# On the Bogomolov-Gieseker inequality for tame Deligne-Mumford surfaces

Yunfeng Jiang and Promit Kundu  Department of Mathematics
University of Kansas
405 Snow Hall 1460 Jayhawk Blvd
Lawrence KS 66045 USA<EMAIL_ADDRESS>Department of Mathematics
University of Kansas
405 Snow Hall 1460 Jayhawk Blvd
Lawrence KS 66045 USA<EMAIL_ADDRESS>Department of Mathematics,
Shanghai Normal University, Shanghai 200234, People’s Republic of China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We generalize the Bogomolov-Gieseker inequality for semistable coherent
sheaves on smooth projective surfaces to smooth Deligne-Mumford surfaces. We
work over positive characteristic $p>0$ and generalize Langer’s method to
smooth Deligne-Mumford stacks. As applications we obtain the Bogomolov
inequality for semistable coherent sheaves on a Deligne-Mumford surface in
characteristic zero, and the Bogomolov inequality for semistable sheaves on a
root stack over a smooth surface which is equivalent to the Bogomolov
inequality for the rational parabolic sheaves on a smooth surface $S$. In a
joint appendix with Hao Max Sun, we generalize the Bogomolov inequality
formula to Simpson Higgs sheaves on tame Deligne-Mumford stacks.

###### Contents

  1. 1 Introduction
    1. 1.1 Outline
    2. 1.2 Convention
  2. 2 Preliminaries on modified stability and some new results
    1. 2.1 Notations
    2. 2.2 Modified stability
    3. 2.3 Notations in characteristic $p$
    4. 2.4 fdHN property
  3. 3 Bogomolov-Gieseker inequality
    1. 3.1 Condition $\star$
    2. 3.2 Two dimensional case
    3. 3.3 Bogomolov inequality for parabolic sheaves
      1. 3.3.1 Root surfaces
      2. 3.3.2 Parabolic sheaves
      3. 3.3.3 Equivalence of categories
      4. 3.3.4 Parabolic Bogomolov inequality
  4. A Higher dimension case
    1. A.1 $\operatorname{Thm}^{1}(\mathop{\rm rk})$ implies $\operatorname{Thm}^{5}(\mathop{\rm rk})$
    2. A.2 $\operatorname{Thm}^{5}(\mathop{\rm rk})$ implies $\operatorname{Thm}^{3}(\mathop{\rm rk})$
    3. A.3 $\operatorname{Thm}^{3}(\mathop{\rm rk})$ implies $\operatorname{Thm}^{4}(\mathop{\rm rk})$
    4. A.4 $\operatorname{Thm}^{4}(\mathop{\rm rk})$ implies $\operatorname{Thm}^{2}(\mathop{\rm rk})$
    5. A.5 $\operatorname{Thm}^{2}(\mathop{\rm rk}-1)$ implies $\operatorname{Thm}^{1}(\mathop{\rm rk})$
    6. A.6 Restriction theorems
  5. B Bogomolov’s inequality for Higgs sheaves

##  1\. Introduction

Let $X$ be a smooth projective surface over an algebraically closed field
$\kappa$ of characteristic zero. The Bogomolov-Gieseker inequality is a famous
formula for slope semistable torsion free coherent sheaves on $X$, which says
that the discriminant $\Delta(E)=2\mathop{\rm rk}(E)c_{2}(E)-(\mathop{\rm
rk}(E)-1)c_{1}(E)^{2}\geq 0$ if $E$ is slope semistable. This formula has many
important applications such as the construction of Bridgeland stability
conditions for surfaces.

If the base field $\kappa$ is of positive characteristic $p>0$, and $X$ is a
smooth projective surface over $\kappa$. Let $F:X\to X$ be the absolute
Frobenius morphism of $X$. A torsion free coherent sheaf $E$ is called
strongly slope semistable if any Frobenius pullback $F^{*}E$ is slope
semistable. For a strongly slope semistable torison free coherent sheaf $E$ on
$X$, the Bogomolov-Gieseker inequality

$\Delta(E)=2\mathop{\rm rk}(E)c_{2}(E)-(\mathop{\rm rk}(E)-1)c_{1}(E)^{2}\geq
0$

still holds, see [13, Theorem 3.2]. In general if $E$ is just slope
semistable, the inequality has a correction term depending on the prime number
$p$, see [13, Theorem 3.3]. The Bogomolov inequality formula in positive
characteristics has applications to prove the boundedness for the moduli
functor of semistable coherent sheaves on $X$, and the restriction theorem of
slope stable torsion free coherent sheaves on $X$ to a divisor $D$ inside $X$.

In this paper we prove the Bogomolov-Gieseker inequality for slope semistable
torsion free coherent sheaves on a smooth projective two dimensional Deligne-
Mumford stack ${\mathcal{X}}$ (called a surface Deligne-Mumford stack) in
characteristic $p>0$. We work on tame surface Deligne-Mumford stacks which
means that the orders of the local stabilizer groups of ${\mathcal{X}}$ are
all coprime to the character $p$. We use the modified slope semistability of
Nironi [21] defined by generating sheaves $\Xi$ on ${\mathcal{X}}$. One
motivation for our study on the Bogomolov-Gieseker inequality for slope
semistable torsion free coherent sheaves is the Vafa-Witten theory for
projective surfaces and surface Deligne-Mumford stacks in [26], [7], where the
Bogomolov-Gieseker inequality for the modified semistable sheaf $E$ will make
the moduli space of Gieseker stable sheaves on a root stack surface
${\mathcal{X}}$ empty for $c_{2}(E)<0$. The Vafa-Witten theory for surface
Deligne-Mumford stacks has applications to prove the S-duality conjecture in
[29] which is a functional duality for the generating functions counting
$\text{SU}(r)$ and
${}^{L}\text{SU}(r)=\text{SU}(r)/{\mathbb{Z}}_{r}$-instantons, see [9], [8],
[10]. On the other hand, the Bogomolov-Gieseker inequality for slope
semistable torsion free coherent sheaves on a surface Deligne-Mumford stack
${\mathcal{X}}$ is interesting in its own since it will prove some restriction
theorem of slope semistable sheaves on ${\mathcal{X}}$ to a large degree
divisor inside ${\mathcal{X}}$. This will have applications to the reduction
of the moduli of stable Higgs bundles on surfaces to the moduli space of
stable Higgs bundles on curves, which is related to the Langlands duality and
mirror symmetry between the moduli spaces of $\text{SL}_{r}$ and
$\text{PGL}_{r}$-Higgs bundles on curves.

Let us first state our main result. We fix ${\mathcal{X}}$ to be a surface
Deligne-Mumford stack, and a polarization $(\Xi,{\mathcal{O}}_{X}(1)=H)$ where
$\Xi$ is a generating sheaf on ${\mathcal{X}}$ and ${\mathcal{O}}_{X}(1)$ a
polarization on the coarse moduli space $X$. We choose the generating sheaf
$\Xi$ to satisfy the condition that its restriction to any codimension one
component in $I{\mathcal{X}}_{1}$ is a direct sum of locally free sheaves of
the same rank. Here $I{\mathcal{X}}_{1}\subset I{\mathcal{X}}$ is the
components in the inertia stack $I{\mathcal{X}}$ consisting of codimension one
components. Then in this case the modified slope of a torsion free coherent
sheaf $E$ is given by:

$\mu_{\Xi}(E)=\frac{\deg(E)}{\mathop{\rm rk}(\Xi)\mathop{\rm rk}(E)}$

where $\deg(E)=c_{1}(E)\cdot H$. The modified slope semistability of a torsion
free sheaf $E$ is equivalent to the orbifold semistability of the surface
Deligne-Mumford stack ${\mathcal{X}}$ where the slope is given by
$\mu(E)=\frac{\deg(E)}{\mathop{\rm rk}(E)}$.

We can not prove a Bogomolov-Gieseker inequality formula for a modified slope
semistable torsion free sheaf $E$ for an arbitrary general generating sheaf
$\Xi$, where the modified slope $\mu_{\Xi}(E)$ is $\frac{\mathop{\rm
rk}(\Xi)\deg(E)}{\mathop{\rm rk}(E)}$ plus the contributions from codimension
one components $I{\mathcal{X}}_{1}\subset I{\mathcal{X}}$. We do not know if
such an inequality holds for general modified slope semistable torsion free
sheaves, except that we can show it holds for root stacks and a special choice
of generating sheaf. This has applications to the calculation of Vafa-Witten
invariants for root stack surfaces in [7]. For a root stack ${\mathcal{X}}$
(associated to a pair $(X,D)$ where $D\subset X$ is a normal crossing divisor)
over a smooth surface $X$, a choice of generating sheaf will gives the
equivalence between the category of coherent sheaves on ${\mathcal{X}}$ and
the category of rational parabolic sheaves on $(X,D)$.

Our main result is:

######  Theorem 1.1.

(Theorem 3.1) Let ${\mathcal{X}}$ be a two dimensional smooth tame projective
Deligne-Mumford stack and let $\Xi$ be a generating sheaf such that its
restriction to every component in $I{\mathcal{X}}_{1}$ is a direct sum of
locally free coherent sheaves of the same rank. Then if $E$ is a strongly
modified slope semistable torsion free sheaf on ${\mathcal{X}}$, we have

$\Delta(E)\geq 0.$

We prove Theorem 1.1 by a method of Moriwaki which is generalized to surface
Deligne-Mumford stacks, plus the calculation using orbifold Grothendieck-
Riemann-Roch formula for the surface Deligne-Mumford stack. In order to prove
the theorem we review the basic knowledge of coherent sheaves over a tame
Deligne-Mumford stack ${\mathcal{X}}$, the Frobenius morphisms and the basic
properties of slope modified semistable torsion free sheaves on
${\mathcal{X}}$. In particular we generalize one of Langer’s inequality
involving maximal and minimal modified slopes in the Harder-Narasimhan
filtration of a torsion free sheaf $E$ to smooth Deligne-Mumford stacks.

As the first application of the Bogomolov-Gieseker inequality in Theorem 1.1,
the Bogomolov-Gieseker inequality for surface Deligne-Mumford stacks in
characteristic zero holds by the standard method of taking limit. The second
application is to prove the Bogomolov-Gieseker inequality for rational
parabolic semistable torsion free sheaves on $(X,D)$ by relating the rational
parabolic sheaves on $(X,D)$ to torsion free sheaves on the root stack
${\mathcal{X}}$ associated with $(X,D)$. The result in Theorem 1.1 can also be
used to construct Bridgeland stability conditions on surface Deligne-Mumford
stacks.

We generalize Langer’s results (the four theorems in [13, §3]) for higher
dimensional smooth Deligne-Mumford stacks ${\mathcal{X}}$ for a special choice
of generating sheaf such that the modified slope is equivalent to the orbifold
slope of ${\mathcal{X}}$. We also provide one restriction theorem for slope
stable torsion free sheaves to a large degree divisor in ${\mathcal{X}}$. The
proof is similar to [13, §3], and we put these arguments in the appendix. The
restriction theorem will have applications to the reduction of the moduli
space of Higgs sheaves on a surface or a surface Deligne-Mumford stack to the
moduli space of Higgs bundles to a large degree curve inside the surface or
the surface Deligne-Mumford stack.

We also prove the Bogomolov inequality for semistable Simpson Higgs sheaves on
tame Deligne-Mumford stacks in Appendix B generalizing the method in [14]. We
should point out the Higgs sheaves here are not the Higgs sheaves in [26],
[7], where the Bogomolov inequality may involve correction terms.

###  1.1. Outline

Here is the short outline of the paper. In §2 we review the modified slope
stability and Frobenius morphisms for torsion free sheaves on a tame
projective Deligne-Mumford stack ${\mathcal{X}}$, and prove some inequalities
for torsion free sheaves on ${\mathcal{X}}$. We prove the Bogomolov-Gieseker
inequality formula Theorem 1.1 in §3, and the Bogomolov inequality for
rational parabolic sheaves for $(X,D)$. Finally in Appendix A we generalize
the results in [13, §3, §5] to higher dimensional smooth tame Deligne-Mumford
stacks, and joint with Hao Sun we prove the Bogomolov inequality for Simpson
Higgs sheaves on tame Deligne-Mumford stacks in Appendix B.

###  1.2. Convention

We work over an algebraically closed field $\kappa$ of characteristic $p>0$
throughout of the paper. All Deligne-Mumford stacks are tame. We denote by
${{\mathbb{G}}_{\mbox{\tiny\rm m}}}$ the multiplicative group over $\kappa$.
We use Roman letter $E$ to represent a coherent sheaf on a tame projective
Deligne-Mumford stack ${\mathcal{X}}$.

### Acknowledgments

The authors would like to thank A. Langer for useful email correspondences and
valuable discussions. This work is partially supported by NSF and a Simons
Foundation.

##  2\. Preliminaries on modified stability and some new results

In this section we review the modified semistability for Deligne-Mumford
surfaces, and Langer’s notations for semistability in characteristic $p$. Some
new results of Shapherd-Barron for Deligne-Mumford surfaces are proved.

###  2.1. Notations

We fix some notations for a smooth projective Deligne-Mumford stack
${\mathcal{X}}$ of dimension $d$. The Deligne-Mumford stack ${\mathcal{X}}$ is
called $tame$, if the stabilizer groups of the Deligne-Mumford stack
${\mathcal{X}}$ are all linearly reductive groups. Equivalently this means the
order of the stabilizer group at any geometric point of $\pi:{\mathcal{X}}\to
X$ is relatively prime to $p$.

Let ${\mathcal{I}}$ be the index set of the components of the inertia stack
$I{\mathcal{X}}$ such that

$I{\mathcal{X}}=\bigsqcup_{g\in{\mathcal{I}}}{\mathcal{X}}_{g}.$

We always use ${\mathcal{X}}_{0}={\mathcal{X}}$ to represent the trivial
component. For example if ${\mathcal{X}}=[Z/G]$ is a global quotient stack,
where $Z$ is a quasi-projective scheme and $G$ is a finite group acting
diagonally on $Z$, then $I{\mathcal{X}}=\bigsqcup_{(g)}[Z^{g}/C(g)]$. Any
component ${\mathcal{X}}_{g}\subset{\mathcal{X}}$ in the inertia stack
$I{\mathcal{X}}$ is a closed substack of ${\mathcal{X}}$. We denote by
$I{\mathcal{X}}_{1}\subset I{\mathcal{X}}$ be the substack of $I{\mathcal{X}}$
consisting of components ${\mathcal{X}}_{g}$ such that their codimension in
${\mathcal{X}}$ is one. Let $\mathop{\rm
pr}\nolimits:I{\mathcal{X}}\to{\mathcal{X}}$ be the map from the inertia stack
$I{\mathcal{X}}$ to ${\mathcal{X}}$.

For ${\mathcal{X}}$, we write

$H_{\operatorname{CR}}^{*}({\mathcal{X}})=H^{*}(I{\mathcal{X}})=\bigoplus_{g\in{\mathcal{I}}}H^{*}({\mathcal{X}}_{g})$

to be the Chen-Ruan cohomology with ${\mathbb{Q}}$-coefficients. For any
torsion free coherent sheaf $E$ on ${\mathcal{X}}$, we use $c_{i}(E)$ to
represent the Chern classes of $E$ on ${\mathcal{X}}$, and $c_{i}(E)\in
H^{2i}({\mathcal{X}})$.

On the component ${\mathcal{X}}_{g}\subset I{\mathcal{X}}$, at a point
$(x,g)\in{\mathcal{X}}_{g}$, let

$T_{x}{\mathcal{X}}=\bigoplus_{0\leq
f<1}\left(T_{x}{\mathcal{X}}\right)_{g,f}$

be the eigenspace decomposition of $T_{x}{\mathcal{X}}$ with respect to the
stabilizer action and $g$ acts on $\left(T_{x}{\mathcal{X}}\right)_{g,f}$ by
$e^{2\pi if}$.

Let $E\in\mbox{Coh}({\mathcal{X}})$ be a coherent sheaf on ${\mathcal{X}}$, we
have an eigenbundle decomposition of $\mathop{\rm pr}\nolimits^{*}E$ and on
$\mathop{\rm pr}\nolimits^{*}E|_{{\mathcal{X}}_{g}}$ we have

$\mathop{\rm pr}\nolimits^{*}E|_{{\mathcal{X}}_{g}}=\bigoplus_{0\leq
f<1}(\mathop{\rm pr}\nolimits^{*}E)_{g,f}$

with respect to the action of the stabilizer of ${\mathcal{X}}_{g}$, where the
element $g$ acts on $(\mathop{\rm pr}\nolimits^{*}E)_{g,f}$ by $e^{2\pi if}$.
Then the orbifold Chern character is:

(2.1.1)
$\widetilde{\operatorname{Ch}}(E)=\bigoplus_{g\in{\mathcal{I}}}\sum_{0\leq
f<1}e^{2\pi if}\operatorname{Ch}((\mathop{\rm pr}\nolimits^{*}E)_{g,f}),$

where $\operatorname{Ch}$ is the general Chern character. Let $l_{g,f}$ be the
rank of $(\mathop{\rm pr}\nolimits^{*}E)_{g,f}$. The orbifold Todd class of
$T{\mathcal{X}}$ is given by

(2.1.2)
$\widetilde{\operatorname{Td}}(T{\mathcal{X}})=\bigoplus_{g\in{\mathcal{I}}}\prod_{\begin{subarray}{c}0\leq
f<1\\\ 1\leq i\leq r_{g,f}\end{subarray}}\frac{1}{1-e^{-2\pi
if}e^{-x_{g,f,i}}}\prod_{f=0}\frac{x_{g,0,i}}{1-e^{-x_{g,0,i}}},$

where $(\mathop{\rm pr}\nolimits^{*}T{\mathcal{X}})_{g,f}$ has rank $r_{g,f}$
and $x_{g,f,i}$ are Chern roots.

For any coherent sheaf $E$ on ${\mathcal{X}}$, orbifold Riemann-Roch theorem
[28] gives:

(2.1.3)
$\chi({\mathcal{X}},E)=\int_{I{\mathcal{X}}}\widetilde{\operatorname{Ch}}(E)\cdot\widetilde{\operatorname{Td}}(T{\mathcal{X}}).$

###  2.2. Modified stability

Let ${\mathcal{X}}$ be a smooth tame projective Deligne-Mumford stack of
dimension $d$. We choose the polarization ${\mathcal{O}}_{X}(1)$ on its coarse
moduli space $\pi:{\mathcal{X}}\to X$. Let $H:=c_{1}({\mathcal{O}}_{X}(1))$.
Recall from [21],

######  Definition 2.1.

A locally free sheaf $\Xi$ on ${\mathcal{S}}$ is $p$-very ample if for every
geometric point of ${\mathcal{S}}$ the representation of the stabilizer group
at that point contains every irreducible representation of the stabilizer
group. We call $\Xi$ a generating sheaf.

Let $\Xi$ be a locally free (generating) sheaf on ${\mathcal{X}}$. We define a
functor

$F_{\Xi}:D\mbox{Coh}_{{\mathcal{X}}}\to D\mbox{Coh}_{X}$

by

$F\mapsto\pi_{*}\mathscr{H}om_{{\mathcal{O}}_{X}}(\Xi,F)$

and a functor

$G_{\Xi}:D\mbox{Coh}_{X}\to D\mbox{Coh}_{{\mathcal{X}}}$

by

$F\mapsto\pi^{*}F\otimes\Xi.$

From [23], the functor $F_{\Xi}$ is exact since the dual $\Xi^{\vee}$ is
locally free and the pushforward $p_{*}$ is exact. The functor $G_{\Xi}$ is
not exact unless $p$ is flat. For instance, if $p$ is a flat gerbe or a root
stack, it is flat.

Let us fix a generating sheaf $\Xi$ on ${\mathcal{X}}$. We call the pair
$(\Xi,{\mathcal{O}}_{S}(1))$ a polarization of ${\mathcal{X}}$. Let $E$ be a
coherent sheaf on ${\mathcal{X}}$, we define the support of $E$ to be the
closed substack associated with the ideal

$0\to{\mathcal{I}}\to{\mathcal{O}}_{{\mathcal{X}}}\to\mathscr{E}nd_{{\mathcal{O}}_{{\mathcal{X}}}}(F).$

So $\dim(\operatorname{Supp}F)$ is the dimension of the substack associated
with the ideal ${\mathcal{I}}\subset{\mathcal{O}}_{{\mathcal{X}}}$ since
${\mathcal{X}}$ is a Deligne-Mumford stack. A pure sheaf of dimension $d$ is a
coherent sheaf $E$ such that for every non-zero subsheaf $E^{\prime}$ the
support of $E^{\prime}$ is of pure dimension $d$. For any coherent sheaf $E$,
we have the torsion filtration:

$0\subset T_{0}(E)\subset\cdots\subset T_{l}(E)=E$

where every $T_{i}(E)/T_{i-1}(E)$ is pure of dimension $i$ or zero, see [6,
§1.1.4].

######  Definition 2.2.

The modified Hilbert polynomial of a coherent sheaf $E$ on ${\mathcal{X}}$ is
defined as:

$H_{\Xi}(E,m)=\chi({\mathcal{X}},E\otimes\Xi^{\vee}\otimes\pi^{*}{\mathcal{O}}_{X}(m))=H(F_{\Xi}(E)(m))=\chi(X,F_{\Xi}(E)(m)).$

Let $E$ be of dimension $l$, then we can write:

$H_{\Xi}(E,m)=\sum_{i=0}^{l}\alpha_{\Xi,i}(E)\frac{m^{i}}{i!}$

which is induced by the case of schemes. The modified Hilbert polynomial is
additive on short exact sequences since the functor $F_{\Xi}$ is exact. If we
don’t choose the generating sheaf $\Xi$, the Hilbert polynomial $H$ on
${\mathcal{X}}$ will be the same as the Hilbert polynomial on the coarse
moduli space $S$. The reduced modified Hilbert polynomial for the pure sheaf
$E$ is defined as

$h_{\Xi}(E)=\frac{H_{\Xi}(E)}{\alpha_{\Xi,d}(E)}.$

Let $E$ be a pure coherent sheaf. We call $E$ Gieseker semistable if for every
proper subsheaf $E^{\prime}\subset E$,

$h_{\Xi}(E^{\prime})\leq h_{\Xi}(E).$

We call $E$ stable if $\leq$ is replaced by $<$ in the above inequality.

######  Definition 2.3.

([21, Definition 3.13]) We define the slope of $E$ by

$\mu_{\Xi}(E)=\frac{\alpha_{\Xi,l-1}(E)}{\alpha_{\Xi,l}(E)}.$

Then $E$ is modified slope (semi)stable if for every proper subsheaf $F\subset
E$,

$\mu_{\Xi}(F)(\leq)<\mu_{\Xi}(E).$

The notion of $\mu$-stability and semistability is related to the Gieseker
stability and semistability in the same way as schemes, i.e.,

$\mu-\text{stable}\Rightarrow\text{Gieseker stable}\Rightarrow\text{Gieseker
semistable}\Rightarrow\mu-\text{semistable}$

######  Remark 2.4.

Recall that $I{\mathcal{X}}_{1}\subset I{\mathcal{X}}$ is the substack of
$I{\mathcal{X}}$ consisting of components such that the codimension of
${\mathcal{X}}_{g}\subset{\mathcal{X}}$ is one. If our Deligne-Mumford stack
${\mathcal{X}}$ is a global quotient stack, which means ${\mathcal{X}}=[Z/G]$
where $Z$ is a quasi-projective scheme and $G$ is a group scheme acting
diagonally. Assume that we can choose the generating sheaf $\Xi$ on
${\mathcal{X}}$ such that its restriction on any component in
$I{\mathcal{X}}_{1}$ is a sum of locally free sheaves of the same rank. If the
sheaf $E$ has dimension $d$, then [21, Proposition 3.18] shows that

$\deg_{\Xi}(E)=\frac{1}{\mathop{\rm
rk}(\Xi)}\alpha_{\Xi,d-1}(E)-\frac{\mathop{\rm rk}(E)}{\mathop{\rm
rk}(\Xi)}\alpha_{\Xi,d-1}({\mathcal{O}}_{{\mathcal{X}}}).$

Here $\alpha_{\Xi,d-1}(E)=\int_{{\mathcal{X}}}c_{1}(E)\cdot H$.

###  2.3. Notations in characteristic $p$

We first fix some notations in characteristic $p>0$. We assume ${\mathcal{X}}$
a smooth tame projective Deligne-Mumford stack of dimension $d$. Let

${\mathcal{X}}^{(i)}={\mathcal{X}}\times_{\mathop{\rm
Spec}\nolimits\kappa}\mathop{\rm Spec}\nolimits(\kappa)$

be the Deligne-Mumford stack obtained from ${\mathcal{X}}$ by applying the
$i$-th power of absolute Frobenius morphism on $\mathop{\rm
Spec}\nolimits\kappa$. The geometric Frobenius morphism
$F_{g}:{\mathcal{X}}\to{\mathcal{X}}^{(1)}$ is defined by the fiber product
${\mathcal{X}}^{(1)}={\mathcal{X}}\times_{\mathop{\rm
Spec}\nolimits\kappa}\mathop{\rm Spec}\nolimits(\kappa)$ and the absolute
Frobenius morphism $F:{\mathcal{X}}\to{\mathcal{X}}$.

We review a bit on the coherent sheaves with connections in [12], where the
theory is for schemes, but in étale topology it works for Deligne-Mumford
stacks. Following N.Katz [12], a connection on a quasi-coherent sheaf $E$ is a
homomorphism

$\nabla:E\to E\otimes\Omega_{{\mathcal{X}}}$

such that $\nabla(ge)=g\nabla(e)+e\otimes dg$ where $g$ and $e$ are sections
of ${\mathcal{O}}_{{\mathcal{X}}}$ and $E$ respectively over an open subset of
${\mathcal{X}}$, and $dg$ denotes the image of $g$ under the canonical
exterior differentiation
$d:{\mathcal{O}}_{{\mathcal{X}}}\to\Omega_{{\mathcal{X}}}$. The kernel
$K:=K(E,\nabla)$ of $\nabla$ is the ${\mathcal{O}}_{{\mathcal{X}}}$-linear map
$K=\nabla_{1}\circ\nabla:E\to E\otimes\Omega_{{\mathcal{X}}}^{2}$ where
$\nabla_{1}:E\otimes\Omega_{{\mathcal{X}}}\to
E\otimes\Omega_{{\mathcal{X}}}^{2}$ is the map defined by
$\nabla_{1}(e\otimes\omega)=e\otimes d\omega-\nabla(e)\wedge\omega$. A
connection $\nabla:E\to E\otimes\Omega_{{\mathcal{X}}}$ is $integrable$ if its
kernel $K=0$. Then let $\operatorname{\textbf{MIC}}({\mathcal{X}})$ be the
category of pairs $(E,\nabla)$ where $E$ is a quasi-coherent
${\mathcal{O}}_{{\mathcal{X}}}$-module and $\nabla$ is an integrable
connection.

The $p$-curvature of an integrable connection $\nabla$ is given by a morphism

$\psi:\mbox{Der}_{\kappa}({\mathcal{X}})\to\operatorname{End}(E),$

by

$\psi(D)=(\nabla(D))^{p}-\nabla(D^{p}),$

where $D$ is a section of $\mbox{Der}_{\kappa}({\mathcal{X}})$.

For the geometric Frobenius morphism
$F_{g}:{\mathcal{X}}\to{\mathcal{X}}^{(1)}$, in [12, Theorem 5.1], N. Katz
constructed a canonical connection $\nabla_{\operatorname{can}}$ on
$F_{g}^{*}E$ which is determined in the following Cartier theorem:

There is an equivalence of categories between the category of quasi-coherent
sheaves on ${\mathcal{X}}^{(1)}$ and the full subcategory of
$\operatorname{\textbf{MIC}}({\mathcal{X}})$ consisting of objects
$(E,\nabla)$ whose $p$-curvature is zero. The equivalence is given by:
$E\mapsto(F_{g}^{*}E,\nabla_{\operatorname{can}})$ and $(E,\nabla)\mapsto
E^{\nabla}$. Here the unique $\nabla_{\operatorname{can}}$ on $F_{g}^{*}E$
makes $E\cong(F_{g}^{*}E)^{\nabla_{\operatorname{can}}}$, and for a sheaf $E$,
$E^{\nabla}$ is the kernel $\ker(\nabla)$ of $\nabla$.

We first have a result of generating sheaves under Frobenius pullbacks:

######  Lemma 2.5.

Let $F_{g}:{\mathcal{X}}\to{\mathcal{X}}^{(1)}$ be the geometric Frobenius
morphism. Then a locally free sheaf $\Xi$ is a generating sheaf on
${\mathcal{X}}^{(1)}$ if and only if $F_{g}^{*}\Xi$ is a generating sheaf on
${\mathcal{X}}$.

###### Proof.

That $\Xi$ is a generating sheaf means it contains all the irreducible
representations of the stabilizer group of ${\mathcal{X}}^{(1)}$. Then the
pullback $F_{g}^{*}\Xi$ contains all the irreducible representations of the
stabilizer group of ${\mathcal{X}}$ since $p$ is prime to all the orders of
the stabilizer groups of ${\mathcal{X}}$. ∎

Thus from the Cartier’s theorem above, the following is a generalization of
[13, Proposition 2.2]:

######  Proposition 2.6.

A coherent sheaf $E$ on ${\mathcal{X}}^{(1)}$ is slope semistable with respect
to $(H,\Xi)$ if and only of $F_{g}^{*}E$ is slope semistable with respect to
$(F_{g}^{*}H,F_{g}^{*}\Xi)$.

###### Proof.

This is from the modified slope of $E$ is the product of some $p$-th power
with the modified slope of $F_{g}^{*}E$. ∎

The following lemma is the generalization of [13, Lemma 2.3]. We recall that a
sheaf $E$ is $\nabla$-semistable if the inequality of modified slopes is
satisfied for all nonzero $\nabla$-preserved subsheaves of $E$.

######  Lemma 2.7.

Consider a torsion free sheaf $E$ on ${\mathcal{X}}$ with a
$\kappa$-connection $\nabla$, and assume that $E$ is $\nabla$-semistable. Let
$0=E_{0}\subset E_{1}\subset\cdots\subset E_{m}=E$ be the usual Harder-
Narasimhan filtration, then the induced morphisms
$E_{i}\to(E/E_{i})\otimes\Omega_{{\mathcal{X}}}$ are nonzero
${\mathcal{O}}_{{\mathcal{X}}}$-morphisms.

###### Proof.

If the induced morphisms $E_{i}\to(E/E_{i})\otimes\Omega_{{\mathcal{X}}}$ are
zero, then it induces a Harder-Narasimhan filtration for $\nabla$-connections
which contradicts the $\nabla$-semistability. ∎

Now we fix some notations for the slopes of a coherent sheaf $E$ on
${\mathcal{X}}$ with respect to the polarization $(\Xi,H)$. Let

$0=E_{0}\subset E_{1}\subset\cdots\subset E_{m}:=E$

be the Harder-Narasimhan filtration of $E$ with respect to modified slope
stability. We denote by $\mu_{\Xi,\max}(E),\mu_{\Xi,\min}(E)$ the maximal and
minimal modified slope of $E$ respectively. We let

$L_{\Xi,\max}(E)=\lim_{k\to\infty}\frac{\mu_{\Xi,\max}((F^{k})^{*}E)}{p^{k}};\quad
L_{\Xi,\min}(E)=\lim_{k\to\infty}\frac{\mu_{\Xi,\min}((F^{k})^{*}E)}{p^{k}}.$

The sequence $\frac{\mu_{\Xi,\max}((F^{k})^{*}E)}{p^{k}}$ is weakly increasing
and $\frac{\mu_{\Xi,\min}((F^{k})^{*}E)}{p^{k}}$ is weakly decreasing. Also
$L_{\Xi,\max}(E)\geq\mu_{\max}(E)$ and $L_{\Xi,\min}(E)\leq\mu_{\min}(E)$.
Suppose that $E$ is slope semistable, then $L_{\Xi,\max}(E)=\mu_{\Xi}(E)$ (or
$L_{\Xi,\min}(E)=\mu_{\Xi}(E)$) if and only if $E$ is strongly slope
semistable.

First we have:

######  Lemma 2.8.

Let $E$ be a slope semistable torsion-free sheaf on ${\mathcal{X}}$ such that
$F^{*}E$ is unstable. Then in the Harder-Narasimhan filtration $0=E_{0}\subset
E_{1}\subset\cdots\subset E_{m}:=F^{*}E$, the
${\mathcal{O}}_{{\mathcal{X}}}$-morphisms
$E_{i}\to(F^{*}E/E_{i})\otimes\Omega_{{\mathcal{X}}}$ induced by
$\nabla_{\operatorname{can}}$ are nonzero.

###### Proof.

Note that this is Proposition 1 in [24] where Shepherd-Barron dealt with
general slope stability for schemes. The result for modified slope stability
is from Proposition 2.6 and Lemma 2.7. ∎

For any divisors $H,A$, $H^{i}\cdot A^{d-i}$ denotes the integration
$\int_{{\mathcal{X}}}H^{i}\cdot A^{d-i}$. Here We still denote $H$ and $A$ for
$\pi^{*}H,\pi^{*}A$ where $\pi:{\mathcal{X}}\to X$ is the map to its coarse
moduli space.

######  Corollary 2.9.

Let us define

$\alpha_{\Xi}(E):=\max(L_{\Xi,\max}(E)-\mu_{\Xi,\max}(E),\mu_{\Xi,\min}(E)-L_{\Xi,\min}(E)).$

Let $L\in\operatorname{Pic}({\mathcal{X}})$ be a $\pi$-ample line bundle on
${\mathcal{X}}$ such that there exists $m\in{\mathbb{Z}}_{>0}$ making
$\bigoplus_{i=1}^{m-1}L^{\otimes i}$ a generating sheaf. Let $A$ be a nef
divisor for $X$, such that $\pi_{*}(T_{X}\otimes\sum_{k=1}^{m-1}(L^{k}))(A)$
is globally generated, we have

$\alpha_{\Xi}(E)\leq\frac{\mathop{\rm rk}(E)-1}{p-1}\left(\max_{1\leq k\leq
m-1}(\mu_{\Xi}(A\otimes L^{k}))\right),$

where

$\max_{1\leq k\leq m-1}(\mu_{\Xi}(A\otimes L^{k}))=\max\\{\mu_{\Xi}(A\otimes
L),\mu_{\Xi}(A\otimes L^{2}),\cdots,\mu_{\Xi}(A\otimes L^{m-1})\\}.$

###### Proof.

This is similar to [13, Cor 2.5]. We modify our case by observing that
$\sum_{k=1}^{m-1}L^{k}$ is a generating sheaf on ${\mathcal{X}}.$ So we have

$\pi^{*}(\pi_{*}(T{\mathcal{X}}\otimes\sum_{k=1}^{m-1}L^{k}))\otimes\sum_{k=1}^{m-1}L^{-k}\to
T{\mathcal{X}}$

which is a surjection. From our assumption
$\pi_{*}(T{\mathcal{X}}\otimes\sum_{k=1}^{m-1}L^{k})(A)$ is globally
generated. Using the right exactness of pullback we obtain,

$(({\mathcal{O}}_{{\mathcal{X}}}(-A))^{r}\otimes\sum_{k=1}^{m-1}L^{-k})\to\pi^{*}\pi_{*}(T{\mathcal{X}}\otimes\sum_{k=1}^{m-1}L^{k})\otimes\sum_{k=1}^{m-1}L^{-k}\to
T{\mathcal{X}}$

which is again a surjection.
Dualising we obtain,

$0\to\Omega_{{\mathcal{X}}}\to(({\mathcal{O}}_{{\mathcal{X}}}(A))^{r}\otimes\sum_{k=1}^{m-1}L^{k})$

Recall the Harder-Narasimhan filtration $0=E_{0}\subset
E_{1}\subset\cdots\subset E_{m}:=F^{*}E$ for $F^{*}E$, tensoring by
$F^{*}(E)/E_{i}$ and taking $\mu_{\Xi,max}$ we obtain,

$\mu_{\Xi,\max}((F^{*}(E)/E_{i})\otimes\Omega_{{\mathcal{X}}})\leq\mu_{\Xi,\max}(((F^{*}(E)/E_{i})\otimes({\mathcal{O}}_{{\mathcal{X}}}(A))^{r}\otimes\sum_{k=1}^{m-1}L^{k}))$

We now use

$\mu_{\Xi,\max}(E\oplus F)=max(\mu_{\Xi,\max}(E),\mu_{\Xi,\max}(F))$

where $E,F$ are torsion free coherent sheaves.
Using the above the right hand side reduces to the following,

$\max_{1\leq k\leq m-1}(\mu_{\Xi,\max}(F^{*}(E)/E_{i}\otimes A\otimes
L^{k})).$

Now we estimate,

$\mu_{\Xi,\max}(F^{*}E/E_{i}\otimes A\otimes
L^{k})=\mu_{\Xi}(E_{i+1}/E_{i})+\mu_{\Xi}(A\otimes L^{k}).$

Taking max into account we obtain,

$\max_{1\leq k\leq m-1}(\mu_{\Xi,\max}(F^{*}(E)/E_{i}\otimes A\otimes
L^{k}))=\max_{1\leq k\leq m-1}(\mu_{\Xi}(A\otimes
L^{k}))+\mu_{\Xi}(E_{i+1}/E_{i}).$

Also from Lemma 2.8,
$\mu_{\Xi,\min}(E_{i})\leq\mu_{\Xi,\max}((F^{*}E/E_{i})\otimes\Omega_{{\mathcal{X}}})$.
Thus from above we get:

$\mu_{\Xi}(E_{i}/E_{i-1})-\mu_{\Xi}(E_{i+1}/E_{i})\leq\max_{1\leq k\leq
m-1}(\mu_{\Xi}(A\otimes L^{k}))$

for each $i=1,\cdots,m-1$. Thus summing over above inequalities we obtain

$\mu_{\Xi,\max}(F^{*}(E))-\mu_{\Xi,\min}(F^{*}E)\leq(\mathop{\rm
rk}(E)-1)\max_{1\leq k\leq m-1}(\mu_{\Xi}(A\otimes L^{k})).$

Note that $\frac{\mu_{\Xi,\max}((F^{k})^{*}E)}{p^{k}}$ is weakly increasing,
and $\mu_{\Xi,\max}(E)\geq\frac{\mu_{\Xi,\min}(F^{*}E)}{p}$, we have by
induction that

$\frac{\mu_{\Xi,\max}((F^{k})^{*}E)}{p^{k}}-\mu_{\Xi,\max}(E)\leq\frac{\mathop{\rm
rk}(E)-1}{p-1}\max_{1\leq k\leq m-1}(\mu_{\Xi}(A\otimes L^{k}))$

Taking limit we get the inequality for $L_{\Xi,\max}(E)-\mu_{\Xi,\max}(E)$,
and the corresponding inequality for $\mu_{\Xi,\min}(E)-L_{\Xi,\min}(E))$ can
be similarly proved. ∎

###  2.4. fdHN property

As in [13, §2.6], a torsion free coherent sheaf $E$ on ${\mathcal{X}}$ has an
fdHN property (finite determinacy of the Harder-Narasimhan filtration) if
there exists a $k_{0}$ such that all quotients in the Harder-Narasimhan
filtration of $(F^{k_{0}})^{*}E$ are strongly modified slope semistable with
respect to $\Xi$. $E$ is fdHN of it has an fdHN property. Similar to the proof
in [13, Theorem 2.7], we have

######  Proposition 2.10.

Let $E$ be a torsion free sheaf on ${\mathcal{X}}$, the $E$ is fdHN.

###### Proof.

Let

$0=E_{0}\subset E_{1}\subset\cdots\subset E_{m}=E$

be the Harder-Narasimhan filtration of $E$. We have the HNP (Harder-Narasimhan
polygon) associated with $E$ defined by connecting the points
$p(E_{0}),p(E_{1}),\cdots,p(E_{m})$ by successively line segments and
connecting the last one with the first one. Here
$p(G)=(\alpha_{\Xi,d-1}(G),\alpha_{\Xi,d}(G))$. Since the base field $\kappa$
has characteristic $p>0$, there is a sequence of polygons $HNP_{k}(E)$, where
$HNP_{k}(E)$ is defined by contracting $HPN((F^{k})^{*}E)$ along the
$\alpha_{\Xi,d-1}$ axis by the factor $1/p^{k}$. All of the polygons are
bounded and $HNP_{k}(E)$ is contained in $HNP_{k+1}(E)$. Then the proof of
[13, Theorem 2.7] goes through without major changes. We omit the details. ∎

##  3\. Bogomolov-Gieseker inequality

In this section we always fix ${\mathcal{X}}$ as a smooth tame projective
Deligne-Mumford stack of dimension $2$. Interesting two dimensional Deligne-
Mumford stacks are reviewed in [7], where the name surface Deligne-Mumford
stack is used. We always fix a generating sheaf $\Xi$ on ${\mathcal{X}}$ and
talk about modified semi-stable sheaves in characteristic p and assume the
existence of a $\pi$ ample line bundle on ${\mathcal{X}}$.

###  3.1. Condition $\star$

We say a generating sheaf $\Xi$ on ${\mathcal{X}}$ satisfying Condition
$\star$ if its restriction on any component in $I{\mathcal{X}}_{1}$ is a sum
of locally free sheaves of the same rank as in [21, Proposition 3.18] and
Remark 2.4.

We have

$\deg_{\Xi}(E)=\frac{1}{\mathop{\rm
rk}(\Xi)}\alpha_{\Xi,d-1}(E)-\frac{\mathop{\rm rk}(E)}{\mathop{\rm
rk}(\Xi)}\alpha_{\Xi,d-1}({\mathcal{O}}_{{\mathcal{X}}}).$

Here $\deg(E)=\int_{{\mathcal{X}}}c_{1}(E)\cdot H^{d-1}=\alpha_{\Xi,d-1}(E)$.
Also $\alpha_{\Xi,d}(E)=\mathop{\rm rk}(E)\mathop{\rm rk}(\Xi)H^{d}$. Since
$\frac{\mathop{\rm rk}(E)}{\mathop{\rm
rk}(\Xi)}\alpha_{\Xi,d-1}({\mathcal{O}}_{{\mathcal{X}}})$ is constant for a
fixed polarization ${\mathcal{O}}_{X}(1)=H,\Xi$, it is reasonable to use
$\mu_{\Xi}(E)=\frac{\mathop{\rm rk}(\Xi)\deg_{\Xi}(E)}{\mathop{\rm rk}(E)}$ as
the definition of modified slope. Note that in this case the modified slope
stability is the same as orbifold stability defined by
$\mu(E)=\frac{\deg(E)}{\mathop{\rm rk}(E)}$.

For any torsion free coherent sheaf $E$ on ${\mathcal{X}}$ of rank
$\mathop{\rm rk}$, recall the Chern character morphism in (2.1.1), we rewrite
it here

$\widetilde{\operatorname{Ch}}(E)=\bigoplus_{g\in{\mathcal{I}}}\sum_{0\leq
f<1}e^{2\pi if}\operatorname{Ch}((\mathop{\rm pr}\nolimits^{*}E)_{g,f}).$

We set:

$\Delta(E):=\operatorname{Ch}_{1}(E)^{2}-2\operatorname{Ch}_{0}(E)\cdot\operatorname{Ch}_{2}(E)=2\mathop{\rm
rk}(E)c_{2}(E)-(\mathop{\rm rk}(E)-1)c_{1}^{2}(E).$

###  3.2. Two dimensional case

In this section we prove the following result:

######  Theorem 3.1.

Let ${\mathcal{X}}$ be a two dimensional smooth projective Deligne-Mumford
stack. Then if $E$ be a strongly modified slope semistable torsion free sheaf
on ${\mathcal{X}}$, we have

$\Delta(E)\geq 0.$

We use the method of [20]. Let us first prove a lemma.

######  Lemma 3.2.

Suppose that $E$ is strongly modified slope semistable torsion free coherent
sheaf of rank $\mathop{\rm rk}(E)$ on ${\mathcal{X}}$ and $c_{1}(E)\cdot H=0$.
Let $L$ be a line bundle on ${\mathcal{X}}$, then there exists a positive
number $M$ such that

$h^{0}({\mathcal{X}},(F^{n})^{*}(E)\otimes L)\leq M\cdot p^{n}$

for sufficiently large integers $n$.

###### Proof.

First we take a positive integer $m$ such that $L\cdot H-mH^{2}<0$. Therefore
we have $c_{1}((F^{n})^{*}(E)\otimes L\otimes H^{-m})<0$. Since
$(F^{n})^{*}(E)\otimes L\otimes H^{-m}$ is modified slope semistable, we have
$H^{0}({\mathcal{X}},(F^{n})^{*}(E)\otimes L\otimes H^{-m})=0$ (otherwise it
will contradicts with the modified semistability). Then we choose a general
element $C\in|mH|$ and consider ${\mathcal{C}}=\pi^{-1}(C)$, and the exact
sequence

$0\to{\mathcal{O}}_{{\mathcal{X}}}(-{\mathcal{C}})\longrightarrow{\mathcal{O}}_{{\mathcal{X}}}\longrightarrow{\mathcal{O}}_{{\mathcal{C}}}\to
0.$

Tensor with $(F^{n})^{*}(E)\otimes L$ we get an inequality

$h^{0}({\mathcal{X}},(F^{n})^{*}(E)\otimes L)\leq
h^{0}({\mathcal{C}},(F_{{\mathcal{C}}}^{n})^{*}(E|_{{\mathcal{C}}})\otimes
L|_{{\mathcal{C}}})$

where $F_{{\mathcal{C}}}$ is the absolute Frobenius morphism of
${\mathcal{C}}$, and $E|_{{\mathcal{C}}},L|_{{\mathcal{C}}}$ are the
restrictions to ${\mathcal{C}}$.

We show that there exists a positive number $M$ such that

$h^{0}({\mathcal{C}},(F_{{\mathcal{C}}}^{n})^{*}(E|_{{\mathcal{C}}})\otimes
L|_{{\mathcal{C}}})\leq M\cdot p^{n}.$

We prove it for any vector bundle $E$ on ${\mathcal{C}}$ of rank $\mathop{\rm
rk}(E)$ and a line bundle $L$ on ${\mathcal{C}}$. The rank one case of $E$ is
obvious since $(F^{n})^{*}E=E^{\otimes p^{n}}$ and by orbifold Riemann-Roch
theorem [28]. General case is proved by induction on the rank $\mathop{\rm
rk}(E)$. Consider an exact sequence

$0\to G\to E\to Q\to 0$

where $G$ is a line bundle and $Q$ is a rank $\mathop{\rm rk}-1$ vector
bundle, and the exact sequence:

$0\to(F^{n})^{*}G\otimes L\longrightarrow(F^{n})^{*}E\otimes
L\longrightarrow(F^{n})^{*}Q\otimes L\to 0.$

Thus we have

$h^{0}({\mathcal{C}},(F_{{\mathcal{C}}}^{n})^{*}(E)\otimes L)\leq
h^{0}({\mathcal{C}},(F_{{\mathcal{C}}}^{n})^{*}(G)\otimes
L)+h^{0}({\mathcal{C}},(F_{{\mathcal{C}}}^{n})^{*}(Q)\otimes L).$

We are done. ∎

### Proof of Theorem 3.1

We claim that we can assume $c_{1}(E)=0$. Let $\mathop{\rm rk}:=l\cdot p^{j}$
for some integers $l$ and $j$, then $l$ is prime to $p$. Then from [2, Lemma
2.1], since $X$ is a scheme, there is a separable finite morphism $\phi:Y\to
X$ such that $c_{1}(\phi^{*}E)$ is divisible by $l$. Note that [2, Lemma 2.1]
proved the statement for smooth schemes, but since we are in two dimensional,
the argument works for surfaces with at most of quotient singularities. The
coarse moduli space $X$ always has quotient singularities. Let $f:=\phi\circ
F^{j}$, then $c_{1}(f^{*}E)$ is divisible by $\mathop{\rm rk}$. We form the
cartesian diagram:

(3.2.1)
$\textstyle{{\mathcal{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{f}}$$\scriptstyle{\pi_{Y}}$$\textstyle{{\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X}$

where ${\mathcal{Y}}$ is the fibre product. Then ${\mathcal{Y}}$ is a Deligne-
Mumford stack with the stacky locus all pullback from the stacky locus of
${\mathcal{X}}$. We set:

$\widetilde{E}:=\widetilde{f}^{*}(E)\otimes{\mathcal{O}}_{{\mathcal{Y}}}\left(-\frac{c_{1}(\widetilde{f}^{*}(E))}{\mathop{\rm
rk}}\right).$

Then we calculate

$c_{1}(\widetilde{E})=0;\quad
c_{2}(\widetilde{E})=\widetilde{f}^{*}\left(c_{2}(E)-\frac{\mathop{\rm
rk}-1}{2\mathop{\rm rk}}c_{1}(E)^{2}\right).$

Since $E$ is strongly modified slope semistable with the slope given by
$\mu_{\Xi}(E)=\frac{\deg(E)}{\mathop{\rm rk}(E)\cdot\mathop{\rm rk}(\Xi)}$,
from [5, Lemma 1.1], $\widetilde{f}^{*}E$ is modified slope semistable.
Therefore $c_{2}(\widetilde{E})\geq 0$ implies that $\Delta(E)\geq 0$.

Then we assume $c_{1}(E)=0$. By Lemma 3.2, there exists positive numbers
$M_{1},M_{2}$ such that

$h^{0}({\mathcal{X}},(F^{n})^{*}(E))\leq M_{1}\cdot p^{n};\quad
h^{0}({\mathcal{X}},(F^{n})^{*}(E^{\vee})\otimes\omega_{{\mathcal{X}}})\leq
M_{2}\cdot p^{n}$

for sufficiently large integers $n$. In the inertia stack $I{\mathcal{X}}$,
${\mathcal{I}}_{1}$ denotes the index set such that
${\mathcal{X}}_{g}\subset{\mathcal{X}}$ has codimension one for
$g\in{\mathcal{I}}_{1}$. We let ${\mathcal{I}}_{2}$ denotes the index set such
that ${\mathcal{X}}_{g}\subset{\mathcal{X}}$ has codimension two, i.e., the
stacky locus consisting of points in ${\mathcal{X}}$. Then by orbifold
Riemann-Roch theorem (2.1.3) from [28],

$\displaystyle\chi({\mathcal{X}},(F^{n})^{*}(E))$
$\displaystyle=\int_{I{\mathcal{X}}}\widetilde{\operatorname{Ch}}((F^{n})^{*}(E))\cdot\widetilde{\operatorname{Td}}(T_{{\mathcal{X}}})$
$\displaystyle=\int_{{\mathcal{X}}}\operatorname{Ch}((F^{n})^{*}(E))\cdot\operatorname{Td}(T_{{\mathcal{X}}})+$
$\displaystyle+\sum_{g\in{\mathcal{I}}_{1}}\int_{{\mathcal{X}}_{g}}\sum_{0\leq
f<1}e^{2\pi
if}\operatorname{Ch}((F^{n})^{*}(E)_{g,f})\cdot\operatorname{Td}(T_{{\mathcal{X}}})_{g,f}$
$\displaystyle+\sum_{g\in{\mathcal{I}}_{2}}\int_{{\mathcal{X}}_{g}}\sum_{0\leq
f<1}e^{2\pi
if}\operatorname{Ch}((F^{n})^{*}(E)_{g,f})\cdot\operatorname{Td}(T_{{\mathcal{X}}})_{g,f}$
$\displaystyle=-c_{2}((F^{n})^{*}(E))+\int_{{\mathcal{X}}}\mathop{\rm
rk}\cdot\operatorname{Td}(T_{{\mathcal{X}}})$
$\displaystyle+\sum_{g\in{\mathcal{I}}_{1}}\left(\int_{{\mathcal{X}}_{g}}\sum_{0\leq
f<1}e^{2\pi if}c_{1}((F^{n})^{*}(E)_{g,f})+\int_{{\mathcal{X}}_{g}}\sum_{0\leq
f<1}e^{2\pi if}\cdot\operatorname{Td}(T_{{\mathcal{X}}})_{g,f}\right)$
$\displaystyle+\sum_{g\in{\mathcal{I}}_{2}}\int_{{\mathcal{X}}_{g}}\sum_{0\leq
f<1}e^{2\pi if}\cdot\operatorname{Td}(T_{{\mathcal{X}}})^{0}_{g,f}$

where $\operatorname{Td}(T_{{\mathcal{X}}})^{0}_{g,f}$ is the constant term of
$\operatorname{Td}(T_{{\mathcal{X}}})_{g,f}$. By the Frobenius pullback
property of Chern classes we have

$\displaystyle\chi({\mathcal{X}},(F^{n})^{*}(E))$
$\displaystyle=-p^{2n}c_{2}(E)+\int_{{\mathcal{X}}}\mathop{\rm
rk}\cdot\operatorname{Td}(T_{{\mathcal{X}}})$
$\displaystyle+\sum_{g\in{\mathcal{I}}_{1}}\left(p^{n}\int_{{\mathcal{X}}_{g}}\sum_{0\leq
f<1}e^{2\pi if}c_{1}((E)_{g,f})+\int_{{\mathcal{X}}_{g}}\left(\sum_{0\leq
f<1}e^{2\pi if}\right)\cdot\operatorname{Td}(T_{{\mathcal{X}}})_{g,f}\right)$
$\displaystyle+\sum_{g\in{\mathcal{I}}_{2}}\int_{{\mathcal{X}}_{g}}\left(\sum_{0\leq
f<1}e^{2\pi if}\right)\cdot\operatorname{Td}(T_{{\mathcal{X}}})^{0}_{g,f}$
$\displaystyle\leq(M_{1}+M_{2})p^{n}$

Hence for large $n$, to ensure the above inequality we must have $c_{2}(E)\geq
0$. We are done. $\square$.

######  Remark 3.3.

If the base field $\kappa$ has character zero, and $E$ is a modified
semistable torsion free sheaf on ${\mathcal{X}}$, then the Bogomolov
inequality $\Delta(E)\geq 0$ also holds by the standard method of taking
limit, see [13]. We omit the details here, and note that in the character zero
case Bogomolov inequality for orbifold semistable torsion free sheaves is
proved in [11, Lemma 2.5] for surface Deligne-Mumford stacks with only
isolated quotient singularities.

###  3.3. Bogomolov inequality for parabolic sheaves

In this section we give an application of the Bogomolov inequality in Theorem
3.1 to rational parabolic sheaves on a surface $X$.

####  3.3.1. Root surfaces

Let $X$ be a smooth projective surface and $D\subset X$ is a simple normal
crossing Cartier divisor. Let $r\in{\mathbb{Z}}_{>0}$ be a positive integer.
The line bundle with the section $({\mathcal{O}}_{X}(D),s_{D})$ defines a
morphism

$S\to[{\mathbb{A}}^{1}/{{\mathbb{G}}_{\mbox{\tiny\rm m}}}].$

Let $\Theta_{r}:[{\mathbb{A}}^{1}/{{\mathbb{G}}_{\mbox{\tiny\rm
m}}}]\to[{\mathbb{A}}^{1}/{{\mathbb{G}}_{\mbox{\tiny\rm m}}}]$ be the morphism
of stacks given by the morphism

$x\in{\mathbb{A}}^{1}\mapsto x^{r}\in{\mathbb{A}}^{1};\quad
t\in{{\mathbb{G}}_{\mbox{\tiny\rm m}}}\mapsto
t^{r}\in{{\mathbb{G}}_{\mbox{\tiny\rm m}}},$

which sends $({\mathcal{O}}_{X}(D),s_{D})$ to $({\mathcal{O}}_{X}(D)^{\otimes
r},s^{r}_{D})$.

######  Definition 3.4.

Let ${\mathcal{X}}:=\sqrt[r]{(X,D)}$ be the stack obtained by the fibre
product

$\textstyle{\sqrt[r]{(X,D)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{[{\mathbb{A}}^{1}/{{\mathbb{G}}_{\mbox{\tiny\rm
m}}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{r}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathcal{O}}_{X}(D),s_{D})}$$\textstyle{[{\mathbb{A}}^{1}/{{\mathbb{G}}_{\mbox{\tiny\rm
m}}}].}$

We call ${\mathcal{X}}=\sqrt[r]{(X,D)}$ the root stack obtained from $X$ by
the $r$-th root construction.

The stack ${\mathcal{X}}=\sqrt[r]{(X,D)}$ is a smooth Deligne-Mumford stack
with stacky locus ${\mathcal{D}}:=\pi^{-1}(D)$, and ${\mathcal{D}}\to D$ is a
$\mu_{r}$-gerbe over $X$ coming from the line bundle
${\mathcal{O}}_{S}(D)|_{D}$.

######  Remark 3.5.

The general root stacks over a logarithmic scheme $X$ is constructed in [3],
and the pair $(X,D)$ defines a canonical log structure on $X$. Since we don’t
need the abstract language of log schemes we refer the reader to [3] for
details.

Theorem 3.1 implies the following result:

######  Proposition 3.6.

Let ${\mathcal{X}}=\sqrt[r]{(X,D)}$ be the $r$-th root stack corresponding to
the pair $(X,D)$, and let $E$ be a strongly slope semistable torsion free
coherent sheaf on ${\mathcal{X}}$ with respect to the polarization
$({\mathcal{O}}_{{\mathcal{X}}},{\mathcal{O}}_{X}(1))$. Then we have

$\Delta(E)=2\mathop{\rm rk}(E)c_{2}(E)-(\mathop{\rm rk}(E)-1)c_{1}(E)^{2}\geq
0.$

####  3.3.2. Parabolic sheaves

######  Definition 3.7.

([18]) Let $E$ be a torsion-free coherent sheaf on $X$. A parabolic structure
on $E$ is given by a length $d$-filtration

$E=F_{1}(E)\supset F_{2}(E)\supset\cdots\supset F_{r}(E)\supset
F_{r+1}(E)=E(-D),$

together with a system of weights

$0\leq\alpha_{1},\alpha_{1},\cdots,\alpha_{r}<1.$

We call $E_{\bullet}=(E,F_{i}(E))$ a rational parabolic sheaf associated with
the divisor $D$ if all the weights $\alpha_{1},\alpha_{1},\cdots,\alpha_{r}$
are all rational. Let $G_{i}(E)=F_{i}(E)/F_{i+1}(E)$. The Hilbert polynomial
$\chi(G_{i}(E)(m))$ is called the $i$-th multiplicity polynomial of the weight
$\alpha_{i}$.

As in [18, Definition 1.8], the parabolic Euler characteristic
$\operatorname{pa}-\chi(E_{\bullet})$ of $E_{\bullet}$ is defined as:

(3.3.1) $\chi(E(-D))+\sum_{i=0}^{d-1}\alpha_{i}\chi(G_{i}).$

The polynomial $\operatorname{pa}-\chi(E_{\bullet}(m))$ is called the
parabolic Hilbert polynomial of $E_{\bullet}$ and the polynomial
$\operatorname{pa}-\chi(E_{\bullet}(m))/\mathop{\rm rk}(E)$ is denoted by
$\operatorname{pa}-p_{E_{\bullet}}(m)$.

######  Definition 3.8.

The parabolic sheaf of $E_{\bullet}$ is said to be parabolic Gieseker stable
(resp. parabolic semistable) if for every parabolic subsheaf $F_{\bullet}$ of
$E_{\bullet}$ with

$0<\mathop{\rm rk}(F)<\mathop{\rm rk}(E)$

we have

$\operatorname{pa}-p_{F_{\bullet}}(m)<\operatorname{pa}-p_{E_{\bullet}}(m),\quad(\text{resp.}\operatorname{pa}-p_{F_{\bullet}}(m)\leq\operatorname{pa}-p_{E_{\bullet}}(m)).$

The parabolic degree $\operatorname{pa-deg}(E_{\bullet})$ is defined by

$\operatorname{pa-
deg}(E_{\bullet})=\int_{0}^{1}\deg(E_{\alpha})d\alpha+\mathop{\rm
rk}(E)\cdot\deg(D).$

We set the parabolic slope as
$\operatorname{pa}-\mu(E_{\bullet})=\frac{\operatorname{pa-
deg}(E_{\bullet})}{\mathop{\rm rk}(E)}$. Then $E_{\bullet}$ is parabolic slope
stable (resp. parabolic semistable) if for every parabolic subsheaf
$F_{\bullet}\subset E_{\bullet}$ with

$0<\mathop{\rm rk}(F)<\mathop{\rm rk}(E)$

we have

$\operatorname{pa}-\mu(F_{\bullet})<\operatorname{pa}-\mu(E_{\bullet}),\quad(\text{resp.}\operatorname{pa}-\mu(F_{\bullet})\leq\operatorname{pa}-\mu(E_{\bullet}).$

####  3.3.3. Equivalence of categories

For the root stack ${\mathcal{X}}=\sqrt[r]{(X,D)}$, we choose the generating
sheaf
$\Xi=\oplus_{i=0}^{r}{\mathcal{O}}_{{\mathcal{X}}}({\mathcal{D}}^{\frac{i}{r}})$.
Let $\mbox{Coh}({\mathcal{X}})$ be the abelian category of coherent sheaves on
${\mathcal{X}}$, and $\mathop{\rm Par}\nolimits_{\frac{1}{r}}(X,D)$ the
abelian category of rational parabolic sheaves on $(X,D)$ with length $r$.
There exist two functors:

${\mathcal{F}}_{{\mathcal{X}}}:\mbox{Coh}({\mathcal{X}})\to\mathop{\rm
Par}\nolimits_{\frac{1}{r}}(X,D);\quad
E\mapsto{\mathcal{F}}_{{\mathcal{X}}}(E)$

where ${\mathcal{F}}_{{\mathcal{X}}}(E)_{l}=\pi_{*}(\mathop{\sf
E}\nolimits\otimes{\mathcal{O}}_{{\mathcal{X}}}(-l{\mathcal{D}}))$; and

$G_{{\mathcal{X}}}:\mathop{\rm
Par}\nolimits_{\frac{1}{r}}(X,D)\to\mbox{Coh}({\mathcal{X}});\quad
E_{\bullet}\mapsto\int^{{\mathbb{Z}}}g_{{\mathcal{X}}}(E_{\bullet})(l,l)$

where $\int^{{\mathbb{Z}}}g_{{\mathcal{X}}}(E_{\bullet})(l,l)$ is the colimit
of wedges:

$\textstyle{g_{{\mathcal{X}}}(E_{\bullet})(l,m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{d,m}}$$\scriptstyle{h_{l,m}}$$\textstyle{g_{{\mathcal{X}}}(E_{\bullet})(l,l)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w(l)}$$\textstyle{g_{{\mathcal{X}}}(E_{\bullet})(m,m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w(m)}$$\textstyle{\mathcal{G}}$

where

  1. (1)

$g_{{\mathcal{X}}}(E_{\bullet}):{\mathbb{Z}}^{0}\times{\mathbb{Z}}\to\mbox{Coh}({\mathcal{X}})$
is a map given by:

$(l,m)\mapsto{\mathcal{O}}_{{\mathcal{X}}}(l{\mathcal{D}})\otimes p^{*}E_{m};$

  2. (2)

$m\geq l$ is an arrow in ${\mathbb{Z}}$, and the arrow $h_{l,m}$ is induced by
the canonical section of the divisor, the arrow $f_{l,m}$ is induced by the
filtration $p^{*}E_{\bullet}$, the arrow $w(l)$ is a dinatural transformation
and ${\mathcal{G}}$ is a sheaf in $\mbox{Coh}({\mathcal{X}})$.

We define that a parabolic sheaf $E_{\bullet}\in\mathop{\rm
Par}\nolimits_{\frac{1}{r}}(X,D)$ to be torsion free if $E_{0}$ is torsion
free. Then we have:

######  Theorem 3.9.

([3, Theorem 6.1]) The functor $G_{{\mathcal{X}}}$ maps torsion free sheaves
on $X$ to torsion free sheaves on ${\mathcal{X}}$. Moreover,
${\mathcal{F}}_{{\mathcal{X}}}$ and $G_{{\mathcal{X}}}$ are inverse to each
other when applied to torsion free sheaves.

####  3.3.4. Parabolic Bogomolov inequality

In this section we apply Proposition 3.6 and Kawamata cover of $(X,D)$
constructed in [1] to deduce the parabolic Bogomolov inequality for parabolic
semistable sheaves.

Let $Y\to X$ be the Galois cover with Galois group
$G=\operatorname{Gal}(\operatorname{Rat}(Y)/\operatorname{Rat}(X))$ as in [1].
Let $r||G|$, then by base change of the root stack in the fibre product
diagram

$\textstyle{{\mathcal{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{f}}$$\scriptstyle{\pi_{Y}}$$\textstyle{{\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X}$

in Diagram 3.2.1, the Deligne-Mumford stack ${\mathcal{Y}}$ is the root stack
of the pullback line bundle $(f^{*}{\mathcal{O}}_{X}(D),f^{*}s_{D})$. Since
the degree of $f^{*}{\mathcal{O}}_{X}(D)$ is divided by $r$, the root stack
associated this line bundle is trivial. Therefore we have ${\mathcal{Y}}\cong
Y$ is a scheme and we have the following diagram:

$\textstyle{{\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{f}}$$\scriptstyle{f}$$\textstyle{X}$

and the quotient stack $[Y/G]\cong{\mathcal{X}}$.

We are ready to state the parabolic Bogomolov inequality. We set up some
notations. First let $W$ be a coherent sheaf on ${\mathcal{X}}$. Then from
Theorem 3.9, there exists a rational parabolic sheaf $(E,F_{*},\alpha_{*})$
such that
$F_{i}=\pi_{*}(W\otimes{\mathcal{O}}_{{\mathcal{X}}}(-i{\mathcal{D}}))$ and
$\alpha_{i}=\frac{i}{r}$.

Let $D=\sum_{\lambda=1}^{h}D_{\lambda}$ be the decomposition of $D$ into
smooth irreducible components. Let $\widetilde{D}=(f^{*}D)_{\mathop{\rm
red}\nolimits}$ which is normal crossing. Then
$f^{*}D_{\lambda}=k_{\lambda}r(f^{*}D_{\lambda})_{\mathop{\rm red}\nolimits}$,
where $1\leq\lambda\leq h$, and $k_{\lambda}\geq 1$ are integers. The torsion
free coherent sheaf $W$ on ${\mathcal{X}}=[Y/G]$ gives a $G$-equivariant
torsion free coherent sheaf on $Y$ which we still denote by $W$. Let

$\iota:\widetilde{D}\to Y$

be the inclusion and $\iota^{\lambda}:\widetilde{D}_{\lambda}\to Y$ be the
inclusion of the component $\widetilde{D}_{\lambda}$, where
$\widetilde{D}_{\lambda}=(f^{*}D_{\lambda})_{\mathop{\rm red}\nolimits}$. Let
$H_{i}=E|_{D}/F_{i}(E|_{D})$ be the sheaf on $D$. Then

$H_{j}=\bigoplus_{\lambda=1}^{h}\overline{\iota}_{*}^{\lambda}H_{j}^{\lambda},$

where $\overline{\iota}^{\lambda}:D_{\lambda}\hookrightarrow D$ denotes the
inclusion. Define $G_{i,\lambda}=H^{\lambda}_{i}/H^{\lambda}_{i+1}$ and

$\widetilde{G}_{i,\lambda}:=(f\iota^{\lambda})^{*}G_{i,\lambda}.$

Then from [1, Formula (3.15)] we have:

(3.3.2)
$W=f^{*}E+\sum_{i=1}^{r}\sum_{\lambda=1}^{h}\sum_{j=1}^{k_{\lambda}m_{i}}\iota_{*}^{\lambda}\left(\widetilde{G}_{i,\lambda}\otimes
N_{\widetilde{D}_{\lambda}}^{j}\right)$

in the K-theory $K_{0}(Y)$, where
$N_{\widetilde{D}_{\lambda}}={\mathcal{O}}_{Y}(\widetilde{D}_{\lambda})|_{\widetilde{D}_{\lambda}}$
is the normal bundle to the divisor $\widetilde{D}_{\lambda}$.

From equivalence between the categories $\mbox{Coh}({\mathcal{X}})$ and
$\mathop{\rm Par}\nolimits_{\frac{1}{r}}(X,D)$ in Theorem 3.9, the the main
result in [3], by choosing the generating sheaf
$\Xi=\bigoplus_{i=0}^{r}{\mathcal{O}}_{{\mathcal{X}}}(i{\mathcal{D}})$, $W$ is
modified semistable if and only if the corresponding rational parabolic sheaf
$(E,F_{*},\alpha_{*})$ is parabolic semistable. But from [1, Lemma 3.13], the
rational parabolic sheaf $(E,F_{*},\alpha_{*})$ is parabolic semistable with
respect to ${\mathcal{O}}_{X}(1)$ if and only if the corresponding sheaf $W$
is orbifold semistable with respect to $f^{*}{\mathcal{O}}_{X}(1)$. From
Proposition 3.6, if $W$ is strongly semistable, then

$\Delta(W)\geq 0.$

Thus we have

######  Theorem 3.10.

([1]) Let $W$ be an orbifold strongly semistable torsion free coherent sheaf
on the root stack ${\mathcal{X}}$ such that its corresponding rational
parabolic sheaf $(E,F_{*},\alpha_{*})$ is parabolic strongly semistable, then
we have

$\displaystyle
c_{2}(E)+c_{1}(E)\cup\left(\sum_{i=1}^{r}\sum_{\lambda=1}^{h}\alpha_{i}\cdot
r_{i\lambda}[D_{\lambda}]\right)+\frac{1}{2}\left(\sum_{i=1}^{r}\sum_{\lambda=1}^{h}\alpha_{i}\cdot
r_{i\lambda}[D_{\lambda}]\right)^{2}$
$\displaystyle-\sum_{i=1}^{r}\sum_{\lambda=1}^{h}\left(\frac{\alpha_{i}^{2}\cdot
r_{i\lambda}\cdot D_{\lambda}\cdot D_{\lambda}}{2}+\alpha_{i}\cdot
d_{i\lambda}\right)$ $\displaystyle\geq\frac{\mathop{\rm
rk}(E)-1}{2\mathop{\rm
rk}(E)}\left(c_{1}(E)+\sum_{i=1}^{r}\sum_{\lambda=1}^{h}\alpha_{i}\cdot
r_{i\lambda}\cdot[D_{\lambda}]\right)^{2}$

where $r_{i\lambda}=\mathop{\rm rk}(G_{i,\lambda})$ and
$d_{i\lambda}=\deg(G_{i,\lambda})$.

###### Proof.

The Chern class formula $c_{2}(W),c_{1}(W)$ are calculated in [1, §4]. Plug
these into the inequality $\Delta(W)\geq 0$ we get the formula in the theorem.
∎

######  Remark 3.11.

[1] proves that the orbifold semistability of $W$ is equivalent to the
parabolic semistability of the corresponding parabolic sheaf
$(E,F_{*},\alpha_{*})$, and the parabolic semistability is equivalent to the
modified semistability of $W$ again for the generating sheaf
$\Xi=\oplus_{i=0}^{r}{\mathcal{O}}_{{\mathcal{X}}}({\mathcal{D}}^{\frac{i}{r}})$.

##  Appendix A Higher dimension case

In the appendix we generalize Langer’s argument to higher dimension case. For
simplicity of the calculation of modified slopes, we restrict to a special
case of smooth Deligne-Mumford stacks ${\mathcal{X}}=[Z/G]$ which is a
quotient stack such that the action of $G$ is diagonalizable. We still let the
generating sheaf $\Xi$ on ${\mathcal{X}}$ satisfying Condition 3.1. Still let
$d=\dim({\mathcal{X}})$ be the dimension of ${\mathcal{X}}$.

Let $F:{\mathcal{X}}\to{\mathcal{X}}$ be the absolute Frobenius map and it is
identity in characteristic zero. Note that a coherent sheaf $E$ on
${\mathcal{X}}$ is strongly modified slope semistable if and only if it is
semistable under Frobenius pullbacks. §2.3 implies that $L_{\Xi,\max}(E)$ and
$L_{\Xi,\min}(E)$ are well defined rational numbers. Let
$L\in\operatorname{Pic}({\mathcal{X}})$ be a $\pi$-ample line bundle on
${\mathcal{X}}$ such that $\bigoplus_{k=1}^{m-1}L^{\otimes k}$ is a generating
sheaf. Recall that $\pi:{\mathcal{X}}\to X$ is the map to its coarse moduli
space. We choose a nef divisor $A$ on the scheme $X$ and let:

$\beta_{\mathop{\rm rk}}:=\beta_{\mathop{\rm
rk}(E)}(A,H)=\left(\frac{\mathop{\rm rk}(E)\cdot(\mathop{\rm
rk}(E)-1)(\max_{1\leq k\leq m-1}(\mu_{\Xi}(A\otimes
L^{k})))}{p-1}\right)^{2}.$

We state several theorems generalizing Langer [13, §3].

######  Theorem A.1.

Let $D_{1}$ be a very ample divisor on $X$ and
${\mathcal{D}}_{1}:=\pi^{-1}(D_{1})$ and ${\mathcal{D}}=\pi^{-1}(D)$ for $D$ a
general element $D\in|D_{1}|$ . If the restriction of a coherent sheaf $E$ on
${\mathcal{X}}$ to ${\mathcal{D}}$ is not modified slope semistable with
respect to $H|_{D}$ and $\Xi|_{D}$, then let $\mu_{\Xi,i},r_{i}$ denote the
modified slopes and ranks respectively in the Harder-Narasimhan filtration of
$E|_{D}$, we have

(A.0.1) $\sum_{i<j}r_{i}r_{j}(\mu_{\Xi,i}-\mu_{\Xi,j})^{2}\leq
H^{d}\Delta(E)+2\mathop{\rm
rk}(E)^{2}(L_{\Xi,\max}(E)-\mu_{\Xi}(E),\mu_{\Xi}(E)-L_{\Xi,\min}(E)).$

######  Theorem A.2.

If a torsion free sheaf $E$ on ${\mathcal{X}}$ is strongly modified slope
semistable, we have

$\Delta(E)\cdot H^{d-2}\geq 0.$

######  Theorem A.3.

If a torsion free sheaf $E$ on ${\mathcal{X}}$ is just modified slope
semistable, then we have

$H^{d}\cdot\Delta(E)\cdot H^{d-2}+\mathop{\rm rk}(\Xi)^{2}\beta_{\mathop{\rm
rk}(E)}\geq 0.$

Before stating the last theorem, we introduce some notations. First for
torsion free sheaves $G^{\prime},G$ on ${\mathcal{X}}$, we set

$\xi^{\Xi}_{G^{\prime},G}=\frac{c_{1}(G^{\prime})}{\mathop{\rm
rk}(\Xi)\mathop{\rm rk}(G^{\prime})}-\frac{c_{1}(G)}{\mathop{\rm
rk}(\Xi)\mathop{\rm rk}(G)}.$

We also set

$K^{+}:=\\{D\in\text{Num}(X)|D^{2}H^{d-2}>0,DH^{d-1}\geq
0\text{\leavevmode\nobreak\ for all nef\leavevmode\nobreak\ }H\\}.$

where $\text{Num}(X)=\operatorname{Pic}(X)\otimes{\mathbb{R}}/\sim$ and $\sim$
is an equivalence relation meaning $L_{1}\sim L_{2}$ if and only if
$L_{1}AH^{d-2}=L_{2}AH^{d-2}$ for all divisors $A$ on $X$.

######  Theorem A.4.

If we have

$H^{d}\cdot\Delta(E)\cdot H^{d-2}+\mathop{\rm rk}(\Xi)^{2}\beta_{\mathop{\rm
rk}(E)}<0,$

then there exists a saturated subsheaf $E^{\prime}\subset E$ such that
$\xi^{\Xi}_{E^{\prime},E}\in K^{+}$.

We prove these theorems by induction on the rank $\mathop{\rm rk}(E)$, and
following Langer’s method. We only state the parts of the proof which are
different to Langer’s method in smooth case and refer to [13, §3] for detailed
arguments in the proof which is the same as Langer. For the induction process,
let $\operatorname{Thm}^{i}(\mathop{\rm rk})$ represent the statement that
Theorem 4.i holds for ranks $\leq\mathop{\rm rk}$ for $i=1,2,3,4$ and
$\operatorname{Thm}^{5}(\mathop{\rm rk})$ represents that Theorem A.2 holds
for $\mathop{\rm rk}(E)\leq\mathop{\rm rk}$.

###  A.1. $\operatorname{Thm}^{1}(\mathop{\rm rk})$ implies
$\operatorname{Thm}^{5}(\mathop{\rm rk})$

Suppose that the torsion free sheaf $E$ is strongly modified slope semistable
with respect to $(H,\Xi)$, and $\Delta(E)\cdot H^{d-2}<0$. We have
$L_{\Xi,\max}(E)=L_{\Xi,\min}=\mu_{\Xi}(E)$. Theorem
$\operatorname{Thm}^{1}(\mathop{\rm rk})$ implies that the restriction of $E$
to $H$ is still modified slope semistable. Since $E$ is strongly modified
slope semistable, $(F^{k})^{*}E$ is also strongly modified slope semistable,
and its restriction to a very general element in $|H|$ is strongly modified
slope semistable. Therefore by induction the restriction of $(F^{k})^{*}E$ to
a very general element in $H_{1}\cap\cdots\cap H_{d-1}$ for
$H_{1},\cdots,H_{d-1}\in|H|$ is strongly modified slope semistable. Therefore
we are reduced to the two dimensional Deligne-Mumford stack case. Then this is
Theorem 3.1.

###  A.2. $\operatorname{Thm}^{5}(\mathop{\rm rk})$ implies
$\operatorname{Thm}^{3}(\mathop{\rm rk})$

First note that in this case,

$\beta_{\mathop{\rm rk}(E)}=\left(\frac{\mathop{\rm rk}(E)\cdot(\mathop{\rm
rk}(E)-1)(\max_{1\leq k\leq m-1}(\mu_{\Xi}(A\otimes
L^{k})))}{p-1}\right)^{2}.$

Our polarization is $(H,\Xi)$, we first have the following inequality:

(A.2.1) $H^{d}\cdot\Delta(E)H^{d-2}+\mathop{\rm rk}(E)^{2}\mathop{\rm
rk}(\Xi)^{2}(L_{\Xi,\max}(E)-\mu_{\Xi}(E))(\mu_{\Xi}(E)-L_{\Xi,\min}(E))\geq
0$

To prove this inequality, first from the finite property fdHPN in §2.4 there
exists a positive integer $k$ such that all the quotients in the Harder-
Narasimhan filtration of $(F^{k})^{*}E$ are strongly modified slope
semistable. Consider the Harder-Narasimhan filtration

$0=E_{0}\subset E_{1}\subset\cdots\subset E_{m}=(F^{k})^{*}E$

and let $F_{i}=E_{i}/E_{i-1}$, $r_{i}=\mathop{\rm rk}(F_{i})$,
$\mu_{i}=\mu_{\Xi}(F_{i})$. The Hodge index theorem (holds for smooth Deligne-
Mumford stacks) implies that

$\displaystyle\frac{\Delta((F^{k})^{*}E)H^{d-2}}{\mathop{\rm rk}(E)}$
$\displaystyle=\sum_{i}\frac{\Delta(F_{i})H^{d-2}}{r_{i}}-\frac{1}{\mathop{\rm
rk}(E)}\sum_{i<j}r_{i}r_{j}\left(\frac{c_{1}(F_{i})}{r_{i}}-\frac{c_{1}(F_{j})}{r_{j}}\right)^{2}H^{d-2}$
$\displaystyle\geq\sum_{i}\frac{\Delta(F_{i})H^{d-2}}{r_{i}}-\frac{\mathop{\rm
rk}(\Xi)^{2}}{H^{d}\mathop{\rm
rk}(E)}\sum_{i<j}r_{i}r_{j}(\mu_{i}-\mu_{j})^{2}$

$\operatorname{Thm}^{5}(\mathop{\rm rk})$ implies that
$\Delta(F_{i})H^{d-2}\geq 0$. Therefore by [13, Lemma 1.4], we have

$\frac{H^{d}\cdot\Delta(E)H^{d-2}}{\mathop{\rm rk}(E)}\geq\\\ -\mathop{\rm
rk}(E)\mathop{\rm
rk}(\Xi)^{2}\left(\mu_{\Xi,\max}((F^{k})^{*}E)-\mu_{\Xi}((F^{k})^{*}E)\right)\left(\mu_{\Xi}((F^{k})^{*}E)-\mu_{\Xi,\min}((F^{k})^{*}E)\right)$

Both sides are divided by $p^{2k}$, we get:

$H^{d}\cdot\Delta(E)H^{d-2}+\mathop{\rm rk}(E)^{2}\mathop{\rm
rk}(\Xi)^{2}(L_{\Xi,\max}(E)-\mu_{\Xi}(E))(\mu_{\Xi}(E)-L_{\Xi,\min}(E))\geq
0.$

It is ready to prove $\operatorname{Thm}^{3}(\mathop{\rm rk})$. Suppose that
$E$ is just modified slope semistable. We aim to use (A.2.1) and Corollary
2.9. The method is the same as in [13, §3.6], and we take an ample divisor $D$
on $X$ and set $H(t)=H+tD$. Similar method shows that the Harder-Narasimhan
filtration of $E$ with respect to $(H(t),\Xi)$ is independent of $t$ when $t$
is positively small. Let $0=E_{0}\subset E_{1}\subset\cdots\subset E_{m}=E$ be
the Harder-Narasimhan filtration with respect to $(H(t),\Xi)$. We have (since
$E$ is modified slope semistable)

$\mu_{\Xi,H}(E)\geq\mu_{\Xi,H}(E_{1})=\lim_{t\to
0}\mu_{\Xi,H(t)}(E_{1})\geq\lim_{t\to 0}\mu_{\Xi,H(t)}(E)=\mu_{\Xi,H}(E).$

Hence

$\lim_{t\to 0}\mu_{\Xi,\max,H(t)}(E)=\lim_{t\to
0}\mu_{\Xi,\max,H(t)}(E_{1})=\mu_{\Xi,H}(E),$

and similarly,

$\lim_{t\to 0}\mu_{\Xi,\min,H(t)}(E)=\mu_{\Xi,H}(E).$

Thus we can apply (A.2.1) and Corollary 2.9, we get the result

$H^{d}\cdot\Delta(E)H^{d-2}+\mathop{\rm rk}(\Xi)^{2}\beta_{\mathop{\rm
rk}(E)}\geq 0.$

###  A.3. $\operatorname{Thm}^{3}(\mathop{\rm rk})$ implies
$\operatorname{Thm}^{4}(\mathop{\rm rk})$

If we have the condition in $\operatorname{Thm}^{4}(\mathop{\rm rk})$, i.e.,
$H^{d}\cdot\Delta(E)H^{d-2}+\mathop{\rm rk}(\Xi)^{2}\beta_{\mathop{\rm
rk}(E)}<0$. Then from $\operatorname{Thm}^{3}(\mathop{\rm rk})$, $E$ is not
modified slope semistable. Let $E^{\prime}\subset E$ be the maximal
destabilizing subsheaf of $E$ and set $E^{\prime\prime}=E/E^{\prime}$,
$r^{\prime}=\mathop{\rm rk}(E^{\prime})$, $r^{\prime\prime}=\mathop{\rm
rk}(E^{\prime\prime})$. First we calculate:

$\frac{\Delta(E)H^{d-2}}{\mathop{\rm
rk}(E)}+\frac{rr^{\prime}\xi^{2}_{E^{\prime},E}H^{d-2}}{r^{\prime\prime}}=\frac{\Delta(E^{\prime})H^{d-2}}{r^{\prime}}+\frac{\Delta(E^{\prime\prime})H^{d-2}}{r^{\prime\prime}}.$

We also have $\mathop{\rm rk}(\Xi)^{2}\frac{\beta_{\mathop{\rm
rk}(E)}}{\mathop{\rm rk}(E)}\geq\mathop{\rm
rk}(\Xi)^{2}(\frac{\beta_{r^{\prime}}}{r^{\prime}}+\frac{\beta_{r^{\prime\prime}}}{r^{\prime\prime}})$.
Since $H^{d}\cdot\Delta(E)H^{d-2}+\mathop{\rm rk}(\Xi)^{2}\beta_{\mathop{\rm
rk}(E)}<0$, and we require $H^{d}>0$, either $\xi^{2}_{E^{\prime},E}>0$ or at
least one of the $H^{d}\cdot\Delta(E^{\prime})H^{d-2}+\mathop{\rm
rk}(\Xi)^{2}\beta_{r^{\prime}}$ and
$H^{d}\cdot\Delta(E^{\prime\prime})H^{d-2}+\mathop{\rm
rk}(\Xi)^{2}\beta_{r^{\prime\prime}}$ is negative. Therefore the same argument
as in [6, Theorem 7.3.3] gives the result.

###  A.4. $\operatorname{Thm}^{4}(\mathop{\rm rk})$ implies
$\operatorname{Thm}^{2}(\mathop{\rm rk})$

Suppose that $\Delta(E)H^{d-2}<0$. The condition in
$\operatorname{Thm}^{4}(\mathop{\rm rk})$, applying to $(F^{l})^{*}E$ (since
$E$ is strongly modified slope semistable by the condition in
$\operatorname{Thm}^{2}(\mathop{\rm rk})$), is:

$H^{d}\cdot\Delta((F^{l})^{*}E)H^{d-2}+\mathop{\rm
rk}(\Xi)^{2}\beta_{\mathop{\rm rk}(E)}<0$

which is equivalent to

$l>\frac{1}{2}\log_{p}\left(-\frac{\mathop{\rm rk}(\Xi)^{2}\beta_{\mathop{\rm
rk}(E)}}{H^{d}\cdot\Delta(E)H^{d-2}}\right).$

Then for large $l$, there exists a saturated torsion free subsheaf
$E^{\prime}\subset(F^{l})^{*}E$ such that $\xi_{E^{\prime},(F^{l})^{*}E}\in
K^{+}$. By “self-duality” property of $K^{+}$ we have
$\xi_{E^{\prime},(F^{l})^{*}E}H^{d-1}>0$, which means that the sheaf $E$ is
not strongly modified semistable, a contradiction.

###  A.5. $\operatorname{Thm}^{2}(\mathop{\rm rk}-1)$ implies
$\operatorname{Thm}^{1}(\mathop{\rm rk})$

We use $\Pi=|H|$ to denote the complete linear system, and let
$Z:=\\{(D,x)\in\Pi\times X:x\in D\\}$ be the incidence variety. Let

$p:Z\to\Pi;\quad q:Z\to X$

be the corresponding projections. For each $s\in\Pi$, let $Z_{s}$ be the
scheme theoretic fiber of $p$ over the point $s$. Consider the following
cartisian diagram:

$\textstyle{{\mathcal{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p\circ\pi}$$\scriptstyle{q}$$\scriptstyle{\pi}$$\textstyle{{\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\Pi}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{X}$

Then ${\mathcal{Z}}$ is a Deligne-Mumford stack which is given by
$\\{(\pi^{-1}(D),x):(D,x)\in\Pi\times X,x\in D\\}$. The generating sheaf $\Xi$
on ${\mathcal{X}}$ is pullback under $q$ and gives a generating sheaf
$q^{*}\Xi$ on ${\mathcal{Z}}$ which is relative to $\Pi$.

We work on the sheaf $q^{*}E$ for a torsion free sheaf $E$ on ${\mathcal{X}}$,
and let

$0=E_{0}\subset E_{1}\subset\cdots\subset E_{m}=q^{*}E$

be the relative Harder-Narasimhan filtration with respect to $p\circ\pi$. This
means that there exists an open subset $U\subset\Pi$ such that all
$F_{i}=E_{i}/E_{i-1}$ are flat over $U$ and for each $s\in U$ the fibers
$(E_{\bullet})_{s}$ is the Harder-Narasimhan filtration of
$E_{s}=q^{*}E|_{{\mathcal{Z}}_{s}}$ for ${\mathcal{Z}}_{s}=\pi^{-1}(Z_{s})$.
From the proof of in [13, §3.9], the relative Harder-Narasimhan filtration is
actually the Harder-Narasimhan filtration of $q^{*}E$ with respect to

$(p^{*}{\mathcal{O}}_{\Pi}(1)^{\dim(\Pi)}q^{*}H,q^{*}\Xi).$

By the finite property in §2.4, for the sheaf $q^{*}E$, there exists a
positive integer $k$ such that all the quotients in the Harder-Narasimhan
filtration of $(F^{k})^{*}(q^{*}E)=q^{*}((F^{k})^{*}E)$ are strongly modified
semistable. We will prove the inequality (A.0.1), and from [13, Lemma 1.5],
when applying to the polygons of the Harder-Narasimhan filtration for modified
slopes, we just prove the case that all the graded pieces $F_{i}$’s are
strongly modified slope semistable with respect to
$(p^{*}{\mathcal{O}}_{\Pi}(1)^{\dim(\Pi)}q^{*}H,q^{*}\Xi)$.

We perform the same argument as in [13, §3.9], and let $\Lambda\subset\Pi$ be
a pencil. Set $Y=p^{-1}(\Lambda)$, and
${\mathcal{Y}}=(p\circ\pi)^{-1}(\Lambda)\subset{\mathcal{Z}}$. Since
$q|_{Y}:Y\to X$ is the blow up of $X$ along the base locus $B$ of $\Lambda$,
we can view $q|_{{\mathcal{Y}}}:{\mathcal{Y}}\to{\mathcal{X}}$ to be the
stacky blow up of ${\mathcal{X}}$ along the locus ${\mathcal{B}}=\pi^{-1}(B)$.
If $d\geq 3$, then $B$ is a smooth connected variety. So ${\mathcal{B}}$ is a
smooth connected substack, and there is only one exceptional divisor
${\mathcal{N}}$ for $q|_{{\mathcal{Y}}}$. We write down

$c_{1}(F_{i}|_{Y})=q|_{{\mathcal{Y}}}^{*}{\mathcal{M}}_{i}+b_{i}{\mathcal{N}}$

where ${\mathcal{M}}_{i}$ are divisors on ${\mathcal{X}}$ which are pullbacks
of divisors $M_{i}$ on $X$ and $b_{i}$ are rational numbers. If the dimension
$d=2$, then $B$ consists of $N=H^{d}$ distinct points and ${\mathcal{B}}$
consists of $N$ distinct stacky points. Let
${\mathcal{N}}_{1},\cdots,{\mathcal{N}}_{N}$ be the exceptional divisors of
$q|_{{\mathcal{Y}}}$. There exist rational numbers $b_{ij}$ and divisors
${\mathcal{M}}_{i}$ such that

$c_{1}(F_{i}|_{Y})=q|_{{\mathcal{Y}}}^{*}{\mathcal{M}}_{i}+\sum_{j}b_{ij}{\mathcal{N}}_{j}.$

Let $b_{i}=(\sum_{j}b_{ij})/N$. We have

$\mu_{\Xi,i}=\frac{c_{1}(F_{i}|_{Y})p^{*}{\mathcal{O}}_{\Pi}(1)q^{*}H^{d-2}}{r_{i}\mathop{\rm
rk}(\Xi)}=\frac{{\mathcal{M}}_{i}\cdot H^{d-1}+b_{i}N}{r_{i}\mathop{\rm
rk}(\Xi)}.$

$\operatorname{Thm}^{2}(\mathop{\rm rk}-1)$ implies that
$\Delta(F_{j}|_{Y})p^{*}{\mathcal{O}}_{\Pi}(1)q^{*}H^{d-2}\geq 0$ for every
$j$. We calculate

$\displaystyle\frac{N\Delta(E)q|_{{\mathcal{Y}}}^{*}H^{d-2}}{\mathop{\rm
rk}(E)}$
$\displaystyle=\sum_{i}\frac{N\Delta(F_{i}|_{{\mathcal{Y}}})q|_{{\mathcal{Y}}}^{*}H^{d-2}}{r_{i}}-\frac{N}{\mathop{\rm
rk}(E)}\sum_{i<j}r_{i}r_{j}\left(\frac{c_{1}(F_{i}|_{{\mathcal{Y}}})}{r_{i}}-\frac{c_{1}(F_{j}|_{{\mathcal{Y}}})}{r_{j}}\right)^{2}\cdot
q|_{{\mathcal{Y}}}^{*}H^{d-2}$ $\displaystyle\geq\frac{N}{\mathop{\rm
rk}(E)}\sum_{i<j}r_{i}r_{j}\left(N\left(\frac{b_{i}}{r_{i}}-\frac{b_{j}}{r_{j}}\right)^{2}-\left(\frac{{\mathcal{M}}_{i}}{r_{i}}-\frac{{\mathcal{M}}_{j}}{r_{j}}\right)^{2}H^{d-2}\right)$
$\displaystyle\geq\frac{1}{\mathop{\rm
rk}(E)}\sum_{i<j}r_{i}r_{j}\left((N)^{2}\left(\frac{b_{i}}{r_{i}}-\frac{b_{j}}{r_{j}}\right)^{2}-\left(\frac{{\mathcal{M}}_{i}H^{d-1}}{r_{i}}-\frac{{\mathcal{M}}_{j}H^{d-1}}{r_{j}}\right)^{2}\right).$

The last inequality is from Hodge index theorem for smooth Deligne-Mumford
stacks, and from the slope $\mu_{\Xi,i}$, the last expression above gives

$2\sum_{i}Nb_{i}\mu_{\Xi,i}-\frac{1}{\mathop{\rm
rk}(E)}\sum_{i<j}r_{i}r_{j}(\mu_{\Xi,i}-\mu_{\Xi,j})^{2}.$

To prove the claim, first
$(q|_{{\mathcal{Y}}})_{*}(E_{i}|_{{\mathcal{Y}}})\subset E$ implies that

$\frac{\sum_{j\leq i}{\mathcal{M}}_{j}H^{d-1}}{\mathop{\rm rk}(\Xi)\sum_{j\leq
i}r_{j}}\leq\mu_{\Xi,\max}(E)$

which gives the inequality:

(A.5.1) $\sum_{j\leq i}b_{j}N\geq\sum_{j\leq i}\mathop{\rm
rk}(\Xi)r_{j}(\mu_{\Xi,j}-\mu_{\Xi,\max}(E))$

Therefore

$\displaystyle\sum_{i}Nb_{i}\mu_{\Xi,i}$
$\displaystyle=\sum_{i}\left(\sum_{i<j}Nb_{j}\right)(\mu_{\Xi,i}-\mu_{\Xi,i+1})$
$\displaystyle\geq\sum_{i}\left(\sum_{j\leq i}\mathop{\rm
rk}(\Xi)r_{j}(\mu_{\Xi,j}-\mu_{\Xi,\max}(E))\right)(\mu_{\Xi,i}-\mu_{\Xi,i+1})$
$\displaystyle=\mathop{\rm
rk}(\Xi)\cdot\sum_{i<j}\frac{r_{i}r_{j}}{\mathop{\rm
rk}(E)}(\mu_{\Xi,i}-\mu_{\Xi,j})^{2}+\mathop{\rm
rk}(E)(\mu_{\Xi}(E)-\mu_{\Xi,\max}(E))(\mu_{\Xi}(E)-\mu_{\Xi,\min}(E)).$

So we get:

$\frac{N\Delta(E)q|_{{\mathcal{Y}}}^{*}H^{d-2}}{\mathop{\rm
rk}(E)}\geq\sum_{i<j}\frac{2\mathop{\rm rk}(\Xi)-1}{\mathop{\rm
rk}(E)}r_{i}r_{j}(\mu_{\Xi,i}-\mu_{\Xi,j})^{2}+2\mathop{\rm
rk}(E)(\mu_{\Xi}(E)-\mu_{\Xi,\max}(E))(\mu_{\Xi}(E)-\mu_{\Xi,\min}(E)).$

###  A.6. Restriction theorems

We generalize the restriction theorem of Langer to smooth Deligne-Mumford
stacks in higher dimensions. We give a general statement for the (A.2.1).
Recall from Corollary 2.9,

$\alpha_{\Xi}(E):=\max(L_{\Xi,\max}(E)-\mu_{\Xi,\max}(E),\mu_{\Xi,\min}(E)-L_{\Xi,\min}(E)).$

Let $L\in\operatorname{Pic}({\mathcal{X}})$ be a $\pi$-ample line bundle on
${\mathcal{X}}$ such that there exists $m\in{\mathbb{Z}}_{>0}$ making
$\bigoplus_{i=1}^{m-1}L^{\otimes i}$ a generating sheaf. Let $A$ be a nef
divisor for $X$, such that $\pi_{*}(T_{X}\otimes\sum_{k=1}^{m-1}(L^{k}))(A)$
is globally generated. Then we have

$\alpha_{\Xi}(E)\leq\frac{\mathop{\rm rk}(E)-1}{p-1}(\max_{1\leq k\leq
m-1}\\{\mu_{\Xi}(A\otimes(L^{k}))\\}).$

######  Theorem A.5.

Let $E$ be a torsion free sheaf on a smooth Deligne-Mumford stack
${\mathcal{X}}$. Then we have

(A.6.1) $H^{d}\cdot\Delta(E)H^{d-2}+\mathop{\rm rk}(E)^{2}\mathop{\rm
rk}(\Xi)^{2}(L_{\Xi,\max}(E)-\mu_{\Xi}(E))(\mu_{\Xi}(E)-L_{\Xi,\min}(E))\geq
0$

and

(A.6.2) $H^{d}\cdot\Delta(E)H^{d-2}+\mathop{\rm rk}(E)^{2}\mathop{\rm
rk}(\Xi)^{2}(\mu_{\Xi,\max}(E)-\mu_{\Xi}(E))(\mu_{\Xi}(E)-\mu_{\Xi,\min}(E))\geq
0$

###### Proof.

The proof of (A.6.1) is the same as in Claim (A.2.1). The proof of formula
(A.6.2) is the same as [13, Theorem 5.1]. ∎

######  Theorem A.6.

Let $E$ be a torsion free sheaf of rank $\mathop{\rm rk}(E)\geq 2$ on a smooth
Deligne-Mumford stack ${\mathcal{X}}$. Suppose that $E$ is slope modified
stable with respect to $(\Xi,{\mathcal{O}}_{X}(1)=H)$. Let $D\subset|mH|$ be a
normal divisor such that $E|_{{\mathcal{D}}}$ has no torsion where
${\mathcal{D}}=\pi^{-1}(D))$. If

$m>\lfloor\frac{\mathop{\rm rk}(E)-1}{\mathop{\rm
rk}(E)}\Delta(E)H^{d-2}+\frac{1}{H^{d}\mathop{\rm rk}(E)(\mathop{\rm
rk}(E)-1)}+\frac{(\mathop{\rm rk}(E)-1)\beta_{\mathop{\rm
rk}}}{H^{d}\mathop{\rm rk}(E)}\rfloor$

then $E|_{{\mathcal{D}}}$ is slope modified stable with respect to
$(\Xi|_{{\mathcal{D}}},H|_{{\mathcal{D}}})$.

###### Proof.

The proof is the same as [13, Theorem 5.2]. ∎

##  Appendix B Bogomolov’s inequality for Higgs sheaves

By Yunfeng Jiang, Promit Kundu and Hao Max Sun

Let $k$ be an algebraically closed field of characteristic $p\geq 0$ and
$\mathcal{X}$ be a smooth tame projective Deligne-Mumford stack of dimension
$d$ over $k$ with coarse moduli space $\pi:\mathcal{X}\rightarrow X$. Let
$\Xi$ be a generating sheaf on $\mathcal{X}$ satisfying Condition $\star$ in
3.1, and let $H$ an ample divisor on $X$.

######  Definition B.1.

A Higgs sheaf $(E,\theta)$ is a pair consisting of a coherent sheaf
$E\in\mbox{Coh}(\mathcal{X})$ and an $\mathcal{O}_{\mathcal{X}}$-homomorphism
$\theta:E\rightarrow E\otimes\Omega_{\mathcal{X}}$ satisfying the
integrability condition $\theta\wedge\theta=0$. We say a Higgs sheaf
$(E,\theta)$ a system of Hodge sheaves if there is a decomposition $E=\oplus
E^{i}$ such that $\theta:E^{i}\rightarrow
E^{i-1}\otimes_{\mathcal{O}_{\mathcal{X}}}\Omega_{\mathcal{X}}$

  1. (1)

We say that $(E,\theta)$ is slope semistable if
$\mu_{\Xi}(E^{\prime})\leq\mu_{\Xi}(E)$ for every Higgs subsheaf
$(E^{\prime},\theta^{\prime})$ of $(E,\theta)$.

  2. (2)

A system of Hodge sheaves $(E,\theta)$ is slope semistable if the inequality
$\mu_{\Xi}(E^{\prime})\leq\mu_{\Xi}(E)$ is satisfied for every subsystem of
Hodge sheaves $(E^{\prime},\theta^{\prime})$ of $(E,\theta)$.

We recall the main results of Ogus and Vologodsky [22], where the theory is
for schemes, but in étale topology it works for Deligne-Mumford stacks.

Assume that $p>0$. Let $S$ be a scheme over $k$ and $f:\mathcal{X}\rightarrow
S$ be a morphism of stacks over $k$. A lifting of $\mathcal{X}/S$ modulo
$p^{2}$ is a morphism
$\widetilde{f}:\widetilde{\mathcal{X}}\rightarrow\widetilde{S}$ of flat
$\mathbb{Z}/p^{2}\mathbb{Z}$-stacks such that $f$ is the base change of
$\widetilde{f}$ by the closed embedding $S\rightarrow\widetilde{S}$ defined by
$p$. Let $\mbox{MIC}_{p-1}(\mathcal{X}/S)$ be the category of
$\mathcal{O}_{\mathcal{X}}$-modules with an integrable connection whose
$p$-curvature is nilpotent of level $\leq p-1$. Let
$\mbox{HIG}_{p-1}(\mathcal{X}^{(1)}/S)$ denote the category of Higgs
$\mathcal{O}_{\mathcal{X}^{(1)}}$-modules with a nilpotent Higgs field of
level $\leq p-1$. We have the following theorem of Ogus and Vologodsky ([22,
Theorem 2.8])

######  Theorem B.2.

If $f:\mathcal{X}\rightarrow S$ is a smooth morphism with a lifting
$\widetilde{\mathcal{X}}^{(1)}\rightarrow\widetilde{S}$ of
$\mathcal{X}^{(1)}\rightarrow S$ modulo $p^{2}$, then the Cartier transform

$C_{\mathcal{X}/S}:\mbox{MIC}_{p-1}(\mathcal{X}/S)\rightarrow\mbox{HIG}_{p-1}(\mathcal{X}^{(1)}/S)$

defines an equivalence of categories with quasi-inverse

$C^{-1}_{\mathcal{X}/S}:\mbox{HIG}_{p-1}(\mathcal{X}^{(1)}/S)\rightarrow\mbox{MIC}_{p-1}(\mathcal{X}/S).$

######  Lemma B.3.

Let $(E,\theta)\in\mbox{HIG}_{p-1}(\mathcal{X}^{(1)}/S)$. Then we have
$[C^{-1}_{\mathcal{X}/S}(E)]=F_{g}^{*}[E]$, where $[\cdot]$ denotes the class
of a coherent sheaf in the Grothendieck group $K_{0}(\mathcal{X})$.

###### Proof.

See [14, Lemma 2]. ∎

######  Corollary B.4.

Assume $S=\mathop{\rm Spec}\nolimits k$, and let
$(E,\theta)\in\mbox{HIG}_{p-1}(\mathcal{X}^{(1)}/S)$. Then $(E,\theta)$ is
slope semistable with respect to $(H,\Xi)$ iff $C^{-1}_{\mathcal{X}/S}(E)$ is
slope $\nabla$-semistable with respect to $(F_{g}^{*}H,F_{g}^{*}\Xi)$.

###### Proof.

The proof is the same as that of [14, Corollary 1]. ∎

######  Lemma B.5.

Let $(E,\theta)$ be a torsion free slope semistable Higgs sheaf on
$\mathcal{X}$. Then there exists an $\mathbb{A}^{1}$-flat family of Higgs
sheaves $(\widetilde{E},\widetilde{\theta})$ on
$\mathcal{X}\times\mathbb{A}^{1}$ such that the restriction
$(\widetilde{E}_{t},\widetilde{\theta}_{t})$ to the fiber over any closed
point $t\in\mathbb{A}^{1}$ is isomorphic to $(E,\theta)$ and
$(E_{0},\theta_{0})$ is a slope semistable system of Hodge sheaves.

###### Proof.

See [15, Corollary 5.7]. ∎

######  Proposition B.6.

Let $V$ be a torsion free sheaf on $\mathcal{X}$, then

$H^{d-2}\Delta(V)\geq-\frac{(\mathop{\rm rk}V\mathop{\rm
rk}\Xi)^{2}}{H^{d}}(L_{\Xi,\max}(V)-\mu_{\Xi}(V))(\mu_{\Xi}(V)-L_{\Xi,\min}(V))$

###### Proof.

The proposition follows from Theorem A.2 by the same arguments as in the proof
of Theorem A.3. ∎

######  Theorem B.7.

Assume $p=0$. Let $(E,\theta)$ be a slope semistable Higgs sheaf with respect
to $(H,\Xi)$. Then we have $H^{d-2}\Delta(E)\geq 0$.

###### Proof.

Deforming $(E,\theta)$ to a system of Hodge sheaves (see Lemma B.5) we can
assume that $(E,\theta)$ is nilpotent. Now we use the standard reduction to
positive characteristic technique, which we recall for the convenience of the
reader (see [19, section 2]). There exists a finitely generated
$\mathbb{Z}$-algebra $R\subset k$ and a tame smooth Deligne-Mumford stack
$\widetilde{\mathcal{X}}\rightarrow S=\mathop{\rm Spec}\nolimits R$ such that
$\mathcal{X}=\widetilde{\mathcal{X}}\times_{S}\mathop{\rm Spec}\nolimits k$.
Let $\widetilde{\pi}:\widetilde{\mathcal{X}}\rightarrow\widetilde{X}$ be its
coarse moduli space. We can assume that $X=\widetilde{X}\times_{S}\mathop{\rm
Spec}\nolimits k$, $\pi$ is induced by $\widetilde{\pi}$ after base change,
and there exists an ample divisor $\widetilde{H}$ on $\widetilde{X}$ extending
$H$ and a generating sheaf $\widetilde{\Xi}$ on $\widetilde{\mathcal{X}}$
extending $\Xi$ such that its restriction to every component in
$I\widetilde{\mathcal{X}}_{1}$ is a direct sum of locally free coherent
sheaves of the same rank. We can also assume that there exists an $S$-flat
family of Higgs sheaves $(\widetilde{E},\widetilde{\theta})$ on
$\widetilde{\mathcal{X}}$ extending $(E,\theta)$.

Shrinking $S$, by openness of semistability we can assume that
$(\widetilde{E}_{s},\widetilde{\theta}_{s})$ is slope semistable with respect
to $(\widetilde{\Xi}_{s},\widetilde{H}_{s})$ for any $s\in S$. Choose a closed
point $s\in S$ such that the characteristic $q$ of the residue field $k(s)$ is
$\geq\mathop{\rm rk}E$. Then the stack
$\widetilde{\mathcal{X}}\times_{S}\mathop{\rm Spec}\nolimits(R/m_{s}^{2})$ is
a lifting of $\widetilde{\mathcal{X}}_{s}$ modulo $q^{2}$. By Corollary B.4,
one can associate to $(\widetilde{E}_{s},\widetilde{\theta}_{s})$ a slope
$\nabla$-semistable sheaf with integrable connection $(V_{s},\nabla_{s})$ with
respect to $(F_{g}^{*}\widetilde{H}_{s},F_{g}^{*}\widetilde{\Xi}_{s})$. From
Lemma B.3, it follows that

(B.0.1)
$\widetilde{H}_{s}^{d-2}\Delta(V_{s})=q^{2}\widetilde{H}_{s}^{d-2}\Delta(\widetilde{E}_{s}).$

Let $0=V_{0}\subset V_{1}\subset\cdots\subset V_{m}=V_{s}$ be the usual
Harder-Narasimhan filtration of $V_{s}$, then by [7, Lemma 2.7], the induced
morphisms
$V_{i}\rightarrow(V_{s}/V_{i})\otimes\Omega_{\widetilde{\mathcal{X}}_{s}}$ are
nonzero $O_{\widetilde{\mathcal{X}}_{s}}$-morphisms. Take a nef divisor $A$ on
$\widetilde{X}_{s}$ such that
$\pi_{*}(T_{\widetilde{\mathcal{X}}_{s}}\otimes\widetilde{\Xi}_{s})(A)$ is
globally generated. From [7, Proposition 2.10 and Corollary 2.11], it follows
that

$\mu_{\widetilde{\Xi}_{s},\max}(V_{s})-\mu_{\widetilde{\Xi}_{s},\min}(V_{s})\leq(\mathop{\rm
rk}V_{s}-1)\left(\frac{\widetilde{H}_{s}^{d-1}A}{\widetilde{H}_{s}^{d}}+M\right)$

and

$\max(L_{\widetilde{\Xi}_{s},\max}(V_{s})-\mu_{\widetilde{\Xi}_{s},\max}(V_{s}),\mu_{\widetilde{\Xi}_{s},\min}(V_{s})-L_{\widetilde{\Xi}_{s},\min}(V_{s}))\leq\frac{\mathop{\rm
rk}V_{s}-1}{q-1}\left(\frac{\widetilde{H}_{s}^{d-1}A}{\widetilde{H}_{s}^{d}}+M\right),$

for some positive constant $M$. They imply that

$(L_{\widetilde{\Xi}_{s},\max}(V_{s})-L_{\widetilde{\Xi}_{s},\min}(V_{s}))\leq\frac{\mathop{\rm
rk}V_{s}-1}{1-\frac{1}{q}}\left(\frac{\widetilde{H}_{s}^{d-1}A}{\widetilde{H}_{s}^{d}}+M\right).$

Hence Proposition B.6 gives

$q^{2}\widetilde{H}_{s}^{d-2}\Delta(\widetilde{E}_{s})=\widetilde{H}_{s}^{d-2}\Delta(V_{s})\geq-\frac{(\mathop{\rm
rk}V_{s}\mathop{\rm rk}\Xi)^{2}}{\widetilde{H}_{s}^{d}}(\frac{\mathop{\rm
rk}V_{s}-1}{1-\frac{1}{q}})^{2}\left(\frac{\widetilde{H}_{s}^{d-1}A}{\widetilde{H}_{s}^{d}}+M\right)^{2}.$

Taking sufficiently large $q$, one obtains

$H^{d-2}\Delta(E)=\widetilde{H}_{s}^{d-2}\Delta(\widetilde{E}_{s})\geq 0.$

∎

Using the same argument in [14, section 6], one can recover [4, Theorem 1.1]:

######  Theorem B.8.

Assume that $p=0$, $d=2$, and the canonical line bundle $K_{\mathcal{X}}$ is
nef, then we have

$c^{2}_{1}(T_{\mathcal{X}})\leq 3c_{2}(T_{\mathcal{X}}).$

###

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