# Albanese morphism of log smooth klt compact Kähler manifold with nef log
anticanonical divisor

Xiaojun WU

###### Abstract

Let $(X,\omega)$ be an n-dimensional compact Kähler manifold. Let
$D=\sum(1-\beta_{j})Y_{j}=\sum(1-\beta_{j})[s_{j}=0]$ a divisor with simple
normal crossings with $\beta_{j}\in]0,1[$ such that $-(K_{X}+D)$ is nef. We
show that its Albanese map is submersion outside an analytic set of
codimension larger than two with connected fibres.

##  1 Regularity of Monge-Ampère equation with conic singularities

The following result on the regularities of the Monge-Ampère equation is well-
known to experts. Since we have not found a precise reference of some result,
we provide some proof here. Let us fix some notations. Let $(X,D)$ be a log
smooth klt pair, i.e. $X$ is a compact Kähler manifold, and
$D=\sum(1-\beta_{k})Y_{k}$ is a $\mathbb{R}$-divisor with simple normal
crossing support such that $\beta_{k}\in]0,1[$ for all $k$. The coefficients
$(\beta_{k})$ are called standard orbifold is $\beta_{k}=1/n_{k}$ for some
$n_{k}\in\mathbb{N}^{*}$ for all $k$. We denote by $\omega_{\beta}$ the
standard cone metric attached with $(\mathbb{C}^{n},D)$, i.e.

$\omega_{\beta}:=i\sum_{k=1}^{d}\frac{dz_{k}\wedge
d\bar{z}_{k}}{|z_{k}|^{2(1-\beta_{k})}}+i\sum_{k=d+1}^{n}dz_{k}\wedge
d\bar{z}_{k}.$

A metric $\omega$ is said to have conic singularities if it is quasi-isometric
to the model metric with conic singularities: more precisely, near each point
$p\in\mathrm{Supp}(D)$ where $\mathrm{Supp}(D)$ is defined by the equation
$\\{z_{1}\cdots z_{d}=0\\}$ for some holomorphic system of coordinates
$(z_{i})$, we want $\omega$ to satisfy

$C^{-1}\omega_{\beta}\leq\omega\leq C\omega_{\beta}$

for some constant $C>0$ and some $\beta$ (near $p$). We have the following
result due to [CGP13].

######  Proposition 1.

Let $X$ be a compact Kähler manifold and
$D=\sum(1-\beta_{j})Y_{j}=\sum(1-\beta_{j})[s_{j}=0]$ a divisor with simple
normal crossings with $\beta_{j}\in]0,1[$. Let
$\mu_{D}=dV_{\omega}/\prod_{j}|s_{j}|^{2(1-\beta_{j})}$ be a volume form with
conic singularities along $D$, $\mu\in\mathbb{R}$, and $\omega$ a Kähler form
on $X$. Then any (bounded) solution $\varphi$ of

$(\omega+i\partial{\overline{\partial}}\varphi)^{n}=e^{\mu\varphi}\mu_{D}$

is Hölder-continuous and the metric
$\omega+\mathrm{id}{\overline{\partial}}\varphi$ has conic singularities along
$D$.

In the following section, we start by considering the standard orbifold case
to illustrate the proof. In this case, we have a more precise regularity of
$\varphi$, and we adapt the notations of [CGP13] in this case. As written at
the beginning of this section, the following Proposition 3 is well-known to
experts.

We fix $\mathbb{D}^{n}\subset\mathbb{C}^{n}$ the unit polydisk centered at the
origin, and a divisor $D=\sum_{k=1}^{d}(1-1/n_{k})D_{k}$ where
$D_{k}=\\{z_{k}=0\\}$ for all $k$, and $d\leq n$.

Consider the branched cover

$\begin{array}[]{cccc}\pi:&\mathbb{D}^{d}\times\mathbb{D}^{n-d}&\longrightarrow&\mathbb{D}^{d}\times\mathbb{D}^{n-d}\\\
&(z_{1},\ldots,z_{d},z_{d+1},\ldots,z_{n})&\longmapsto&(z_{1}^{n_{1}},\ldots,z_{d}^{n_{d}},z_{d+1},\ldots,z_{n}).\end{array}$

If we denote by $w$ the coordinates upstairs,

$\pi^{*}\omega_{\beta}=i\sum_{k=1}^{d}|n_{k}|^{2}dw_{k}\wedge
d\bar{w}_{k}+i\sum_{k=d+1}^{n}dw_{k}\wedge d\bar{w}_{k}.$

In particular, $\pi^{*}\omega_{\beta}$ is a genuine metric upstairs. Recall
that in the standard orbifold case, a function $f$ is said to have orbifold
$C^{k,\alpha}$ regularity (denoted by $C^{k,\alpha,\beta}$) if its pull-back
$\pi^{*}f$ by the ramified cover $\pi$ is $C^{k,\alpha}$ in the usual sense.
More generally, let $\tau$ (resp. $\sigma$) be a bounded $(1,0)$-form (resp.
$(1,1)$-form) on $\mathbb{D}^{n}\setminus D$. Then we say that $\tau\in
C^{\alpha,\beta}$ if for all $k$, we have
$\pi^{*}\tau(\frac{\partial}{\partial w_{k}})\in C^{\alpha}$ in the usual
sense (resp. $\sigma\in C^{\alpha,\beta}$ if for all $k$ and $l$, we have
$\pi^{*}\sigma(\frac{\partial}{\partial
w_{k}},\frac{\partial}{\partial\overline{w}_{l}})\in C^{\alpha}$ in the usual
sense). Define

$C^{2,\alpha,\beta}:=\\{f\in L^{\infty}(\mathbb{D}^{n});f,\partial
f,{\overline{\partial}}\partial f\in C^{\alpha,\beta}\\}$

with the associated natural norm. It is checked in Lemma 7.2 [CGP13] that this
definition coincides with the definition of Donaldson [Don12].

Cover $X$ be open sets $U_{i}$ so that each open set is biholomorphic to the
unit polydisk with ramified cover associated with $D$. Denote $\tilde{U}_{i}$
the ramified cover of $\pi_{i}:\tilde{U_{i}}\to U_{i}$. Let $N_{i}$ be the
ramified order. We have a more precise regularity result than Proposition 1,
also stated in [CGP13].

######  Proposition 2.

Let $(X,D)$ as in Proposition 1 with standard orbifold coefficients, and let
$\varphi\in L^{\infty}(X)$ be any solution of

$(\omega+i\partial{\overline{\partial}}\varphi)^{n}=e^{\mu\varphi}\mu_{D}$

Then $\varphi$ belongs to the class $C^{2,\alpha,\beta}$.

More precisely, in the standard orbifold case, by elliptic regularity, one can
show that $\pi^{*}\varphi$ is in fact smooth.

######  Proposition 3.

Let $(X,D)$ as in Proposition 1 with standard orbifold coefficients, and let
$\varphi\in L^{\infty}(X)$ be any solution of

$(\omega+i\partial{\overline{\partial}}\varphi)^{n}=e^{\mu\varphi}\mu_{D}$

Then $\pi^{*}\varphi$ is smooth in each ramified cover.

###### Proof.

Note that $\pi^{*}d\nu_{D}$ is a smooth volume from upstairs. Without loss of
generality, assume that $\omega=i\partial{\overline{\partial}}\psi$ for some
smooth function locally. Locally the pullback of the Monge-Ampère equation on
the ramified cover can be written as

$\mathrm{det}(\frac{\partial^{2}}{\partial
w_{i}\partial\overline{w}_{j}}\pi^{*}(\psi+\varphi))=ge^{\mu\pi^{*}\varphi}$

for some nowhere vanishing smooth function $g$. By Proposition 2,
$\pi^{*}(\omega+i\partial{\overline{\partial}}\varphi)$ is a genuine metric
with $C^{0,\alpha}$ coefficients. Denote locally

$\pi^{*}(\omega+i\partial{\overline{\partial}}\varphi)=\sum_{i,j}h_{ij}\sqrt{-1}dw^{i}\wedge
d\overline{w}^{j}.$

From the Monge-Ampère equation, we have an a priori positive lower bound on
the determinant of $h_{ij}$. In particular, if $h_{ij}\in C^{k,\alpha}$ for
some $k\geq 0$, then $\mathrm{det}(h_{ij})^{-1}\in C^{k,\alpha}$ (hence
$h^{ij}\in C^{k,\alpha}$). Take $\frac{\partial}{\partial
w_{l}}\mathrm{log}(\forall l)$ for both sides of equation

$h^{ij}\frac{\partial^{2}}{\partial
w_{i}\partial\overline{w}_{j}}\frac{\partial}{\partial
w_{l}}\pi^{*}(\omega+i\partial{\overline{\partial}}\varphi)=\frac{\partial}{\partial
w_{l}}\mathrm{log}g+\mu\frac{\partial}{\partial w_{l}}\pi^{*}\varphi.$

Inductively for $k\geq 0$, by interior Schauder estimates for linear elliptic
equations with coefficients in $C^{k,\alpha}$, we obtain the a priori
$C^{k+3,\alpha}$ norm estimate of $\pi^{*}\varphi$ in terms of the
$C^{k+1,\alpha}$ norm of $\pi^{*}\varphi$. Thus $\pi^{*}\varphi$ is smooth in
any ramified cover by bootstrapping. ∎

We now recall the definition of a singular metric on a vector bundle according
to [Paun16].

######  Definition 1.

A singular Hermitian metric $h$ on $E$ is given locally by a measurable,
possibly unbounded map with values in the set of semi-positive Hermitian
matrices, such that $0<\mathrm{det}h<\infty$ almost everywhere.

By this definition, a solution of the Monge-Ampère equation with conic
singularities defines a singular metric on $T_{X}$. In particular, the
solution also induces a singular metric on any quotient bundle of $T_{X}$. We
observe that by the Monge-Ampère equation, the Ricci curvature of the singular
metric is well defined as a current. However, one can notice that the
curvature tensor of $T_{X}$ is not necessarily well-defined as a current with
values in semi-positive, possibly unbounded Hermitian matrices.

Now we return to the existence and regularity of the Monge-Ampère equation for
the general coefficient case.

In fact, the work of [Gue13] and [CGP13] gives the following weak estimate for
the following type of Monge-Ampère equation. The theorem is not essentially
used for the following section, but the discussion after the theorem also
applies to this slight generalisation. So we still state it explicitly.

######  Theorem 1.

Let $X$ be an n-dimensional compact Kähler manifold, and let
$D=\sum_{i}a_{i}D_{i}$, $E=\sum_{j}b_{j}E_{j}$ be two effective
$\mathbb{R}$-divisors with simple normal crossing support, such that for all
$1\leq i\leq r$, $0<a_{i}<1$. Assume that $D$ and $E$ have no common
irreducible component. Let $\omega$ be a Kähler metric on $X$, $dV$ a smooth
volume form, and let $\varepsilon>0$. Then the weak solution of the Monge-
Ampère equation

$\langle(\omega+\frac{i}{2\pi}\partial{\overline{\partial}}\varphi)^{n}\rangle=e^{\varepsilon\varphi}\frac{\prod|t_{j}|^{2b_{j}}dV}{\prod|s_{i}|^{2a_{i}}}$

exists, which is smooth on $X\smallsetminus(D\cup E)$ and has an upper bound
by a metric with conic singularity along $D$. Here $\langle\bullet\rangle$ is
the positive intersection product defined in [BEGZ10]. Here $s_{i}$(resp.
$t_{j}$) is the canonical section of $\mathcal{O}(D_{i})$ (resp.
$\mathcal{O}(E_{j})$) and $|s_{i}|^{2}$ (resp. $|t_{j}|^{2}$) is the norm of
$s_{i}$ (resp. $t_{j}$) with respect to some smooth metric.

We observe that the existence of a solution is proved in [BEGZ10]. As a
consequence of Theorem 1, there exists $C>0$ such that the solution has on
$X\smallsetminus(D\cup E)$ an upper bound

$\omega+\frac{i}{2\pi}\partial{\overline{\partial}}\varphi\leq\frac{C\omega}{\prod_{i}|s_{i}|^{2a_{i}}}.$

By the Monge-Ampère equation, we find on $X\smallsetminus(D\cup E)$ a lower
bound

$\omega+\frac{i}{2\pi}\partial{\overline{\partial}}\varphi\geq
e^{\varepsilon\varphi}\frac{\prod|t_{j}|^{2b_{j}}\omega}{\prod|s_{i}|^{2a_{i}}}(\frac{C}{\prod_{i}|s_{i}|^{2a_{i}}})^{-(n-1)}.$

Notice that since the solution is smooth on $X\smallsetminus(D\cup E)$, the
above inequalities are satisfied pointwise. By the result of [BEGZ10],
$|\varphi|$ is uniformly bounded on $X$. In particular, we have

$\omega+\frac{i}{2\pi}\partial{\overline{\partial}}\varphi\geq\frac{C\prod|t_{j}|^{2b_{j}}\omega}{\prod|s_{i}|^{2a_{i}}}(\frac{C}{\prod_{i}|s_{i}|^{2a_{i}}})^{-(n-1)}.$

In conclusion outside $D\cup E$, the solution
$\omega+\frac{i}{2\pi}\partial{\overline{\partial}}\varphi$ viewed as a
Hermitian form over $T_{X}$ with respect to $\omega$ has positive eigenvalues
bounded from above by $\frac{C}{\prod_{i}|s_{i}|^{2a_{i}}}$ and bounded from
below by
$\frac{C\prod|t_{j}|^{2b_{j}}}{\prod|s_{i}|^{2a_{i}}}(\frac{C}{\prod_{i}|s_{i}|^{2a_{i}}})^{-(n-1)}.$

Let us observe that for the singular metric on the determinant line bundle of
the quotient bundle $Q$ given by a short exact sequence of vector bundles

$0\to S\to T_{X}\to Q\to 0,$

the curvature form is well-defined as a current. We detail the argument below.
Suppose we are in the situation of Theorem 1, with the same notation above.
Since the metric is smooth outside $D\cup E$, we only need to study the
neighbourhood of $D\cup E$. By a $C^{\infty}$ splitting of the exact sequence,
we can view $Q$ as a subbundle of $T_{X}$.
$\omega+\frac{i}{2\pi}\partial{\overline{\partial}}\varphi$ thus induces a
Hermitian form over $Q$ which we will denote by
$\omega+\frac{i}{2\pi}\partial{\overline{\partial}}\varphi|_{Q}$. By the
minimax principle, for the induced Hermitian form on $Q$, the eigenvalues are
bounded from above by $\frac{C}{\prod_{i}|s_{i}|^{2a_{i}}}$ and bounded from
below by
$\frac{C\prod|t_{j}|^{2b_{j}}}{\prod|s_{i}|^{2a_{i}}}(\frac{C}{\prod_{i}|s_{i}|^{2a_{i}}})^{-(n-1)}.$
To prove that the curvature of $\mathrm{det}(Q)$ is well-defined as a current
(not necessarily positive), it is enough to prove that
$\mathrm{log}(\mathrm{det}(\omega+\frac{i}{2\pi}\partial{\overline{\partial}}\varphi|_{Q}))\in
L^{1}_{\mathrm{loc}}$.
$\mathrm{det}(\omega+\frac{i}{2\pi}\partial{\overline{\partial}}\varphi|_{Q})$
is the product of all eigenvalues of the Hermitian form
$\omega+\frac{i}{2\pi}\partial{\overline{\partial}}\varphi|_{Q}$. Thus we get
the estimate for the potential

$|\mathrm{log}(\mathrm{det}(\omega+\frac{i}{2\pi}\partial{\overline{\partial}}\varphi|_{Q}))|\leq\sum_{i}C_{i}\mathrm{log}|s_{i}|^{2}+\sum_{j}C_{j}\mathrm{log}|t_{j}|^{2}+C$

for some $C_{i}>0$, $C_{j}>0$ and $C>0$. In the following, we will refer to
this type of control as potentials possessing at most logarithmic poles along
$D\cup E$. Notice also that for any $i$, $\mathrm{log}|z_{i}|$ is locally
integrable with respect to the euclidean metric. In particular, the curvature
of the induced metric on $\mathrm{det}(Q)$ is well defined as a current since
locally it is the $i\partial{\overline{\partial}}$ of some
$L^{1}_{\mathrm{loc}}$ function.

For any global potential $\psi$ (defined on $X$) possessing at most
logarithmic poles along $D\cup E$. As analogy to Monge-Ampère operator in the
sense of Bedford-Taylor [BT82], we want to define
$i\partial{\overline{\partial}}\psi\wedge\omega_{\varphi}^{n-1}$. Fix
$a\in[0,1[$, the integration

$\int\mathrm{log}(r)r^{1-2a}=\frac{1}{2-2a}(\mathrm{log}(r)-\frac{1}{2-2a})r^{2-2a}$

is finite over $[0,1]$. In particular, $\mathrm{log}|z||z|^{-2a}$ as a
function $z\in\mathbb{C}$ is locally integrable near 0 with respect to the
Lebesgue measure. Thus the coefficients of $\psi\wedge\omega_{\varphi}^{n-1}$
are locally integrable with respect to the Lebesgue measure. Define

$i\partial{\overline{\partial}}\psi\wedge\omega_{\varphi}^{n-1}:=i\partial{\overline{\partial}}(\psi\wedge\omega_{\varphi}^{n-1}).$

By Stokes theorem, we have that

$\int_{X}i\partial{\overline{\partial}}\psi\wedge\omega_{\varphi}^{n-1}=0.$

##  2 Albanese morphism

In this section, we start by generalising the results of [Cao13] to log smooth
cases with standard orbifold coefficients. To start with, we need the
following result.

######  Theorem 2.

Let $(X,\omega)$ be an n-dimensional compact Kähler manifold. Let
$D=\sum(1-1/n_{j})Y_{j}=\sum(1-1/n_{j})[s_{j}=0]$ a divisor with simple normal
crossings with $n_{j}\in\mathbb{N}^{*}$ such that $-(K_{X}+D)$ is nef. Let

$0=\mathcal{E}_{0}\subset\mathcal{E}_{1}\subset\cdots\subset\mathcal{E}_{s}=T_{X}$

be a filtration of torsion-free subsheaves such that
$\mathcal{E}_{i+1}/\mathcal{E}_{i}$ is an $\omega$-stable torsion-free
subsheaf of $T_{X}/\mathcal{E}_{i}$ of maximal slope. Then for any $i$, the
slope of $\mathcal{E}_{i+1}/\mathcal{E}_{i}$ with respect to $\omega^{n-1}$,
namely

$\mu(\mathcal{E}_{i+1}/\mathcal{E}_{i}):=\int_{X}c_{1}(\mathcal{E}_{i+1}/\mathcal{E}_{i})\wedge\omega^{n-1},$

is positive.

###### Proof.

By the stability condition, to prove the theorem, it is sufficient to prove
that for any $i$

$\int_{X}c_{1}(T_{X}/\mathcal{E}_{i})\wedge\omega^{n-1}\geq 0.$

Let $r$ be the generic rank of $T_{X}/\mathcal{E}_{i}$ for some fixed $i$. The
naturally induced morphism
$\wedge^{r}T_{X}\to\mathrm{det}(T_{X}/\mathcal{E}_{i})$ corresponds to a
section $\tau\in
H^{0}(X,\mathrm{det}(T_{X}/\mathcal{E}_{i})\otimes\Omega^{r}_{X})$. $\tau$ is
non-vanishing outside a closed analytic set of codimension at least two on
which $T_{X}/\mathcal{E}_{i}$ is locally free. Fix an arbitrary smooth metric
$h$ on $\mathrm{det}(T_{X}/\mathcal{E}_{i})$.

The critical step is the existence of positive closed $(1,1)$-current in a
Kähler class which is smooth outside an SNC divisor and whose Ricci curvature
can be taken “arbitrary small” outside the divisor using the theorems in
[GP16] and [CGP13].

Let $T_{\varepsilon}$ be a sequence of smooth forms in $c_{1}(-(K_{X}+D))$
such that $T_{\varepsilon}\geq-\varepsilon\omega$ whose existence is ensured
by the nefness condition. To get the lower bound, we want to solve the
following Kähler-Einstein type of equation

$\mathrm{Ric}(\omega_{\varphi_{\varepsilon}})=-\varepsilon\omega_{\varphi_{\varepsilon}}+\varepsilon\omega+T_{\varepsilon}+[D]$

where
$\omega_{\varphi_{\varepsilon}}:=\omega+i\partial{\overline{\partial}}\varphi_{\varepsilon}$
is the unknown. Notice that both sides belong to the class $c_{1}(-K_{X})$. In
order to solve the Kähler-Einstein type of equation, we thus solve the
following Monge-Ampère equation by Theorem 1. Let $\gamma$ be a smooth
representative of the class $\\{[D]\\}$, which is induced from the curvature
forms of some smooth metrics $(\mathcal{O}(Y_{i}),h_{i})$. By the
$\partial{\overline{\partial}}$-lemma, there exists $f_{\varepsilon}\in
C^{\infty}(X)$ such that
$T_{\varepsilon}+\gamma=\mathrm{Ric}(\omega)+\frac{i}{2\pi}\partial{\overline{\partial}}f_{\varepsilon}$.
The Monge-Ampère equation equivalent to the Kähler-Einstein type of equation
can be written as

$\omega_{\varphi_{\varepsilon}}^{n}=\frac{\omega^{n}e^{\varepsilon\varphi_{\epsilon}-f_{\varepsilon}}}{\prod_{i}|s_{i}|_{h_{i}}^{2(1-1/n_{i})}}.$

Take the same notations as after Proposition 1. Note that
$\pi^{*}\omega_{\varphi_{\varepsilon}}$ is a smooth metric when pulling back
onto the ramified cover by Proposition 3. Calculate on the ramified cover

$i\partial{\overline{\partial}}\pi^{*}\mathrm{log}|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}=\frac{i\\{D^{\prime}\pi^{*}\tau,D^{\prime}\pi^{*}\tau\\}}{\pi^{*}|\tau|^{2}}-\frac{i\\{D^{\prime}\pi^{*}\tau,\pi^{*}\tau\\}\wedge\\{\pi^{*}\tau,D^{\prime}\pi^{*}\tau\\}}{\pi^{*}|\tau|^{4}}$
$-\pi^{*}i\Theta(\mathrm{det}(T_{X}/\mathcal{E}_{i}),h)-\frac{\\{\pi^{*}\tau,i\Theta(\pi^{*}\omega_{\varphi_{\epsilon}})\pi^{*}\tau\\}}{\pi^{*}|\tau|^{2}}$

where $\\{\\}$ is a canonical sesquilinear pairing

$C^{\infty}(\wedge^{p}T^{*}_{X}\otimes\Omega_{X}^{r}\otimes\mathrm{det}(T_{X}/\mathcal{E}_{i}))\times
C^{\infty}(\wedge^{q}T^{*}_{X}\otimes\Omega_{X}^{r}\otimes\mathrm{det}(T_{X}/\mathcal{E}_{i}))\to
C^{\infty}(\wedge^{p+q}T^{*}_{X}).$

Note that the right-handed term on the first line is positive in the sense of
currents by Schwarz inequality. Observe that the right-handed term is locally
integrable (and hence well defines its product with
$\pi^{*}\omega_{\varphi_{\epsilon}}^{n-1}$ in the sense of currents on the
ramified cover). Since $\pi^{*}\omega_{\varphi_{\epsilon}}$ is smooth, the
coefficients of
$\frac{\\{\pi^{*}\tau,i\Theta(\pi^{*}\omega_{\varphi_{\epsilon}})\pi^{*}\tau\\}}{\pi^{*}|\tau|^{2}}$
is locally bounded. On the other hand,
$|D^{\prime}\pi^{*}\tau|^{2}/|\pi^{*}\tau|^{2}$ is locally integrable with
respect to the Lebesgue measure. To see this, let $\mathcal{I}$ be the (local)
ideal defined by the coefficients of $\pi^{*}\tau$. Without loss of
generality, it is enough to consider the case that $\mathcal{I}$ is non-
trivial. By Hironaka’s resolution of singularities [Hir64], there exists a
modification $p:U^{\prime}\to\tilde{U}$ such that
$\mathcal{I}\cdot\mathcal{O}_{U^{\prime}}=\mathcal{O}(-\sum_{i}\lambda_{i}E_{i})$
with $\sum_{i}\lambda_{i}E_{i}$ an effective SNC divisor. Note that since
$\tau$ is non-vanishing outside a codimension 2 set, $p$ is not an identity
map. Denote by $D\mathcal{I}$ the (local) ideal defined by the differentials
of the coefficients of $\pi^{*}\tau$. Then
$D\mathcal{I}\cdot\mathcal{O}_{U^{\prime}}$ is contained in
$\mathcal{O}(-\sum_{i}(\lambda_{i}-1)E_{i})$.
$K_{U^{\prime}/\tilde{U}}=\sum_{i}\nu_{i}E_{i}$ with $\nu_{i}\geq 1$. In
particular, the pullback of Lebesgue measure can be written as
$\prod_{i}|s_{E_{i}}|^{2\nu_{i}}$ times a smooth nowhere vanishing form. Thus
the coefficient of the product of
$p^{*}(|D^{\prime}\pi^{*}\tau|^{2}/|\pi^{*}\tau|^{2})$ with the pullback of
Lebesgue measure is locally bounded on $U^{\prime}$. In particular,
$|D^{\prime}\pi^{*}\tau|^{2}/|\pi^{*}\tau|^{2}$ is locally integrable.

On the ramified cover, the Chern curvature, which is also the Levi-Civita
curvature, satisfies

$\mathrm{Ric}(\pi^{*}\omega_{\varphi_{\varepsilon}})=-\varepsilon\pi^{*}\omega_{\varphi_{\varepsilon}}+\pi^{*}(T_{\epsilon}+\epsilon\omega)\geq-\varepsilon\pi^{*}\omega_{\varphi_{\varepsilon}}.$

Consider

$i\partial{\overline{\partial}}\pi^{*}\mathrm{log}|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}\wedge\pi^{*}\omega_{\varphi_{\epsilon}}^{n-1}\geq(-\pi^{*}i\Theta(\mathrm{det}(T_{X}/\mathcal{E}_{i}),h)-\frac{\\{\pi^{*}\tau,i\Theta(\pi^{*}\omega_{\varphi_{\epsilon}})\pi^{*}\tau\\}}{\pi^{*}|\tau|^{2}})\wedge\pi^{*}\omega_{\varphi_{\epsilon}}^{n-1}.$

On the other hand, by Proposition 2.7 [Cao13], with local curvature
calculations, we have

$\frac{\\{\pi^{*}\tau,i\Theta(\pi^{*}\omega_{\varphi_{\epsilon}})\pi^{*}\tau\\}}{\pi^{*}|\tau|^{2}}\wedge\pi^{*}\omega_{\varphi_{\epsilon}}^{n-1}\leq\epsilon\pi^{*}\omega_{\varphi_{\epsilon}}^{n}.$

Note that $\mathrm{log}|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}$ possesses
at most logarithmic poles along $D$. In fact, the dual metric on $T_{X}^{*}$
could only vanish along $D$ when the initial metric on $T_{X}$ has conic
singularities. In particular,
$i\partial{\overline{\partial}}\mathrm{log}|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}\wedge\omega_{\varphi_{\epsilon}}^{n-1}$
is well-defined as discussed after Theorem 1.

Let $\theta_{i}$ be a partition of unity associated with the chosen open
cover.

$0=\sum_{i}\int_{X}\theta_{i}i\partial{\overline{\partial}}\mathrm{log}|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}\wedge\omega_{\varphi_{\epsilon}}^{n-1}=\sum_{i}\int_{X}i\partial{\overline{\partial}}\theta_{i}\mathrm{log}|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}\wedge\omega_{\varphi_{\epsilon}}^{n-1}$
$=\sum_{i}\int_{\tilde{U}_{i}}\frac{1}{N_{i}}\pi_{i}^{*}(i\partial{\overline{\partial}}\theta_{i}\mathrm{log}|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}\wedge\omega_{\varphi_{\epsilon}}^{n-1})=\sum_{i}\int_{\tilde{U}_{i}}\frac{1}{N_{i}}\pi_{i}^{*}(\theta_{i}i\partial{\overline{\partial}}\mathrm{log}|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}\wedge\omega_{\varphi_{\epsilon}}^{n-1}).$

Thus we have

$\int_{X}c_{1}(T_{X}/\mathcal{E}_{i})\wedge\omega^{n-1}=\sum_{i}\int_{\tilde{U}_{i}}\frac{1}{N_{i}}\pi_{i}^{*}(\theta_{i}i\partial{\overline{\partial}}\mathrm{log}|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}\wedge\omega_{\varphi_{\epsilon}}^{n-1})+\int_{X}i\Theta(\mathrm{det}(T_{X}/\mathcal{E}_{i}),h)\wedge\omega_{\varphi_{\epsilon}}^{n-1}$
$\geq-\epsilon\sum_{i}\int_{\tilde{U}_{i}}\frac{1}{N_{i}}\pi_{i}^{*}(\theta_{i}\omega_{\varphi_{\epsilon}}^{n})=-\epsilon\int_{X}\omega_{\varphi_{\epsilon}}^{n}=-\epsilon\int_{X}\omega^{n}.$

The conclusion follows by taking $\epsilon\to 0+$ ∎

Now we consider the general coefficients case following a suggestion of Mihai
Păun.

######  Theorem 3.

Let $(X,\omega)$ be an n-dimensional compact Kähler manifold. Let
$D=\sum(1-b_{j})D_{j}=\sum(1-b_{j})[s_{j}=0]$ a divisor with simple normal
crossings with $b_{j}\in]0,1[$ such that $-(K_{X}+D)$ is nef. Let

$0=\mathcal{E}_{0}\subset\mathcal{E}_{1}\subset\cdots\subset\mathcal{E}_{s}=T_{X}$

be a filtration of torsion-free subsheaves such that
$\mathcal{E}_{i+1}/\mathcal{E}_{i}$ is an $\omega$-stable torsion-free
subsheaf of $T_{X}/\mathcal{E}_{i}$ of maximal slope. Then for any $i$, the
slope of $\mathcal{E}_{i+1}/\mathcal{E}_{i}$ with respect to $\omega^{n-1}$,
namely

$\mu(\mathcal{E}_{i+1}/\mathcal{E}_{i}):=\int_{X}c_{1}(\mathcal{E}_{i+1}/\mathcal{E}_{i})\wedge\omega^{n-1},$

is positive.

###### Proof.

Construct as above proof of Theorem 2 a section $\tau\in
H^{0}(X,\mathrm{det}(T_{X}/\mathcal{E}_{i})\otimes\Omega^{r}_{X})$. As in the
above theorem 2, construct a sequence of metrics $\omega_{\varphi_{\epsilon}}$
with conic singularities by solving Monge-Ampère equations. Consider local
coordinate $(z^{1},\cdots,z^{n})$ such that the support of $D$ is given by
$\\{z^{1}\cdots z^{d}=0\\}$. In the general case, we do a cut-off in a tubular
neighbourhood of $D$ of radius $\delta$ and study the asymptotic behavior as
$\delta\to 0+$ to get the estimate of the slope from the integration of
$i\partial{\overline{\partial}}\mathrm{log}|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}\wedge\omega_{\varphi_{\epsilon}}^{n-1}$
over $X\setminus D$. Locally

$\tau=\sum_{I,|I|=r}\tau_{I}dz^{I}$

with multi-index $I=\\{i_{1},\cdots,i_{r}\\}$ of length $r$ and
$|\tau|^{2}_{\omega_{\varphi_{\epsilon}}}$ by the conic singularities is
locally equivalent to $\sum_{I,|I|=r}\prod_{j,1\leq i_{j}\leq
d}|z_{i_{j}}|^{2-2b_{i_{j}}}|\tau_{I}|^{2}$. Near the origin,

$O(1)\leq-\mathrm{log}|\tau|^{2}_{\omega_{\varphi_{\epsilon}}}\leq-\mathrm{log}(\sum_{I,|I|=r}\prod_{j,1\leq
i_{j}\leq d}|z_{i_{j}}|^{2}|\tau_{I}|^{2})+O(1)$

where $O(1)$ means a bounded term. Thus to study the local integrability of
$-\mathrm{log}|\tau|^{2}_{\omega_{\varphi_{\epsilon}}}$ with respect to some
positive locally finite measure, it is enough to study the local integrability
of $-\mathrm{log}(\sum_{I,|I|=r}\prod_{j,1\leq i_{j}\leq
d}|z_{i_{j}}|^{2}|\tau_{I}|^{2})$ with respect to that measure which has the
same singularities as some local ideals. Note that outside the support of $D$
(where the metric is smooth), pointwise, we have

$i\partial{\overline{\partial}}\mathrm{log}|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}\wedge\omega_{\varphi_{\epsilon}}^{n-1}\geq(-i\Theta(\mathrm{det}(T_{X}/\mathcal{E}_{i}),h)-\epsilon\omega_{\varphi_{\epsilon}})\wedge\omega_{\varphi_{\epsilon}}^{n-1}.$

However, $-\mathrm{log}(\sum_{I,|I|=r}\prod_{j,1\leq i_{j}\leq
d}|z_{i_{j}}|^{2}|\tau_{I}|^{2})$ may have strict positive generic Lelong
number along the support of $D$ which may not be integrable with respect to
metric with Poincaré type singularities along $D$. (For example, in the
Poincaré type singularity case, the local integrability is equivalent to the
fact that $1/(r|\mathrm{log}(r)|^{\alpha})$ is locally integrable near 0 if
and only if $\alpha>1$.)

Thus we need to rewrite $-\mathrm{log}(\sum_{I,|I|=r}\prod_{j,1\leq i_{j}\leq
d}|z_{i_{j}}|^{2}|\tau_{I}|^{2})$ in a golbal form and subtract the divisorial
part along the support of $D$.

Let $\omega_{\lceil D\rceil}$ be a smooth metric on $X\setminus D$ which is
locally quasi-isometric near every point in $\lceil D\rceil=\\{z_{1}\cdots
z_{d}=0\\}$ to

$\omega_{\lceil D\rceil}:=i\sum_{k=1}^{d}\frac{dz_{k}\wedge
d\bar{z}_{k}}{|z_{k}|^{2}}+i\sum_{k=d+1}^{n}dz_{k}\wedge d\bar{z}_{k}.$

(For example, this kind of metric can be constructed as follows. Cover $X$ by
open coordinate charts such that $\lceil D\rceil$ can be written as zeros of
local coordinates. Assume furthermore that the coordinate charts on which
$\lceil D\rceil$ is not irreducible do not intersect each other, up to taking
some further refinement of the open cover. Construct $\omega_{\lceil D\rceil}$
by glueing local ones via a partition of unity. Let us check that the glueing
metric has desired properties.

Assume that $U_{z}$ is a coordinate chart such that $\lceil D\rceil\cap
U_{z}=\\{z_{1}\cdots z_{d}=0\\}$ and that $U_{w}$ is a coordinate chart such
that $\lceil D\rceil\cap U_{w}=\\{w_{1}=0\\}$ with non-empty intersection.
Over $U_{z}\cap U_{w}$, $z_{1}/w_{1}\in\mathcal{O}^{*}(U_{z}\cap U_{w})$ since
they define the same irreducible divisor on the intersection. Thus for some
$C>0$ large enough, $idz_{1}\wedge
d\overline{z}_{1}/|z_{1}|^{2}+Ci\sum_{k=1}^{n}dz_{k}\wedge d\bar{z}_{k}$ is
equivalent to $idw_{1}\wedge
d\overline{w}_{1}/|w_{1}|^{2}+i\sum_{k=2}^{n}dw_{k}\wedge d\bar{w}_{k}$ over
any chosen relative compact subset of $U_{z}\cap U_{w}$. Thus after the
partition of unity, the glueing metric has desired properties. Of course, the
glueing metric is not Kähler on $X$.)

Let $\mathcal{I}$ be the ideal sheaf defined (locally) by the coefficients of
type $\prod_{j,1\leq i_{j}\leq d}z_{i_{j}}\tau_{I}$ to which
$\mathrm{log}|\tau|^{2}_{\omega_{\lceil D\rceil}}$ is equivalent to. Note that
$\mathcal{I}$ is locally defined, but its integral closure is globally
defined. In fact, the germ of the integral closure is the holomorphic function
germs $f$ such that locally

$\mathrm{log}|f|^{2}\leq\mathrm{log}|\tau|^{2}_{\omega_{\lceil
D\rceil}}+O(1).$

We denote this globally defined integral closure by $\overline{\mathcal{I}}$.
Consider the divisorial valuation associated with $D_{i}$

$\mu(\overline{\mathcal{I}},D_{i}):=\max\\{m\in\mathbb{N},\overline{\mathcal{I}}\subset\mathcal{I}_{D_{i}}^{m}\\}$

where $\mathcal{I}_{D_{i}}$ is the identification of $\mathcal{O}(-D_{i})$ as
an ideal sheaf. Consider

$\sigma:=\prod_{i}s_{D_{i}}^{\mu(\overline{\mathcal{I}},D_{i})}$

which is a global holomorphic section on $X$ where $s_{D_{i}}$ are the
canonical sections of $\mathcal{O}(D_{i})$. Consider
$\mathrm{log}(|\tau|_{\omega_{\lceil D\rceil},h}^{2}/|\sigma|^{2})$ such that
any $D_{i}$ is not an irreducible component of its pole set.

Now outside the support of $D$ (where the metric is smooth), pointwise, we
have

$i\partial{\overline{\partial}}\mathrm{log}(|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}/|\sigma|^{2})\wedge\omega_{\varphi_{\epsilon}}^{n-1}\geq(-i\Theta(\mathrm{det}(T_{X}/\mathcal{E}_{i}),h)-\epsilon\omega_{\varphi_{\epsilon}}+\sum_{j}i\mu(\overline{\mathcal{I}},D_{j})\Theta(\mathcal{O}_{X}(D_{j})))\wedge\omega_{\varphi_{\epsilon}}^{n-1}.$

Note that by choice of $\sigma$, the poles of
$\mathrm{log}(|\tau|_{\omega_{\lceil D\rceil},h}^{2}/|\sigma|^{2})$ contain no
irreducible component of $D$. In local coordinates,
$\mathrm{log}(|\tau|_{\omega_{\lceil D\rceil},h}^{2}/|\sigma|^{2})$ has the
same singularities as ideal quotient
$\mathcal{J}:=(\mathcal{I}:\prod_{i}\mathcal{I}_{D_{i}}^{\mu(\overline{\mathcal{I}},D_{i})})$.
In the following, we study the local integrability of
$\mathrm{log}(|\tau|_{\omega_{\lceil D\rceil},h}^{2}/|\sigma|^{2})$ with
respect to some volume form of mixed type of Poincaré type and conic type.

Let $\chi$ be a cut-off function $[0,\infty[\to[0,1]$. Following section 9 of
[CGP13], define

$\rho(x):=\mathrm{log}(\mathrm{log}\frac{1}{\prod_{i}|s_{D_{i}}|^{2}}).$

Note that $-i\partial{\overline{\partial}}\rho$ is bounded from above by
Poincaré type singularities along the support of $D$, which in local
coordinates can be written as

$i\sum_{k=1}^{d}\frac{dz_{k}\wedge
d\bar{z}_{k}}{|z_{k}|^{2}\mathrm{log}^{2}(|z_{k}|)}+i\sum_{k=d+1}^{n}dz_{k}\wedge
d\bar{z}_{k}.$

We claim that

$-\mathrm{log}(|\tau|_{\omega_{\lceil
D\rceil},h}^{2}/|\sigma|^{2})i\partial{\overline{\partial}}\rho\wedge\omega_{\varphi_{\epsilon}}^{n-1}$

is locally integrable. It is enough to show that near the support of $D$ in
local coordinates, for any $i_{0}\in[1,d]$,

$-\mathrm{log}(|\tau|_{\omega_{\lceil
D\rceil},h}^{2}/|\sigma|^{2})\frac{id(z_{i_{0}})\wedge
d(\overline{z_{i_{0}}})}{|z_{i_{0}}|^{2}\mathrm{log}^{2}(|z_{i_{0}}|)}\wedge\prod_{i\neq
i_{0},i\leq d}\frac{id(z_{i})\wedge
d(\overline{z_{i}})}{|z_{i}|^{2(1-\beta_{i})}}\wedge\prod_{i\geq
d+1}idz_{i}\wedge d\overline{z_{i}}$

is locally integrable. To simplify the notations, in the following, we assume
that $i_{0}=1$.

Consider the branched cover

$\begin{array}[]{cccc}\pi:&\mathbb{D}^{d}\times\mathbb{D}^{n-d}&\longrightarrow&\mathbb{D}^{d}\times\mathbb{D}^{n-d}\\\
&(z_{1},z_{2},\ldots,z_{d},z_{d+1},\ldots,z_{n})&\longmapsto&(z_{1},z_{2}^{n_{2}},\ldots,z_{d}^{n_{d}},z_{d+1},\ldots,z_{n}).\end{array}$

with $n_{i}(\forall i\geq 2)$ large enough such that $\beta_{i}n_{i}\geq 1$.
The local integrability is equivalent to the fact that

$\pi^{*}\big{(}-\mathrm{log}(|\tau|_{\omega_{\lceil
D\rceil},h}^{2}/|\sigma|^{2})\frac{id(z_{1})\wedge
d(\overline{z_{1}})}{|z_{1}|^{2}\mathrm{log}^{2}(|z_{1}|)}\wedge\prod_{d\geq
i\geq 2}\frac{id(z_{i})\wedge
d(\overline{z_{i}})}{|z_{i}|^{2(1-\beta_{i})}}\wedge\prod_{i\geq
d+1}idz_{i}\wedge d\overline{z_{i}}\big{)}$

is locally integrable.

Let $f_{j}(1\leq j\leq r)$ be the local generators of $\pi^{*}\mathcal{J}$. A
sufficient condition is that

$-\mathrm{log}(\sum_{j}|f_{j}|^{2})\frac{id(z_{1})\wedge
d(\overline{z_{1}})}{|z_{1}|^{2}\mathrm{log}^{2}(|z_{1}|)}\wedge\prod_{i\geq
2}id(z_{i})\wedge d(\overline{z_{i}})$

is locally integrable with the choice of branched cover.

By the construction of $\mathcal{J}$, the restrictions of $f_{j}$ on
$\\{z_{1}=0\\}$ are not all identically zero. By continuity, on
$\mathbb{D}_{r_{0}}\times\mathbb{D}^{n-1}$ for small radius $r_{0}$, the
restrictions of $f_{j}$ on $\\{z_{1}=s\\}(\forall|s|<r_{0})$ are not all
identically zero.

Consider $-\mathrm{log}(-\mathrm{log}(\sum_{j}|f_{j}|^{2}))$ as a continuous
family of psh functions on $\mathbb{D}^{n-1}$ parametrized by
$z_{1}\in\mathbb{D}_{r_{0}}$. Note that the Lelong number of any psh function
in this family at any point in $\mathbb{D}^{n-1}$ is 0.

By uniform Skoda integrability theorem (cf. e.g. Theorem 2.50 [GZ17]), for any
$s\in\mathbb{D}$ such that $|s|<r_{0}$,

$\int_{\\{s\\}\times\mathbb{D}^{n-1}}-\mathrm{log}(\sum_{j}|f_{j}|^{2})\prod_{i\geq
2}id(z_{i})\wedge d(\overline{z_{i}})\leq C$

for some $C>0$ independent of $s$ by vanishing Lelong number. Since the
Poincaré metric in one variable is locally integrable, we finish the proof of
the claim about the local integrability by the Fubini theorem.

Similarly, by Schwarz inequalities, we can show that

$-\mathrm{log}(|\tau|_{\omega_{\lceil
D\rceil},h}^{2}/|\sigma|^{2})i\partial\rho\wedge{\overline{\partial}}\rho\wedge\omega_{\varphi_{\epsilon}}^{n-1}$

is locally integrable.

For each $\delta>0$, let $\theta_{\delta}:[0,\infty[\to[0,1]$ be a smooth
function which is equal to zero on the interval $[0,1/\delta]$, and which is
equal to 1 on the interval $[1+1/\delta,\infty]$. One may for example define
$\theta_{\delta}(x)=\chi(x-1/\delta)$. Consider $\chi_{\delta}:X\to[0,1]$
defined by

$\chi_{\delta}(x):=1-\theta_{\delta}(\rho(x)).$

Consider Stokes’s formula

$\int_{X}\chi_{\delta}i\partial{\overline{\partial}}(\mathrm{log}(|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}/|\sigma|^{2}))\wedge\omega_{\varphi_{\epsilon}}^{n-1}=\int_{X}\partial\chi_{\delta}\wedge
i{\overline{\partial}}\mathrm{log}(|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}/|\sigma|^{2})\wedge\omega_{\varphi_{\epsilon}}^{n-1}$
$=-\int_{X}{\overline{\partial}}\chi_{\delta}\wedge
i\partial\mathrm{log}(|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}/|\sigma|^{2})\wedge\omega_{\varphi_{\epsilon}}^{n-1}=\int_{X}i\partial{\overline{\partial}}\chi_{\delta}\mathrm{log}(|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}/|\sigma|^{2})\wedge\omega_{\varphi_{\epsilon}}^{n-1}.$

Note that the left-handed side term is pointwise bigger than
$\chi_{\delta}(-i\Theta(\mathrm{det}(T_{X}/\mathcal{E}_{i}),h)-\epsilon\omega_{\varphi_{\epsilon}}+\sum_{j}i\mu(\overline{\mathcal{I}},D_{j})\Theta(\mathcal{O}_{X}(D_{j})))\wedge\omega_{\varphi_{\epsilon}}^{n-1}$.
Moreover, the integration of the lower bound over $X$ tends to

$\int_{X}(-c_{1}(T_{X}/\mathcal{E}_{i})-\epsilon\omega+\sum_{j}\mu(\overline{\mathcal{I}},D_{j})c_{1}(\mathcal{O}(D_{j})))\wedge\omega^{n-1}$

as $\delta\to 0+$.

On the order hand,
$i\partial{\overline{\partial}}\chi_{\delta}=-\theta_{\delta}^{\prime}(\rho)i\partial{\overline{\partial}}\rho-\theta_{\delta}^{\prime\prime}(\rho)i\partial\rho\wedge{\overline{\partial}}\rho.$
Note that $|\theta_{\delta}^{\prime\prime}|,|\theta_{\delta}^{\prime}|$ is
uniformly bounded independent of $\delta$. Thus
$\mathrm{log}(|\tau|_{\omega_{\varphi_{\epsilon},h}}^{2}/|\sigma|^{2})i\partial{\overline{\partial}}\chi_{\delta}\wedge\omega_{\varphi_{\epsilon}}^{n-1}$
is uniformly bounded by some integrable function by the previous claim.

By Lebesgue’s dominated convergence, the limit of the right-handed term on the
second line is 0 as $\delta\to 0+$.

Taking $\delta\to 0+$ gives

$0\geq\int_{X}(-c_{1}(T_{X}/\mathcal{E}_{i})-\epsilon\omega+\sum_{j}\mu(\overline{\mathcal{I}},D_{j})c_{1}(\mathcal{O}(D_{j})))\wedge\omega^{n-1}$

which finishes the proof by taking $\epsilon\to 0+$. ∎

Notice that a similar result is shown in Theorem 0.4 of [Ou17] under the
hypothesis that $-K_{X}$ is nef and that $X$ is projective with mild
singularity.

To illustrate the local integrability in the proof of Theorem 3, we give the
following elementary example.

######  Example 1.

Take the same notations as in the proof of Theorem 3. Assume that the complex
dimension of $X$ is two and the support of $D$ has two irreducible components
intersecting transversally at $x\in X$. Assume that $\mathcal{J}$ is the
maximal ideal at $x$. Denote $(z_{1},z_{2})$ local coordinates near $x$ such
that the support of $D$ is defined by $\\{z_{1}z_{2}=0\\}$ where $x$
corresponds to the origin.

The local integrability near $x$ is equivalent to showing that in local
coordinates, for any $i_{0}\in[1,2]$,

$-\mathrm{log}(|\tau|_{\omega_{\lceil
D\rceil},h}^{2}/|\sigma|^{2})\frac{id(z_{i_{0}})\wedge
d(\overline{z_{i_{0}}})}{|z_{i_{0}}|^{2}\mathrm{log}^{2}(|z_{i_{0}}|)}\wedge\prod_{i\neq
i_{0},i\leq 2}\frac{id(z_{i})\wedge
d(\overline{z_{i}})}{|z_{i}|^{2(1-\beta_{i})}}$

is locally integrable.

Let $\pi$ be the blow-up of $X$ at $x$. It is equivalent to showing that

$\pi^{*}\big{(}-\mathrm{log}(|z_{1}|^{2}+|z_{2}|^{2})\frac{id(z_{i_{0}})\wedge
d(\overline{z_{i_{0}}})}{|z_{i_{0}}|^{2}\mathrm{log}^{2}(|z_{i_{0}}|)}\wedge\prod_{i\neq
i_{0},i\leq 2}\frac{id(z_{i})\wedge
d(\overline{z_{i}})}{|z_{i}|^{2(1-\beta_{i})}}\big{)}$

is locally integrable near the exceptional divisor. Without loss of
generality, assume that $i_{0}=1$. In local coordinates near $x$, $\pi$ is
given by

$\pi(w_{1},w_{2})=(w_{1}w_{s},\cdots,w_{s-1}w_{s},w_{s},w_{s+1}w_{s},\cdots,w_{2}w_{s}).$

In local coordinates, it is enough to show that

$-\mathrm{log}|w_{s}|^{2}\pi^{*}(\frac{id(z_{1})\wedge
d(\overline{z_{1}})}{|z_{1}|^{2}\mathrm{log}^{2}(|z_{1}|)}\wedge\frac{id(z_{2})\wedge
d(\overline{z_{2}})}{|z_{2}|^{2(1-\beta_{2})}})$

is locally integrable near $\\{w_{s}=0\\}$.

If $s=2$, we have an upper bound for

$-\mathrm{log}|w_{2}|^{2}\frac{id(w_{1}w_{2})\wedge
d(\overline{w_{1}w_{2}})}{|w_{1}w_{2}|^{2}\mathrm{log}^{2}(|w_{1}w_{2}|)}\wedge\frac{id(w_{2})\wedge
d(\overline{w_{2}})}{|w_{2}|^{2(1-\beta_{2})}}$

whose potential with respect to the Lebesgue measure is bounded by

$-\mathrm{log}|w_{2}|^{2}\times 1/(|w_{1}|^{2}\mathrm{log}^{2}(w_{1}))\times
1/(|w_{2}|^{2(1-\beta_{2})})$

which is locally integrable.

If $s=1$, we have an upper bound for

$-\mathrm{log}|w_{1}|^{2}\frac{id(w_{1})\wedge
d(\overline{w_{1}})}{|w_{1}|^{2}\mathrm{log}^{2}(|w_{1}|)}\wedge\frac{id(w_{1}w_{2})\wedge
d(\overline{w_{1}w_{2}})}{|w_{1}w_{2}|^{2(1-\beta_{2})}}$

whose potential with respect to the Lebesgue measure is bounded by

$-\mathrm{log}|w_{1}|^{2}\times
1/(|w_{1}|^{2}\mathrm{log}^{2}(w_{1}))\times|w_{1}|^{2\beta_{2}}/(|w_{2}|^{2(1-\beta_{2})})$

which is locally integrable.

Now the arguments of Proposition 5.1 of [Cao13] give the following corollary.

######  Corollary 1.

Let $(X,\omega)$ be an n-dimensional compact Kähler manifold. Let
$D=\sum(1-\beta_{j})Y_{j}=\sum(1-\beta_{j})[s_{j}=0]$ a divisor with simple
normal crossings with $\beta_{j}\in]0,1[$ such that $-(K_{X}+D)$ is nef. Then
the Albanese morphism $\alpha_{X}$ is surjective with connected fibres. In
fact, the Albanese map is submersion outside an analytic set of codimension
larger than 2.

###### Proof.

The proof in [Cao13] only uses the fact that the slopes with respect to
$\omega^{n-1}$ of the sheaves obtained as graded pieces of the Harder-
Narasimhan filtration are positive. Hence using Theorem 3, the result is a
direct consequence of his arguments. For the convenience of the readers, we
just give here proof of the fact that the fibres of the Albanese map are
connected. We follow the arguments in the Proposition 3.9 of [DPS94].

Let $X\to Y\to\mathrm{Alb}(X)$ be the Stein decomposition of the Albanese map
with $Y=\mathrm{Spec}\;\alpha_{X*}\mathcal{O}_{X}$. Since $X$ is smooth, $Y$
is normal. We claim that the map $f:Y\to\mathrm{Alb}(X)$ is étale. The reason
is as follows. By the arguments in [Cao13], there exists $Z$ an analytic
subset in $\mathrm{Alb}(X)$ with codimension at least 2 such that
$X\smallsetminus\alpha_{X}^{-1}(Z)\to\mathrm{Alb}(X)\smallsetminus Z$ is
submersion (thus a fibration). Thus $Y\smallsetminus
f^{-1}(Z)\to\mathrm{Alb}(X)\smallsetminus Z$ is étale. We denote by $F$ the
fibre of the fibration $f|_{Y\smallsetminus f^{-1}(Z)}$ which is finite. By
the long exact sequence associated with a fibration, we have

$\pi_{1}(F)\to\pi_{1}(Y\smallsetminus
f^{-1}(Z))\to\pi_{1}(\mathrm{Alb}(X)\smallsetminus Z)\to\pi_{0}(F)$

where $\pi_{1}(F)=0$ and $\pi_{0}(F)$ is finite. In particular,
$\pi_{1}(Y\smallsetminus f^{-1}(Z))$ is a free Abelian group of rank
$2q:=2\mathrm{dim}_{\mathbb{C}}\mathrm{Alb}(X)$. Notice that by the
codimension condition, we have $\pi_{1}(\mathrm{Alb}(X)\smallsetminus
Z)\cong\pi_{1}(\mathrm{Alb}(X))$. $\mathrm{Alb}(X)$ is isomorphic to the
quotient of the universal cover $\mathbb{C}^{q}$ of $\mathrm{Alb}(X)$ under
the group action $\pi_{1}(\mathrm{Alb}(X))$. Define $T$ to be the quotient of
$\mathbb{C}^{q}$ under the group action $\pi_{1}(Y\smallsetminus f^{-1}(Z))$
with the natural cover $p:T\to\mathrm{Alb}(X)$. By the homotopy lifting
property, there exists a map $g:Y\smallsetminus f^{-1}(Z)\to T$ such that
$p\circ g=f|_{Y\smallsetminus f^{-1}(Z)}$. Remark that $g$ is holomorphic
since it is given by the composition of $f$ with the holomorphic local inverse
of $p$. Since $Y\smallsetminus f^{-1}(Z)\to\mathrm{Alb}(X)\smallsetminus Z$ is
finite, $f^{-1}(Z)$ is of codimension at least 2. Since $Y$ is normal, $g$
extends to a morphism $g:Y\to T$. Now $g$ is a generically injective morphism
between $Y$ and $T$. Since $T$ is smooth, the inverse map of $T\smallsetminus
p^{-1}(Z)\to Y$ also extends across $p^{-1}(Z)$ which gives the inverse
morphism of $g$. In conclusion $g$ is a biholomorphism between $T$ and $Y$
which proves that $f$ is étale. Another way to prove that $f$ is étale is
using the purity of branch locus. In fact, $f:Y\to\mathrm{Alb}(X)$ is a
ramified finite cover with $Y$ normal and $\mathrm{Alb}(X)$ smooth. Then by
the purity of the branch locus, the branch locus is either of codimension one
or empty. Since $f$ is a submersion outside an analytic set of
$\mathrm{Alb}(X)$ of codimension at least 2, the branch locus has to be empty.

In particular, $Y$ is a finite étale cover of the torus $\mathrm{Alb}(X)$, so
$Y$ itself is a torus. By the universality of the Albanese morphism, there
exists a morphism $h:\mathrm{Alb}(X)\to Y$ such that the morphism $X\to Y$
factorises through $h$. Since the morphisms $X\to Y$ and $\alpha_{X}$ are
surjective, we have $h\circ f=id_{Y}$ and $f\circ h=id_{\mathrm{Alb}(X)}$.
Thus $f$ is a biholomorphism, and the Albanese morphism has connected fibres.
∎

The arguments of [Cao13] combined with Theorem 3 also give the following
affirmation of a conjecture of Mumford. The general conjecture of Mumford
states that a projective or compact Kähler manifold $X$ is rationally
connected if and only if $H^{0}(X,(T_{X}^{*})^{\otimes m})=0$ for any $m\geq
1$.

######  Corollary 2.

Let $(X,\omega)$ be an n-dimensional compact Kähler manifold. Let
$D=\sum(1-\beta_{j})Y_{j}=\sum(1-\beta_{j})[s_{j}=0]$ a divisor with simple
normal crossings with $\beta_{j}\in]0,1[$ such that $-(K_{X}+D)$ is nef. Then
the following properties are equivalent:

(1) $X$ is projective and rationally connected.

(2)$H^{0}(X,(T_{X}^{*})^{\otimes m})=0$ for any $m\geq 1$.

(3) For every $m\geq 1$ and every finite étale cover $\tilde{X}$ of $X$, one
has $H^{0}(\tilde{X},\Omega_{\tilde{X}}^{m})=0$.

Acknowledgement I thank Jean-Pierre Demailly, my PhD supervisor, for his
guidance, patience and generosity. I would like to thank my post-doc mentor
Mihai Păun, for many supports and very useful suggestions on this objective. I
would like to thank Junyan Cao, Henri Guenancia for some very interesting
discussions on this objective. This work is supported by DFG Projekt Singuläre
hermitianische Metriken für Vektorbündel und Erweiterung kanonischer
Abschnitte managed by Mihai Păun.

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# ConferencingSpeech 2022 Challenge: Non-intrusive Objective Speech Quality
Assessment (NISQA) Challenge for Online Conferencing Applications

##  1 Abstract

With the advances in speech communication systems such as online conferencing
applications, we can seamlessly work with people regardless of where they are.
However, during online meetings, speech quality can be significantly affected
by background noise, reverberation, packet loss, network jitter, etc. Because
of its nature, speech quality is traditionally assessed in subjective tests in
laboratories and lately also in crowdsourcing following the international
standards from ITU-T Rec. P.800 series. However, those approaches are costly
and cannot be applied to customer data. Therefore, an effective objective
assessment approach is needed to evaluate or monitor the speech quality of the
ongoing conversation. The ConferencingSpeech 2022 challenge targets the non-
intrusive deep neural network models for the speech quality assessment task.
We open-sourced a training corpus with more than 86K speech clips in different
languages, with a wide range of synthesized and live degradations and their
corresponding subjective quality scores through crowdsourcing. 18 teams
submitted their models for evaluation in this challenge. The blind test sets
included about 4300 clips from wide ranges of degradations. This paper
describes the challenge, the datasets, and the evaluation methods and reports
the final results.

Index Terms: speech quality, deep learning, non-intrusive model

##  2 Introduction

With the popularity of remote conferencing, voice-based human-computer
interaction has become mainstream. Environment noise, room reverberation,
digital signal processing, and network transmission can all degrade the
quality of the speech signal. In these applications, speech quality assessment
is in high demand. So far, the above fields have made great progress. In ITU-T
Rec. P.800 [1], the international telecommunication union develops subjective
evaluation procedures to assess the speech quality, which is the most
preferred approach to evaluate the speech quality. However, it must be
performed under controlled conditions, which is often time consuming and
expensive. Meanwhile, the perceptual evaluation of speech quality (PESQ) [2]
and perceptual objective listening quality analysis (POLQA) [3] are designed
to objectively evaluate speech quality. However, they need clean reference
speech signals as comparison input. In order to non-intrusively assess the
speech quality, ITU-T Rec. P.563 [4] was developed but only target narrow-band
applications. As deep learning shines in various fields, deep neural networks
have been developed to address the non-intrusive speech quality assessment
problem recently [5, 6, 7, 8, 9, 10, 11, 12]. Nevertheless, most of these
methods adopt PESQ or POLQA as the speech quality label, which can not really
represent the subjective ratings in all impairments. Only a few datasets with
subjective scores have been published, which limits the application of deep
learning in the above problem. Therefore, a large dataset with subjective
speech quality scores and a non-intrusive speech quality assessment method,
which can better reflect perceived subjective feeling, are urgently needed.

The ConferencingSpeech 2022 challenge aims to stimulate research in the above-
mentioned areas. We provided comprehensive training and test datasets that
contain at least 200 hours of speech samples with subjective test scores. We
hope this challenge helps facilitate idea exchanges and discussions in this
special session. Meanwhile, this challenge has the following features: 1) We
aim for non-intrusive models for evaluating the speech quality (i.e., without
reference speech signals), which is more practical in online conferencing
applications. 2) With the continuous expansion of bandwidth in voice
communication systems, the existing standardized non-intrusive objective
speech quality assessment method for narrowband speech such as defined in
ITU-T P.563 is no longer applicable. Therefore, this challenge aims to
effectively evaluate the speech quality for signals with broader bandwidth. 3)
To truly reflect subjective opinion on speech quality, the training and test
datasets contain the mean opinion score (MOS), which is obtained through
subjective absolute category rating tests in crowdsourcing and in accordance
with the ITU-T Rec P.808 [13] using its open-sourced implementation [14]. 4)
As far as we know, this is the first challenge on non-intrusive objective
speech quality assessment in online conferencing. We provide a training speech
corpus of more than 200 hours of speech samples with corresponding subjective
MOS which are covering most of the impairment scenarios users might face in
on-line speech communication. It is believed that this will also promote the
development of non-intrusive objective speech quality assessment methods.

Table 1: Proportion of the degradations applied in Tencent Corpus.

Impairment | Percentage
---|---
White noise |  | $10\%$
Nonstationary background noise |  | $60\%$
High-pass/low-pass filtering |  | $3.75\%$
Amplitude clipping |  | $1.25\%$
AMR [15]/Opus [16] codec |  | $5\%$
Nonstationary background noise + AMR/Opus codec |  | $5\%$
White noise + AMR/Opus codec |  | $5\%$
High-pass/low-pass filtering + AMR/Opus codec |  | $5\%$
Amplitude clipping + non-stationary background noise |  | $5\%$

##  3 Task Description

In this challenge, comprehensive training datasets with ground truth MOS were
provided to each registered team. It is anticipated that the participating
teams use only the impaired speech signals to design corresponding algorithms
or models, so that the output prediction scores are close to the real MOS. The
final ranking of this challenge will be determined by the accuracy of the
predicted MOS from the submitted model or algorithm on the evaluation test
dataset, in terms of root mean squared error (RMSE) and Pearson correlation
coefficient (PCC).

It is worth noting that there are no restrictions on the source of the
training and development test datasets in this challenge. Participants can use
any dataset that is beneficial to the designed algorithm or model for
development. However, if additional data is used in training, then an ablation
study should be included that shows the benefit to the test set. Meanwhile,
the time-consumption and causality of the proposed algorithm or model are not
within the scope of this challenge.

Table 2: Proportion of the second step simulated impaired speech in Tencent
Corpus.

Impairment | Percentage
---|---
Only first step impairments |  | $60\%$
First step + noise suppression |  | $10\%$
First step + noise suppression + packet loss concealment |  | $1.25\%$
Clean speech |  | $3.75\%$
Clean speech + packet loss concealment |  | $1.25\%$

##  4 Data Description

In this challenge, we provided the participants with four voice datasets along
with MOS labels, namely Tencent Corpus, NISQA Corpus, IU Bloomington Corpus,
and PSTN Corpus. Among them, except for the NISQA Corpus, the other three
datasets are all made public for the first time. Each dataset will be
described in detail below.

###  4.1 Tencent Corpus

This dataset includes speech conditions with reverberation and without
reverberation. In the without reverberation condition, there are about 10000
Chinese speech clips and all speech clips experience the simulated impairments
which is very often in online conference. In the with reverberation condition,
simulated impairments and live recording speech clips are both considered and
totally count about 4000.

In the without reverberation condition, the selected source speech clips were
artificially added with some damage to simulate the voice impairment scenario
that may be encountered in the online meeting scene. In order to prevent the
possible speaker-dependent behavior of the trained model, the original speech
data was selected from three publicly available datasets Magic data [17], ST
Mandarin [18] and AIshell_100h [19]. Each speech clip in the source data was
processed with one type of impairment and only one type. The different
impairment types and the corresponding percentage of the speech clips applied
with each impairment type are listed in Table 1. Based on the speech clips
processed in the above step, we applied another speech processing step
including noise suppression [20] and packet loss concealment [21] to simulate
more realistic online communication, and also some clean speech clips were
added to form the final speech dataset. Those processing and corresponding
percentage in the final dataset are listed in Table 2.

In order to make the subjective database more comprehensive, 4000 speech clips
with reverberation were added to the dataset. 28% of them were generated with
simulated reverberation and 72% were recorded in realistic reverberant rooms.
In the simulated reverberation condition, the source data came from the
purchased king-asr-166 dataset. Meanwhile, various room sizes and
reverberation delays were considered. The subjective scoring procedure was
conducted in a crowdsourcing way similar to ITU-T P.808 including the
qualification - training - rating step, except that the training step was
simplified due to the scoring platform we were using. Each clip was rated by
more than 24 listeners. After data cleaning, more than 20 subjective scores
were obtained for each speech clip and averaged to obtain the final MOS score.

###  4.2 NISQA Corpus

The NISQA Corpus includes more than 14000 speech samples with simulated (e.g.,
codecs, packet-loss, background noise) and live (e.g., mobile phone, Zoom,
Skype, WhatsApp) conditions. The corpus is already publicly available
therefore we only included it in the training and development test sets in the
competition. Subjective ratings were collected through an extension of the
P.808 Toolkit [14] in which participants rated the overall quality and the
quality dimensions noisiness, coloration, discontinuity, and loudness. Each
clip has on average 5 valid votes.

Further details about this corpus are provided in [22]. We also created a new
test dataset (TUB hereafter) using unimpaired signals of 136 conversation
tests. We selected a portion of speech with no overlapping between two
speakers, at least 55% active speech, and added leading and trailing silences.
That leads to 865 source clips, from which a basic clustering algorithm could
detect seven different clusters. Meanwhile, we created 62 synthetic
degradation conditions (different codecs, bandwidths, single or multiple
background noises, packet lost scenarios, etc.). Each condition was applied on
seven randomly selected source clips (one per cluster). Finally, we collected
on average 18 subjective ratings per clip using the P.808 Framework [14].

###  4.3 IU Bloomington Corpus

There are 36000 speech clips extracted from COSINE [23] and VOiCES [24]
datasets. The speech clips are truncated between 3 to 6 seconds long, with a
total length of around 45 hours. For the VOiCES dataset, 4 versions of each
speech utterance were provided, including reference (i.e., foreground speech),
anchor (i.e., low-pass filtered reference), and two reverberant stimuli. The
approximated speech-to-reverberation ratios are between -4.9 to 4.3 dB. Three
versions of each speech utterance were provided for the COSINE dataset,
including reference (i.e., close-talking mic), anchor, and noisy (i.e., chest
or shoulder mic) stimuli. The approximate signal-to-noise ratios (SNRs) range
from -10.1 to 11.4 dB. We crowdsourced our listening tests on Amazon
Mechanical Turk by publishing 700 human intelligence tasks following ITU-R
BS.1534 [25]. For more details, please refer to [26].

###  4.4 PSTN Corpus

The clean reference files used for the phone calls are derived from the public
audiobook dataset Librivox. Because many of the recordings are of poorer
quality, the files have been filtered according to their quality as described
in [27], leaving in a total 441 hours from 2150 speakers of good quality
speech. These audiobook chapters were then segmented into 10 seconds clips and
filtered for having a speech activity of at least $50\%$. Since, in practice,
there are often environmental sounds present during phone calls, we used the
DNS Challenge 2021 [27] to add background noise. The noise clips are taken
from Audioset [28], Freesound, and the DEMAND [29] corpus and added to the
clean files with an SNR between $0-40$ dB. The perceived speech quality of the
training and test sets were annotated in a listening experiment on AMT,
according to P.808. Each training set file was rated by 5 participants, while
the test set files were rated by 30 participants to ensure a low confidence
interval of the MOS values for the model evaluation. For more details, please
refer to [30].

###  4.5 Dataset Division

The training, development, and evaluation test sets in this challenge are all
originated from the above-mentioned datasets. It is worth noting that the IU
Bloomington corpus differs from the Tencent, NISQA and PSTN corpora that used
ITU-T P.808 for subjective testing, where the IU Bloomington corpus adopted
ITU-R BS.1534 for subjective testing, which resulted in a rating range of
0$\sim$100 instead of 1$\sim$5\. Thus, the IU Bloomington corpus will only be
provided to participants as additional materials, speech clips from IU
Bloomington corpus will not appear in the evaluation test set of the
challenge. Participants can decide whether to use it according to their needs.

Due to the imbalanced size of the datasets, 80% of Tencent Corpus and 95% of
PSTN Corpus are used for training and development. The rest 20% of Tencent
Corpus, 5% of PSTN Corpus, and newly created TUB corpus are used for
evaluation test in this challenge. We aim to make the impairment situation and
score distribution in the divided dataset as even as possible.

In summary, there are about 86000 speech clips for training and development,
and 4372 clips for the evaluation test in this challenge. They are composed of
Chinese, English, and German, and consider background noise, speech
enhancement system, reverberation, codecs, packet-loss and other possible
online conference voice impairment scenarios.

##  5  Baseline System

This challenge provided two baseline systems, including Baseline System 1 and
Baseline System 2 [22]. The Baseline System 1 is a simplified version of the
model in [31]. It is made of a deep feed forward network followed by long
short-term memory and average pooling. The Baseline System 2 is the complete
model from [22]. It is based on a convolutional neural network (CNN) and
attention mechanism. The log-mel-spectrograms of the speech signal is provided
as input to the CNN network to extract the speech quality features at
different times. The estimated per-frame quality values are then aggregated
over time by using an attention model and the final score is predicted by the
attention pooling block. Both Baseline Systems were trained on all datasets
provided in this challenge.

##  6 Challenge Results

###  6.1 Evaluation Setup and Results

According to ITU-T P.1401 [32], we calculated RMSE to evaluate the accuracy,
the outlier ratio (OR) for consistency and PCC for linearity.

The RMSE is calculated according to the following equation:

$\operatorname{RMSE}=\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}\operatorname{Perror}(i)^{2}},$
(1)

where $\mathrm{N}$ denotes the total number of speech utterances, $i$
indicates the $i$-th speech signal. Perror represents the prediction error
which is defined as the difference between the measured and predicted MOS:

$\operatorname{Perror}(i)=\operatorname{MOS}(i)-\operatorname{MOS}_{p}(i).$
(2)

The OR represents the number of outlier-points to the total number of speech
utterances. An outlier is defined as a point for which the prediction error is
larger than the 95% confidence interval of the MOS value. PCC, on the other
hand, measures the linear relationship between a model’s estimation and the
subjective data. Meanwhile, due to bias or offsets, different gradients and
different qualitative rank orders are always present in subjective
evaluations. The statistical uncertainty always exists in the collected MOS.
Therefore, a mapping function is recommended to compensate for the possible
variance between several subjective experiments. In this challenge, a third-
order polynomial function is applied, which can be derived from
$y^{\prime\prime}=a+by+cy^{2}+dy^{3}$, where rmse
$\left(x,y^{\prime\prime}\right)\rightarrow\min$ and $f(y)=$ monotonous
between $y^{\prime\prime}_{\min}$ and $y^{\prime\prime}_{\max}$.

Figure 1:  Distribution of MOS values in three blind test sets.

For each model, we create one mapping function per test dataset. Then the
mapped predictions were used to calculated RMSE$\\_$MAP and OR. In this
challenge, we decided to rank models based on their accuracy, i.e.,
RMSE$\\_$MAP. The descriptive statistics on subjective ratings per blind test
set are provided in Table 3 and Figure 1.

Table 3: Descriptive statistics on subjective ratings in blind test sets.

Dataset | Average No. | Average | MOS
---|---|---|---
| ratings p. clip | 95%CI | min | max
Tencent Corpus | 28 | 0.20 | 1.00 | 5.00
TUB Corpus | 18 | 0.40 | 1.00 | 4.37
PSTN Corpus | 24 | 0.35 | 1.74 | 4.29

(a) mean of all datasets result analysis

(b) Tencent Corpus result analysis

(c) TUB Corpus result analysis

(d) PSTN Corpus result analysis

Figure 2:  Challenge result analysis

A total of 18 teams from different countries submitted results. Based on the
submitted results, their PCC, RMSE, RMSE$\\_$MAP, and OR are calculated. The
specific results are shown in Figure 2.

###  6.2 Key takeaways

  * •

In Figure 2 (a), it can be observed that the prediction results of all teams
are better than the results of the Baseline System 1. It suggests that the
generalization of Baseline System 1 is not good enough to effectively cover
different datasets. Meanwhile 11 teams achieved better results than the
Baseline System 2, accounting for 61$\%$ of the teams who submitted their
results.

  * •

The rank-order of models change based on the dataset and the criteria to use.
For any future benchmarking, we recommend to consider multiple blind test sets
created by different laboratories. PCC is strongly dependents to how well the
MOS values in the test set are distributed: For Tencent Corpus team reach a
PCC in range of [0.88 , 0.97] whereas in the other two datasets the achieved
PCC was strongly smaller (cf. Figure 1). The OR is directly influenced by the
95% CI, therefore OR values from one dataset cannot be directly compare to
another dataset.

  * •

We did not observe a significant difference between mapped RMSE values of top
2, 8, and 9 models in datasets Tencent, TUB, and PSTN, respectively. However,
team #1 is consistently in the top-three for all datasets.

##  7 Conclusion

The ConferencingSpeech 2022 Challenge was organized to help researchers from
academia and industry to facilitate the development of non-intrusive objective
speech quality assessment for online conferencing applications. We open-
sourced several large training datasets with subjective scores. We recommend
considering different blind test sets, created by multiple groups, for any
similar challenges or benchmarking tasks.

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