# On the Log Abundance for Compact Kähler threefolds II

Omprokash Das  School of Mathematics
Tata Institute of Fundamental Research
Homi Bhabha Road, Navy Nagar
Colaba, Mumbai 400005, India<EMAIL_ADDRESS><EMAIL_ADDRESS>and
Wenhao Ou  Institute of Mathematics
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Beijing 100190, CHINA<EMAIL_ADDRESS>
###### Abstract.

In this article we show that if $(X,\Delta)$ is a log canonical compact Kähler
threefold pair such that $K_{X}+\Delta$ is nef of numerical dimension
$\nu(X,K_{X}+\Delta)=2$, then $K_{X}+\Delta$ is semi-ample. This result
combined with our previous work shows that the log abundance holds for log
canonical compact Kähler threefold pairs.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Complex analytic varieties
    2. 2.2 Index-one cover
    3. 2.3 MMP for complex analytic varieties
    4. 2.4 Holomorphic foliations
  3. 3 Miayoka’s inequality on complex orbifolds
    1. 3.1 Complex orbifolds
    2. 3.2 Stable orbifold sheaves
    3. 3.3 Bimeromorphic morphisms of complex orbifolds
    4. 3.4 Miyaoka’s inequality
  4. 4 Comparison of Chern classes
    1. 4.1 Relative de Rham cohomology
    2. 4.2 Relative Chern classes
    3. 4.3 Borel-Moore homology
    4. 4.4 Comparison with $\mathbb{Q}$-Chern classes.
    5. 4.5 The case of complex analytic threefolds.
  5. 5 Positivity of logarithmic cotangent sheaves
    1. 5.1 $\mathbb{P}^{1}$-fibirations
    2. 5.2 Positivity of foliated canonical divisors
    3. 5.3 The case of non-uniruled threefolds
  6. 6 Lower bounds on Euler characteristics
    1. 6.1 Elementary properties
    2. 6.2 Existence of lower bound for the Euler characteristics
  7. 7 Construction of bimeromorphic models
  8. 8 Proof of the main theorem

##  1\. Introduction

The Minimal Model Program (MMP) is an important tool in birational
classifications of projective varieties $Z$. When $\operatorname{dim}Z=3$, the
MMP was fully established in the 90’s, and one has the following
classification. For a smooth projective threefold, there is a mildly singular
birational model $Z\dashrightarrow X$, such that either there is a Fano
fibration $X\to Y$, or the canonical class $K_{X}$ is nef. In the second case,
the canonical class $K_{X}$ induces a fibration $X\to Y$, thanks to the
following theorem, known as the abundance theorem (see [Miy87, Miy88b, Miy88a]
and [Kaw92]).

######  Theorem 1.1.

Let $X$ be a projective threefold with terminal singularities. If $K_{X}$ is
nef, then it is semi-ample.

The MMP was extended to the category of compact Kähler threefolds in [HP16]
and [HP15]. Particularly, we can recover the similar bimeromorphic
classification. When $Z$ is a smooth compact Kähler threefold, there is a
mildly singular bimeromorphic model $Z\dashrightarrow X$ such that either
there is a Fano fibration $X\to Y$, or the canonical class $K_{X}$ is nef.

It is natural to expect Theorem 1.1 holds for Kähler threefolds. If $X$ is not
algebraic, then the numerical dimension $\nu(X,K_{X})$ is at most $2$. When
$\nu(X,K_{X})=0$, it follows from the non-vanishing theorem of [DP03, Theorem
0.3] that $mK_{X}\cong\mathcal{O}_{X}$ for some $m>0$. If $\nu(X,K_{X})=1$,
then one can adapt the method for projective varieties to conclude, see
[CHP16, Section 8.A]. If $\nu(X,K_{X})=2$, it is claimed in [CHP16, Theorem
8.2] that $K_{X}$ is semi-ample. However, the authors have acknowledged a gap
in the proof of [CHP16]. The issue arises in Step 1 of the proof of [CHP16,
Theorem 8.2], where a result of Kawamata, namely [Kaw88, Theorem 9.6] is
incorrectly applied. A possible approach to the solution is suggested in
[CHP23].

In this paper, we focus on this case and prove the following theorem.

######  Theorem 1.2.

Let $(X,\Delta)$ be a lc pair such that $X$ is a compact Kähler threefold.
Assume that $K_{X}+\Delta$ is nef of numerical dimension $2$. Then it is semi-
ample.

We note that the conjectural solution suggested in [CHP23] is to first prove a
Bogolomov-Gieseker type inequality on a compact Kähler threefold with klt
singularities. However, we take a different approach here. For the specific
compact Kähler threefold $X$ with klt singularities we work with, we first
construct a Kähler orbifold $Y$ and a bimeromorphic morphism $\pi:Y\to X$ with
nice properties (see Lemma 6.12), and then we use Faulk’s Bogomolov-Gieseker
type inequality (see [Fau22]) on $Y$. Afterward, we mirror the arguments of
[Kol92, Chapter 14]. The crux of our techniques lies in Section 6, and the
necessary cohomological tools are developed in Section 4.

Combining the main theorem of [DO23] with the theorem above, we complete the
log abundance for log canonical Kähler threefolds.

######  Corollary 1.3.

Let $(X,\Delta)$ be a lc pair such that $X$ is a compact Kähler threefold. If
$K_{X}+\Delta$ is nef, then it is semi-ample.

As an application, we have the following corollary, as shown in [CHP16,
Theorem 1.2]. It is one of the steps towards the three dimensional Kodaira
problem in [Lin17].

######  Corollary 1.4.

Let $X$ be a simple compact Kähler threefold with klt singularities. Then $X$
is bimeromorphic to a quotient $Y/G$, where $Y$ is a complex torus and $G$ is
a finite group acting on $Y$.

This article is organized in the following manner. After reviewing some
preliminaries in Section 2, we will recall the notion of complex orbifolds and
prove an orbifold Miyaoka inequality in Section 3. In Section 4, we introduce
relative Chern classes, which compare orbifold Chern classes and Chern classes
on a desingularization. We also show that relative second Chern classes can be
computed by reducing to surfaces. In Section 5, we prove some positivity
results on second Chern classes of cotangent sheaves. With these in hand, we
complete the proof of the main theorem in the last three sections.

Acknowledgments We would like thank Junyan Cao, Mitchell Faulk, Patrick Graf,
Christopher Hacon, Andreas Höring, Chiu-Chu Melissa Liu, Mihai Păun, Burt
Totaro and Zheng Xu for important and valuable communications. W. Ou was
supported by the National Key R&D Program of China (No. 2021YFA1002300). O.
Das was partially supported by the Start–Up Research Grant(SRG), Grant No. #
SRG/2020/000348 of the Science and Engineering Research Board (SERB), Govt. Of
India, during the preparation of this article.

##  2\. Preliminaries

We will fix some notation and recall some elementary results in this section.

###  2.1. Complex analytic varieties

A _complex analytic variety_ is a reduced and irreducible complex space. For a
complex analytic variety $X$, we will denote by
$X_{\operatorname{\textsubscript{\rm sm}}}$ its smooth locus and by
$X_{\operatorname{\textsubscript{\rm sing}}}$ its singular locus. We also call
a smooth complex analytic variety a complex manifold.

When $X$ is a normal complex analytic variety and $B$ a reduced Weil divisor
on $X$, we define $\Omega^{[1]}_{X}(\log\,B)$, the reflexive logarithmic
cotangent sheaf as follows. Let $r\colon Y\to X$ be a log resolution of
$(X,B)$ and let $\Gamma=r_{*}^{-1}B$. Then the logarithmic cotangent bundle
$\Omega_{Y}^{1}(\log\,\Gamma)$ is well defined. We define
$\Omega^{[1]}_{X}(\log B):=(r_{*}\Omega_{Y}^{1}(\log\,\Gamma))^{**}$. We note
that such a definition is independent of the resolution $Y$.

Following [Gra62], a complex analytic variety $X$ is called _Kähler_ , if
there exists a closed positive $(1,1)$-form on $X$ such that the following
holds: for every point $x\in X$, there exists an open neighborhood $x\in U$, a
closed embedding $U\to V$ into an open subset $V\subseteq\mathbb{C}^{N}$, and
a strictly plurisubharmonic $\mathcal{C}^{\infty}$ function $f:V\to\mathbb{R}$
such that $\omega|_{U\cap X_{\operatorname{\textsubscript{\rm
sm}}}}=(\sqrt{-1}\partial\bar{\partial}f)|_{U\cap
X_{\operatorname{\textsubscript{\rm sm}}}}$. We also refer to [Dem85, Section
1] for general $(p,q)$-forms on analytic varieties.

For a normal compact Kähler variety $(X,\omega)$ with rational singularities,
a class in $H^{2}(X,\mathbb{R})$ is called _nef_ if it can be represented by a
smooth form $\alpha$ with local potentials such that for every $\epsilon>0$,
there exists a $\mathcal{C}^{\infty}$ function $f_{\epsilon}$ such that
$\alpha+\sqrt{-1}\partial\bar{\partial}f_{\epsilon}\geq-\epsilon\omega$. For
more explanations on the notation, we refer to [DO23, Definition 2.1] and the
references therein.

Let $f:X\to Y$ be a proper surjective morphism of normal complex analytic
varieties. A divisor $D$ is called $f$-exceptional if
$\operatorname{codim}_{Y}f(\operatorname{Supp}(D))\geq 2$. The following lemma
is a consequence of Hironaka’s flattening theorem.

######  Lemma 2.1.

Let $f:X\to Y$ be a proper surjective morphism between compact complex normal
analytic varieties. Then there exist projective bimeromorphic morphisms
$g\colon Y^{\prime}\to Y$ and $g^{\prime}\colon X^{\prime}\to X$ with the
following commutative diagram, such that

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f^{\prime}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y}$

  1. (1)

$Y^{\prime}$ is smooth, and

  2. (2)

for any proper bimeromorphic morphism $\varphi\colon W\to X^{\prime}$, every
$(f^{\prime}\circ\varphi)$-exceptional divisor is also
$(g^{\prime}\circ\varphi)$-exceptional.

###### Proof.

By [Hir75, Corollary 1], there is a projective bimeromorphic morphism
$Y_{1}\to Y$ such that the natural morphism $f_{1}\colon X_{1}\to Y_{1}$ is
flat, where $X_{1}$ is the main component of $X\times_{Y}Y_{1}$. In
particular, $f_{1}$ is equidimensional. Let $Y^{\prime}\to Y_{1}$ be a
desingularization, and let $X^{\prime}$ be the normalization of the main
component of $X_{1}\times_{Y_{1}}Y^{\prime}$. Then the induced morphism
$f^{\prime}\colon X^{\prime}\to Y^{\prime}$ is also equidimensional. Let
$D\subseteq X^{\prime}$ be a prime Weil divisor. Since $f^{\prime}$ is
equidimensional, either $f^{\prime}(D)=Y^{\prime}$ or $f^{\prime}(D)$ is a
prime Weil divisor on $Y^{\prime}$. In particular, for any proper
bimeromorphic morphism $\varphi\colon W\to X^{\prime}$, every
$(f^{\prime}\circ\varphi)$-exceptional divisor is in fact
$\varphi$-exceptional, and hence it is $(g^{\prime}\circ\varphi)$-exceptional.
This completes our proof. ∎

###  2.2. Index-one cover

Let $X$ be a normal variety and $\mathscr{L}$ a reflexive sheaf of rank $1$.
Then $\mathscr{L}$ is called $\mathbb{Q}$-Cartier, if there is a positive
integer $m>0$ such that $(\mathscr{L}^{\otimes m})^{**}$ is locally free. The
smallest such $m$ is called the Cartier index of $\mathscr{L}$.

Let $(o\in X)$ be a germ of normal complex analytic variety, and $\mathcal{L}$
a $\mathbb{Q}$-Cartier reflexive sheaf of rank 1 on $X$. Let $m>0$ be an
integer such that $(\mathcal{L}^{\otimes m})^{**}$ is locally free. Shrinking
$X$, we assume that there is an isomorphism
$\varphi\colon(\mathcal{L}^{\otimes m})^{**}\cong\mathcal{O}_{X}$. Such an
isomorphism induces a cyclic cover $f\colon Y\to X$, which is a finite
surjective morphism of degree $m$ of normal varieties, such that
$(f^{*}\mathcal{L})^{**}\cong\mathcal{O}_{Y}$. Furthermore, $f$ is étale in
codimension one. When $m$ is the Cartier index of $\mathcal{L}$, the morphism
$f$ is called the _index-one cover_ of $\mathcal{L}$. For more explanations,
we refer to [KM98, Definition 2.52]. In the next two lemmas, we prove some
universal properties on cyclic covers.

######  Lemma 2.2.

With the notation above, up to isomorphism, $f$ is locally unique in the
following sense. If $\varphi^{\prime}\colon(\mathcal{L}^{\otimes
m})^{**}\cong\mathcal{O}_{X}$ is another isomorphism, and if $f^{\prime}\colon
Y^{\prime}\to X$ the the cyclic cover induced by $\varphi^{\prime}$, then up
to shrinking $X$ around $o$, there is an isomorphism $Y\cong Y^{\prime}$ over
$X$.

###### Proof.

There is some nowhere vanishing holomorphic function $u$ on $X$ such that
$\varphi^{\prime}=u\cdot\varphi$. Shrinking $X$ if necessary, we may assume
that $u$ has a $m$-th root $v$. We set
$\mathcal{L}^{[i]}=(\mathcal{L}^{\otimes i})^{**}$ for all integer $i$. Then
from the construction of [KM98, Definition 2.52], we see that
$Y=\mathrm{Spec}_{X}\,\bigoplus_{i\geq 0}\mathcal{L}^{[-i]}$ such that the
$\mathcal{O}_{X}$-algebra structure on $\bigoplus_{0\leq
i<m}\mathcal{L}^{[-i]}$ is given by

$\mathcal{L}^{[-i]}\times\mathcal{L}^{[-j]}\overset{\varphi^{-1}}{\longrightarrow}\mathcal{L}^{[-i-j+m]}.$

If we denote this algebra by $\mathcal{A}$, and the one corresponding to
$Y^{\prime}$ by $\mathcal{A}^{\prime}$, then the multiplication by $v^{-1}$ on
$\mathcal{L}$ induces an isomorphism of $\mathcal{O}_{X}$-algebras from
$\mathcal{A}$ to $\mathcal{A}^{\prime}$. This proves that $Y\cong Y^{\prime}$.
∎

######  Lemma 2.3.

Let $\mathcal{L}$ be a reflexive sheaf of rank 1 on a germ of normal complex
analytic variety $(o\in X)$. Let $m,n,d$ be positive integers such that $n=dm$
and that $(\mathcal{L}^{\otimes m})^{**}\cong\mathcal{O}_{X}$. If $f\colon
Y\to X$ and $g\colon Z\to X$ are the cyclic covers defined as above, with of
degree $m$ and $n$ respectively, then, by shrinking $X$ if necessary, $Z$ is
isomorphic to a disjoint union of copies of $Y$.

In particular, every such kind of cyclic cover is locally isomorphic to a
disjoint union of copies of the index-one cover of $\mathcal{L}$.

###### Proof.

We denote by $\varphi\colon(\mathcal{L}^{\otimes m})^{**}\to\mathcal{O}_{X}$
and $\gamma\colon(\mathcal{L}^{\otimes n})^{**}\to\mathcal{O}_{X}$ the
isomorphisms. Then there is a nowhere vanishing holomorphic function $u$ on
$X$ such that $\gamma=u\cdot\varphi^{d}$. If $u=1$, then we see that $Z$ is
just the disjoint union of $d$ copies of $Y$.

In general, shrinking $X$ if necessary, we may assume that $u$ has a $d$-th
root $v$. The we consider the isomorphism $\varphi^{\prime}=v\cdot\varphi$. It
induces a cyclic cover $Y^{\prime}\to X$ which is isomorphic to $Y$ by Lemma
2.2. We notice that $(\varphi^{\prime})^{d}=\gamma$. Hence $Z$ is just a
disjoint union of copies of $Y^{\prime}$. This completes the proof of the
lemma. ∎

An integral divisor $\Delta$ on $(o\in X)$ is $\mathbb{Q}$-Cartier if and only
if the reflexive sheaf $\mathcal{O}_{X}(\Delta)$ is $\mathbb{Q}$-Cartier. If
it is the case, then the Cartier index of $\Delta$ is equal to the one of
$\mathcal{O}_{X}(\Delta)$. We define the index-one cover of $\Delta$ over
$(o\in X)$ as the index-one cover $f\colon Y\to X$ of
$\mathcal{O}_{X}(\Delta)$. Then the integral divisor $f^{*}\Delta$ is Cartier.

###  2.3. MMP for complex analytic varieties

Since the log canonical abundance is known for projective threefolds (see
[Kol92] and [KMM94]), in this paper, we mainly focus on compact Kähler
threefolds which are not algebraic. Particularly, if $X$ is a uniruled non-
algebraic compact Kähler threefold with $\mathbb{Q}$-factorial klt
singularities, then the base of the MRC fibration has dimension $2$ by [DO23,
Lemma 3.10]. The following theorems follow from [DO23, Theorem 9.1 and Theorem
9.3].

######  Theorem 2.4.

Let $(X,\Delta)$ be a $\mathbb{Q}$-factorial compact Kähler threefold dlt
pair. Assume that $X$ is non-algebraic and that the base of the MRC fibration
for $X$ has dimension $2$. Then the following assertions are equivalent:

  1. (1)

$K_{X}+\Delta\sim_{\mathbb{Q}}D\geq 0$ for some $\mathbb{Q}$-divisor $D$;

  2. (2)

$K_{X}+\Delta$ is pseudo-effective;

  3. (3)

$(K_{X}+\Delta)\cdot F\geq 0$ for a general fiber $F$ of the MRC fibration of
$X$.

######  Theorem 2.5.

Let $(X,\Delta)$ be a $\mathbb{Q}$-factorial compact Kähler threefold dlt
pair. If $K_{X}+\Delta$ is pseudo-effective, then it is
$\mathbb{Q}$-effective.

The MMP for a three-dimensional compact Kähler pair $(X,\Delta)$ was studied
in [HP16], [HP15], [CHP16], [DO23], [DH20] and [DHP22]. We summarize the
following statement.

######  Theorem 2.6.

Let $(X,\Delta)$ be a dlt pair such that $X$ is a $\mathbb{Q}$-factorial
compact Kähler threefold. Then the cone theorem, the contraction theorem and
the existence of flips hold for $(X,\Delta)$. Furthermore, any
$(K_{X}+\Delta)$-MMP terminates.

###### Proof.

If $K_{X}+\Delta$ is pseudoeffective, then the cone theorem follows from
[DO23, Theorem 10.12]; if $K_{X}+\Delta$ is not pseudoeffective, then the cone
theorem is established in [DHP22, Theorem 5.2]. The contraction theorem
follows from [DHP22, Corollary 5.3 and Corollary 5.6]. For the existence of
flips, we refer to [CHP16, Theorem 4.3]. Finally, for the termination of
flips, see [DO23, Theorem 3.3]. ∎

We can now introduce the following special MMP (see also [KMM94, Lemma 5.1]
for projective threefolds).

######  Lemma 2.7.

Let $(X,\Delta)$ be a dlt pair such that $X$ is a $\mathbb{Q}$-factorial
compact Kähler threefold. Suppose that $K_{X}+\Delta$ is not nef, and that
there is an effective divisor $H$ such that $K_{X}+\Delta+H$ is nef. Then a
$(K_{X}+\Delta)$-MMP, trivial with respect to $K_{X}+\Delta+H$, is a sequence
of bimeromorphic maps

$(X,\Delta)=(X_{0},\Delta_{0})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots\dashrightarrow(X_{n},\Delta_{n}),$

such that the following properties hold:

  1. (1)

For each $0\leq i\leq n-1$, there is a $(K_{X_{i}}+\Delta_{i})$-negative
extremal ray $R_{i}$ such that $(K_{X_{i}}+\Delta_{i}+H_{i})\cdot R_{i}=0$,
and $X_{i}\dashrightarrow X_{i+1}$ is the divisorial contraction or flip
corresponding to $R_{i}$.

  2. (2)

If $K_{X}+\Delta+\lambda H$ is pseudoeffective for some $\lambda<1$, then
there is a smallest integer $n\geq 0$ such that
$K_{X_{n}}+\Delta_{n}+(1-\epsilon)H_{n}$ is nef for all $\epsilon>0$
sufficiently small.

  3. (3)

If $K_{X}+\Delta+\lambda H$ is not pseudoeffective for any $\lambda<1$, then
there is a $(K_{X_{n}}+\Delta_{n})$-Mori fibration $f\colon X_{n}\to Y$ such
that $K_{X_{n}}+\Delta_{n}+H_{n}$ is $f$-relatively numerically trivial.

###### Proof.

With the help of Theorem 2.6, the same argument of [DO23, Theorem 4.5] shows
that such a MMP always exists. ∎

###  2.4. Holomorphic foliations

Let $X$ be a normal complex analytic variety. A _foliation_ on $X$ is a
saturated subsehaf $\mathcal{F}$ of the reflexive tangent sheaf $T_{X}$, which
is closed under the Lie bracket. For a dominant morphism $f\colon Y\to X$ of
normal complex analytic varieties, the pullback foliation $f^{-1}\mathcal{F}$
of $\mathcal{F}$ is the foliation on $Y$ induced by the image of
$f^{*}\mathcal{F}$ in $T_{Y}$, via the differential map of $f$.

Assume that $X$ is a complex manifold and that $\mathcal{F}$ a holomorphic
foliation on $X$. We define

$K_{\mathcal{F}}:=-c_{1}(\mathcal{F})$

and use the following notation. For a $\mathbb{Q}$-divisor $\Delta$ on $X$,
for an integer $m$ such that $m\Delta$ is integral, we set

$\mathcal{O}_{X}(m(K_{\mathcal{F}}+\Delta)):=(\det\mathcal{F})^{\otimes(-m)}\otimes\mathcal{O}_{X}(m\Delta).$

##  3\. Miayoka’s inequality on complex orbifolds

The goal of this section is to prove a Miyaoka type inequality on compact
Kähler threefolds with quotient singularities, see Proposition 3.11. Complex
analytic varieties with quotient singularities are closely related to complex
orbifolds. Hence, in this section, we will first recall the notion of complex
orbifolds and orbifold coherent sheaves. Then we will introduce blowups of
complex orbifolds, which enable us to reduce the study of the second Chern
class of orbifold torsion-free sheaves to the one of orbifold vector bundles.
With these tools in hand, we can then prove the main result of the section.

###  3.1. Complex orbifolds

Smooth ($\mathcal{C}^{\infty}$) orbifolds or V-manifolds were first introduced
in [Sat56] and later on were studied in various contexts. A complex orbifold
is a smooth orbifold that every local chart is a complex manifold and every
local group action is holomorphic. We briefly recall its definition by
adapting the notion of [CR02, Section 4].

######  Definition 3.1.

We take the following definitions.

  1. (1)

Let $U$ be a connected Hausdorff space. An orbifold chart (or a uniformization
system) of $U$ is a triple $(\widetilde{U},G,\pi)$ such that $\widetilde{U}$
is an open domain in some $\mathbb{C}^{n}$, that $G$ is a finitie group acting
holomorphically on $\widetilde{U}$, and that $\pi\colon\widetilde{U}\to U$ is
a continuous map inducing a homeomorphism from $\widetilde{U}/G$ to $U$. We
denote by $\mathrm{ker}\,G\subseteq G$ the maximal subgroup acting trivially
on $U$. The chart is called effective if $\mathrm{ker}\,G$ is the trivial
subgroup. It is called standard if the action of $G$ is free in codimension
one.

  2. (2)

Two orbifold chart $(\widetilde{U}_{1},G_{1},\pi_{1})$ and
$(\widetilde{U}_{2},G_{2},\pi_{2})$ of $U$ are isomorphic if there are
isomorphisms $\varphi\colon\widetilde{U}_{1}\to\widetilde{U}_{2}$ and
$\lambda\colon G_{1}\to G_{2}$ such that $\pi_{1}=\pi_{2}\circ\varphi$ and
that $\varphi$ is equivariant with respect to $\lambda$.

  3. (3)

Let $i\colon U^{\prime}\hookrightarrow U$ be an open subset, and
$(\widetilde{U}^{\prime},G^{\prime},\pi^{\prime})$ an orbifold chart of
$U^{\prime}$. An injection (or embedding) from
$(\widetilde{U}^{\prime},G^{\prime},\pi^{\prime})$ to $(\widetilde{U},G,\pi)$
consists of an open embedding
$\varphi\colon\widetilde{U}^{\prime}\to\widetilde{U}$ and a group monomorphism
$\lambda\colon G^{\prime}\to G$ such that $\varphi$ is equivariant with
respect to $\lambda$, that $\lambda$ induces an isomorphism from
$\mathrm{ker}\,G^{\prime}$ to $\mathrm{ker}\,G$, and that
$i\circ\pi^{\prime}=\pi\circ\varphi$.

  4. (4)

Let $X$ be a connected second countable Hausdorff space. An  orbifold atlas on
$X$ is a collection $\mathcal{U}=\\{(\widetilde{U},G,\pi)\\}$ of orbifold
charts of open subsets of $X$ which covers $X$ and is compatible in the
following sense. For any two orbifold charts $(\widetilde{U},G,\pi)$ and
$(\widetilde{U}^{\prime},G^{\prime},\pi^{\prime})$ in $\mathcal{U}$, of open
subsets $U$ and $U^{\prime}$ respectively, for any point $x\in U\cap
U^{\prime}$, there is an open neighborhood $V$ of $x$ with an orbifold chart
$(\widetilde{V},H,\rho)$ such that there are injections from
$(\widetilde{V},H,\rho)$ to $(\widetilde{U},G,\pi)$ and
$(\widetilde{U}^{\prime},G^{\prime},\pi^{\prime})$.

  5. (5)

An orbifold atlas $\mathcal{U}$ on $X$ is a refinement of another one
$\mathcal{V}$ if every orbifold chart of $\mathcal{U}$ admits an injection
into an orbifold chart of $\mathcal{V}$.

  6. (6)

Two orbifold atlas on $X$ are equivalent if they admit a common refinement. An
orbifold atlas is maximal if it contains any atlas equivalent to it. From
Zorn’s lemma, we see that every equivalent class of orbifold atlas contains a
unique maximal element.

  7. (7)

A complex orbifold is a connected second countable Hausdorff space $X$
equipped with an equivalent class of orbifold atlas. We write
$X_{\mathrm{orb}}=\\{(V_{i},G_{i},\pi_{i})\\}$ or simply
$X_{\mathrm{orb}}=\\{(V_{i},G_{i})\\}$ for such an orbifold structure with an
orbifold atlas $\\{(V_{i},G_{i},\pi_{i})\\}$. The underlying spcae $X$ is
called the quotient space of $X_{\mathrm{orb}}$. The dimension of
$X_{\mathrm{orb}}$ is the complex dimension of any orbifold chart.

  8. (8)

A complex orbifold is called effective or reduced, if every chart is
effective. It is called standard if every chart is standard.

  9. (9)

Let $X_{\mathrm{orb}}=\\{(V_{i},G_{i})\\}$ be a complex orbifold. An orbifold
subvariety $Z_{\mathrm{orb}}$ is defined by a collection $\\{W_{i}\subseteq
V_{i}\\}$ of $G_{i}$-invariant subvarieties which is compatible on the
overlaps, in the following sense. If there are injections of orbifold charts
$\varphi_{i}\colon V\to V_{i}$ and $\varphi_{j}\colon V\to V_{j}$, then
$\varphi_{i}^{-1}(W_{i})=\varphi_{j}^{-1}(W_{j}).$ The subvariety
$Z_{\mathrm{orb}}$ is called smooth if every $W_{i}$ is a submanifold of
$V_{i}$. When $Z_{orb}$ is smooth with irreducible image $Z$ in $X$, we obtain
an induced orbifold structure $Z_{\mathrm{orb}}=\\{(W_{i},G_{i})\\}$.

  10. (10)

An orbifold differential form $\sigma$ on an orbifold
$X_{\mathrm{orb}}=\\{(V_{i},G_{i})\\}$ is given by a collection
$\\{\sigma_{i}\\}$ of $G_{i}$-invariant differential forms on the collection
of charts $\\{V_{i}\\}$ which is compatible on the overlaps.

In [Sat56, Theorem 1], Satake proved the de Rham isomorphism theorem for
effective oriented $\mathcal{C}^{\infty}$ orbifolds, in particular for
effective complex orbifolds: for every integer $p\geq 0$, there is a natural
isomorphism

(3.1) $H^{p}_{\rm dR}(X_{\mathrm{orb}},\mathbb{R})\cong H^{p}(X,\mathbb{R}).$

######  Remark 3.2.

If $X_{\mathrm{orb}}=\\{(V_{i},G_{i})\\}$ is a non effective orbifold, then
there is a canonically induced effective orbifold structure
$X_{\mathrm{orb}}^{\prime}=\\{(V_{i},G^{\prime}_{i})\\}$, where
$G_{i}^{\prime}=G_{i}/\mathrm{ker}\,G_{i}$. We note that an orbifold
differential form on $X_{\mathrm{orb}}$ is an orbifold differential form on
$X_{\mathrm{orb}}^{\prime}$, and vice versa. Therefore, the isomorphism (3.1)
still holds for non effective orbifold.

######  Definition 3.3.

We also use the following notions of orbifold vector bundles and Hermitian
metrics, etc. Let $X_{\mathrm{orb}}=\\{(V_{i},G_{i})\\}$ be a complex
orbifold, possibly non effective.

  1. (1)

An orbifold vector bundle $\mathcal{E}_{\mathrm{orb}}=\\{\mathcal{E}_{i}\\}$
on $X_{\mathrm{orb}}=\\{(V_{i},G_{i})\\}$ is a collection of holomorphic
$G_{i}$-linearized vector bundles $\mathcal{E}_{i}$ on $V_{i}$ which is
compatible along the overlaps, in the following sense. For any chart
$(V,G,\pi)$ in the maximal atlas of $X_{\mathrm{orb}}$, there is a
$G$-linearized vector bundle $\mathcal{E}$ on $V$ satisfying the following
properties. If $(V,G)$ is equal to some $(V_{i},G_{i})$, then
$\mathcal{E}=\mathcal{E}_{i}$. If there is an injection
$\varphi\colon(V,G,\pi)\to(V^{\prime},G^{\prime},\pi^{\prime})$, then there is
an isomorphism
$\Psi_{\varphi}\colon\mathcal{E}\cong\varphi^{*}\mathcal{E}^{\prime}$.
Furthermore, if
$\iota\colon(V^{\prime},G^{\prime},\pi^{\prime})\to(V^{\prime\prime},G^{\prime\prime},\pi^{\prime\prime})$
is another injection, then we have
$\Psi_{\iota\circ\varphi}=(\varphi^{*}\Psi_{\iota})\circ\Psi_{\varphi}$.

  2. (2)

When $X_{\mathrm{orb}}$ is effective, orbifold vetcor bundles can be defined
equivalently as follows. There is an open dense subset $X^{\circ}\subseteq X$
such that each projection map $\pi_{i}\colon V_{i}\to X$ is étale over
$X^{\circ}$. Hence $X^{\circ}$ has a natural structure of complex manifold.
There is a holomorphic vector bundle $\mathcal{E}^{\circ}$ on $X^{\circ}$ such
that $\pi_{i}^{*}\mathcal{E}^{\circ}$ extends to the vector bundle
$\mathcal{E}_{i}$ on $V_{i}$.

  3. (3)

An orbifold coherent sheaf $\mathcal{E}_{\mathrm{orb}}=\\{\mathcal{E}_{i}\\}$
on $X_{\mathrm{orb}}$ is defined in the same manner.

  4. (4)

An orbifold coherent sheaf $\mathcal{E}_{\mathrm{orb}}$ is called torsion-free
(resp. reflexive) if every data $\mathcal{E}_{i}$ on $V_{i}$ is so.

  5. (5)

For an orbifold vector bundle $\mathcal{E}_{\mathrm{orb}}$, a Hermitian metric
$h_{\mathrm{orb}}$ on $\mathcal{E}_{\mathrm{orb}}$ is a collection
$\\{h_{i}\\}$ of $G_{i}$-invariant Hermitian metrics on $\mathcal{E}_{i}$
which are compatible on the overlap.

  6. (6)

For a Hermitian orbifold vector bundle
$(\mathcal{E}_{\mathrm{orb}},h_{\mathrm{orb}})$, we can define the orbifold
Chern classes $\hat{c}_{i}(\mathcal{E}_{\mathrm{orb}})$ using the curvature
tensors $c_{i}(h_{\mathrm{orb}})$, which are automatically orbifold
differential forms. They are uniquely determined in the orbifold de Rham
cohomology groups of $X_{\mathrm{orb}}$, hence by (3.1) in the singular
cohomology groups of $X$.

  7. (7)

The complex orbifold $X_{\mathrm{orb}}$ is called  Kähler if there is a closed
positive orbifold $(1,1)$-form $\omega$. A cohomology class $\alpha\in
H^{2}(X,\mathbb{R})$ is called an orbifold Kähler class on $X_{\mathrm{orb}}$
if there is an orbifold Kähler form $\omega$ whose cohomology class is
$\alpha$.

######  Remark 3.4.

Complex orbifolds are closely related to complex analytic varieties with
quotient singularities. We summarize some properties as follows.

  1. (1)

From [Car57, Théorème 4], we see that the quotient space $X$ of an effective
complex orbifold $X_{\mathrm{orb}}$ has a natural structure of complex
analytic variety, with quotient singularities. Furthermore, if $(V_{i},G_{i})$
is an orbifold chart of $X_{\mathrm{orb}}$, then the quotient map
$\pi_{i}\colon V_{i}\to\pi_{i}(V_{i})$ is holomorphic. The holomorphic
functions on any open subset $U\subseteq\pi_{i}(V_{i})$ are exactly the
$G_{i}$-invariant holomorphic functions on $\pi_{i}^{-1}(U)\subseteq V_{i}$.

  2. (2)

Conversely, given a complex analytic variety $X$ of dimension $n$ with
quotient singularities, it admits a unique standard orbifold structure after
[Pri67, Theorem 1]. We call it the standard orbifold structure on $X$.

  3. (3)

A same quotient space $X$ may have non isomorphic orbifold structures. For
example, we take $X=\mathbb{C}$. Let $(\mathbb{C},\\{\mathrm{Id}\\},\pi_{1})$
and $(\mathbb{C},\\{\mathrm{Id},-\mathrm{Id}\\},\pi_{2})$ be two orbifold
charts of $X$, where $\pi_{1}$ is the identity map and $\pi_{2}$ is the map
sending a complex number $z$ to $z^{2}$. Then these two charts induces
different orbifold structures on $X$.

  4. (4)

Let $X_{\mathrm{orb}}=\\{(V_{i},G_{i})\\}$ be an effective complex orbifold
and $X$ its quotient space. Assume that $Y$ is a complex analytic space and
there are $G_{i}$-equivariant morphisms $q_{i}\colon V_{i}\to Y$, where the
actions of the $G_{i}$’s on $Y$ are trivial. Assume that the $q_{i}$’s are
compatible on the overlap. Then they induces a holomorphic map $f\colon X\to
Y$. Indeed, by the definition of topological quotient spaces, the map is well-
defined and continuous. To prove that it is holomorphic, it is enough to show
that for any local holomorphic function $\varphi$ on $Y$, the function
$\varphi\circ f$ is holomorphic. By definition, each $\varphi\circ q_{i}$ is
$G_{i}$-invariant and holomorphic. Hence we can conclude by (1) of Remark 3.4.

  5. (5)

Let $X_{\mathrm{orb}}=\\{(V_{i},G_{i},\pi_{i})\\}$ be a standard complex
orbifold, and let $X$ be its quotient space. Then there is a correspondence
between reflexive coherent sheaves $\mathcal{E}$ on $X$ and orbifold relfexive
sheaves $\mathcal{E}_{\mathrm{orb}}=\\{\mathcal{E}_{i}\\}$ on
$X_{\mathrm{orb}}$. Indeed, if $\mathcal{E}$ is a reflexive coherent sheaf on
$X$, then the reflexive pullbacks $(\pi_{i}^{*}\mathcal{E})^{**}$ define an
orbifold reflexive sheaf on $X_{\mathrm{orb}}$. Conversely, if
$\mathcal{E}_{i}$ is a $G_{i}$-equivariant reflexive coherent sheaf on
$V_{i}$, then the $G_{i}$-invariant pushforward
$((\pi_{i})_{*}\mathcal{E}_{i})^{G_{i}}$ is a reflexive sheaf on
$\pi_{i}(V_{i})\subseteq X$, see for example [GKKP11, Lemma A.4].

  6. (6)

Let $X_{\mathrm{orb}}$ be a standard complex orbifold and $X$ its quotient
space. Assume that $\mathcal{L}$ is a reflexive sheaf on $X$ of rank one such
that $\mathcal{N}=(\mathcal{L}^{\otimes b})^{**}$ is a line bundle of some
$b>0$. Then we claim that

$b\cdot\hat{c}_{1}(\mathcal{L})=\hat{c}_{1}(\mathcal{N})=c_{1}(\mathcal{N})\in
H^{2}(X,\mathbb{R}).$

Indeed, by taking an orbifold Hermitian metric, we can deduce that
$\hat{c}_{1}(\mathcal{N})|_{X_{\operatorname{\textsubscript{\rm
sm}}}}=c_{1}(\mathcal{N})|_{X_{\operatorname{\textsubscript{\rm sm}}}}$. Since
$X_{\operatorname{\textsubscript{\rm sing}}}$ has real codimension at least 4
in $X$, this implies that $\hat{c}_{1}(\mathcal{N})=c_{1}(\mathcal{N})\in
H^{2}(X,\mathbb{R})$. This explains that the orbifold first Chern class
coincides with the first Chern class for a $\mathbb{Q}$-Cartier reflexive
sheaf or a $\mathbb{Q}$-Cartier divisor. For this reason, we may use
$\hat{c}_{1}$ and $c_{1}$ interchangeably.

After (5) of Remark 3.4, we will use the following notion. Assume that $X$ is
a complex analytic variety with quotient singularities, and that $\mathcal{E}$
a reflexive coherent sheaf on $X$. If $\mathcal{E}$ induces an orbifold vector
bundle $\mathcal{E}_{\mathrm{orb}}$ on the standard orbifold structure, we set

$\hat{c}_{i}(\mathcal{E}):=\hat{c}_{i}(\mathcal{E}_{\mathrm{orb}}).$

The following lemma from [Wu23, Lemma 1] (see also [CGN+23, Construction 6.1])
relates Kähler orbifolds and Kähler varieties with quotient singularities. For
the reader’s convenience, we include a proof.

######  Lemma 3.5.

Let $(X,\omega)$ be a compact Kähler variety with quotient singularities. Let
$X_{\mathrm{orb}}$ be the standard orbifold structure on $X$. Then there is an
orbifold Kähler form on $X_{\mathrm{orb}}$ of the shape

$\omega_{\mathrm{orb}}=pr_{X}^{*}\omega+\sqrt{-1}\partial\overline{\partial}\varphi,$

for some $\mathcal{C}^{\infty}$ orbifold real function $\varphi$, where
$pr_{X}^{*}\omega$ is the orbifold form on $X_{\mathrm{orb}}$ defined by
pulling back $\omega$ on every orbifold chart. In particular, the class of
$\omega$ in $H^{2}(X,\mathbb{R})$ is an orbifold Kähler class on
$X_{\mathrm{orb}}$.

###### Proof.

We will adapt the method of [Dem92, Lemma 3.5]. Let $n=\operatorname{dim}X$.
There is a finite open covering $\\{U_{i}\\}$ of $X$, such that for each $i$,
there is an integer $N_{i}$ and a relatively compact open subset
$W_{i}\subseteq\mathbb{C}^{N_{i}}$ with the following properties. There is a
closed embedding $\iota_{i}$ from $U_{i}$ into $W_{i}$, there is a Kähler form
$\Omega_{i}$ on a neighborhood the closure $\overline{W_{i}}$, such that
$\omega|_{U_{i}}=\iota^{*}\Omega_{i}.$

By considering bump functions supported in the $W_{i}$’s, we can obtain a
partition of the unity $\\{\eta_{i}\\}$ subordinate to $U_{i}$, so that there
is a $\mathcal{C}^{\infty}$ function $\rho_{i}$, compactly supported in
$W_{i}$, such that $\eta_{i}=\rho_{i}\circ\iota_{i}$. Since $\Omega_{i}$ is
positive, by compactness, there are real numbers $A_{i}>0$ such that, on every
$W_{i}$,

$\sqrt{-1}(\rho_{i}\partial\overline{\partial}\rho_{i}-\partial\rho_{i}\wedge\overline{\partial}\rho_{i})\geq-
A_{i}\Omega_{i}$

Up to refining the covering $\\{U_{i}\\}$, we can assume that there is an
orbifold chart $(V_{i},G_{i})$ for each $U_{i}$, such that
$V_{i}\subseteq\mathbb{C}^{n}$ is a relatively compact open neighborhood of
the origin. We denote by $p_{i}\colon V_{i}\to W_{i}$ the natural morphism,
which is a holomorphic map. Let $\theta_{i}=\rho_{i}\circ p_{i}$. Then
$\\{\theta_{i}\\}$ is a $\mathcal{C}^{\infty}$ orbifold partition of the
unity, subordinate to $\\{V_{i}\\}$. The inequality of the previous paragraph
implies that, on each $V_{i}$,

$\sqrt{-1}(\theta_{i}\partial\overline{\partial}\theta_{i}-\partial\theta_{i}\wedge\overline{\partial}\theta_{i})\geq-
A_{i}\cdot\pi_{i}^{*}\Omega_{i}=-A_{i}\cdot pr_{X}^{*}\omega.$

For each $i$, we define $w_{i}({x})=\sum_{g\in G_{i}}||g({x})||^{2}$ for any
${x}\in V_{i}\subseteq\mathbb{C}^{n}$, where $||\cdot||$ is the standard
Euclidean norm on $\mathbb{C}^{n}$. Then $w_{i}$ is $G_{i}$-invariant and
$\sqrt{-1}\partial\overline{\partial}w_{i}$ is a $G_{i}$-invariant Kähler form
on $V_{i}$. We set

$\gamma:=\sqrt{-1}\sum_{i}\theta_{i}\partial\overline{\partial}w_{i}.$

Then $\gamma$ is an orbifold positive $(1,1)$-form on $X_{\mathrm{orb}}$. We
note that each $w_{i}$ is bounded on $V_{i}$. Hence there are constants
$C_{i}$ such that, for any $x\in V_{i}$, we have

$w_{i}(x)\leq C_{i}+\sup_{\ k\neq i,\ V_{k}\ni x}w_{k}(x),$

in the following sense. Let $y$ be the image of $x$ in $X$. Then we say that
$x\in V_{k}$ if and only if $y\in U_{k}$. If this is the case, then we set
$w_{k}(x):=w_{k}(x^{\prime})$, where $x^{\prime}\in V_{k}$ is any point lying
over $y$.

Let

$K=2\sum_{i}A_{i}\exp(C_{i})$

and

$w=\log\,\left(\sum_{i}\theta_{i}^{2}\exp(w_{i})\right).$

Then $w$ is a $\mathcal{C}^{\infty}$ orbifold function on $X_{\mathrm{orb}}$.
The same calculation as in [Dem92, Lemma 3.5] shows that

$\sqrt{-1}\partial\overline{\partial}w\geq\gamma-K\cdot pr_{X}^{*}\omega.$

We set $\varphi=\frac{1}{K}w$. It follows that

$pr_{X}^{*}\omega+\sqrt{-1}\partial\overline{\partial}\varphi\geq\frac{1}{K}\gamma,$

which is positive. This completes the proof of the lemma. ∎

###  3.2. Stable orbifold sheaves

We can extend the notion of slope stability for torsion-free sheaves to the
setting of orbifolds.

######  Definition 3.6.

Let $X_{\mathrm{orb}}$ be a compact complex orbifold of dimension $n$ and
$\mathcal{E}_{\mathrm{orb}}$ a torsion-free orbifold coherent sheaf of rank
$r$. Let $\alpha\in H^{2}(X,\mathbb{R})$ be any class. We can define the slope
of $\mathcal{E}_{\mathrm{orb}}$ with respect to $\alpha^{n-1}$ as

$\mu_{\alpha^{n-1}}(\mathcal{E}_{\mathrm{orb}}):=\frac{\hat{c}_{1}(\mathcal{E}_{\mathrm{orb}})\cdot\alpha^{n-1}}{r},$

where $\hat{c}_{1}(\mathcal{E}_{\mathrm{orb}})$ is defined as the orbifold
first Chern class of the determinant orbifold line bundle
$(\bigwedge^{r}\mathcal{E}_{\mathrm{orb}})^{**}$. The sheaf
$\mathcal{E}_{\mathrm{orb}}$ is said to be $\alpha^{n-1}$-semistable if for
any non zero subsheaf
$\mathcal{F}_{\mathrm{orb}}\subseteq\mathcal{E}_{\mathrm{orb}}$, we have
$\mu_{\alpha^{n-1}}(\mathcal{E}_{\mathrm{orb}})\geq\mu_{\alpha^{n-1}}(\mathcal{F}_{\mathrm{orb}})$.
It is said to be $\alpha^{n-1}$-stable if the inequality is strict for every
non zero proper subsheaf $\mathcal{F}_{\mathrm{orb}}$.

Harder-Narasimhan filtrations and Jordan-Hölder filtrations exist for orbifold
coherent sheaves, with respect to orbifold Kähler classes. To see this, we can
adapt the classic argument (see for example [Kob14, Section V.7]) to the
orbifold setting. The first step is to construct an appropriate orbifold
metric on the locally free part of a torsion-free orbifold coherent sheaf.

######  Lemma 3.7.

Let $(X_{\mathrm{orb}},\omega)$ be a compact Kähler orbifold and
$\mathcal{E}_{\mathrm{orb}}$ a torsion-free coherent sheaf on
$X_{\mathrm{orb}}$. Then there is an orbifold Hermitian metric $h$ on the
locally free locus of $\mathcal{E}_{\mathrm{orb}}$, satisfying the following
property. For any point $x\in X$ in the quotient space, there is an open
neighborhood $x\in U$ with an orbifold chart $(V,G)$, there is an equivariant
inclusion of coherent sheaves
$\mu\colon\mathcal{E}_{\mathrm{orb}}|_{V}\to\mathcal{H}^{\prime}_{V}$ on $V$,
such that $\mathcal{H}^{\prime}_{V}$ is free, that $\mu$ is a subbundle on the
locally free locus of $\mathcal{E}_{\mathrm{orb}}|_{V}$, and that $h|_{V}$ is
equal to the restriction on $\mathcal{E}_{\mathrm{orb}}|_{V}$ of some smooth
$G$-invariant metric $h^{\prime}$ on $\mathcal{H}^{\prime}_{V}$.

In particular, if $\Omega$ is the Chern curvature of $h$ and
$K=\Lambda_{\omega}\Omega$ is the contraction of $\Omega$ with $\omega$, then
$||K||$ is integrable.

###### Proof.

The first step is to embed $\mathcal{E}_{\mathrm{orb}}$ locally equivariantly
into a free sheaf. Since $X$ is compact, there is a finite orbifold atlas of
charts $\\{(V_{i},G_{i},\pi_{i})\\}$ such that the following properties hold
for each $i$. We set $\mathcal{E}_{i}=\mathcal{E}_{\mathrm{orb}}|_{V_{i}}$.
The dual sheaf $\mathcal{E}_{i}^{*}$ is generated by global sections
$\sigma_{i}^{1},...,\sigma_{i}^{a_{i}}\in H^{0}(V_{i},\mathcal{E}_{i}^{*})$.
Let $S_{i}\subseteq H^{0}(V_{i},\mathcal{E}_{i}^{*})$ be the linear subspace
generated by the global sections $g^{*}\sigma_{i}^{j}$, where $g\in G_{i}$ and
$j=1,...,a_{i}$. Then $S_{i}$ is invariant for the natural action of $G_{i}$,
and we obtain a $G_{i}$-equivariant surjective morphism

$\mathcal{O}_{V_{i}}\otimes_{\mathbb{C}}S_{i}\to\mathcal{E}_{i}^{*},$

where $G_{i}$ acts trivially on the factor $\mathcal{O}_{V_{i}}$ of
$\mathcal{O}_{V_{i}}\otimes_{\mathbb{C}}S_{i}$. By taking the duality, we
obtain a $G_{i}$-equivariant injective morphism of coherent sheaves
$\mathcal{E}_{i}^{**}\to\mathcal{H}_{i},$ where
$\mathcal{H}_{i}:=(\mathcal{O}_{V_{i}}\otimes_{\mathbb{C}}S_{i})^{*}$. Hence
we deduce a $G_{i}$-equivariant injective morphism
$\mathcal{E}_{i}\to\mathcal{H}_{i}$. Furthermore, $\mathcal{E}_{i}$ is a
subbundle of $\mathcal{H}_{i}$ on the locally free locus of $\mathcal{E}_{i}$.
This completes the proof of the first step.

Let $l^{\prime}_{i}$ be a smooth $G_{i}$-invariant Hermitian metric on
$\mathcal{H}_{i}$ and let $l_{i}$ be its restriction on the locally free locus
of $\mathcal{E}_{i}$. Let $\\{\rho_{i}\\}$ be an orbifold partition of the
unity subordinate to $\\{V_{i}\\}$. We set $h=\sum\rho_{i}l_{i}$.

We will show that $h$ satisfies the property of the lemma. Let $x\in X$ be a
point and $U$ a neighborhood of $x$. Up to shrinking $U$, we may assume that
there is an orbifold chart $(V,G)$ of $U$, such that $V$ admits an injection
into $V_{i}$ whenever $x\in\pi_{i}(V_{i})$. We set

$\mathcal{H}^{\prime}_{V}=\bigoplus_{x\in\pi_{i}(V_{i})}\mathcal{H}_{i}|_{V}.$

We also define
$\mu\colon\mathcal{E}_{\mathrm{orb}}|_{V}\to\mathcal{H}^{\prime}_{V}$ as the
diagonal morphism, and define $h^{\prime}$ as the direct sum metric
$\oplus\rho_{i}l^{\prime}_{i}$. Then we have
$h|_{V}=h^{\prime}|_{\mathcal{E}_{\mathrm{orb}}|_{V}}$.

If $\Omega^{\prime}$ is the Chern curvature of $h^{\prime}$ and if
$K^{\prime}=\Lambda_{\omega}\Omega^{\prime}$, then we have, on $V$,

$||K||\leq||K^{\prime}||.$

Since $||K^{\prime}||$ is $\mathcal{C}^{\infty}$ on $V$, it is integrable. We
deduce that $||K||$ is integrable on $V$. This completes the proof of the
lemma. ∎

With this orbifold Hermitian metric, we can deduce the following assertion.

######  Lemma 3.8.

Let $(X_{\mathrm{orb}},\omega)$ be a compact Kähler orbifold of dimension $n$
and $\mathcal{E}_{\mathrm{orb}}$ a torsion-free coherent sheaf on
$X_{\mathrm{orb}}$.

  1. (1)

Then there is a constant $C$ such that for any non zero coherent orbifold
subsheaf $\mathcal{F}_{\mathrm{orb}}\subseteq\mathcal{E}_{\mathrm{orb}}$, we
have

$\hat{c}_{1}(\mathcal{F}_{\mathrm{orb}})\cdot\omega^{n-1}\leq C.$

  2. (2)

Let $E_{\mathrm{orb}}$ be a smooth orbifold subvariety of $X_{\mathrm{orb}}$
of codimension one. Then there is a constant $C^{\prime}$, such that for any
non zero coherent saturated orbifold subsheaf
$\mathcal{F}_{\mathrm{orb}}\subseteq\mathcal{E}_{\mathrm{orb}}$, we have

$\hat{c}_{1}(\mathcal{F}_{\mathrm{orb}})\cdot\hat{c}_{1}(\mathcal{O}_{X_{\mathrm{orb}}}(E_{\mathrm{orb}}))\cdot\omega^{n-2}\leq
C^{\prime}.$

###### Proof.

For item (1), we can adapt the method of [Kob14, Lemma V.7.16]. Since the rank
of $\mathcal{E}_{\mathrm{orb}}$ is finite, by taking the determinant orbifold
line bundle of $\mathcal{F}_{\mathrm{orb}}$, and by replacing
$\mathcal{E}_{\mathrm{orb}}$ with
$(\bigwedge^{d}\mathcal{E}_{\mathrm{orb}})^{**}$, where $d$ is the rank of
$\mathcal{F}_{\mathrm{orb}}$, we may assume that $\mathcal{F}_{\mathrm{orb}}$
is an orbifold line bundle. Replacing $\mathcal{F}_{\mathrm{orb}}$ by its
saturation in $\mathcal{E}_{\mathrm{orb}}$, we may assume that it is a
saturated subsheaf. We equip $\mathcal{E}_{\mathrm{orb}}$ with an orbifold
metric $h$ as in Lemma 3.7. Then we deduce that

$\hat{c}_{1}(\mathcal{F}_{\mathrm{orb}})\cdot\omega^{n-1}\leq\frac{1}{2n\pi}\cdot\int_{X_{\mathrm{orb}}}||K||\omega^{n}.$

The upper bound is independent of the subsheaf $\mathcal{F}_{\mathrm{orb}}$.
This completes the proof item (1).

Now we treat item (2). Without loss of generality, we may assume that the
image $E$ of $E_{\mathrm{orb}}$ in $X$ is irreducible. As in the previous
paragraph, we may assume that $\mathcal{E}_{orb}$ is reflexive and
$\mathcal{F}_{\mathrm{orb}}$ is an orbifold line bundle. We observe that
$\mathcal{E}_{\mathrm{orb}}|_{E_{\mathrm{orb}}}$ and
$\mathcal{F}_{\mathrm{orb}}|_{E_{\mathrm{orb}}}$ are orbifold coherent sheaves
on the orbifold $E_{\mathrm{orb}}$. Let
$(\mathcal{E}^{\prime}_{\mathrm{orb}})=(\mathcal{E}_{\mathrm{orb}}|_{E_{\mathrm{orb}}})/\mathrm{torsion}$
and
$\mathcal{F}^{\prime}_{\mathrm{orb}}=\mathcal{F}_{\mathrm{orb}}|_{E_{\mathrm{orb}}}$.
Since $\mathcal{F}_{\mathrm{orb}}$ is saturated in
$\mathcal{E}_{\mathrm{orb}}$, the induced morphism
$\mathcal{F}^{\prime}\to\mathcal{E}^{\prime}$ is non zero. It follows that
$\mathcal{F}^{\prime}$ is an orbifold subsheaf of $\mathcal{E}^{\prime}$. The
metric $h$ on $\mathcal{E}_{\mathrm{orb}}$ as in Lemma 3.7 induces a metric
$h^{\prime}$ on $\mathcal{E}^{\prime}$, on an open dense subset
$E_{\mathrm{orb}}^{\circ}$ of $E_{\mathrm{orb}}$. Indeed, we may take
$E_{\mathrm{orb}}^{\circ}$ as the intersection of $E_{\mathrm{orb}}$ and the
locally free locus of $\mathcal{E}$. Since $\mathcal{E}$ is reflexive, it is
locally free in codimension two of $X_{\mathrm{orb}}$. It follows that the
complement of $E_{\mathrm{orb}}^{\circ}$ in $E_{\mathrm{orb}}$ has codimension
at least two.

The property of $h$ implies that, for any point $x$ in the quotient space $E$
of $E_{\mathrm{orb}}$, there is an orbifold chart $(V^{\prime},G^{\prime})$ of
$E_{\mathrm{orb}}$ over $x$, an equivariant Hermitian vector bundle
$(\mathcal{G}^{\prime},l^{\prime})$ on $V^{\prime}$, an equivariant injection
$\mathcal{E}^{\prime}_{\mathrm{orb}}|_{V^{\prime}}\to\mathcal{G}^{\prime}$,
such that $h^{\prime}|_{V^{\prime}}$ is equal to the restriction of
$l^{\prime}$ on $\mathcal{E}^{\prime}|_{V^{\prime}}$. Therefore, the same
argument as in the first paragraph implies that, there is a constant
$C^{\prime}$ such that

$\hat{c}_{1}(\mathcal{F}^{\prime}_{\mathrm{orb}})\cdot(\omega|_{E_{\mathrm{orb}}})^{n-2}\leq
C^{\prime}.$

Finally, we observe that

$\hat{c}_{1}(\mathcal{F}_{\mathrm{orb}})\cdot\hat{c}_{1}(\mathcal{O}_{X_{\mathrm{orb}}}(E_{\mathrm{orb}}))\cdot\omega^{n-2}=\hat{c}_{1}(\mathcal{F}^{\prime}_{\mathrm{orb}})\cdot(\omega|_{E_{\mathrm{orb}}})^{n-2}.$

This completes the proof of the lemma. ∎

Now we can conclude the existence of Harder-Narasimhan filtrations and Jordan-
Hölder filtrations.

######  Lemma 3.9.

Let $X_{\mathrm{orb}}$ be a compact complex orbifold of dimension $n$ and
$\omega$ an orbifold Kähler class on $X_{\mathrm{orb}}$. Let
$\mathcal{E}_{\mathrm{orb}}$ be a torsion-free orbifold coherent sheaf on
$X_{\mathrm{orb}}$. Then the following assertions hold.

  1. (1)

There is a unique filtration of saturated orbifold subsheaves

$0=\mathcal{F}^{0}_{\mathrm{orb}}\subseteq\cdots\subseteq\mathcal{F}^{k}_{\mathrm{orb}}=\mathcal{E}_{\mathrm{orb}}$

such that the subquotients
$\mathcal{F}^{j}_{\mathrm{orb}}/\mathcal{F}^{j-1}_{\mathrm{orb}}$ are
$\omega^{n-1}$-semistable with strictly decreasing slopes.

  2. (2)

Assume that $\mathcal{E}_{\mathrm{orb}}$ is $\omega^{n-1}$-semistable. Then
there is a filtration of saturated orbifold subsheaves

$0=\mathcal{G}^{0}_{\mathrm{orb}}\subseteq\cdots\subseteq\mathcal{G}^{k}_{\mathrm{orb}}=\mathcal{E}_{\mathrm{orb}}$

such that the subquotients
$\mathcal{G}^{j}_{\mathrm{orb}}/\mathcal{G}^{j-1}_{\mathrm{orb}}$ are
$\omega^{n-1}$-stable with equal slopes.

###### Proof.

Following the strategy of [Kob14, Theorem V.7.15 and Theorem V.7.18], the
lemma follows from item (1) of Lemma 3.8. ∎

###  3.3. Bimeromorphic morphisms of complex orbifolds

We will define the blowup of an orbifold at a smooth center. The objective of
this operation is to reduce the study of orbifold coherent torsion-free sheaf
to that of orbifold vector bundle.

Let $Y_{\mathrm{orb}}=\\{(V_{i},G_{i})\\}$ be a compact complex orbifold, and
let $Z_{\mathrm{orb}}$ be a smooth orbifold subvariety of $Y_{\mathrm{orb}}$
which consists of a family $\\{W_{i}\subseteq V_{i}\\}$ of $G_{i}$-invariant
closed complex submanifolds. We assume that the image $Z$ of
$Z_{\mathrm{orb}}$ in $Y$ is irreducible.

We can define the blowup

$f_{\mathrm{orb}}\colon X_{\mathrm{orb}}\to Y_{\mathrm{orb}}$

of $Y_{\mathrm{orb}}$ with center $Z_{\mathrm{orb}}$ as follows. Let
$\varphi_{i}\colon\widehat{V}_{i}\to V_{i}$ be the blowup with center $W_{i}$.
There is an induced action of $G_{i}$ on $\widehat{V}_{i}$ such that
$\varphi_{i}$ is $G_{i}$-equivariant. Then
$X_{\mathrm{orb}}=\\{(\widehat{V}_{i},G_{i})\\}$ is a complex compact orbifold
and the family of morphisms $\\{\varphi_{i}\\}$ induces the morphism
$f_{\mathrm{orb}}$ of orbifolds. We call it the blowup of $Y_{\mathrm{orb}}$
at the smooth center $Z_{\mathrm{orb}}$. It induces a holomorphic morphism
$f\colon X\to Y$ of the quotient spaces.

Let $F_{i}\subseteq\widehat{V}_{i}$ be the exceptional locus of $\varphi_{i}$.
They define a smooth orbifold subvariety $E_{\mathrm{orb}}$ of
$X_{\mathrm{orb}}$. Each morphsim $F_{i}\to W_{i}$ is a
$\mathbb{P}^{d}$-bundle, where $d+1$ is the codimension of $Z$ in $Y$. For any
integer $n$, we can define the orbifold line bundle
$\mathcal{O}_{X_{\mathrm{orb}}}(nE_{\mathrm{orb}}):=\\{\mathcal{O}_{\widehat{V}_{i}}(nF_{i})\\}$.
We note that, for $\mathcal{O}_{\widehat{V}_{i}}(-F_{i})$, its restriction on
every fiber of $\varphi_{i}$ is isomorphic to
$\mathcal{O}_{\mathbb{P}^{d}}(1)$, and its restriction on
$\widehat{V}_{i}\setminus{F_{i}}$ is trivial. A similar argument as in [Voi02,
Proposition 3.24] implies that, if $\alpha$ is an orbifold Kähler class on
$Y_{\mathrm{orb}}$, then there is a rational number $b>0$, such that

$f^{*}\alpha-b\hat{c}_{1}(\mathcal{O}_{X_{\mathrm{orb}}}(E_{\mathrm{orb}}))$

is an orbifold Kähler class on $X_{\mathrm{orb}}$.

Alternatively, let $E\subseteq X$ be the exceptional locus of $f\colon X\to
Y$, which is equal to the image of $E_{\mathrm{orb}}$ in $X$. Then there is a
positive integer $k$ such that
$\hat{c}_{1}(\mathcal{O}_{X}(E))=k\hat{c}_{1}(\mathcal{O}_{X_{\mathrm{orb}}}(E_{\mathrm{orb}})$.
We then deduce that

$f^{*}\alpha-a\hat{c}_{1}(\mathcal{O}_{X}(E))$

is an orbifold Kähler class for some rational number $a>0$.

More generally, we can consider the composition $f_{\mathrm{orb}}\colon
X_{\mathrm{orb}}\to Y_{\mathrm{orb}}$ of a sequence of $k$ blowups at smooth
centers. There are $k$ irreducible components $E_{1},...,E_{k}$ of the
exceptional locus of the induced morphism $f\colon X\to Y$. If $\omega$ is an
orbifold Kähler class on $Y_{\mathrm{orb}}$, then there are rational numbers
$a_{1},...,a_{k}>0$ such that

(3.2) $f^{*}\alpha-\sum_{i=1}^{k}a_{j}\hat{c}_{1}(\mathcal{O}_{X}(E_{j}))$

is an orbifold Kähler class on $X_{\mathrm{orb}}$.

The following lemma is a consequence of the equivariant version of resolution
of singularities.

######  Lemma 3.10.

Let $Y_{\mathrm{orb}}={(V_{i},G_{i})}$ be a compact complex orbifold, and
$\mathcal{E}_{orb}=\\{\mathcal{E}_{i}\\}$ an orbifold coherent torsion-free
sheaf on $Y_{\mathrm{orb}}$. Then by successively blowing up smooth centers,
we can obtain a morphism $f_{\mathrm{orb}}\colon X_{\mathrm{orb}}\to
Y_{\mathrm{orb}}$ so that the following properties hold.

  1. (1)

There is an orbifold atlas $X_{\mathrm{orb}}={(\widehat{V}_{i},G_{i})}$ so
that $f_{\mathrm{orb}}$ is induced by the morphisms
$\varphi_{i}\colon\widehat{V}_{i}\to V_{i}$, which are successive blowups of
$G_{i}$-invariant smooth centers.

  2. (2)

The orbifold coherent sheaf
$(f_{\mathrm{orb}}^{*}\mathcal{E}_{orb})/\mathrm{torsion}$, which is defined
by the coherent sheaves $(\varphi_{i}^{*}\mathcal{E}_{i})/\mathrm{torsion}$ on
$\widehat{V}_{i}$, is an orbifold vector bundle.

  3. (3)

For each irreducible components $E$ of the exceptional locus of $f\colon X\to
Y$, it corresponds to a smooth orbifold subvariety of $X_{\mathrm{orb}}$.

###  3.4. Miyaoka’s inequality

We will prove the following proposition, which is a variant of [CHP16, Theorem
7.11], and its argument goes back to [Miy87, Theorem 6.1]. It will play an
important role in the positivity results on cotangent sheaves in Section 5.

######  Proposition 3.11.

Let $(X,\omega)$ be a compact Kähler threefold with quotient singularities
only. Let $\mathcal{E}$ be a reflexive sheaf on $X$ which induces an orbifold
vector bundle $\mathcal{E}_{\rm orb}$ on the standard orbifold structure
$X_{\rm orb}$ of $X$. Assume that the following conditions hold:

  1. (1)

$\alpha$ is a nef class on $X$ such that $\alpha^{2}\not\equiv 0$ and
$\alpha^{3}=0$.

  2. (2)

$\mathcal{E}$ is generically nef with respect to $\alpha_{\epsilon}$ for all
$0<\epsilon\ll 1$, where $\alpha_{\epsilon}=\alpha+\epsilon\omega$.

  3. (3)

$\hat{c}_{1}(\mathcal{E})=\alpha-\beta$ such that $\alpha\cdot\beta\equiv 0$,

Then $\hat{c}_{2}(\mathcal{E})\cdot\alpha\geq 0$.

We recall that $\mathcal{E}$ is generically nef with respect to a class
$\gamma$ if for any torsion-free quotient $\mathcal{E}\to\mathcal{Q}$ of
coherent sheaves, we have $\hat{c}_{1}(\mathcal{Q})\cdot\gamma^{2}\geq 0$. In
the proof of [Miy87, Theorem 6.1], one needs to consider second Chern classes
of torsion-free sheaves. To deal with this problem in the setting of
orbifolds, we will use the following assertion, which reduce the situation to
Chern classes of orbifold vector bundles.

######  Lemma 3.12.

Let $(Y_{\mathrm{orb}},\alpha)$ be a compact Kähler orbifold of dimension $3$.
Let $\mathcal{E}_{\mathrm{orb}}$ be a torsion-free orbifold coherent sheaf of
rank $r$ on $Y_{\mathrm{orb}}$ which is $\alpha^{2}$-stable. Assume that
$f_{\mathrm{orb}}\colon X_{\mathrm{orb}}\to Y_{\mathrm{orb}}$ is a composition
of blowups at smooth centers as in Lemma 3.10. In particular,
$\mathcal{G}_{\mathrm{orb}}=(f_{\mathrm{orb}}^{*}\mathcal{E}_{\mathrm{orb}})/{\mathrm{torsion}}$
is an orbifold vector bundle. Then we have

$\left(\hat{c}_{2}(\mathcal{G}_{\mathrm{orb}})-\frac{r-1}{2r}\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}})\right)\cdot
f^{*}\alpha\geq 0,$

where $f\colon X\to Y$ is the induced morphism.

The idea of the proof is to approximate $f^{*}\alpha$ by a family of orbifold
Kähler classes $\\{\alpha_{\epsilon}\\}$ on $X_{\mathrm{orb}}$. In order to
carry out this method, we need to show that $\mathcal{G}_{\mathrm{orb}}$ is
$(\alpha_{\epsilon})^{2}$-stable for all $\epsilon>0$ small enough. Before
giving the proof of this lemma, we will first establish a few preparatory
results.

######  Lemma 3.13.

Let $(Y_{\mathrm{orb}},\alpha)$ be a compact Kähler orbifold of dimension $3$,
and let $f_{\mathrm{orb}}\colon X_{\mathrm{orb}}\to Y_{\mathrm{orb}}$ be the
composition of blowups at smooth centers. Let $f\colon X\to Y$ be the induced
morphism on quotient spaces. Assume that $\omega$ is an orbifold Kähler class
on $X_{\mathrm{orb}}$ such that

$\omega=f^{*}\alpha-\sum_{k}a_{k}\hat{c}_{1}(\mathcal{O}_{X}(E_{k})),$

where the $E_{k}$’s are the irreducible components of $f$-exceptional locus,
and the $a_{k}$’s are positive rational numbers. Assume that each $E_{k}$
corresponds to a smooth orbifold subvarieties $E_{k,\mathrm{orb}}$ of
$X_{\mathrm{orb}}$.

Let $\mathcal{G}_{\mathrm{orb}}$ be an orbifold coherent torsion-free sheaf on
$X_{\mathrm{orb}}$. Then the following set is bounded from above,

$\displaystyle\left\\{\hat{c}_{1}(\mathcal{F}_{\mathrm{orb}})\cdot(f^{*}\alpha)^{2},\
\ \hat{c}_{1}(\mathcal{F}_{\mathrm{orb}})\cdot f^{*}\alpha\cdot\omega,\ \
\hat{c}_{1}(\mathcal{F}_{\mathrm{orb}})\cdot\omega^{2}\right\\}$

where $\mathcal{F}_{\mathrm{orb}}\subseteq\mathcal{G}_{\mathrm{orb}}$ runs
through all orbifold coherent subsheaves.

###### Proof.

Since $f^{*}\alpha$ and $\omega$ are both nef classes, it is enough to prove
the statement for saturated subsheaves
$\mathcal{F}_{\mathrm{orb}}\subseteq\mathcal{G}_{\mathrm{orb}}$. Furthermore,
replacing $\mathcal{F}_{\mathrm{orb}}$ and $\mathcal{G}_{\mathrm{orb}}$ by
their reflexive hulls, we may assume that $\mathcal{G}_{\mathrm{orb}}$ is
reflexive. We define

$m(\beta,\beta^{\prime}):=\sup_{\mathcal{F}_{\mathrm{orb}}}\,\\{\hat{c}_{1}(\mathcal{F}_{\mathrm{orb}})\cdot\beta\cdot\beta^{\prime}\\}\in\mathbb{R}\cup\\{+\infty\\}$

for any classes $\beta,\beta^{\prime}\in H^{2}(X,\mathbb{R})$, where
$\mathcal{F}_{\mathrm{orb}}$ runs through all non zero saturated orbifold
coherent subsheaves of $\mathcal{G}_{\mathrm{orb}}$. Then, by item (1) of
Lemma 3.8, we see that $m(\omega,\omega)$ is a real number. By applying the
same argument to the torsion-free sheaf
$(f_{\mathrm{orb}})_{*}\mathcal{G}_{\mathrm{orb}}$ on $Y_{\mathrm{orb}}$, we
deduce that $m(f^{*}\alpha,f^{*}\alpha)$ is finite as well. We remark that

$\hat{c}_{1}(\mathcal{F}_{\mathrm{orb}})\cdot
f^{*}\alpha\cdot\omega=\hat{c}_{1}(\mathcal{F}_{\mathrm{orb}})\cdot(\omega^{2}+(\sum_{k}a_{k}\hat{c}_{1}(\mathcal{O}_{X}(E_{k})))\cdot\omega).$

Thus, it suffices to show that
$m(\hat{c}_{1}(\mathcal{O}_{Y_{\mathrm{orb}}}(E_{k})),\omega)$ is finite for
each $k$.

We notice that there are positive constants $b_{k}$ such that

$\hat{c}_{1}(\mathcal{O}_{X}(E_{k}))=b_{k}\hat{c}_{1}(\mathcal{O}_{X_{\mathrm{orb}}}(E_{k,orb})).$

Hence by item (2) of Lemma 3.8, we deduce that
$m(\hat{c}_{1}(\mathcal{O}_{X}(E_{k})),\omega)$ is finite for every $k$. This
completes the proof of the lemma. ∎

######  Lemma 3.14.

Let
$f_{\mathrm{orb}}\colon(X_{\mathrm{orb}},\omega)\to(Y_{\mathrm{orb}},\alpha)$
be as in Lemma 3.13. Let $\alpha_{\epsilon}=f^{*}\alpha+\epsilon\omega$ for
all $\epsilon>0$. Assume that $\mathcal{G}_{\mathrm{orb}}$ is an orbifold
coherent torsion-free sheaf on $X_{\mathrm{orb}}$ which is
$(f^{*}\alpha)^{2}$-stable. Then it is $(\alpha_{\epsilon})^{2}$-stable for
all $\epsilon>0$ small enough.

###### Proof.

We claim that there is a number $C>0$, such that

$\mu_{(f^{*}\alpha)^{2}}(\mathcal{F}_{\mathrm{orb}})\leq\mu_{(f^{*}\alpha)^{2}}(\mathcal{G}_{\mathrm{orb}})-C.$

for any non zero proper orbifold subsheaf
$\mathcal{F}_{\mathrm{orb}}\subseteq\mathcal{G}_{orb}$.

Indeed, assume that $X_{\mathrm{orb}}=\\{(V_{i},G_{i})\\}$ is a finite atlas
of charts. Let $q$ be the lcm of the cardinalities of the $G_{i}$’s. Then for
any orbifold coherent sheaf $\mathcal{L}_{\mathrm{orb}}$ on
$X_{\mathrm{orb}}$, we have

$\hat{c}_{1}(\mathcal{L}_{\mathrm{orb}})\in H^{2}(X,\frac{1}{q}\mathbb{Z}).$

It follows that

$\\{\mu_{(f^{*}\alpha)^{2}}(\mathcal{F}_{\mathrm{orb}})\ :\
0\neq\mathcal{F}_{\mathrm{orb}}\subseteq\mathcal{G}_{\mathrm{orb}}\\}$

is a discrete subset of $\mathbb{R}$. We may hence deduce the existence of $C$
from the stability of $\mathcal{G}_{\mathrm{orb}}$. Once we obtain such a
number $C>0$, the lemma follows from Lemma 3.13. ∎

We can now deduce Lemma 3.12.

###### Proof of Lemma 3.12.

There is an orbifold Kähler class on $X_{\mathrm{orb}}$ of the shape

$\omega=f^{*}\alpha-\sum a_{k}\hat{c}_{1}(\mathcal{O}_{X}(E_{k})),$

where the $E_{k}$’s are the irreducible components of $f$-exceptional locus,
and the $a_{k}$’s are positive rational numbers. Let
$\alpha_{\epsilon}=f^{*}\alpha+\epsilon\omega$ for $\epsilon>0$. Then
$\alpha_{\epsilon}$ is an orbifold Kähler class on $X_{\mathrm{orb}}$, and
$\mathcal{G}_{\mathrm{orb}}$ is $(\alpha_{\epsilon})^{2}$-stable for
$\epsilon>0$ small enough by Lemma 3.14. By [Fau22, Theorem 1],
$\mathcal{G}_{\mathrm{orb}}$ admits an orbifold Hermitian-Einstein metric with
respect to $\alpha_{\epsilon}$. Hence by a similar argument as in Lübke’s
theorem in [Lüb82] (see also [Kob14, Theorem 4.4.7]), we deduce that

$\left(\hat{c}_{2}(\mathcal{G}_{\mathrm{orb}})-\frac{r-1}{2r}\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}})\right)\cdot\alpha_{\epsilon}\geq
0.$

We can then conclude the lemma by taking the limit as $\epsilon\to 0$. ∎

Now we are ready to prove Proposition 3.11.

###### Proof of Proposition 3.11.

We first remark the following estimate,

$\frac{1}{\alpha_{\epsilon}^{3}}\cdot(\alpha-\beta)\cdot\alpha_{\epsilon}^{2}=\frac{2\epsilon\cdot\alpha^{2}\omega+\epsilon^{2}\cdot(\alpha-\beta)\omega^{2}}{3\epsilon\cdot\alpha^{2}\omega+3\epsilon^{2}\cdot\alpha\omega^{2}+\epsilon^{3}\cdot\omega^{3}}\sim_{\epsilon\to
0}\frac{2\epsilon\cdot\alpha^{2}\omega}{3\epsilon\cdot\alpha^{2}\omega}=\frac{2}{3}.$

Therefore, if $\epsilon>0$ is sufficiently small, we have

$\frac{1}{\alpha_{\epsilon}^{3}}\cdot(\alpha-\beta)\cdot\alpha_{\epsilon}^{2}\leq
1.$

We fix some $\epsilon>0$ sufficiently small so that the previous inequality
holds. Since $\alpha_{\epsilon}$ is a Kähler class on $X$, it is an orbifold
Kähler class on $X_{\mathrm{orb}}$ by Lemma 3.5. By taking the Harder-
Narasimhan filtration of $\mathcal{E}_{\mathrm{orb}}$, and then taking a
Jordan-Hölder filtration on each subquotient, we obtain a filtration of
orbifold reflexive coherent sheaves

$0=\mathcal{F}^{0}_{\mathrm{orb}}\subseteq\cdots\subseteq\mathcal{F}^{k}_{\mathrm{orb}}=\mathcal{E}_{\mathrm{orb}}$

such that each subquotient
$\mathcal{G}^{j}_{\mathrm{orb}}:=\mathcal{F}^{j}_{\mathrm{orb}}/\mathcal{F}^{j-1}_{\mathrm{orb}}$
is $(\alpha_{\epsilon})^{2}$-stable. Furthermore, if $\delta_{\epsilon,j}$ is
the slope of $\mathcal{G}_{\mathrm{orb}}^{j}$ with respect to
$(\alpha_{\epsilon})^{2}$, then we have

$\delta_{\epsilon,1}\geq\cdots\geq\delta_{\epsilon,k}.$

Since $X_{\mathrm{orb}}$ is standard, $\mathcal{F}^{k-1}_{\mathrm{orb}}$
induces a coherent subsheaf $\mathcal{F}^{k-1}\subseteq\mathcal{E}$, such that
$\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{k})=\hat{c}_{1}(\mathcal{E})-\hat{c}_{1}(\mathcal{F}^{k-1})$,
see (5) of Remark 3.4. Since $\mathcal{E}$ is generically nef with respect to
$\alpha_{\epsilon}$, we deduce that $\delta_{\epsilon,k}\geq 0$.

By applying Lemma 3.10 to the subquotients $\mathcal{G}^{j}_{\mathrm{orb}}$
successively, we obtain an orbifold $X^{\prime}_{\mathrm{orb}}$ with the
natural projection $f_{\mathrm{orb}}\colon X^{\prime}_{\mathrm{orb}}\to
X_{\mathrm{orb}}$ so that we have an induced filtration

$0=\mathcal{F}^{\prime
0}_{\mathrm{orb}}\subseteq\cdots\subseteq\mathcal{F}^{\prime
k}_{\mathrm{orb}}=\mathcal{E}^{\prime}_{\mathrm{orb}}=f_{\mathrm{orb}}^{*}\mathcal{E}_{\mathrm{orb}}$

such that each subquotient $\mathcal{G}^{\prime
j}_{\mathrm{orb}}=\mathcal{F}^{\prime j}_{\mathrm{orb}}/\mathcal{F}^{\prime
j-1}_{\mathrm{orb}}$ is an orbifold vector bundle. By Lemma 3.12, we have

$\left(\hat{c}_{2}(\mathcal{G}^{\prime
j}_{\mathrm{orb}})-\frac{r_{j}-1}{2r_{j}}\hat{c}_{1}(\mathcal{G}^{\prime
j}_{\mathrm{orb}})^{2}\right)\cdot f^{*}\alpha_{\epsilon}\geq 0,$

where $r_{j}$ is the rank of $\mathcal{G}^{j}_{\mathrm{orb}}$, and $f\colon
X^{\prime}\to X$ is the induced morphism on quotient spaces.

As a consequence, we get

$\displaystyle\hat{c}_{2}(\mathcal{E})\cdot\alpha_{\epsilon}$ $\displaystyle=$
$\displaystyle\hat{c}_{2}(f^{*}_{\mathrm{orb}}\mathcal{E}_{\mathrm{orb}})\cdot
f^{*}\alpha_{\epsilon}$ $\displaystyle=$
$\displaystyle\left(\sum_{j}\hat{c}_{2}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}j})+\sum_{j<l}\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}j})\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}l})\right)\cdot
f^{*}\alpha_{\epsilon}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\left(\sum_{j}\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}j})^{2}+2\sum_{j<l}\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}j})\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}l})-\sum_{j}\frac{1}{r_{j}}\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}j})^{2}\right)\cdot
f^{*}\alpha_{\epsilon}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\hat{c}_{1}(f_{\mathrm{orb}}^{*}\mathcal{E}_{\mathrm{orb}})^{2}-\sum_{j}\frac{1}{r_{j}}\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}j})^{2}\right)\cdot
f^{*}\alpha_{\epsilon}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\hat{c}_{1}(\mathcal{E})^{2}\cdot\alpha_{\epsilon}-\frac{1}{2}\left(\sum_{j}\frac{1}{r_{j}}\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}j})^{2}\right)\cdot
f^{*}\alpha_{\epsilon}$

Applying the Hodge index theorem (see Lemma 3.15 below) on $X^{\prime}$, and
noting that $(f^{*}\alpha_{\epsilon})^{3}=\alpha_{\epsilon}^{3}$, we obtain
that

$\alpha_{\epsilon}^{3}\cdot\left(\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}j})^{2}\cdot
f^{*}\alpha_{\epsilon}\right)\leq\left(\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}j})\cdot(f^{*}\alpha_{\epsilon})^{2}\right)^{2}.$

Hence we deduce that

$\displaystyle\left(\sum_{j}\frac{1}{r_{j}}\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}j})^{2}\right)\cdot
f^{*}\alpha_{\epsilon}$ $\displaystyle\leq$
$\displaystyle\frac{1}{\alpha_{\epsilon}^{3}}\sum_{j}\frac{1}{r_{j}}\left(\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}j})\cdot(f^{*}\alpha_{\epsilon})^{2}\right)^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{\alpha_{\epsilon}^{3}}\sum_{j}r_{j}\delta_{\epsilon,j}^{2}$
$\displaystyle\leq$
$\displaystyle\frac{\delta_{\epsilon,1}}{\alpha_{\epsilon}^{3}}\sum_{j}r_{j}\delta_{\epsilon,j}$
$\displaystyle=$
$\displaystyle\frac{\delta_{\epsilon,1}}{\alpha_{\epsilon}^{3}}\cdot\hat{c}_{1}(\mathcal{E}_{\mathrm{orb}})\cdot\alpha_{\epsilon}^{2}$
$\displaystyle=$
$\displaystyle\frac{\delta_{\epsilon,1}}{\alpha_{\epsilon}^{3}}\cdot(\alpha-\beta)\cdot\alpha_{\epsilon}^{2}.$

Since $\epsilon>0$ is small enough, from the first paragraph, we obtain that

$\left(\sum_{j}\frac{1}{r_{j}}\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}j})^{2}\right)\cdot
f^{*}\alpha_{\epsilon}\leq\delta_{\epsilon,1}\cdot\frac{(\alpha-\beta)\cdot\alpha_{\epsilon}^{2}}{\alpha_{\epsilon}^{3}}\leq\delta_{\epsilon,1}\cdot
1=\delta_{\epsilon,1}.$

Since
$\sum_{j}r_{j}\delta_{\epsilon,j}=\hat{c}_{1}(\mathcal{E})\cdot\alpha_{\epsilon}^{2}$
and since $\delta_{\epsilon,j}\geq 0$ for $j=1,...,k$, we deduce that
$r_{1}\delta_{\epsilon,1}\leq\hat{c}_{1}(\mathcal{E})\cdot\alpha_{\epsilon}^{2}$.
Hence

$\left(\sum_{j}\frac{1}{r_{j}}\hat{c}_{1}(\mathcal{G}_{\mathrm{orb}}^{{}^{\prime}j})^{2}\right)\cdot
f^{*}\alpha_{\epsilon}\leq\frac{1}{r_{1}}\hat{c}_{1}(\mathcal{E})\cdot\alpha_{\epsilon}^{2}\leq\hat{c}_{1}(\mathcal{E})\cdot\alpha_{\epsilon}^{2}.$

Therefore, we get

$\hat{c}_{2}(\mathcal{E})\cdot\alpha_{\epsilon}=\hat{c}_{2}(f_{\mathrm{orb}}^{*}\mathcal{E}_{\mathrm{orb}})\cdot
f^{*}\alpha_{\epsilon}\geq\frac{1}{2}\hat{c}_{1}(\mathcal{E})^{2}\cdot\alpha_{\epsilon}-\frac{1}{2}\hat{c}_{1}(\mathcal{E})\cdot\alpha_{\epsilon}^{2}$

By tending $\epsilon$ to zero, we conclude that

$\hat{c}_{2}(\mathcal{E})\cdot\alpha\geq\frac{1}{2}\hat{c}_{1}(\mathcal{E})^{2}\cdot\alpha-\frac{1}{2}\hat{c}_{1}(\mathcal{E})\cdot\alpha^{2}=0.$

This completes the proof of the proposition. ∎

The following version of Hodge index theorem was used in the previous proof.

######  Lemma 3.15.

Let $X$ be a compact Kähler variety of dimension $n$ with rational
singularities and let $\alpha\in H^{2}(X,\mathbb{R})$ be a nef class. Then for
any $\beta\in H^{2}(X,\mathbb{R})$, we have

$\alpha^{n}\cdot(\beta^{2}\cdot\alpha^{n-2})\leq(\beta\cdot\alpha^{n-1})^{2}.$

###### Proof.

Let $r\colon\widetilde{X}\to X$ be a desingularization such that there is a
Kähler class $\omega$ on $\widetilde{X}$. Let
$\alpha_{\epsilon}=r^{*}\alpha+\epsilon\omega$ for $\epsilon>0$. Then each
$\alpha_{\epsilon}$ is a Kähler class on $\widetilde{X}$. The classic Hodge
index theorem implies that

$\alpha_{\epsilon}^{n}\cdot((r^{*}\beta)^{2}\cdot\alpha_{\epsilon}^{n-2})\leq(r^{*}\beta\cdot\alpha_{\epsilon}^{n-1})^{2}.$

Then the lemma follows by taking the limit as $\epsilon\to 0$. ∎

##  4\. Comparison of Chern classes

Consider a standard complex orbifold $X_{\mathrm{orb}}$ and its quotient space
$X$. We have seen in (5) of Remark 3.4 that every orbifold vector bundle
$\mathcal{E}_{\mathrm{orb}}$ induces a reflexive sheaf $\mathcal{E}$ on $X$.
The objective of this section is to compare the orbifold Chern classes of
$\mathcal{E}_{\mathrm{orb}}$ and the Chern classes of
$\widetilde{\mathcal{E}}$, where $\widetilde{\mathcal{E}}$ is a vector bundle
on a desingularization $r\colon\widetilde{X}\to X$ such that
$(r_{*}\widetilde{\mathcal{E}})^{**}=\mathcal{E}$.

###  4.1. Relative de Rham cohomology

Let $M$ be a $\mathcal{C}^{\infty}$ manifold. We denote by
$\mathscr{D}^{\prime i}$ the sheaf of real currents of degree $i$, i.e.
differential $i$-forms with real distribution coefficients. Then there is a
complex

$\cdots\to D^{i}(M)\overset{\mathrm{d}}{\to}D^{i+1}(M)\to\cdots,$

where $D^{i}(M)=\Gamma(M,\mathscr{D}^{\prime i}(M))$ is the group of
$\mathcal{C}^{\infty}$ global sections of the sheaf $\mathscr{D}^{\prime i}$.
The cohomology groups of the previous complex are denoted by
$H^{i}_{\mathscr{D}^{\prime}}(M,\mathbb{R})$. Then there are natural
isomorphisms $H^{i}_{\mathscr{D}^{\prime}}(M,\mathbb{R})\cong
H^{i}(M,\mathbb{R})$ known as the de Rham isomorphisms. We consider the
following complex

$\cdots\to D^{i}(M,U)\overset{\mathrm{d}}{\to}D^{i+1}(M,U)\to\cdots$

where $U\subseteq M$ is an open set, $D^{i}(M,U)=D^{i}(M)\oplus D^{i-1}(U)$
and the differentials are defined in the following way:

$\mathrm{d}(\omega,\eta)=(\mathrm{d}\omega,\omega|_{U}-\mathrm{d}\eta).$

We denote the cohomology groups of the previous complex by
$H^{i}_{\mathscr{D}^{\prime}}(M,U,\mathbb{R})$. Then there is a long exact
sequence

$\cdots\to H^{i}_{\mathscr{D}^{\prime}}(M,U,\mathbb{R})\to
H^{i}_{\mathscr{D}^{\prime}}(M,\mathbb{R})\overset{\beta}{\to}H^{i}_{\mathscr{D}^{\prime}}(U,\mathbb{R})\to\cdots$

such that $\beta$ is the restriction map. We refer to [BT82, Page 78-79] for
detailed discussion on relative de Rham cohomology. We note that [BT82] only
treats de Rham cohomology of differential forms. But the proof of [BT82, Claim
6.48] also holds for de Rham cohomology of currents in our setting. By
applying the five lemma on the long exact sequences of relative cohomology, we
obtain natural isomorphisms

$H^{i}_{\mathscr{D}^{\prime}}(M,U,\mathbb{R})\cong H^{i}(M,U,\mathbb{R}).$

###  4.2. Relative Chern classes

We aim to compare orbifold Chern classes and Chern classes on a
desingularization. Similar topics have been studied before, especially for
surfaces, see for example [Wah93], [Bla96] and [Lan00]. Inspired by their
works, we will introduce the relative Chern classes in any dimension.

Let $X$ be a normal complex analytic variety with quotient singularities, and
$r\colon\widetilde{X}\to X$ a desingularization which is an isomorphism over
$X_{\operatorname{\textsubscript{\rm sm}}}$. We denote by
$E\subseteq\widetilde{X}$ the exceptional locus of $r$, and define
$\widetilde{X}^{\circ}=\widetilde{X}\setminus E$.

We denote the standard orbifold structure of $X$ with an atlas of charts by
$X_{\mathrm{orb}}=\\{(V_{i},G_{i})\\}$. Let $\mathcal{E}$ be a reflexive sheaf
on $X$ which induces an orbifold vector bundle
$\mathcal{E}_{\mathrm{orb}}=\\{\mathcal{E}_{i}\\}$ on $X_{\mathrm{orb}}$.
Assume that there is a vector bundle $\widetilde{\mathcal{E}}$ on
$\widetilde{X}$ such that $(r_{*}(\widetilde{\mathcal{E}}))^{**}=\mathcal{E}$.

Let $h_{X_{\operatorname{\textsubscript{\rm sm}}}}$ be a Hermitian metric on
the vector bundle $\mathcal{E}|_{X_{\operatorname{\textsubscript{\rm sm}}}}$
which induces an orbifold Hermitian metric $\\{h_{i}\\}$ on
$\mathcal{E}_{\mathrm{orb}}=\\{\mathcal{E}_{i}\\}$. On one hand, by [Bla96,
Lemma 1.15], the metric Chern classes
$c_{i}(h_{X_{\operatorname{\textsubscript{\rm sm}}}})$ induce closed currents
$r^{*}c_{i}(h)$ on $\widetilde{X}$. On the other hand, by definition,
$c_{i}(h_{X_{\operatorname{\textsubscript{\rm sm}}}})$ defines the orbifold
Chern classes
$\hat{c}_{i}(\mathcal{E})=\hat{c}_{i}(\mathcal{E}_{\mathrm{orb}})$. See
[Bla96, Section 2] for more details.

Let $\tilde{h}$ be a Hermitian metric on $\widetilde{\mathcal{E}}$, and
$c_{i}(\tilde{h})$ the metric Chern classes. Then
$c_{i}(\tilde{h})-r^{*}c_{i}(h)$ is exact on $\widetilde{X}^{\circ}$. That is,
there is some $\eta\in D^{2i-1}(\widetilde{X}^{\circ})$ such that

$(c_{i}(\tilde{h})-r^{*}c_{i}(h))|_{\widetilde{X}^{\circ}}=\mathrm{d}\eta.$

Hence it corresponds to the relative de Rham current cohomology class

$(c_{i}(\tilde{h})-r^{*}c_{i}(h),\eta)\in
H^{2i}_{\mathscr{D}^{\prime}}(\widetilde{X},\widetilde{X}^{\circ},\mathbb{R}).$

We notice that this class is independent of the choices of the metrics. We
define the relative Chern class $c_{i}(\widetilde{\mathcal{E}},r)$ as the
relative cohomology class

(4.1)
$c_{i}(\widetilde{\mathcal{E}},r):=c_{i}(\tilde{h})-r^{*}c_{i}(h)=c_{i}(\widetilde{\mathcal{E}})-r^{*}\hat{c}_{i}(\mathcal{E})\in
H^{2i}(\widetilde{X},\widetilde{X}^{\circ},\mathbb{R})$

via the de Rham isomorphism.

Let $T_{\widetilde{X}}$ be the tangent bundle of $\widetilde{X}$. Then
$(r_{*}T_{\widetilde{X}})^{**}=T_{X}$. We set
$\hat{c}_{i}(X):=\hat{c}_{i}(T_{X})$. Then we can define the following
relative Chern classes

(4.2) $c_{i}(\widetilde{X},r):=c_{i}(T_{\widetilde{X}},r).$

We also define the following classes

$c_{1}^{2}(\widetilde{\mathcal{E}},r):=(c_{1}(\widetilde{X},\widetilde{\mathcal{E}}))^{2}-(r^{*}\hat{c}_{1}(X,\mathcal{E}))^{2}\mbox{
and
}c_{1}^{2}(\widetilde{X},r):=(c_{1}(\widetilde{X}))^{2}-(r^{*}\hat{c}_{1}(X))^{2}$

as elements in $H^{4}(\widetilde{X},\widetilde{X}^{\circ},\mathbb{R})$, as
well as

(4.3)
$\ell(\widetilde{\mathcal{E}},r):=c_{1}^{2}(\widetilde{\mathcal{E}},r)+c_{2}(\widetilde{\mathcal{E}},r)\mbox{
and
}\ell(\widetilde{X},r):=c_{1}^{2}(\widetilde{X},r)+c_{2}(\widetilde{X},r).$

There are natural isomorphisms

$H^{2i}(\widetilde{X},\widetilde{X}^{\circ},\mathbb{R})\cong
H^{BM}_{2n-2i}(E,\mathbb{R}),$

where $H^{BM}$ is the Borel-Moore homology group, see Subsection 4.3 below for
more details. Particularly, when $\widetilde{X}$ is a surface, we have

$H^{BM}_{0}(E,\mathbb{R})\cong\mathbb{R}^{d},$

where $d$ is the number of singular points in $X$. Furthermore, since
$c_{1}^{2}+c_{2}$ is invariant under blowups of points on smooth surfaces, we
see that $\ell(\widetilde{X},r)$ depends only on the singularities of $X$ when
$X$ is a surface.

###  4.3. Borel-Moore homology

As we have seen in the previous subsection, the relative Chern classes lie in
some groups of relative cohomology, which are canonically isomorphic to
certain groups of Borel-Moore homology. In this subsection, we recall some
elementary results concerning Borel-Moore homology. For more details, we refer
to [Ful98, Chapter 19] and the references therein.

Let $X$ be a complex analytic space of pure dimension $n$. For simplicity, we
always assume that it is an open subset of a compact complex analytic space
$M$. Furthermore, if ${X}^{\prime}$ is the Euclidean closure of $X$ in $M$, we
assume that $({X}^{\prime},{X}^{\prime}\setminus X)$ is homeomorphic to a CW-
complex pair. There are several interpretation of
$H^{BM}_{\bullet}(X,\mathbb{R})$:

  1. (1)

Assume that $\overline{X}$ is a compact space containing $X$ as an open
subset, such that $(\overline{X},\overline{X}\setminus X)$ is homeomorphic to
a CW-complex pair. Then
$H^{BM}_{\bullet}(X,\mathbb{R})=H_{\bullet}(\overline{X},\overline{X}\setminus
X,\mathbb{R}).$

  2. (2)

Assume that $X$ is embedded into an oriented manifold $M$ of real dimension
$d$, as a closed subset. Then
$H^{BM}_{\bullet}(X,\mathbb{R})=H^{d-\bullet}(M,M\setminus X,\mathbb{R}).$

  3. (3)

$H^{BM}_{\bullet}(X,\mathbb{R})$ is dual to the cohomology with compact
support $H^{\bullet}_{c}(X,\mathbb{R})$.

The top Borel-Moore homology group $H^{BM}_{2n}(X,\mathbb{R})$ is generated by
the fundamental classes $[X_{i}]$, where the $X_{i}$’s are the irreducible
components of $X$. The dimension of the $0$-th Borel-Moore homology group
$H^{BM}_{0}(X,\mathbb{R})$ is equal to the number of compact connected
component of $X$. Particularly, when $X$ is compact connected, we fix the
natural isomorphism $H^{BM}_{0}(X,\mathbb{R})\cong\mathbb{R}$ by sending the
class of a point to $1$.

######  Lemma 4.1.

Let $f\colon X\to Y$ be a proper morphism of complex analytic varieties. Let
$V\subseteq Y$ be a connected non empty Euclidean open subset and
$U=f^{-1}(V)$. Assume that $V$ is regular enough, e.g. its closure and
boundary are homeomorphic to finite CW-complexes. Then for any integer $k\geq
0$, we have the following commutative diagram

$\textstyle{H^{BM}_{k}(X,\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{BM}_{k}(U,\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{BM}_{k}(Y,\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{BM}_{k}(V,\mathbb{R})}$

where the horizontal arrows are restriction maps and the vertical arrows are
proper pushforwards.

###### Proof.

We set $S=X\setminus U$ and $T=Y\setminus V$. Let
$\overline{X}=X\cup\\{\infty_{X}\\}$ and $\overline{Y}=Y\cup\\{\infty_{Y}\\}$
be the one-point compactifications. Then $f$ extends naturally to a proper
continuous map from $\overline{X}$ to $\overline{Y}$, mapping $\infty_{X}$ to
$\infty_{Y}$. We have the following natural isomorphisms

$H^{BM}_{k}(X,\mathbb{R})=H_{k}(\overline{X},\\{\infty_{X}\\},\mathbb{R}),\
H^{BM}_{k}(U,\mathbb{R})=H_{k}(\overline{X},S\cup\\{\infty_{X}\\},\mathbb{R}),$

and

$H^{BM}_{k}(Y,\mathbb{R})=H_{k}(\overline{Y},\\{\infty_{Y}\\},\mathbb{R}),\
H^{BM}_{k}(V,\mathbb{R})=H_{k}(\overline{X},T\cup\\{\infty_{X}\\},\mathbb{R}).$

We remark that there is a commutative diagram

$\textstyle{H_{k}(\overline{X},\\{\infty_{X}\\},\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{k}(\overline{X},S\cup\\{\infty_{X}\\},\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{k}(\overline{Y},\\{\infty_{Y}\\},\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{k}(\overline{Y},T\cup\\{\infty_{Y}\\},\mathbb{R})}$

where the horizontal arrows are natural morphisms of relative homology, and
the virtical arrows are pushforwards of singular homology. This implies the
diagram of the lemma. ∎

For a closed analytic subspace $E\subseteq X$ and for any integers $p,q\geq
0$, there is a cap product operation

$\smallfrown\colon H^{p}(X,X\setminus E,\mathbb{R})\times
H^{BM}_{p+q}(X,\mathbb{R})\to H^{BM}_{q}(E,\mathbb{R}).$

We have the following statement.

######  Lemma 4.2.

Let $f\colon X\to Y$ be a proper morphism of complex analytic varieties. Let
$V\subseteq Y$ be a compact analytic subspace. Let $\sigma\in
H^{BM}_{\bullet}(X,\mathbb{R})$ and $\ell\in H^{\bullet}(Y,Y\setminus
V,\mathbb{R})$. Then the following projection formula holds:

$f_{*}\sigma\smallfrown\ell=f_{*}(\sigma\smallfrown f^{*}\ell)\in
H^{BM}_{\bullet}(V,\mathbb{R}).$

###### Proof.

By taking one-point compactifications of $X$ and $Y$, the problem is reduced
to the usual projection formula for homology and cohomology. ∎

Assume that $X$ is a complex manifold of dimension $n$. For any closed
analytic subspace $S\subseteq X$, for each integer $p\geq 0$, there is an
isomorphism

(4.4) $H^{2n-p}(X,X\setminus S,\mathbb{R})\to H^{BM}_{p}(S,\mathbb{R})$

given by the cap product with $[X]$. If $S$ is irreducible of dimension $k$,
then there is a unique cohomology class

$cl^{X}(S)\in H^{2n-2k}(X,X\setminus S,\mathbb{R})$

such that the fundamental class of $[S]$ is given by

$[S]=cl^{X}(S)\smallfrown[X]\in H^{BM}_{2n}(S,\mathbb{R}).$

In particular, when $S$ is a submanifold of $X$, then $cl^{X}(S)$ of just the
Thom class of $S$ in $X$.

Let $E\subseteq X$ be a closed analytic subspace. We set $\Delta=E\cap S$. For
any class $\ell\in H^{q}(X,X\setminus E,\mathbb{R})$, any class $\sigma\in
H^{BM}_{p}(S,\mathbb{R})$, we define

(4.5) $\ell\smallfrown\sigma:=(\ell\smallsmile u)\smallfrown[X].$

as an element in $H^{BM}_{p-q}(\Delta,\mathbb{R}),$ where $u\in
H^{2n-p}(X,X\setminus S,\mathbb{R})$ is the unique element such that
$\sigma=u\smallfrown[X]$, and $\smallsmile$ is the cup product of cohomology.

###  4.4. Comparison with $\mathbb{Q}$-Chern classes.

In this subsection, we will compare the orbifold Chern classes in our context
and the $\mathbb{Q}$-Chern classes defined in [Mum83, Section 3] or [Kol92,
Chapter 10], on projective surfaces $X$ with quotient singularities. Let
$\mathcal{E}$ be a reflexive sheaf on $X$ which is also an orbifold vector
bundle on the standard orbifold structure $X_{\mathrm{orb}}$. We note that the
$\mathbb{Q}$-Chern classes, which we denote by
$\overline{c}_{i}(\mathcal{E})$, of [Kol92, Chapter 10], is a cycle class in
$A_{i}(X)\otimes\mathbb{Q}$, where $A_{i}$ is the Chow group of cycles of
dimension $i$. However, the orbifold Chern classes $\hat{c}_{i}(\mathcal{E})$
are cohomology classes. In order to compare them, we will map them into a same
set. Let $[X]\in H_{4}(X,\mathbb{R})$ be the fundamental class of $X$, and let
$[\overline{c}_{i}(\mathcal{E})]$ be the fundamental class of
$\overline{c}_{i}(\mathcal{E})$ in $H_{2i}(X,\mathbb{R})$.

######  Lemma 4.3.

With the notation above, we have

(4.6)
$[\overline{c}_{i}(\mathcal{E})]=\hat{c}_{i}(\mathcal{E})\smallfrown[X]\in
H_{4-2i}(X,\mathbb{R}).$

As a consequence, we can apply the calculation of [Kol92, Chapter 10] for
$\mathbb{Q}$-Chern classes to orbifold Chern classes in our context for
projective surfaces.

###### Proof.

Let $i\in\\{0,1,2\\}$. We recall the definition in [Kol92, Chapter 10]. There
is a finite cover $p\colon\widetilde{X}\to X$, called global cover in [Mum83,
Section 2], satisfying the following conditions.

  1. (1)

$p$ is Galois of group $G$.

  2. (2)

$\widetilde{X}$ is a normal surface, hence Cohen-Macaulay.

  3. (3)

There is a finite family of orbifold charts
$X_{\mathrm{orb}}=\\{(V_{i},G_{i})\\}$, such that for every
$\tilde{x}\in\widetilde{X}$, there is an open Euclidean neighborhood
$\widetilde{U}$ of $\tilde{x}$, such that $p|_{\widetilde{U}}$ factors through
some $V_{i}$.

As a consequence, we deduce that
$\widetilde{\mathcal{E}}=(p^{*}\mathcal{E})^{**}$ is locally free. We can view
the Chern class of $\widetilde{\mathcal{E}}$, which we denote by
$\tilde{c}_{i}(\widetilde{\mathcal{E}})$, as an element in
$A^{i}(\widetilde{X})$, where $A^{i}$ is the $i$-th Chow cohomology group, in
the sense of [Ful98, Chapter 17]. Note that
$\tilde{c}_{i}(\widetilde{\mathcal{E}})$ is invariant under the natural action
of $G$. The Chern classes in [Kol92, Chapter 10] are defined as

$\overline{c}_{i}(\mathcal{E})=\frac{1}{|G|}\cdot
p_{*}(\tilde{c}_{i}(\widetilde{\mathcal{E}})\smallfrown[\widetilde{X}]^{\prime})\in
A_{2-i}(X)\otimes\mathbb{R},$

where $[\widetilde{X}]^{\prime}\in A_{2}(\widetilde{X})$ is the fundamental
class of $X$ (see also [Mum83, Lemma 3.5]).

Let $\widetilde{Y}\to\widetilde{X}$ be an equivariant desingularization, and
let $Y=\widetilde{Y}/G$. We have the following commutative diagram.

$\textstyle{\widetilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{q}$$\textstyle{\widetilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X}$

Let $h_{\operatorname{\textsubscript{\rm sm}}}$ be a Hermitian metric on
$\mathcal{E}|_{X{\operatorname{\textsubscript{\rm sm}}}}$ which extends to an
orbifold metric on $\mathcal{E}_{\mathrm{orb}}$. Then by the condition (3) of
the global cover, the pullback of $h_{\operatorname{\textsubscript{\rm sm}}}$
under $p\circ f$ extends to a $G$-invariant metric $h^{\prime}$ on
$\widetilde{\mathcal{F}}:=f^{*}\widetilde{\mathcal{E}}$. Since $\widetilde{Y}$
is smooth projective, various definitions of Chern classes are consistent.
More precisely, there is a cycle $T$ of dimension $2-i$ in $\widetilde{Y}$
such that

$[T]=c_{i}(\widetilde{\mathcal{F}},h^{\prime})\smallfrown[\widetilde{Y}]\in
H_{4-2i}(\widetilde{Y},\mathbb{R}),$

and that

$T=\tilde{c}_{i}(\widetilde{\mathcal{F}})\smallfrown[\widetilde{Y}]^{\prime}\in
A_{2-i}(\widetilde{Y})\otimes\mathbb{R}.$

Furthermore, we may assume that $T$ is $G$-invariant.

On the one hand, we note that $(\widetilde{Y},G,q)$ is an orbifold chart of
$Y$, and that $\widetilde{\mathcal{F}}$ is an orbifold vector bundle. With
this orbifold structure, we have

$\hat{c}_{i}(\widetilde{\mathcal{F}})=c_{i}(\widetilde{\mathcal{F}},h^{\prime}).$

Moreover, the condition (3) above implies that
$\hat{c}_{i}(\widetilde{\mathcal{F}})=g^{*}\hat{c}_{i}(\mathcal{E}_{\mathrm{orb}})=g^{*}\hat{c}_{i}(\mathcal{E}).$
By the projection formula, this shows that

$\hat{c}_{i}(\mathcal{E})\smallfrown[X]=g_{*}(\hat{c}_{i}(\widetilde{\mathcal{F}})\smallfrown[{Y}])\in
H_{4-2i}(X,\mathbb{R}).$

Hence we get

$\hat{c}_{i}(\mathcal{E})\smallfrown[X]=g_{*}\left(\frac{1}{|G|}\cdot
q_{*}(c_{i}(\widetilde{\mathcal{F}},h^{\prime})\smallfrown[\widetilde{Y}])\right)=\frac{1}{|G|}\cdot
p_{*}f_{*}[T].$

On the other hand, by the projection formula, we have

$f_{*}T=f_{*}(\tilde{c}_{i}(\widetilde{\mathcal{F}})\smallfrown[\widetilde{Y}]^{\prime})=\tilde{c}_{i}(\widetilde{\mathcal{E}})\smallfrown[\widetilde{X}]^{\prime}\in
A_{2-i}(\widetilde{X})\otimes\mathbb{R}.$

It follows that

$[\overline{c}_{i}(\mathcal{E})]=\left[\frac{1}{|G|}\cdot
p_{*}(\tilde{c}_{i}(\widetilde{\mathcal{E}})\smallfrown[\widetilde{X}]^{\prime})\right]=\frac{1}{|G|}\cdot
p_{*}f_{*}[T]\in H_{4-2i}(X,\mathbb{R}).$

This completes the proof of the lemma. ∎

###  4.5. The case of complex analytic threefolds.

We will focus on the case when $X$ is a complex analytic threefold and study
the relative class
$\ell(\widetilde{X},r)=c_{1}^{2}(\widetilde{X},r)+c_{2}(\widetilde{X},r)$
defined in (4.3). For later applications, we need to compute its pushforward
to $X$. Throughout this subsection, in order to have a better interpretation
of Borel-Moore homology, we will assume that $X$ is a Zariski open subset of a
compact complex analytic threefold.

######  Lemma 4.4.

Let $X$ be a complex analytic threefold with quotient singularities, and
$r\colon\widetilde{X}\to X$ a desingularization such that $r$ is an
isomorphism over $X_{\rm sm}$. We denote by $C_{1},...,C_{r}$ the
$1$-dimensional irreducible components of $X_{\rm sing}$. Then there are real
numbers $a_{i}$ such that for any class $\sigma\in H^{2}_{c}(X,\mathbb{R})$,
we have

$\Big{(}(c_{1}(\widetilde{X}))^{2}+c_{2}(\widetilde{X})\Big{)}\cdot
r^{*}\sigma=\Big{(}(\hat{c}_{1}(X))^{2}+\hat{c}_{2}(X)+\sum_{i=1}^{r}a_{i}[C_{i}]\Big{)}\cdot\sigma$

as real numbers. In other words, we have

$\ell(\widetilde{X},r)\cdot
r^{*}\sigma=\Big{(}\sum_{i=1}^{r}a_{i}[C_{i}]\Big{)}\cdot\sigma.$

The dot products in the lemma are defined as follows. For example, we note
that $H^{6}_{c}(X,\mathbb{R})$ is canonically isomorphic to $\mathbb{R}$. Then
for any $a\in H^{4}(X,\mathbb{R})$, the product $a\cdot\sigma$ is the number
corresponding to $a\smallsmile\sigma\in H^{6}_{c}(X,\mathbb{R})$. For any
$b\in H^{BM}_{2}(X,\mathbb{R})$, the product $b\cdot\sigma$ is then defined by
the duality between $H^{BM}_{2}(X,\mathbb{R})$ and $H^{2}_{c}(X,\mathbb{R})$.

###### Proof.

Let $E:=\operatorname{Ex}(r)$ be the exceptional locus of $r$ and let
$\widetilde{X}^{\circ}=\widetilde{X}\backslash{E}$. Then
$\ell(\widetilde{X},r)\in
H^{4}(\widetilde{X},\widetilde{X}^{\circ},\mathbb{R})$ and

$\ell(\widetilde{X},r)\smallfrown[\widetilde{X}]\in H^{BM}_{2}(E,\mathbb{R}).$

On the one hand, the Poincaré duality shows that

$\ell(\widetilde{X},r)\cdot
r^{*}\sigma=(\ell(\widetilde{X},r)\smallfrown[\widetilde{X}])\cdot
r^{*}\sigma$

as real numbers. On the other hand, since $r$ is proper, there are real
numbers $a_{1},...,a_{r}$ such that

(4.7)
$r_{*}{(\ell(\widetilde{X},r)\smallfrown[\widetilde{X}])}=\sum_{i=1}^{r}a_{i}[C_{i}],$

where $[C_{i}]\in H_{2}^{BM}(X_{\operatorname{\textsubscript{\rm
sing}}},\mathbb{R})$ is the fundamental class of $C_{i}$. This completes the
proof of the lemma. ∎

In the following lemmas, we show that one can compute the numbers $a_{i}$ by
taking a local slice intersecting $C_{i}$ transversally. From [GK20, Lemma
5.8] it follows that for $x\in C_{i}$ a general point, there is an Euclidean
neighborhood $U\subseteq X$ of $x$ such that $U\cong S\times\mathbb{D}$, where
$(o\in S)$ is a germ of surface quotient singularities and
$\mathbb{D}\subseteq\mathbb{C}$ is the open unit disk. Let
$\widetilde{U}=r^{-1}(U)$. Shrinking $U$ if necessary we may assume that
$\widetilde{U}\cong\widetilde{S}\times\mathbb{D}$, where
$\mu\colon\widetilde{S}\to S$ is a desingularization. Let
$\Delta:=\operatorname{Ex}(\mu)$ and let $\pi\colon\widetilde{U}\to\mathbb{D}$
be the natural projection. We denote by
$h\colon\widetilde{S}\hookrightarrow\widetilde{U}$ the inclusion induced by
the zero section of $\mathbb{D}$.

######  Lemma 4.5.

With the notations and hypothesis as above, the number $a_{i}$ represents the
class $h^{*}\ell(\widetilde{U},r)\smallfrown[\widetilde{S}]\in
H_{0}^{BM}(\Delta,\mathbb{R})$ via the natural isomorphism
$H_{0}^{BM}(\Delta,\mathbb{R})\cong\mathbb{R}$.

###### Proof.

Let $\widetilde{T}:=h(\widetilde{S})\subseteq\widetilde{U}$. By [Ful98,
Equation (8), Section 19.1], we see that

$\ell(\widetilde{U},r)\smallfrown[\widetilde{T}]=(\ell(\widetilde{U},r)\smallsmile
cl^{\widetilde{U}}(\widetilde{T}))\smallfrown[\widetilde{U}]=h^{*}\ell(\widetilde{U},r)\smallfrown[\widetilde{S}]\in
H_{0}^{BM}(\Delta,\mathbb{R}).$

Let $\delta\in H^{2}(\mathbb{D},\mathbb{D}\setminus\\{0\\},\mathbb{R})$ be the
Thom class of $\\{0\\}$ in $\mathbb{D}$. Then
$cl^{\widetilde{U}}(\widetilde{T})=\pi^{*}\delta$. Therefore, we have

$\ell(\widetilde{U},r)\smallfrown[\widetilde{T}]=(\ell(\widetilde{U},r)\smallsmile\pi^{*}\delta)\smallfrown[\widetilde{U}]=\pi^{*}\delta\smallfrown(\ell(\widetilde{U},r)\smallfrown[\widetilde{U}]).$

We note that $\ell(\widetilde{U},r)\smallfrown[\widetilde{U}]\in
H^{BM}_{2}(E\cap\widetilde{U},\mathbb{R})$, where $E=\operatorname{Ex}(r)$.

On one hand, by applying the projection formula of Lemma 4.2 to the proper
morphism $\pi\colon E\cap\widetilde{U}\to\mathbb{D}$, we deduce that

(4.8)
$\delta\smallfrown\pi_{*}(\ell(\widetilde{U},r)\smallfrown[\widetilde{U}])=\pi_{*}(\ell(\widetilde{U},r)\smallfrown[\widetilde{T}])\in
H^{BM}_{0}(\\{0\\},\mathbb{R}).$

On the other hand, by Lemma 4.1, we have

$r_{*}(\ell(\widetilde{U},r)\smallfrown[\widetilde{U}])=a_{i}[C_{i}\cap U],$

which implies that

$\pi_{*}(\ell(\widetilde{U},r)\smallfrown[\widetilde{U}])=a_{i}[\mathbb{D}].$

Hence we deduce that the LHS of the equation (4.8) is equal to $a_{i}$. This
shows the statement of the lemma. ∎

######  Lemma 4.6.

With the notations and hypothesis as in Lemma 4.5, we have
$a_{i}=\ell(\widetilde{S},\mu)\smallfrown[\widetilde{S}]$, via the natural
isomorphism $H_{0}^{BM}(\Delta,\mathbb{R})\cong\mathbb{R}$.

###### Proof.

From Lemma 4.5, we see that
$a_{i}=h^{*}\ell(\widetilde{U},r)\smallfrown[\widetilde{S}]$. Since the Chern
classes commute with pullbacks, we obtain that
$h^{*}\ell(\widetilde{U},r)=\ell(h^{*}T_{\widetilde{U}},\mu)$. We note that
$h^{*}T_{\widetilde{U}}\cong
T_{\widetilde{S}}\oplus\mathcal{O}_{\widetilde{S}}$. Thus we have

$h^{*}\ell(\widetilde{U},r)=\ell(h^{*}T_{\widetilde{U}},\mu)=\ell(\widetilde{S},\mu).$

This completes the proof of the lemma. ∎

By the same techniques, we obtain the following lemma on comparison of second
Chern classes.

######  Lemma 4.7.

Let $X$ be a compact analytic threefold with quotient singularities. Let $B$
be a reduced boundary on $X$ such that $(X,B)$ is lc. Assume that
$\Omega_{X}^{[1]}(\mathrm{log}\,B)$ induces orbifold vector bundle on the
standard orbifold structure of $X$. Then there exists an effective $1$-cycle
$C$ in $X$ such that

$\hat{c}_{2}(X)-\hat{c}_{2}(\Omega_{X}^{[1]}(\mathrm{log}\,B))+(K_{X}+B)\cdot
B=[C]\in H^{4}(X,\mathbb{R}).$

###### Proof.

We first assume that $(X,B)$ is log smooth. Then there are exact sequences

$0\to\Omega_{X}^{1}\to\Omega_{X}^{1}(\log\,B)\to\bigoplus\mathcal{O}_{B_{i}}\to
0,$ $0\to\mathcal{O}_{X}(-B_{i})\to\mathcal{O}_{X}\to\mathcal{O}_{B_{i}}\to
0,$

where the $B_{i}$’s are the irreducible components of $B$. Thanks to the
theory of Chern classes of coherent sheaves on compact complex manifolds, see
[Gri10, Section 3], we can deduce that

$c_{2}(X)-c_{2}(\Omega_{X}^{1}(\mathrm{log}\,B))+(K_{X}+B)\cdot B\equiv 0.$

In general, let $r\colon\widetilde{X}\to X$ be a log resolution of $(X,B)$ and
$\widetilde{B}=r^{-1}_{*}(B)$. We consider the relative class

$\displaystyle q(\widetilde{X},r)$ $\displaystyle=$
$\displaystyle\Big{(}{c}_{2}(\widetilde{X})-{c_{2}}(\Omega_{\widetilde{X}}^{1}(\mathrm{log}\,\widetilde{B}))+(K_{\widetilde{X}}+\widetilde{B})\cdot\widetilde{B}\Big{)}$
$\displaystyle-r^{*}\Big{(}\hat{c}_{2}(X)-\hat{c}_{2}(\Omega_{X}^{[1]}(\mathrm{log}\,B))+(K_{X}+B)\cdot
B\Big{)}.$

The same argument of Lemma 4.4 shows that, for any $\sigma\in
H^{2}_{c}(X,\mathbb{R})$,

$q(\widetilde{X},r)\cdot
r^{*}\sigma=\Big{(}\sum_{i=1}^{r}b_{i}C_{i}\Big{)}\cdot\sigma,$

where the $C_{i}$’s are the 1-dimensional irreducible components of
$X_{\operatorname{\textsubscript{\rm sing}}}$. It suffices to show that
$b_{i}\leq 0$ for all $i$. To this end, we use the method of Lemma 4.5 and
Lemma 4.6. Let $x$ be a general point in $C_{i}$. Then by [Kol92, Proposition
16.6], there is a neighborhood $U\subseteq X$ of $x$ such that the following
properties hold. There is a lc surface pair $(S,F)$ such that
$(U,B)\cong(S,F)\times\mathbb{D}$, where $\mathbb{D}$ is the unit disk. There
is a log smooth surface pair $(W,\Theta)$ and a group $G$ acting on it, freely
in codimension 1 of $W$, such that $(W,\Theta)/G\cong(S,F)$.

Let $\widetilde{U}=r^{-1}(U)$. Shrinking $U$ if necessary, we may assume that
$(\widetilde{U},\widetilde{B}\cap\widetilde{U})$ is isomorphic to
$(\widetilde{S},\widetilde{F})\times\mathbb{D}$, where
$\mu\colon\widetilde{S}\to S$ is a log resolution of $(S,F)$ and
$\widetilde{F}=\mu^{-1}_{*}F$. Now by the same arguments of Lemma 4.5 and
Lemma 4.6, we see that $b_{i}$ can be calculated as follows,

$\displaystyle b_{i}$ $\displaystyle=$
$\displaystyle\Big{(}\Big{(}{c}_{2}(\widetilde{S})-{c_{2}}(\Omega_{\widetilde{S}}^{1}(\mathrm{log}\,\widetilde{F}))+(K_{\widetilde{S}}+\widetilde{F})\cdot\widetilde{F}\Big{)}$
$\displaystyle-r^{*}\Big{(}\hat{c}_{2}(S)-\hat{c}_{2}(\Omega_{S}^{[1]}(\mathrm{log}\,F))+(K_{S}+F)\cdot
F\Big{)}\Big{)}\smallfrown[\widetilde{S}]$ $\displaystyle\in$ $\displaystyle
H^{BM}_{0}(\Delta,\mathbb{R}).$

via the natural isomorphism $H^{BM}_{0}(\Delta,\mathbb{R})\cong\mathbb{R}$.

Furthermore, by [Kaw88, Lemma 9.9], we may assume that
$W\subseteq\mathbb{C}^{2}$ is an Euclidean open neighborhood of the origin,
that the action of $G$ extends to a linear action on $\mathbb{C}^{2}$, and
that $\Theta=\Theta^{\prime}\cap U$, where
$\Theta^{\prime}\subseteq\mathbb{C}^{2}$ is a union of coordinates
hyperplanes. Let $(T,\overline{F})$ be a projective compactification of
$(\mathbb{C}^{2}/G,\Theta^{\prime}/G).$ Then there is an Euclidean open
embedding $\iota\colon(S,F)\to(T,\overline{F})$. Let $o\in S$ be the unique
singular point. By taking a partial desingularization, we may assume that
$(T,\overline{F})$ is log smooth outside $\iota(o)$. By abuse of notation, we
still denote by $\mu\colon\widetilde{T}\to T$ a desingularization, such that
$\mu^{-1}(S)=\widetilde{S}$. Since the Chern classes commute with pullbacks,
in order to compute $b_{i}$ we may replace $S$ by $T$ and assume that $S$ is a
projective surface and $(S,F)$ is log smooth outside $o$. Then we can apply
[Kol92, Equation (10.8.8)] to deduce that $b_{i}\leq 0$. This completes the
proof of the lemma. ∎

##  5\. Positivity of logarithmic cotangent sheaves

The main objective of this section is to establish the following proposition.
We will also study its variant for non-uniruled threefolds, see Proposition
5.9 in the last subsection, whose proof is simpler.

######  Proposition 5.1.

Let $Y$ be a non-algebraic uniruled compact Kähler threefold with quotient
singularities. Assume the following conditions hold:

  1. (1)

There is a bimeromorphic morphism $\pi\colon Y\to X$ to a normal compact
Kähler threefold such that $L=k(K_{X}+\Delta)$ is Cartier and nef for some
reduced boundary $\Delta$ and some integer $k>0$.

  2. (2)

$L^{2}\not\equiv 0$ and $L^{3}=0$.

  3. (3)

$\pi^{*}L|_{P}\equiv 0$ for every $\pi$-exceptional divisor $P$.

  4. (4)

If $B=\pi^{-1}_{*}\Delta$, then $K_{Y}+(1-\lambda)B$ is pseudoeffective for
some $\lambda>0$.

  5. (5)

$\Omega_{Y}^{[1]}(\mathrm{log}\,B)$ induces an orbifold vector bundle on the
standard orbifold structure of $Y$.

Then $\hat{c}_{2}(\Omega_{Y}^{[1]}(\mathrm{log}\,B))\cdot\pi^{*}L\geq 0$.

This proposition is a consequence of the generic nefness of the logarithmic
cotangent sheaf $\Omega_{Y}^{[1]}(\mathrm{log}\,B)$, see Lemma 5.8 below. A
key ingredient is the following result.

######  Proposition 5.2.

Let $X$ be a smooth uniruled non-algebraic compact $\mathbb{Q}$-factorial
Kähler threefold. Let $\mathcal{F}$ be the foliation corresponding to the MRC
fibration. Assume that there is a $\mathbb{Q}$-divisor $\Delta$ such that
$(X,\Delta)$ is klt and $K_{X}+\Delta$ is pseudoeffective. Then
$(K_{\mathcal{F}}+\Delta_{\rm hor})\cdot\omega^{2}\geq 0$ for any Kähler class
$\omega$ on $X$, where $\Delta_{\rm hor}$ is the horzontal part of $\Delta$
over the base of the MRC fibration.

If we assume that $X$ is a projective threefold and the base of its MRC
fibration is a surface, then the same conclusion of the previous proposition
still holds, and is a special case of [Dru17, Proposition 4.1]. Indeed, our
proof follows the ideas there, which we sketch as follows. For simplicity, we
assume that the MRC fibration of $X$ is an equidimensional morphism $f\colon
X\to Y$. Then from the positivity results of direct images, we may deduce
certain positivity on $K_{X/Y}+\Delta_{\rm hor}$. We remark that
$K_{X/Y}-K_{\mathcal{F}}$ is an effective divisor, supported in the locus
where $f$ has non reduced fibers. To deduce the positivity of
$K_{\mathcal{F}}$ from the one of $K_{X/Y}$, the proof of [Dru17, Proposition
4.1] applies Kawamata’s cyclic covering trick on $Y$, and reduces the problem
to the case when $f$ has reduced fibers in codimension one. In such a case, we
have $K_{X/Y}=K_{\mathcal{F}}$ and we can conclude. In the situation of
Proposition 5.2, Kawamata’s covering trick is not known for compact Kähler
varieties. Therefore, in this section, we will use the basechange method of
[DO23, Lemma 9.11], and argue with some elementary geometric computation.

###  5.1. $\mathbb{P}^{1}$-fibirations

We will first prove some results on fibrations whose general fibers are
isomorphic to $\mathbb{P}^{1}$.

######  Lemma 5.3.

Let $f:X\to Y$ be a proper fibration of complex manifolds such that the
general fibers of $f$ are isomorphic to $\mathbb{P}^{1}$. Let $\mathcal{L}$ be
a line bundle on $X$ whose restrictions on the general fibers of $f$ are
trivial bundles. Then there is a $f$-exceptional divisor $E_{1}$ and an
effective divisor $E_{2}$ such that
$f^{*}((f_{*}\mathcal{L})^{**})\cong\mathcal{L}\otimes\mathcal{O}_{X}(E_{1}-E_{2})$.

###### Proof.

We observe that $f_{*}\mathcal{L}$ is a torsion-free sheaf of rank 1 on $Y$.
Since $Y$ is smooth, the reflexive hull $(f_{*}\mathscr{L})^{**}$ is a line
bundle on $Y$. Let $U\subseteq Y$ be the largest open subset over which
$f_{*}\mathcal{L}$ is locally free. Then $\operatorname{codim}_{Y}(Y\setminus
U)\geq 2$. Let $V=f^{-1}(U)$. Then every prime Weil divisor contained in
$X\setminus V$ is $f$-exceptional.

On one hand, we notice that the natural morphism
$f^{*}f_{*}\mathcal{L}\to\mathcal{L}$ is generically an isomorphism. Hence the
induced morphism $(f^{*}f_{*}\mathcal{L})^{**}\to\mathcal{L}$ of line bundles
is generically an isomorphism as well. It follows that there is an effective
divisor $E_{2}$, such that

$(f^{*}f_{*}\mathcal{L})^{**}\cong\mathcal{L}\otimes\mathcal{O}_{X}(-E_{2}).$

On the other hand, the natural morphism
$f_{*}\mathcal{L}\to(f_{*}\mathcal{L})^{**}$ induces natural morphisms

$f^{*}f_{*}\mathcal{L}\to f^{*}((f_{*}\mathcal{L})^{**})$

and

$(f^{*}f_{*}\mathcal{L})^{**}\to f^{*}((f_{*}\mathcal{L})^{**})$

which are isomorphisms on $V$. Hence there is a divisor $E_{1}$ with support
in $X\setminus V$ such that

$(f^{*}f_{*}\mathcal{L})^{**}\otimes\mathcal{O}(E_{1})\cong
f^{*}((f_{*}\mathcal{L})^{**}).$

Therefore, we obtain that

$f^{*}((f_{*}\mathcal{L})^{**})\cong\mathcal{L}\otimes\mathcal{O}_{X}(E_{1}-E_{2}).$

Since $E_{1}$ is contained in $X\setminus V$, it is $f$-exceptional. This
completes the proof of the lemma. ∎

######  Lemma 5.4.

Let $f\colon X\to Y$ be a projective fibration from a smooth analytic surface
to a germ $(y\in Y)$ of smooth curve. Assume that the general fibers of $f$
are isomorphic to $\mathbb{P}^{1}$. Let $\mathcal{F}$ be the foliation on $X$
induced by $f$. Suppose that $\Delta$ is a $\mathbb{Q}$-divisor such that
$(X,\Delta)$ has canonical singularities and every component of $\Delta$ is
isomorphic to $Y$ via $f$.

Let $g\colon X^{\prime}\to X$ be the blowup of a point $x\in X$ lying over
$y$, $f^{\prime}\colon X^{\prime}\to Y$ the composite morphism,
$\Delta^{\prime}:=g^{-1}_{*}\Delta$, and $\mathcal{F}^{\prime}$ the foliation
induced by $f^{\prime}$. Then the difference
$K_{\mathcal{F}^{\prime}}+\Delta^{\prime}-g^{*}(K_{\mathcal{F}}+\Delta)$ is a
$g$-exceptional effective $\mathbb{Q}$-divisor. In particular, there is a
natural isomorphism

$g_{*}\mathcal{O}_{X^{\prime}}(m(K_{\mathcal{F}^{\prime}}+\Delta^{\prime}))\cong\mathcal{O}_{X}(m(K_{\mathcal{F}}+\Delta))$

for any positive integer $m$ such that $m\Delta$ is a $\mathbb{Z}$-divisor.

###### Proof.

We claim that $x$ is contained in at most two components of the fiber
$X_{y}:=f^{-1}(y)$. To see this claim we run a $K_{X}$-MMP over $Y$ and end
with a $\mathbb{P}^{1}$-bundle $\varphi:Z\to Y$. Such a MMP exists as we can
assume that $f$ has only fintely many singular fibers, up to shrinking $Y$
around $y$. Then every step of the MMP is a contraction of a component of a
singular fiber. Let $\psi:X\to Z$ be the induced morphism. Then $\psi$ can be
seen as the composition of blowups at smooth points. Hence $x$ is contained in
at most two components of $X_{y}$.

We first assume that $x$ is contained in two irreducible components $F_{1}$
and $F_{2}$ of $X_{y}$. Let $m_{1}$ and $m_{2}$ be the coefficients of $F_{1}$
and $F_{2}$ in the divisor $f^{*}y$. Then locally around $x\in X$ we have

$K_{\mathcal{F}}=K_{X/Y}-(m_{1}-1)F_{1}-(m_{2}-1)F_{2}.$

Let $E\subseteq X^{\prime}$ be the $g$-exceptional divisor. Then locally
around $E$ in $X^{\prime}$ we have

$\displaystyle K_{\mathcal{F}^{\prime}}$ $\displaystyle=$ $\displaystyle
K_{X^{\prime}/Y}-(m_{1}+m_{2}-1)E-(m_{1}-1)g_{*}^{-1}F_{1}-(m_{2}-1)g_{*}^{-1}F_{2}$
$\displaystyle=$ $\displaystyle
g^{*}K_{X/Y}-(m_{1}+m_{2}-2)E-(m_{1}-1)g_{*}^{-1}F_{1}-(m_{2}-1)g_{*}^{-1}F_{2}$
$\displaystyle=$ $\displaystyle
g^{*}K_{X/Y}-(m_{1}-1)g^{*}F_{1}-(m_{2}-1)g^{*}F_{2}$ $\displaystyle=$
$\displaystyle g^{*}K_{\mathcal{F}}.$

Since $g$ is an isomorphism over $X\backslash\\{x\\}$, the equality
$K_{\mathcal{F}^{\prime}}\sim g^{*}K_{\mathcal{F}}$ holds globally on
$X^{\prime}$. Moreover, since every component of $\Delta$ is isomorphic to $Y$
via $f$, the components of $\Delta$ are sections of $f:X\to Y$. In particular,
the components of $\Delta$ do not pass through the singular points of $X_{y}$,
and hence $x$ is not contained in the support of $\Delta$. Therefore
$\Delta^{\prime}=g^{-1}_{*}\Delta=g^{*}\Delta$ and we have

$K_{\mathcal{F}^{\prime}}+\Delta^{\prime}=g^{*}(K_{\mathcal{F}}+\Delta).$

Now we assume that $x$ is contained in exactly one component $F_{1}$ of
$X_{y}$. Let $m_{1}$ be the coefficient of $F_{1}$ in $f^{*}y$. Then a similar
computation as above shows that

$K_{\mathcal{F}^{\prime}}=g^{*}K_{\mathcal{F}}+E.$

Then we have

$K_{\mathcal{F}^{\prime}}+\Delta^{\prime}=g^{*}(K_{\mathcal{F}}+\Delta)+({1-\beta})E,$

where $\beta$ is the multiplicity of $\Delta$ at $x$. Since $(X,\Delta)$ is
canonical, we have $\beta\leq 1$. It follows that

$K_{\mathcal{F}^{\prime}}+\Delta^{\prime}-g^{*}(K_{\mathcal{F}}+\Delta)\geq
0.$

This completes our proof. ∎

######  Lemma 5.5.

Let $f\colon X\to Y$ be a projective fibration between complex manifolds.
Assume that general fibers are isomorphic to $\mathbb{P}^{1}$. Let
$\mathcal{F}$ be the foliation of $f$. Assume that there is a
$\mathbb{Q}$-divisor $\Delta$ on $X$ whose components are all bimeromorphic to
$Y$ via $f$ such that $(X,\Delta)$ is terminal and that $K_{X}+\Delta$ has
intersection number $0$ with the general fibers of $f$. Then for any positive
integer $m$ such that $m\Delta$ is an integral divisor, the natural morphism

$\varphi\colon f_{*}\mathcal{O}_{X}(m(K_{\mathcal{F}}+\Delta))\to
f_{*}\mathcal{O}_{X}(m(K_{X/Y}+\Delta))$

is an isomorphism in codimension $1$.

###### Proof.

Since the question is local on the base $Y$, we may assume that $Y$ is a germ
of complex manifold. Cutting $Y$ by $\operatorname{dim}Y-1$ number of general
sections of $H^{0}(Y,\mathcal{O}_{Y})$, we assume that $Y$ is a smooth curve
and $X$ is a smooth surface. Furthermore, we may also assume that $f$ has only
finitely many singular fibers. Now we can run a $f$-relative
$(K_{X}+\Delta)$-MMP over $Y$ and obtain a pair
$(\widehat{X},\widehat{\Delta})$ such that $K_{\widehat{X}}+\widehat{\Delta}$
is nef over $Y$.

Since all the components of $\Delta$ are horizontal over $Y$, none of them are
contracted by this MMP. Hence, $(\widehat{X},\widehat{\Delta})$ is still
terminal. In particular, $\widehat{X}$ is still a smooth surface. Now we run a
$K_{\widehat{X}}$-MMP over $Y$, trivial with respect to
$K_{\widehat{X}}+\widehat{\Delta}$. Since $K_{\widehat{X}}+\widehat{\Delta}$
is numerically trivial on the general fibers of $\widehat{X}\to Y$, the MMP
terminates with a $\mathbb{P}^{1}$-bundle $f^{\prime}\colon X^{\prime}\to Y$.
Let $\Delta^{\prime}$ be the strict transform of $\widehat{\Delta}$ onto
$X^{\prime}$. Then $(X^{\prime},\Delta^{\prime})$ has canonical singularities,
since the MMP is $(K_{\widehat{X}}+\widehat{\Delta})$-trivial and does not
contract any component of $\widehat{\Delta}$.

Let $g\colon X\to X^{\prime}$ be the induced bimeromorphic morphism. Since
$f^{\prime}\colon X^{\prime}\to Y$ is a $\mathbb{P}^{1}$-bundle, there is a
natural isomorphism

$f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(m(K_{\mathcal{F}^{\prime}}+\Delta^{\prime}))\to
f^{\prime}_{*}\mathcal{O}_{X^{\prime}}(m(K_{X^{\prime}/Y}+\Delta^{\prime})).$

Observe that the morphism $g\colon X\to X^{\prime}$ can be seen as the
composition of blowups of points. Thus by repeated application of Lemma 5.4 we
obtain the following isomorphism

$g_{*}\mathcal{O}_{X}(m(K_{\mathcal{F}}+\Delta))\cong\mathcal{O}_{X^{\prime}}(m(K_{\mathcal{F}^{\prime}}+\Delta^{\prime})).$

Since $(X^{\prime},\Delta^{\prime})$ is canonical, we also get

$g_{*}\mathcal{O}_{X}(m(K_{X/Y}+\Delta))\cong\mathcal{O}_{X^{\prime}}(m(K_{X^{\prime}/Y}+\Delta^{\prime})).$

Hence we deduce that

$f_{*}\mathcal{O}_{X}(m(K_{\mathcal{F}}+\Delta))\cong
f_{*}\mathcal{O}_{X}(m(K_{X/Y}+\Delta)).$

This completes our proof. ∎

###  5.2. Positivity of foliated canonical divisors

We are now ready to prove Proposition 5.2 and Proposition 5.1. We first
consider a special case of Proposition 5.2 in the following lemma.

######  Lemma 5.6.

Let $f\colon X\to Y$ be a projective fibration between normal compact Kähler
varieties such that general fibers of $f$ are isomorphic to $\mathbb{P}^{1}$
and that $X$ is smooth. Assume that there is a $\mathbb{Q}$-divisor $\Delta$
with $\lfloor\Delta\rfloor=0$, such that every component of $\Delta$ is
bimeromorphic to $Y$ via $f$, and that $K_{X}+\Delta$ intersects the general
fibers of $f$ non-negatively. Let $\mathcal{F}$ be the foliation induced by
$f$ and $n$ the dimension of $X$. Then
$(K_{\mathcal{F}}+\Delta)\cdot\omega^{n-1}\geq 0$ for any Kähler class
$\omega$ on $X$.

###### Proof.

Since $\Delta$ is an effective divisor, it is enough to assume that
$K_{X}+\Delta$ is numerically trivial on general fibers of $f$. By using Lemma
2.1, we construct the following commutative diagram

(5.1)
$\textstyle{(X^{\prime},\Delta^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{p}$$\textstyle{(X,\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{Y}$

such that

  1. (1)

$X^{\prime}$ and $Y^{\prime}$ are smooth,

  2. (2)

$p$ and $q$ are projective bimeromorphic,

  3. (3)

every $g$-exceptional divisor is also $p$-exceptional,

  4. (4)

$\Delta^{\prime}$ is the strict transform of $\Delta$ such that the components
of $\Delta^{\prime}$ are smooth and pairwise disjoint; in particular,
$(X^{\prime},\Delta^{\prime})$ has terminal singularities.

Observe that the general fiber of $g$ are isomorphic to $\mathbb{P}^{1}$ and
$K_{X^{\prime}}+\Delta^{\prime}$ is numerically trivial on the general fibers
of $g$. In particular, there is a sufficiently divisible positive integer $m$
such that
$g_{*}\mathcal{O}_{X^{\prime}}(m(K_{X^{\prime}/Y^{\prime}}+\Delta^{\prime}))$
is a rank one torsion-free sheaf on $Y^{\prime}$. Let $\mathcal{G}$ be the
foliation induced by $g$. Then $K_{\mathcal{G}}+\Delta^{\prime}$ is trivial on
the general fibers of $g$. By Lemma 5.5,

(5.2) $g_{*}\mathcal{O}_{X^{\prime}}(m(K_{\mathcal{G}}+\Delta^{\prime}))\to
g_{*}\mathcal{O}_{X^{\prime}}(m(K_{X^{\prime}/Y^{\prime}}+\Delta^{\prime}))$

is an isomorphism in codimension one. Taking reflexive hulls we get the
following isomorphism of line bundles on $Y^{\prime}$

(5.3)
$(g_{*}\mathcal{O}_{X^{\prime}}(m(K_{\mathcal{G}}+\Delta^{\prime})))^{**}\to(g_{*}\mathcal{O}_{X^{\prime}}(m(K_{X^{\prime}/Y^{\prime}}+\Delta^{\prime})))^{**}.$

By [PT18, Corollary 5.2.1], the torsion-free sheaf
$g_{*}\mathcal{O}_{X^{\prime}}(m(K_{X^{\prime}/Y^{\prime}}+\Delta^{\prime}))$
has a singular metric with semipositive curvature current. Thus by [PT18,
Remark 2.4.2(3)], the line bundle
$\mathscr{M}:=(g_{*}\mathcal{O}_{X^{\prime}}(m(K_{X^{\prime}/Y^{\prime}}+\Delta^{\prime})))^{**}$
has a singular metric with semipositive curvature current; in particular, the
line bundle $\mathscr{M}$ is pseudoeffetive, and so is
$\mathscr{L}:=(g_{*}\mathcal{O}_{X^{\prime}}(m(K_{\mathcal{G}}+\Delta^{\prime})))^{**}$.

By Lemma 5.3, there is a $g$-exceptional $\mathbb{Q}$-divisor $E_{1}$ and an
effective divisor $E_{2}$ on $X^{\prime}$ such that

$g^{*}\mathscr{L}\cong\mathcal{O}_{X^{\prime}}(m(K_{\mathcal{G}}+\Delta^{\prime}+E_{1}-E_{2})).$

Also, observe that there is a $p$-exceptional $\mathbb{Q}$-divisor $F$ such
that

$p^{*}\mathcal{O}_{X}(m(K_{\mathcal{F}}+\Delta))\cong\mathcal{O}_{X^{\prime}}(m(K_{\mathcal{G}}+\Delta^{\prime}+F)).$

Thus for any Kähler class $\omega$ on $X$ we have

$\displaystyle(K_{\mathcal{F}}+\Delta)\cdot\omega^{n-1}$ $\displaystyle=$
$\displaystyle p^{*}(K_{\mathcal{F}}+\Delta)\cdot(p^{*}\omega)^{n-1}$
$\displaystyle=$
$\displaystyle(K_{\mathcal{G}}+\Delta^{\prime}+F)\cdot(p^{*}\omega)^{n-1}$
$\displaystyle=$
$\displaystyle(K_{\mathcal{G}}+\Delta^{\prime})\cdot(p^{*}\omega)^{n-1}.$

Since $E_{1}$ is $g$-exceptional, it is also $p$-exceptional by our
construction. Thus we obtain that

$\displaystyle(K_{\mathcal{G}}+\Delta^{\prime})\cdot(p^{*}\omega)^{n-1}$
$\displaystyle=$
$\displaystyle(K_{\mathcal{G}}+\Delta^{\prime}+E_{1})\cdot(p^{*}\omega)^{n-1}$
$\displaystyle\geq$
$\displaystyle(K_{\mathcal{G}}+\Delta^{\prime}+E_{1}-E_{2})\cdot(p^{*}\omega)^{n-1}$
$\displaystyle=$
$\displaystyle\frac{1}{m}c_{1}(\mathcal{L})\cdot(p^{*}\omega)^{n-1}$
$\displaystyle\geq$ $\displaystyle 0,$

This completes the proof of the lemma. ∎

We improve this lemma in the following assertion, which only requires that
every component of $\Delta$ is horizontal over $Y$.

######  Lemma 5.7.

Let $f\colon X\to Y$ be a projective fibration between normal compact Kähler
varieties such that general fibers of $f$ are isomorphic to $\mathbb{P}^{1}$
and that $X$ is smooth. Assume that there is a $\mathbb{Q}$-divisor $\Delta$
with $\lfloor\Delta\rfloor=0$, such that every component of $\Delta$ is
horizontal over $Y$, and that $K_{X}+\Delta$ has non-negative intersection
number with the general fibers of $f$. Let $\mathcal{F}$ be the foliation
corresponding to $f$ and $n$ the dimension of $X$. Then
$(K_{\mathcal{F}}+\Delta)\cdot\omega^{n-1}\geq 0$ for any Kähler class
$\omega$ on $X$.

###### Proof.

The idea is to reduce to the case of Lemma 5.6. By applying [DO23, Lemma 9.11]
to the irreducible components of $\Delta$ successively, and by taking a
desingularization in the end, we can obtain a commutative diagram

$\textstyle{(X_{2},\Delta_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\scriptstyle{p_{2}}$$\textstyle{(X_{1},\Delta_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{p_{1}}$$\textstyle{(X,\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{2}}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{1}}$$\textstyle{Y}$

such that

  1. (1)

$q_{1}$ and $p_{1}$ are finite,

  2. (2)

$Y_{1}$ is normal,

  3. (3)

$X_{1}$ is the normalization of the main component of $X\times_{Y}Y_{1}$,

  4. (4)

$X_{2}$ and $Y_{2}$ are smooth,

  5. (5)

$p_{2}$ and $q_{2}$ are projective bimeromorphic,

  6. (6)

$\Delta_{1}=p^{*}_{1}\Delta$,

  7. (7)

every components of $\Delta_{1}$ is bimeromorphic to $Y_{1}$ via $f_{1}$,

  8. (8)

$\Delta_{2}$ is the strict transform of $\Delta_{1}$.

Let $p_{1}^{-1}\mathcal{F}$ be the pullback foliation on $X_{1}$ and let
$\mathcal{F}_{1}$ be the foliation induced by $f_{1}$. Since
$p_{1}^{-1}\mathcal{F}$ is induced by $q_{1}\circ f_{1}$, it follows that
$\mathcal{F}_{1}\subseteq p_{1}^{-1}\mathcal{F}$, and hence
$\mathcal{F}_{1}=p_{1}^{-1}\mathcal{F}$ as both of them have same rank. Since
$p_{1}$ is étale over a dense Zariski open subset of $X$ of the form
$f^{-1}(U)$, where $U\subseteq Y$ is a dense Zariski open subset of $Y$, it
follows that the ramification divisor, say $R$, of $p_{1}$ is vertical over
$Y_{1}$. Thus $R$ is $\mathcal{F}_{1}$-invariant. From [Dru21, Lemma 3.4] and
[CS20, Proposition 2.2] it follows that
$K_{\mathcal{F}_{1}}=p_{1}^{*}K_{\mathcal{F}}$. Hence we have

$p_{1}^{*}(K_{\mathcal{F}}+\Delta)=p_{1}^{*}K_{\mathcal{F}}+p_{1}^{*}\Delta=K_{\mathcal{F}_{1}}+\Delta_{1}.$

Let $\mathcal{F}_{2}$ be the foliation induced by $f_{2}$ and let $g:X_{2}\to
X$ be the composite morphism. Then from the computation above it follows that
there is a $g$-exceptional $\mathbb{Q}$-divisor $E$ on $X_{2}$ such that

$g^{*}(K_{\mathcal{F}}+\Delta)=K_{\mathcal{F}_{2}}+\Delta_{2}+E.$

Since $(K_{\mathcal{F}_{2}}+\Delta_{2})\cdot(g^{*}\omega)^{n-1}\geq 0$ by
Lemma 5.6, we obtain that

$(K_{\mathcal{F}}+\Delta)\cdot\omega^{n-1}\geq 0.$

This completes our proof. ∎

Now we can deduce Proposition 5.2.

###### Proof of Proposition 5.2.

By Theorem 2.4, $K_{X}+\Delta$ is pseudoeffective if and only if
$K_{X}+\Delta_{\operatorname{\textsubscript{\rm hor}}}$ is pseudoeffective.
Hence, without loss of generality, we may assume that
$\Delta=\Delta_{\operatorname{\textsubscript{\rm hor}}}$. Moreover,
$K_{X}+\Delta$ has nonnegative intersection number with the general fibers of
the MRC fibration. By [HP15, Theorem 1.1], $X$ is bimeromorphic to a Mori
fiber space $\hat{f}\colon\widehat{X}\to\widehat{Y}$ such that $\widehat{Y}$
is a normal compact surface. Furthermore, the general fibers of $\hat{f}$ map
to the general leaves of $\mathcal{F}$. By blowing up the graph of
$X\dashrightarrow\widehat{X}$, we get the following diagram,

$\textstyle{(X^{\prime},\Delta^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{p}$$\textstyle{(X,\Delta)}$$\textstyle{Y^{\prime}}$

such that

  1. (1)

$X^{\prime}$ is smooth and $Y^{\prime}=\widehat{Y}$,

  2. (2)

$p$ is proper bimeromorphic,

  3. (3)

$f^{\prime}$ is projective with general fibers isomorphic to $\mathbb{P}^{1}$,

  4. (4)

general fibers of $f^{\prime}$ map to the leaves of $\mathcal{F}$ via $p$,

  5. (5)

$\Delta^{\prime}$ is the strict transform of $\Delta$,

  6. (6)

$\lfloor\Delta^{\prime}\rfloor=0$.

Let $\omega^{\prime}=p^{*}\omega$ and let $\mathcal{F}^{\prime}$ be the
foliation induced by $f^{\prime}$. Observe that
$\mathcal{F}^{\prime}=p^{-1}\mathcal{F}$. Then there is a $p$-exceptional
$\mathbb{Q}$-divisor $E$ on $X^{\prime}$ such that

$K_{\mathcal{F}^{\prime}}+\Delta^{\prime}+E=p^{*}(K_{\mathcal{F}}+\Delta).$

We note that $p:X^{\prime}\to X$ is an isomorphism around the general fibers
of the MRC fibration of $X$. Thus $K_{X^{\prime}}+\Delta^{\prime}$ has
nonnegative intersection number with the general fibers of $f^{\prime}$. By
Lemma 5.7, we have
$(K_{\mathcal{F}^{\prime}}+\Delta^{\prime})\cdot\omega^{\prime 2}\geq 0$.
Since $E$ is $p$-exceptional, we conclude that
$(K_{\mathcal{F}}+\Delta)\cdot\omega^{2}\geq 0$. ∎

Now we can establish the generic nefness of some logarithmic cotangent sheaves
in the following lemma.

######  Lemma 5.8.

Let $(X,B)$ be a lc pair with a reduced boundary $B$, where $X$ is a uniruled
non-algebraic compact Kähler threefold, with quotient singularities. Assume
that there is some $0<\lambda<1$ such that $K_{X}+(1-\lambda)B$ is
pseudoeffective. Let $\omega$ be any Kähler class on $X$. Then the reflexive
logarithmic cotangent sheaf $\Omega_{X}^{[1]}(\mathrm{log}\,B)$ is
$\omega$-generically nef.

###### Proof.

Let $\mathcal{E}\subseteq\Omega_{X}^{[1]}(\mathrm{log}\,B)$ be a proper non
zero saturated subsheaf. We need to show that

${c}_{1}(\mathcal{E})\cdot\omega^{2}\leq(K_{X}+B)\cdot\omega^{2}.$

Let $r\colon\widetilde{X}\to X$ be a log resolution of $(X,B)$ and let
$\widetilde{B}=r^{-1}_{*}B$. Then $K_{\widetilde{X}}+\lambda\widetilde{B}$ is
pseudoeffective by Theorem 2.4. We remark that

$(r_{*}\Omega_{\widetilde{X}}^{1}(\mathrm{log}\,\widetilde{B}))^{**}=\Omega_{X}^{[1]}(\mathrm{log}\,B).$

Hence, there is a saturated subsheaf
$\widetilde{\mathcal{E}}\subseteq\Omega_{\widetilde{X}}^{1}(\mathrm{log}\,\widetilde{B})$
such that $(r_{*}\widetilde{\mathcal{E}})^{**}=\mathcal{E}$. As a consequence,
by the projection formula, we only need to show that

$c_{1}(\widetilde{\mathcal{E}})\cdot(r^{*}\omega)^{2}\leq(K_{\widetilde{X}}+\widetilde{B})\cdot(r^{*}\omega)^{2}.$

Thus, replacing $(X,B)$ by $(\widetilde{X},\widetilde{B})$ if necessary, we
may assume that $(X,B)$ is log smooth. If the rank of $\mathcal{E}$ is $1$,
then the inequality follows from [CP16, Theorem 1.1]. Now we assume that
$\mathcal{E}$ has rank $2$. Assume by contradiction that

$c_{1}({\mathcal{E}})\cdot\omega^{2}>(K_{X}+B)\cdot\omega^{2}.$

Let $\mathcal{G}=(\Omega^{1}_{X}(\mathrm{log}\,{B})/\mathcal{E})^{*}$. Then
$\mathcal{G}$ is a rank one saturated subsheaf of the logarithmic tangent
bundle $T_{X}(-\log\,{B})$, and we have

$c_{1}(\mathcal{G})\cdot\omega^{2}>0.$

Let $\mathcal{G}\hookrightarrow\mathcal{F}$ be the saturation of $\mathcal{G}$
in the tangent bundle $T_{X}$. Then $\mathcal{F}$ is a foliation on $X$, and
we have

$c_{1}(\mathcal{F})\cdot\omega^{2}\geq c_{1}(\mathcal{G})\cdot\omega^{2}>0.$

Since $\mathcal{F}$ has rank one, it follows that
$c_{1}(\mathcal{F})=-K_{\mathcal{F}}$. By a criterion of Brunella (see [Bru06,
Theorem 1.1]), $\mathcal{F}$ is a foliation by rational curves. Therefore, it
must be the foliation corresponding to the MRC fibration of $X$.

We remark that $\mathcal{G}=\mathcal{F}\cap T_{X}(-\log\,B)$. By [Cla17,
Proposition 2.17], we obtain that

$c_{1}(\mathcal{G})=-(K_{\mathcal{F}}+B_{\operatorname{\textsubscript{\rm
hor}}}),$

where $B_{\operatorname{\textsubscript{\rm hor}}}$ is the horizontal part of
$B$ over the base of the MRC fibration. Since $(X,(1-\lambda)B)$ is klt and
$K_{X}+(1-\lambda)B$ is pseudoeffective, by Proposition 5.2, we obtain that

$(K_{\mathcal{F}}+B_{\operatorname{\textsubscript{\rm
hor}}})\cdot\omega^{2}\geq(K_{\mathcal{F}}+(1-\lambda){B}_{\operatorname{\textsubscript{\rm
hor}}})\cdot\omega^{2}\geq 0.$

Hence $c_{1}(\mathcal{G})\cdot\omega^{2}\leq 0$. We obtain a contradiction. ∎

We can now conclude Proposition 5.1

###### Proof of Proposition 5.1.

By Lemma 5.8, $\Omega_{Y}^{[1]}(\mathrm{log}\,B)$ is generically nef with
respect to any nef class on $Y$. Furthermore, we remark that

$\hat{c}_{1}(\Omega_{Y}^{[1]}(\mathrm{log}\,B))=K_{Y}+B=\pi^{*}(K_{X}+\Delta)-E,$

where $E$ is some $\pi$-exceptional divisor. By assumption, we have
$\pi^{*}L\cdot E\equiv 0$. Thus we can apply Proposition 3.11, with
$\mathcal{E}=\Omega_{Y}^{[1]}(\mathrm{log}\,B)$,
$\alpha=\pi^{*}(K_{X}+\Delta)$ and $\beta=E$, to conclude that
$\hat{c}_{2}(\Omega_{Y}^{[1]}(\mathrm{log}\,B))\cdot\pi^{*}L\geq 0$. ∎

###  5.3. The case of non-uniruled threefolds

We also have the following variant of Proposition 5.1.

######  Proposition 5.9.

Let $Y$ be a compact Kähler threefold with quotient singularities. Assume the
following properties hold:

  1. (1)

There is a bimeromorphic morphism $\pi\colon Y\to X$ to a normal compact
Kähler threefold such that $L=k(K_{X}+\Delta)$ is Cartier and nef for some
reduced boundary $\Delta$ and some integer $k>0$.

  2. (2)

$L^{2}\not\equiv 0$ and $L^{3}=0$.

  3. (3)

$\pi^{*}L|_{P}\equiv 0$ for every $\pi$-exceptional divisor $P$.

  4. (4)

$Y$ is not uniruled.

  5. (5)

If $B=\pi_{*}^{-1}\Delta$, then $\Omega_{Y}^{[1]}(\mathrm{log}\,B)$ induces an
orbifold vector bundle on the standard orbifold structure $Y_{\mathrm{orb}}$.

Then $\hat{c}_{2}(\Omega_{Y}^{[1]}(\mathrm{log}\,B))\cdot\pi^{*}L\geq 0$.

###### Proof.

By Lemma 5.10 below, we see that $\Omega_{Y}^{[1]}(\mathrm{log}\,B)$ is
generically nef with respect to any nef class. There is a $\pi$-exceptional
divisor $E$ such that

$\hat{c}_{1}(\Omega_{Y}^{[1]}(\mathrm{log}\,B))=K_{Y}+B=\pi^{*}(K_{X}+\Delta)-E.$

Hence we can deduce the proposition by applying Proposition 3.11, with
$\mathcal{E}=\Omega_{Y}^{[1]}(\mathrm{log}\,B)$,
$\alpha=\pi^{*}(K_{X}+\Delta)$ and $\beta=E$. ∎

######  Lemma 5.10.

Let $X$ be a compact Kähler threefold with quotient singularities only. Assume
that $X$ is not uniruled. Let $B$ be a reduced divisor. Then for any nef class
$\alpha$ and any torsion-free quotient
$\Omega_{X}^{[1]}(\log\,B)\to\mathcal{Q}$ of coherent sheaves, we have
$\hat{c}_{1}(\mathcal{Q})\cdot\alpha^{2}\geq 0$.

###### Proof.

Let $\mathcal{N}\subseteq\mathcal{Q}$ be the subsheaf induced by the inclusion
$\Omega_{X}^{[1]}\subseteq\Omega_{X}^{[1]}(\log\,B)$. Then

$\hat{c}_{1}(\mathcal{Q})\cdot\alpha^{2}\geq\hat{c}_{1}(\mathcal{N})\cdot\alpha^{2}.$

Hence, it is enough to prove the case when $B=0$.

Assume that $B=0$. Let $\widetilde{X}\to X$ be a desingularization. Since $X$
is not uniruled, $K_{\widetilde{X}}$ is pseudoeffective. By Theorem 2.5, there
is some effective divisor $D\sim_{\mathbb{Q}}K_{\widetilde{X}}$. Hence, by
[Eno88, Theorem 1.4], if $\Omega^{1}_{\widetilde{X}}\to\mathcal{Q}^{\prime}$
is the induced quotient, then we have
$c_{1}(\mathcal{Q}^{\prime})\cdot(r^{*}\alpha)^{2}\geq 0$. This implies that
$\hat{c}_{1}(\mathcal{Q})\cdot\alpha^{2}\geq 0$. ∎

##  6\. Lower bounds on Euler characteristics

In this section we will work under the following setup.

######  Setup 6.1.

Let $X$ be a $\mathbb{Q}$-factorial compact Kähler threefold, and $\Delta$ a
reduced boundary on $X$. Assume that the following properties hold.

  1. (1)

$(X,\Delta)$ is lc.

  2. (2)

$X$ has terminal singularities outside the support of $\Delta$.

  3. (3)

$L:=k(K_{X}+\Delta)$ is Cartier and nef for some integer $k>0$.

  4. (4)

There is an effective $\mathbb{Q}$-divisor $D\sim_{\mathbb{Q}}K_{X}+\Delta$
such that $D_{\operatorname{\textsubscript{\rm red}}}=\Delta$.

  5. (5)

$\nu(X,K_{X}+\Delta)=2$.

  6. (6)

If $C\subseteq X$ is a curve such that $(K_{X}+\Delta)\cdot C>0$, then
$(X,\Delta)$ is dlt near the general points of $C$.

Our goal is to prove the following proposition. When $X$ is projective, this
is an important step in [Kol92, Chapter 14], where the algebraic abundance was
established. In the case of Kähler threefolds, the difficulty is that
Bogomolov-Gieseker type inequalities for klt Kähler varieties are still
unknown. Our solution is to construct a modification $Y$ of $X$, which has
quotient singularities only, see Lemma 6.12. Afterwards, we can apply orbifold
Bogomolov-Gieseker type inequalities on $Y$.

######  Proposition 6.2.

With the notations and hypothesis as in Setup 6.1, we assume further that
$K_{X}+(1-\epsilon)\Delta$ is nef for any $0<\epsilon\ll 1$. Then there is a
constant $\lambda$ such that $\chi(X,\mathscr{O}_{{X}}(nL))\geq\lambda$ for
all $n\geq 0$. Furthermore, if there is a curve $C\subseteq
X_{\operatorname{\textsubscript{\rm sing}}}$ such that $C\cdot L>0$, then
$\chi(X,\mathscr{O}_{{X}}(nL))$ grows at least linearly in $n$.

###  6.1. Elementary properties

Before going into the proof of Proposition 6.2, we first show some
consequences of the hypothesis in Setup 6.1.

######  Lemma 6.3.

Assume that we are in Setup 6.1. Then $L^{2}\cdot\Delta_{i}=0$ for any
irreducible component $\Delta_{i}$ of $\Delta$. If in addition
$K_{X}+(1-\epsilon)\Delta$ is nef for any $0<\epsilon\ll 1$, then
$K_{X}^{2}\cdot L=0$.

###### Proof.

Since $\nu(X,K_{X}+\Delta)=2$, it follows that $L^{2}\cdot D=L^{3}=0.$ Since
$L$ is nef and $D$ is effective, by the property (4), we deduce that
$L^{2}\cdot\Delta_{i}=0$ for any irreducible component $\Delta_{i}$ of
$\Delta$.

Now we assume further that $K_{X}+(1-\epsilon)\Delta$ is nef for any
$0<\epsilon\ll 1$. Then $\nu(X,K_{X}+(1-\epsilon)\Delta)\leq 2$, and thus

$(K_{X}+(1-\epsilon)\Delta)^{2}\cdot(D-\epsilon\Delta)=(K_{X}+(1-\epsilon)\Delta)^{3}=0$

for all $0<\epsilon\ll 1$. Since $(D-\epsilon\Delta)$ is effective for all
$0<\epsilon\ll 1$, we deduce that

$(K_{X}+(1-\epsilon)\Delta)^{2}\cdot\Delta_{i}=0$

for every component $\Delta_{i}$ of $\Delta$. The LHS above is a polynomial in
$\epsilon$ with infinitely many solutions, so its coefficients must all be
$0$:

$K_{X}^{2}\cdot\Delta_{i}=\Delta^{2}\cdot\Delta_{i}=K_{X}\cdot\Delta\cdot\Delta_{i}=0.$

It then follows that $K_{X}^{2}\cdot L=0$. ∎

The following results are extracted from [Kol92, Lemma 14.2].

######  Setup 6.4.

Assume that we are in Setup 6.1. Let $\Gamma$ be an irreducible component of
$\Delta$ such that $\nu(\Gamma,L|_{\Gamma})>0$. By adjunction, there is a
boundary divisor $\Theta$ on $\Gamma$ such that $(\Gamma,\Theta)$ is slc (see
[Fuj00, Remark 1.2(5)]) and

$(K_{X}+\Delta)|_{\Gamma}\sim_{\mathbb{Q}}K_{\Gamma}+\Theta.$

Let $\pi\colon\widehat{\Gamma}\to\Gamma$ be the normalization. Then there is a
boundary $\widehat{\Theta}$ such that $(\widehat{\Gamma},\widehat{\Theta})$ is
lc and that
$K_{\widehat{\Theta}}+\widehat{\Gamma}\sim_{\mathbb{Q}}\pi^{*}(K_{X}+\Delta)$.
By log abundance for Kähler surfaces (see [DO23, Theorem 6.3]),
$K_{\widehat{\Theta}}+\widehat{\Gamma}$ induces a fibration
$\hat{f}\colon\widehat{\Gamma}\to B$.

######  Lemma 6.5.

With the notations and hypothesis of Setup 6.4, assume that a general fiber
$\widehat{F}$ of $\hat{f}$ is a rational curve.

  1. (1)

Then $\Gamma$ is normal around $F:=\pi(\widehat{F})$.

In particular, we have an almost proper meromorphic map
$f\colon\Gamma\dashrightarrow B$ (i.e. the general fibers are proper) induced
by $\hat{f}$. Let $\Theta_{\operatorname{\textsubscript{\rm hor}}}$ be the
horizontal part of $\Theta$ over $B$. We decompose

$\Theta_{\operatorname{\textsubscript{\rm
hor}}}=\lfloor\Theta_{\operatorname{\textsubscript{\rm
hor}}}\rfloor+\Theta_{\operatorname{\textsubscript{\rm hor}}}^{<1}$

into the sum of its integral part and its fractional part. Then the following
properties hold.

  1. (2)