows from the same argument, by using the remark after Lemma 4.7). Let
$s_{i}$ and $s_{E}$ be the canonical section of $F_{i}$ and of $E$,
respectively. One of the important points in our proof is to construct a
sequence $(a_{k},b_{k})\in\mathbb{N}^{2}$ with the following properties.

  1. (1)

We have $(a_{0},b_{0})=(0,0),(a_{N},b_{N})=(a,b)$ and for each $k\geqslant 0$
the successive differences $a_{k+1}-a_{k},b_{k+1}-b_{k}$ are elements of the
set $\\{0,1\\}$.

  2. (2)

The bundle $F_{k}:=F+a_{k}F_{1}+b_{k}F_{2}$ admits a metric $\displaystyle
h_{k}$ whose local weights can be written as

$\varphi_{k}=c\varphi_{F}+c_{1}\log|f_{1}|^{2}+c_{2}\log|f_{2}|^{2}$

where $h_{F}=e^{-\varphi_{F}}$ is the non-singular, semi-positively curved
metric on $F$, the constant $c$ is positive and $c_{i}\in]0,1[$ for $i=1,2$.

Such sequence will be constructed in Lemma 4.7. We denote by

(4.5.5) $s_{k}:=s_{1}^{(a_{k+1}-a_{k})}\otimes s_{2}^{(b_{k+1}-b_{k})}\in
H^{0}(X,(a_{k+1}-a_{k})F_{1}+(b_{k+1}-b_{k})F_{2}).$

The property (1) above shows that (4.5.1) can written as composition of the
maps

(4.5.6) $s_{k}:H^{q}(X,K_{X}+E+F_{k})\rightarrow H^{q}(X,K_{X}+E+F_{k+1}).$

Thus, in order to conclude it is sufficient to prove that (4.5.6) is injective
for every $k$.

Let $\lambda$ be a smooth form in $H^{q}(X,K_{X}+E+F_{k})$ such that

$\lambda\otimes s_{k}=0\in H^{q}(X,K_{X}+E+F_{k+1}).$

In other words there exists a smooth form $\alpha\in
C^{\infty}_{(n,q-1)}(X,E+F_{k+1})$ such that

$s_{k}\otimes\lambda=\bar{\partial}\alpha.$

By property (2) we can equip $F_{k}=F+a_{k}F_{1}+b_{k}F_{2}$ with the metric
$h_{k}$. Now since the weight of the metric $h_{k}$ contains a singular part
$c_{i}\log|f_{i}|^{2}$ for some rational number $0<c_{i}<1$ and it follows
from Lemma 4.6 below that we can find forms $\beta_{1},\beta_{2}$ with log
poles along $E+F$ and $\theta$ with log poles along $E$, such that the
equality

(4.5.7)
$\frac{\lambda}{s_{E}}=\bar{\partial}\Big{(}\frac{\alpha}{s_{E}s_{k}}\Big{)}=D^{\prime}_{h_{k}}(\beta_{1})+\bar{\partial}\theta+i\Theta_{h_{k}}\wedge\beta_{2}$

holds true on $X\setminus(E+F_{1}+F_{2})$. Therefore

(4.5.8)
$\frac{\lambda}{s_{E}}-\bar{\partial}\theta=D^{\prime}_{h_{k}}(\beta_{1})+i\Theta_{h_{k}}\wedge\beta_{2}$

in the complement of $E+F_{1}+F_{2}$. Since $\theta$ is smooth near a generic
point of $F_{1}+F_{2}$, the forms on the LHS of (4.5.8) are non-singular at
the generic point of $F_{1}+F_{2}$, we are precisely in the framework of
Theorem 1.1.

We infer that

$\displaystyle\lambda-s_{E}\bar{\partial}\theta$

is $\bar{\partial}$-exact. Since $\theta$ has by construction only simple
poles along $E$, $\bar{\partial}(\theta)\otimes
s_{E}=\bar{\partial}(\theta\cdot s_{E})$ is $\bar{\partial}$-exact. Then
$\lambda$ is $\bar{\partial}$-exact and (4.5.6) is injective. The theorem is
thus proved, modulo the two lemmas below. ∎

To complete our proof, we establish next the auxiliary statements we used; we
start with the existence of forms $\beta_{i},\theta$.

######  Lemma 4.6.

Let $F_{1}+F_{2}+E$ be a snc divisor on $X$ and let $L$ be a holomorphic line
bundle on $X$ with a possible singular metric $h_{L}$ such that

$i\Theta_{h_{L}}(L)=c_{1}[F_{1}]+c_{2}[F_{2}]+\theta_{L}$

where $c_{1},c_{2}$ are non zero and $\theta_{L}$ is smooth. Let $A$ be a
$L$-valued smooth $(n,0)$-form on $X\setminus(E+F_{1}+F_{2})$ with log poles
along $E+F_{1}+F_{2}$. Then we can find forms $\beta_{1},\beta_{2}$ with at
most log poles along $E+F_{1}+F_{2}$, and $\theta$ with at most log poles
along $E$, such that the equality

$\bar{\partial}A=\bar{\partial}\theta+D^{\prime}_{h_{L}}(\beta_{1})+i\Theta_{h_{L}}\wedge\beta_{2}$

holds in the complement of the support of $E+F_{1}+F_{2}$.

###### Proof.

Let $\\{\Omega_{\alpha}\\}_{\alpha}$ be a cover of $X$ and let
$\theta_{\alpha}$ be a partition of unity with respect to the covering. We
suppose that $F_{1}+F_{2}$ on $\Omega_{\alpha}$ are defined by
$z_{1,\alpha}\cdot z_{2,\alpha}=0$. We define a smooth vector field

$V_{1}:=\sum_{\alpha}\theta_{\alpha}\frac{\partial}{\partial z_{1,\alpha}}\in
C^{\infty}(X,T_{X}).$

Now on any local open set $\Omega\subset X$, we suppose that $F_{1}+F_{2}$ is
defined by $z_{1}\cdot z_{2}=0$. Then $V_{1}$ is of type:

$V_{1}|_{\Omega}=(z_{1}+\mathcal{O}((z_{1})^{2})\frac{\partial}{\partial
z_{1}}+\mathcal{O}(z_{1}z_{2})\frac{\partial}{\partial z_{2}}+\sum_{i\geqslant
3}a_{i}\frac{\partial}{\partial z_{i}}.$

Then there is a constant $c$ such that $\displaystyle
A_{1}:=A-cD^{\prime}_{h_{L}}(V_{1}\rfloor A)$ is smooth on
$X\setminus(E+F_{2})$ and simple pole along $E+F_{2}$. Moreover, by
construction, $V_{1}\rfloor A$ has no poles along $F_{1}$ and it has at most
log poles along $E+F_{2}$. Note that we used the fact that $A$ is a
$(n,0)$-form where $n=\dim X$.

Now

$\bar{\partial}(A)=\bar{\partial}(A_{1})+c\bar{\partial}(D^{\prime}_{h_{L}}(V_{1}\rfloor
A))=\bar{\partial}(A_{1})+D^{\prime}_{h_{L}}\bar{\partial}(cV_{1}\rfloor
A)+i\Theta_{h_{L}}(L)\wedge(cV_{1}\rfloor A)$

pointwise on $X\setminus(E+F_{1}+F_{2})$. By construction,
$\bar{\partial}(cV_{1}\rfloor A)$ and $cV_{1}\rfloor A$ have log poles
$E+F_{2}$.

We repeat the same argument for $A_{1}$ to remove its pole along $F_{2}$. The
resulting forms are $\theta$ with log poles along $E$ and
$\beta_{1},\beta_{2}$ with log poles along $E+F$ so our lemma proved. ∎

Finally, the construction of the sequence $(a_{k},b_{k})_{k=0,\dots,N}$ with
the two properties (1) and (2) can be done as follows.

######  Lemma 4.7.

Let $F$ be a holomorphic line bundle on $X$ and let $m\in\mathbb{N}^{\star}$.
Let $s\in H^{0}(X,mF)$ such that $\mathrm{Div}(s)=aF_{1}+bF_{2}$ for some
$a,b\in\mathbb{N}^{\star}$. We consider a sequence
$\\{(a_{k},b_{k})\\}_{k=1}^{N}\subset\mathbb{N}^{2}$ constructed as follows:

  1. (i)

We define $(a_{0},b_{0})=(0,0)$. For $k\geqslant 1$, if $(a_{k},b_{k})=(a,b)$,
then we stop at this step. If not, we take
$c_{k}:=\min\\{\frac{a_{k}}{a},\frac{b_{k}}{b}\\}$.

  2. (ii)

If $\frac{a_{k}}{a}>c_{k}$, then we set $a_{k+1}:=a_{k}$ and if
$\frac{a_{k}}{a}=c$, we define $a_{k+1}:=a_{k}+1$.

  3. (iii)

We do the same for $b_{k+1}$: if $\frac{b_{k}}{b}>c_{k}$, then
$b_{k+1}:=b_{k}$ and if $\frac{b_{k}}{b}=c$, then we set $b_{k+1}:=b_{k}+1$.

Then for every $k\in\mathbb{N}$, we have

(4.7.1)
$0=\min\\{a_{k}-c_{k}a,b_{k}-c_{k}b\\}\leqslant\max\\{a_{k}-c_{k}a,b_{k}-c_{k}b\\}<1.$

In particular, there exist $\displaystyle
c_{1,k},c_{2,k}\in]0,1[\cap\mathbb{Q}$ and $d_{k}\in\mathbb{Q}^{+}$ such that

(4.7.2) $F+a_{k}F_{1}+b_{k}F_{2}\equiv d_{k}F+c_{1,k}F_{1}+c_{2,k}F_{2}.$

###### Proof.

We prove (4.7.1) by induction. By the definition of $c_{k}$, we have

$\min\\{a_{k}-c_{k}a,b_{k}-c_{k}b\\}=0$

so it is enough to establish the strict inequality for $\max$.

Assume that (4.7.1) holds for $k$. Note first that by construction,
$c_{k+1}>c_{k}$. There are three cases to consider.

$\bullet$ $c_{k}=\frac{a_{k}}{a}=\frac{b_{k}}{b}$. Then
$a_{k+1}-c_{k}a=b_{k+1}-c_{k}b=1$. As $c_{k+1}>c_{k}$, we know that (4.7.1)
holds for $k+1$.

$\bullet$ $c_{k}=\frac{a_{k}}{a}<\frac{b_{k}}{b}$. Then $a_{k+1}=a_{k}+1$ and
$b_{k+1}=b_{k}$. Therefore

$a_{k+1}-c_{k+1}a=a_{k}+1-c_{k+1}a=1+(c_{k}-c_{k+1})a<1,$

and

$b_{k+1}-c_{k+1}b=b_{k}-c_{k+1}b<b_{k}-c_{k}b<1.$

Then (4.7.1) holds for $k+1$.

$\bullet$ $c_{k}=\frac{b_{k}}{b}<\frac{a_{k}}{a}$. We argue as in the second
case.

For the second part of the lemma, thanks to (4.7.1), we have

$F+a_{k}F_{1}+b_{k}F_{2}\equiv(1+c_{k}m)F+(a_{k}-c_{k}a)F_{1}+(b_{k}-c_{k}b)F_{2}.$

Let $\varepsilon\in\mathbb{Q}^{+}$ small enough. Then

$F+a_{k}F_{1}+b_{k}F_{2}\equiv(1+c_{k}m-\varepsilon
m)F+(a_{k}-c_{k}a+\varepsilon a)F_{1}+(b_{k}-c_{k}b+\varepsilon b)F_{2}.$

Thanks to (4.7.1), for $\varepsilon$ small enough, we know that
$(a_{k}-c_{k}a+\varepsilon a)$ and $(b_{k}-c_{k}b+\varepsilon b)\in]0,1[$.
Moreover $(1+c_{k}m-\varepsilon m)>0$. Then (4.7.2) is proved. ∎

In general $(s=0)$ may have several components,
$\mathrm{Div}(s)=\sum_{1\leqslant i\leqslant s}a_{i}F_{i}$. The construction
of the sequence $\\{(a_{1,k},\cdots,a_{s,k})\\}_{k\in\mathbb{N}}$ is
absolutely identical.

Indeed, suppose that $(a_{1,k},\cdots,a_{s,k})$ has already been constructed,
for some $k\geqslant 1$. The next element $(a_{1,k+1},\cdots,a_{s,k+1})$ of
our sequence is obtained as follows: let

$c_{k}=\min\Big{\\{}\frac{a_{1,k}}{a_{1}},\frac{a_{2,k}}{a_{2}},\cdots,\frac{a_{s,k}}{a_{s}}\Big{\\}}.$

We define

$a_{i,k+1}:=a_{i,k}+1$

in case $\displaystyle\frac{a_{i,k}}{a_{i}}=c_{k}$, and

$a_{i,k+1}:=a_{i,k}$

if $\frac{a_{i,k}}{a_{i}}>c_{k}.$ One can easily verify by induction that for
every $k\in\mathbb{N}$ the relation

(4.7.3)
$0=\min\\{a_{i,k}-c_{k}a_{i}\\}_{i=1}^{s}\leqslant\max\\{a_{i,k}-c_{k}a_{i}\\}_{i=1}^{s}<1$

holds, precisely as we did in the case $s=2$.

##  5\. Appendix

We detail here the calculations needed for the evaluation of the limit (4.2.7)
(and discussion of the examples). They are rather long, but after all, nothing
more than Taylor expansions. In connection to these topics, we are referring
to the very nice article [1] and the references therein.

As a ”warm-up” we consider the following particular case.

$(\diamondsuit)$ _Assume that we can find a smooth metric $h_{0}$ on
$\mathcal{O}(Y)$ whose local weights $\varphi_{0}$ are _independent_ on the
$z_{1},\dots,z_{r}$, where the $\displaystyle(z_{i})_{i=1,\dots,n}$ are
coordinates such that $\varphi_{L}(z)=\sum
q_{i}\log|z_{i}|^{2}+\varphi_{L,0}.$_

In such case we argue as follows. Consider the following measure

(5.0.1)
$d\lambda_{f,q}(z):=\frac{f(z)}{\prod_{i=1}^{r}z_{i}^{k_{i}}}\frac{d\lambda(z)}{\prod_{i=1}^{r}|z_{i}|^{2-2\delta_{i}}}$

where the $\delta_{i}$ are strictly positive reals and $k_{i}$ are positive
integers.

Thanks to $(\diamondsuit)$ we immediately see that the quantity

(5.0.2)
$\int_{(\mathbb{C}^{n},0)}\rho_{\varepsilon}^{\prime}\left(\prod|z_{i}|^{2}e^{-\varphi_{0}(z^{\prime})}\right)d\lambda_{f,q}(z)$

where $z^{\prime}=(z_{r+1},\dots z_{n})$ tends to zero, regardless to the size
of the multiplicities $q_{i}$. This is the case, since $f$ is smooth, so
(5.0.2) amounts to the evaluation of

(5.0.3)
$\int_{(\mathbb{C}^{n},0)}g(z^{\prime})\rho_{\varepsilon}^{\prime}\left(\prod|z_{i}|^{2}e^{-\varphi_{0}(z^{\prime})}\right)\frac{\prod_{i=1}^{r}z_{i}^{m_{i}}\overline{z}_{i}^{m_{\overline{i}}}}{\prod_{i=1}^{r}z_{i}^{k_{i}}}\frac{d\lambda(z)}{\prod_{i=1}^{r}|z_{i}|^{2-2\delta_{i}}}$

where $g$ is a smooth function only depending of the variables $z^{\prime}$.

The integral (5.0.3) is _equal_ to zero -for any $\varepsilon>0$ \- unless we
have

(5.0.4) $m_{i}-k_{i}=m_{\overline{i}}\geqslant 0.$

But for such exponents the limit as $\varepsilon\to 0$ of (5.0.3) is equal to
zero, because the pole of order $k_{i}$ disappears, and $\delta_{i}>0$.

####  5.0.1. Taylor expansions

In general, i.e. in the absence of the assumption $(\diamondsuit)$, our
arguments are virtually the same, modulo a few technicalities that we discuss
next.

We will use Taylor formula with integral reminder. In order to illustrate the
type of statement we are after, we first start with a simple case.

Let $\phi$ be a smooth function defined on $\mathbb{R}$, and consider

(5.0.5) $f(x_{1},x_{2}):=\phi(x_{1}+x_{2})$

a function defined say in a ball centred at the origin in $\mathbb{R}^{2}$.

The Taylor expansion of $f$ with respect to $x_{1}$ reads as follows

(5.0.6) $\displaystyle f(x_{1},x_{2})=$
$\displaystyle\phi(x_{2})+x_{1}\phi^{\prime}(x_{2})+\dots+\frac{x_{1}^{k}}{k!}\phi^{(k)}(x_{2})$
(5.0.7) $\displaystyle+$
$\displaystyle\frac{x_{1}^{k+1}}{(k+1)!}\int_{0}^{1}\phi^{(k+1)}(tx_{1}+x_{2})(1-t)^{k}dt.$