# Boundedness of semistable sheaves

Haoyang Guo  Max Planck Institute for Mathematics, Vivatsgasse 7, Bonn, 53115,
Germany<EMAIL_ADDRESS>,  Sanal Shivaprasad  Department of
Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109,
United States<EMAIL_ADDRESS>,  Dylan Spence  Department of Mathematics,
Indiana University, Rawles Hall, 831 East 3rd Street, Bloomington, IN 47405,
United States<EMAIL_ADDRESS>and  Yueqiao Wu  Department of
Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109,
United States<EMAIL_ADDRESS>
###### Abstract.

In this expository article, we follow Langer’s work in [5] to prove the
boundedness of the moduli space of semistable torsion-free sheaves over a
projective variety, in any characteristic.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Bogomolov’s inequality
  4. 4 Restriction to hypersurfaces and Bogomolov’s inequality
  5. 5 Boundedness of torsion-free sheaves

##  1\. Introduction

The study of moduli is one of the oldest branches of algebraic geometry and
forms one of the central pillars of our modern understanding. Historically,
such questions date back to Riemann, who, in his pursuit to understand what we
now call Riemann surfaces, determined that a complex projective curve of genus
$g$ depends on exactly $3g-3$ parameters (which he referred to as ”moduli” -
hence the name). The central question of moduli theory has not evolved much;
we are interested in whether or not various algebraic objects can be
”parametrized” (in some sense) by some other algebraic object. For example,
one might be interested in forming a moduli space of algebraic curves with
some fixed genus, or in our case, a moduli space of torsion-free sheaves with
fixed Hilbert polynomial on a fixed variety $X$.

To construct the moduli space of torsion-free sheaves $\\{E_{\alpha}\\}$ with
fixed Hilbert polynomial $P$ over a given variety $X$, one of the first and
the most fundamental problems is the _boundedness_ of the collection
$\\{E_{\alpha}\\}$. This property is equivalent to the moduli space, if
exists, being a finite type scheme over the base field, and thus a reasonable
geometric object that one can work with. However it was quickly discovered
that even with fixing the Hilbert polynomial, certain pathological examples
prevent such a moduli space from being well-behaved. Mumford, in the case of
curves, introduced the notion of _semistability_ for a torsion-free sheaf as a
solution for this problem, and to demonstrate its efficacy, he showed that the
collection of semistable vector bundles of a fixed rank and degree on a fixed
curve is bounded. As the notion of semistability can be generalized to
torsion-free sheaves over general projective varieties, it is natural to ask
about the boundedness for semistable sheaves on higher dimensional varieties.
Following Langer’s work in [5], the goal of our article is to give a positive
answer to the boundedness problem. Precisely, we prove the following:

######  Theorem 1.

Let $\mathcal{M}^{P}(X)$ be the moduli space of semistable torsion-free
sheaves with fixed Hilbert polynomial $P$. Then it is of finite-type.

Let us now discuss the idea of the proof. Given any (semistable) torsion-free
sheaf $E$ in our collection, we write $E|_{H}$ for the restriction of $E$ to a
general hypersurface $H$, and $\mu_{\max}(E|_{H})$ for the maximal slope in
the Harder-Narasimhan polygon of $E|_{H}$.

Our most important tool in the proof is Kleiman’s criterion (Theorem 14) and
an accompanying inequality (Lemma 15), which, when combined, say that to prove
the boundedness of the family of torsion-free sheaves $\mathcal{M}^{P}(X)$, it
suffices to give a uniform bound of $\mu_{\max}(E|_{H})$. Because of this
fact, most of the work in proving boundedness is producing such a uniform
bound, which we obtain in Corollary 44 and Theorem 45. Explicitly, it can be
understood as follows:

######  Theorem 2.

Let $E$ be a torsion-free sheaf over $X$, with Hilbert polynomial $P$, and
denote the maximal slope of its Harder-Narasimhan polygon be $\mu_{0}$. Then
we have

$\mu_{\max}(E|_{H})\leq C_{1}(P)+C_{2}(P)\mu_{0},$

where $H$ is a general hypersurface of $X$, and $C_{i}(P)$ are constants
determined only by $P$ and $X$.

The key is that this result allows us to bound the slopes $\mu_{\max}(E|_{H})$
of the restrictions $E|_{H}$ by the slopes of the original $E$, plus extra
terms and coefficients that are controlled only by the Hilbert polynomial of
$E$.

The major contents of our article are then devoted to obtaining the inequality
above, following [5, Section 3]. The strategy can be understood as a double
induction on two different results. The first one, Theorem  (Res($r$))., is a
special case of Theorem 2 above, considering the invariants before and after
the restriction of $E$ to a hypersurface, where $E$ is torsion-free of rank
$\leq r$ (c.f. [5, Theorem 3.1]). It relates the slopes of the Harder-
Narasimhan filtration of the restriction $E|_{H}$ to the discriminant and
Harder-Narasimhan filtration of $E$ itself. The second result is a collection
of several various numerical inequalities, which we refer to as _Bogomolov’s
inequalities_ (Theorem  (BI($r$))., c.f. [5, Theorem 3.2-3.4, and
$T^{5}(r)$]), due to their similarity with the well-known inequality in
characteristic zero of the same name.

Up to an extra term, it says that the discriminant $\Delta(E)$ of $E$ is non-
negative when the torsion-free sheaf $E$ is (strongly) semistable, where $E$
is of rank $\leq r$. Using the notations above, Langer’s induction schema can
be summarized as the following two implications (c.f. Section 4):

(1.1) $\begin{cases}BI(r)\Rightarrow Res(r);&\\\ Res(r)+BI(r)\Rightarrow
BI(r+1).&\end{cases}$

We note that both results are true automatically for rank $r=1$. In this way,
the two technical results are proved together, thus so is the inequality in
Theorem 2 and the boundedness in Theorem 1.

We also mention that in the positive characteristic case, a Frobenius pullback
of a semistable sheaf may no longer be semistable. So we define semistable
sheaves whose Frobenius pullbacks are all semistable as being _strongly
semistable_. A priori, for a given sheaf, the Harder-Narasimhan filtrations of
its Frobenius pullbacks could be unrelated to each other. One of Langer’s key
observations is that their differences are controllable: the Harder-Narasimhan
filtrations eventually stabilize with respect to the Frobenius (Theorem 29,
[5, Theorem 2.7]). This allows us to pass between semistable and strongly
semistable sheaves in the proof.

Lastly, we briefly mention the history preceding Langer’s result; a more
detailed introduction can be found in [5]. Theorem 1 was known in
characteristic zero, proven by Barth, Spindler, Maruyama, Forster, Hirschowitz
and Schneider. In positive characteristic however, only the cases for curves
and surfaces were known. At that time, it was known that understanding the
numerical quantities associated to $E$ before and after restricting $E$ to a
hypersurface (often called Grauet–Mülich type results) should allow one to
prove boundedness for a given family, but prior to [5] only a coarser result
by Mehta and Ramanathan was available. On the other hand, Bogomolov showed the
non-negativity of the discriminant of $E$ whenever $E$ is semistable, assuming
$X$ is in characteristic zero. It was unknown as well whether the result is
completely true in positive characteristic, but it was clear that some version
would be needed. Thus Langer’s key contributions were in developing positive
characteristic versions of the above, and also in combining them in a fruitful
way.

### Leitfaden of the article.

We start with Section 2 on necessary preliminaries. The section contains three
subsections, including basics on the stability of coherent sheaves and
additional results in positive characteristic. Moreover, as the proof of main
theorems requires working with polarizations consisting of nef divisors, we
also include a subsection on how to approximate the nef polarizations by ample
polarizations. In Section 3 we introduce several forms of Bogomolov’s
inequalities in positive characteristic, and show that they are equivalent in
Theorem  (BI($r$)).. For the reader’s convenience, this section corresponds to
the implications $T^{5}(r)\Rightarrow T^{3}(r)\Rightarrow T^{4}(r)\Rightarrow
T^{2}(r)\Rightarrow T^{5}(r)$ in [5, $\S$3.6 - $\S$3.8]. In Section 4, we
prove Theorem 41, and complete the major technical part of the article by
proving the induction schema as in (2). This corresponds to [5, $\S$3.5,
$\S$3.9] and [4]. At last, in Section 5, we combine the main technical results
above to show the boundedness of the moduli space $\mathcal{M}^{P}(X)$, thus
finishing the proof of Theorem 1.

### Acknowledgements

We would like to express our gratitude to the organizers of the Stacks Project
Workshop for organizing the event online during the pandemic. We thank Alex
Perry for kindly guiding us through the paper during the week and beyond. We
also thank Faidon Andriopoulos for helpful discussions during the workshop,
and thank Adrian Langer for answering technical questions. At last, we thank
the referee for reading the draft carefully and proposing various comments to
help improve the article.

Haoyang Guo was partially funded by the FRG grant no. DMS-1952399 during the
writing of the project.

##  2\. Preliminaries

In this section, we provide the preliminaries for the article.

### Stability of Coherent Sheaves

In this section we recall some useful facts on the (semi)stability of coherent
sheaves. We refer to [3] for details and proofs. Let $X$ be a smooth
projective variety of dimension $n$ over an algebraically closed field. Fix
$n-1$ ample divisors $D_{1},\ldots,D_{n-1}$ on $X$. Let $E$ be a coherent
sheaf on $X$. Recall that a sheaf $E$ is said to be pure (of dimension
$d\leq\operatorname{dim}X$) if for all nontrivial subsheaves $f\subset E$,
$\operatorname{dim}(F):=\operatorname{dim}\operatorname{supp}(F)=d$.

######  Definition 3.

The _slope_ of $E$, with respect to the polarization $(D_{1},\ldots,D_{n-1})$,
is defined by

$\mu(E):=\frac{\deg(E)}{\mathrm{rk}(E)}:=\frac{c_{1}(E)\cdot D_{1}\cdots
D_{n-1}}{\mathrm{rk}(E)}$

where $\mathrm{rk}(E)$ is the rank of $E$ at the generic point. If
$\mathrm{rk}(E)=0$, then the $\mu(E):=\infty$.

As indicated, the slope of a coherent sheaf does depend on the choice of ample
(or nef) divisors defining a polarization. For the purpose of clarity, we will
not reference this choice of polarization when discussing slope unless there
is a serious risk of confusion.

######  Definition 4.

We say that $E$ is _(semi)stable_ if $E$ is pure and for any nonzero proper
subsheaf $F\subset E$, $\mu(F)<\mu(E)$ (resp. $\mu(F)\leq\mu(E)$).

This notion of (semi)stability is more generally known as
slope-(semi)stability or $\mu$-(semi)stability. An equivalent formulation of
the definition can also be stated in terms of quotient sheaves.

######  Lemma 5.

[3, Proposition 1.2.6] Let $E$ be a coherent sheaf. Then $E$ is semistable if
and only if for all the quotient sheaves $E\to G$, $\mu(E)\leq\mu(G).$

This formulation of the definition gives a quick proof of the following very
useful fact.

######  Lemma 6.

[3, Proposition 1.2.7] Let $E_{1}$ and $E_{2}$ be semistable sheaves with
slopes $\mu_{1}$ and $\mu_{2}$ such that $\mu_{1}>\mu_{2}$. Then,
$\mathrm{Hom}(E_{1},E_{2})=0$.

The definition of semistability also behaves well with regards to exact
sequences, as the following two lemmas indicate.

######  Lemma 7.

Let $E^{\prime}\to E\to E^{\prime\prime}$ be a short exact sequence of
torsion-free coherent sheaves, and let $\mu^{\prime},\mu,\mu^{\prime\prime}$
and $r^{\prime},r,r^{\prime\prime}$ be the slopes and the ranks of
$E^{\prime},E,E^{\prime\prime}$ respectively. Then,
$r\mu=r^{\prime}\mu^{\prime}+r^{\prime\prime}\mu^{\prime\prime}$. In
particular if the slopes of two of $E^{\prime},E,E^{\prime\prime}$ are the
same, then so is the case for the third.

######  Lemma 8.

If $E^{\prime}\to E\to E^{\prime\prime}$ is a short exact sequence and let
$\mu^{\prime},\mu,\mu^{\prime\prime}$ denote the slopes of
$E^{\prime},E,E^{\prime\prime}$ respectively, then

  * •

If $E$ is semistable and either $\mu^{\prime}=\mu$ or
$\mu^{\prime\prime}=\mu$, then $E^{\prime}$ and $E^{\prime\prime}$ are also
semistable.

  * •

If $E^{\prime}$ and $E^{\prime\prime}$ are semistable and
$\mu^{\prime}=\mu^{\prime\prime}$, then $E$ is semistable with
$\mu=\mu^{\prime}=\mu^{\prime\prime}$.

One of the more important technical tools is the Harder-Narasimhan filtration,
which is defined below.

######  Definition 9.

A _Harder-Narasimhan filtration_ for $E$ is an increasing filtration

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

such that the factors $F_{i}:=E_{i}/E_{i-1},i=1,\cdots,d$ are semistable
sheaves with slopes $\mu_{i}$ satisfying

$\mu_{\max}(E):=\mu_{1}>\mu_{2}>\cdots>\mu_{d}=:\mu_{\min}(E).$

######  Proposition 10.

[3, Theorem 1.3.4] Every torsion-free sheaf $E$ has a unique Harder-Narasimhan
filtration.

######  Remark 11.

Just like the slope, the Harder-Narasimhan filtration is also dependent on the
choice of the polarization $(D_{1},\ldots,D_{n-1})$, and could be different
with respect to polarizations.

######  Remark 12.

We should also remark here that in the construction of the Harder-Narasimhan
filtration for a torsion-free sheaf, an important step is establishing the
existence and uniqueness of a _maximal destabilizing subsheaf_. We use this
notion a few times, so we give its definition here. Given a torsion-free sheaf
$E$, then the _maximal destabilizing subsheaf_ $F\subset E$ is a semistable
coherent subsheaf such that $\mu(F)\geq\mu(G)$ for all other subsheaves
$G\subset E$, and moreover if $\mu(F)=\mu(G)$, then $F\supset G$.

Next, we wish to introduce the key tools of the paper, Kleiman’s criterion and
a related inequality. For the conclusion of Kleiman’s criterion to make sense
however, we should remind the reader of the technical property of boundedness.

######  Definition 13.

Let $M$ be a set of coherent sheaves on $X$. Then $M$ is said to be _bounded_
if there is a scheme $B$ of finite-type and a coherent sheaf $F$ on $X\times
B$ with $M\subset\\{F_{b}\,|\,b\in B\text{ closed }\\}$. Here $F_{b}$ is the
pullback of $F$ along $X\times\\{b\\}\to X\times B$.

Kleiman’s criterion provides a very convenient way of determining whether or
not a given family is bounded.

######  Theorem 14 (Kleiman’s criterion).

[3, Theorem 1.7.8] Let $\\{E_{\alpha}\\}$ be a family of coherent sheaves over
$X$ with the same Hilbert polynomial $P$. Then the family is bounded if and
only if there are constants $C_{i}$, for $i=0,\ldots,\deg(P)$, such that for
every $E_{\alpha}$ there exists an $E_{\alpha}$-regular sequence of
hyperplanes $H_{1},\ldots,H_{\deg(P)}$, satisfying

$h^{0}(E_{\alpha}|_{\cap_{j\leq i}H_{j}})\leq C_{i},\forall i.$

Here we recall that a hyperplane $s\in H^{0}(X,\mathcal{O}_{X}(1))$ is said to
be _$E$ -regular_ if the map

$E(-1)\overset{\cdot s}{\to}E$

is injective. A sequence $\\{s_{1},...,s_{l}\\}\subset
H^{0}(X,\mathcal{O}_{X}(1))$ is $E$-regular if $s_{i}$ is
$E/(s_{1},..,s_{i-1})E(-i)$-regular for all $1\leq i\leq l$.

######  Lemma 15.

[3, Lemma 3.3.2] Let $E$ be a torsion-free sheaf of rank $r$. Then for any
$E$-regular sequence of hyperplane sections $H_{1},\cdots,H_{n}$, the
following inequality holds for $i=1,\cdots,n$:

$\frac{h^{0}(X_{i},E_{i})}{r\deg(X)}\leq\frac{1}{i!}\left[\frac{\mu_{\max}(E_{1})}{\deg(X)}+i\right]_{+}^{i},$

where $X_{i}\in|H_{1}|\cap\cdots\cap|H_{n-i}|$, $E_{i}=E|_{X_{i}}$, and
$[x]_{+}=\max\\{0,x\\}$ for any real number $x$.

Combining the previous two results, provided we can get a uniform bound on
$\mu_{\max}$, will prove the boundedness for semistable sheaves with a fixed
Hilbert polynomial. To get such estimates, we will often use the following
lemma.

######  Lemma 16.

If $0\to A\to B\to C\to 0$ is an exact sequence of torsion-free sheaves, then
$\mu_{\max}(B)\leq\max\\{\mu_{\max}(A),\mu_{\max}(C)\\}$.

At the end of the subsection, we introduce the following definition that helps
visualize the Harder-Narasimhan filtration.

######  Definition 17 (Harder-Narasimhan polygon).

Consider a torsion-free sheaf $E$ on $X$ with its Harder-Narasimhan filtration
$0=E_{0}\subset\dots\subset E_{m}=E.$ Let
$p(E_{i})=(\mathrm{rk}(E_{i}),\deg(E_{i}))$. We define the _Harder-Narasimhan
polygon_ of $E$, denoted as $\mathrm{HNP}(E)$, to be the convex hull of the
points $p(E_{0}),\dots,p(E_{m})$ in $\mathbb{R}^{2}$. See Figure 1.

$p(E_{0})=(0,0)$slope = $\mu(E_{1}/E_{0})$$p(E_{1})$slope =
$\mu(E_{2}/E_{1})$$p(E_{2})$$p(E_{m})$slope = $\mu(E)$ Figure 1. An
illustration of the Harder Narasimhan Polygon

In fact, the Harder-Narasimhan polygon is a convex polygon with vertices
$p(E_{0}),\dots,p(E_{m})$, and the slope of the line segment
$\overline{p(E_{i})p(E_{i+1})}$ is $\mu(E_{i}/E_{i+1})$.

######  Remark 18.

If $F\subset E$ is a torsion-free subsheaf $F$ inside of a semistable sheaf
$E$, then the point $p(F)=(\mathrm{rk}(F),\deg(F))$ lies below the polygon
${\mathrm{HNP}}(E)$ i.e. for any $x\in[0,\mathrm{rk}(F)]$, we have

$\sup\\{y\leavevmode\nobreak\ |\leavevmode\nobreak\
(x,y)\in{\mathrm{HNP}}(F)\\}\leq\sup\\{y\leavevmode\nobreak\
|\leavevmode\nobreak\ (x,y)\in{\mathrm{HNP}}(E)\\}.$

### Approximation of nef polarizations

In this subsection, we approximate the Harder-Narasimhan filtration for a
polarization consisting of _nef divisors_ , that will be used later. As this
is a nonstandard setup and does not appear in most of the literature, we
provide the full details of the proof. We assume the existence and uniqueness
of the (absolute) Harder-Narasimhan filtration with respect to an _ample
polarization_ as in last subsection.

Our first main result this subsection is the following approximation result.

######  Theorem 19.

Let $(D_{1},\ldots,D_{n-1})$ be a set of nef divisors, and $H$ an ample
divisor. Assume $E$ is a torsion-free coherent sheaf over $X$. Then there
exists a positive number $\epsilon$, and a (unique) filtration of saturated
subsheaves

$0=E_{0}\subset E_{1}\subset\cdots E_{l}=E,$

such that for any $t\in(0,\epsilon)$, the above is the Harder-Narasimhan
filtration for the ample polarization $(D_{1}+tH,\ldots,D_{n-1}+tH)$.

###### Proof.

Denote $H_{i}(t)$ to be the $\mathbb{R}$-divisor $D_{i}+tH$, which is ample
[6, Corollary 1.4.10]. For a torsion-free sheaf $F$, we denote $\deg_{t}(F)$
and $\mu_{t}(F)$ to be the degree and slope of $F$ with respect to the
polarization $(H_{1}(t),\ldots,H_{n-1}(t))$. Note that when $t=0$, we get
exactly the degree and slope with respect to the polarization
$(D_{1},\cdots,D_{n-1})$, which we write in short as $\deg(F)$ and $\mu(F)$
separately.

We first notice that it suffices to show the following: there exists
$\epsilon>0$ and a saturated subsheaf $E_{1}\subset E$, such that $E_{1}$ is
the maximal destabilizing subsheaf of $E$ for $(H_{1}(t),\ldots,H_{n-1}(t))$
and any $t\in(0,\epsilon)$. The uniqueness of $E_{1}$ follows from the
uniqueness of the Harder-Narasimhan filtration for the ample polarization
$(H_{1}(t),\cdots H_{n-1}(t))$ as in Proposition 10.

To find such $E_{1}$, let $\mathcal{C}$ denote the set of torsion-free
subsheaves of $E$ and consider the following map of sets

$\displaystyle\mu_{t}(-):\mathcal{C}$
$\displaystyle\longrightarrow\\{\text{Degree }n-1\text{ polynomials in }t\\};$
$\displaystyle E^{\prime}$
$\displaystyle\longmapsto\mu_{t}(E^{\prime})=\frac{c_{1}(E^{\prime})H_{1}(t)\cdots
H_{n-1}(t)}{\mathrm{rk}(E^{\prime})}.$

For each $E^{\prime}\subset E$, the image $\mu_{t}(E^{\prime})$ is the
polynomial $a_{0}+a_{1}t+\cdots+a_{n-1}t^{n-1}$, where we have

$\displaystyle a_{0}(E^{\prime})=\mu(E^{\prime});$ $\displaystyle
a_{1}(E^{\prime})=\sum_{1\leq i\leq
n-1}\mu_{(H,D_{1},\ldots,\hat{D}_{i},\ldots,D_{n-1})}(E^{\prime}).$

For general $j\leq n$, the coefficient $a_{j}$ is the sum of slopes of
$E^{\prime}$ with respect to all possible choice of polarizations
$(H,\ldots,H,D_{i_{1}},\ldots,D_{i_{n-1-j}})$, such that each of the first $j$
entries of the polarization are all equal to $H$. Moreover, each polynomial
$\mu_{t}(E^{\prime})$ has coefficients in $\frac{1}{r!}\mathbb{Z}$, whose
coefficients are finite linear combinations of slopes of $E^{\prime}$ for a
fixed, finite number of choices of polarization. In particular, for each
$0\leq i\leq n$, the collection of coefficients
$\\{a_{i}(E^{\prime})\leavevmode\nobreak\ |\leavevmode\nobreak\
E^{\prime}\subset E\\}$ is bounded above. 111To see this, we first note that
since $a_{i}(E^{\prime})$ is a finite positive linear combination of
$\mu_{H^{i},D_{j_{1}},\ldots,D_{j_{n-1-i}}}(E^{\prime})$, it suffices to bound
each slope for the given nef polarization. The latter can be proved by
induction on ranks of $E$ as in the classical case: To check the case when $E$
is a line bundle, it suffices to show the set of slopes of sub line bundles is
bounded by slope of $E$ itself, which is true again by approximation via
adding each $D_{j_{l}}$ by $tH$ and making $t$ approach to zero. Here we
observe that the inequality holds for any $t>0$, so by the continuity, we get
the upper bound. The general case of $E$ follows from the induction hypothesis
on $E^{\prime}$ and $E^{\prime\prime}$ in a short exact sequence of vector
bundles $E^{\prime}\rightarrow E\rightarrow E^{\prime\prime}$.  Here we denote
the subset of polynomials consisting of the image of the map $\mu_{t}(-)$ by
$\mathcal{D}$.

We then define a lexicographic order on elements in $\mathcal{D}$ as follows:

$W_{1}(t)=\sum^{d}a_{i}t^{i}<W_{2}(t)=\sum^{d}b_{i}t^{i}\text{ if }\\\
a_{0}=b_{0},\cdots,a_{i-1}=b_{i-1},a_{i}<b_{i}\text{ for some }i.$

Since the coefficients of polynomials in $\mathcal{D}$ are bounded above, we
can find the maximum polynomial $P(t)$ in $\mathcal{D}$. Moreover, since the
degree of a sheaf is a discrete quantity, we can find a subsheaf $E_{1}$ in
the preimage of $\mu_{t}^{-1}(P(t))$ whose rank is maximal.

Finally, we prove that $E_{1}$ is exactly the maximal rank destabilizing
subsheaf of $E$ with respect to $(H_{1}(t),\ldots,H_{n-1}(t))$ for $t$ small
enough. To show this, it suffices to show the following, whose complete
details are left to the reader:

######  Claim 20.

Let $\mathcal{D}$ be a set of degree $n$ polynomials with coefficients bounded
above, and whose coefficients are in $\frac{1}{N}\mathbb{Z}$ for some positive
integer $N$. Then a polynomial $P(t)\in\mathcal{D}$ is maximum with respect to
the lexicographic order if and only if for small enough $t>0$ $P(t)>Q(t)$ (in
$\mathbb{R}$) for all other $Q(t)\in\mathcal{D}$.

The idea of the claim is the following: as $t$ approaches zero, higher power
terms are dominated by those with lower powers, so the polynomial that has the
largest first several terms under the lexicographic order will also have
maximum value in $\mathbb{R}$ for $t$ small enough. ∎

The above filtration of subsheaves $\\{E_{i}\\}$ in general fails to be the
Harder-Narasimhan filtration with respect to $(D_{1},\ldots,D_{n-1})$ for
$t=0$. However, the filtration is in fact a _weak Harder-Narasimhan
filtration_ with respect to $(D_{1},\ldots,D_{n-1})$, in the sense that each
$E_{i}/E_{i-1}$ is semistable with respect to $(D_{1},\ldots,D_{n-1})$, and we
have inequalities

$\mu(E_{i+1}/E_{i})\geq\mu(E_{i}/E_{i-1}),\,\,1\leq i\leq l-1.$

Note that we do not have strict inequalities here, which is part of the
definition of the Harder-Narasimhan filtration.

The semistability of the subsheaves $E_{i}$ can be seen as follows:

######  Lemma 21.

Let $E^{\prime}$ be a subsheaf of the torsion-free sheaf $E$ such that for
$t>0$ small enough, $E^{\prime}$ is semistable with respect to
$(H_{1}(t),\ldots,H_{n-1}(t))$. Then $E^{\prime}$ is semistable with respect
to $(D_{1},\ldots,D_{n-1})$.

###### Proof.

Assume $E^{\prime}$ admits a subsheaf $E^{\prime}_{1}$ whose slope with
respect to $(D_{1},\ldots,D_{n-1})$ is strictly larger than that of
$E^{\prime}$. By the continuity of the polynomial $\mu_{t}$ in $t$, we get the
inequality $\mu_{t}(E^{\prime}_{1})>\mu_{t}(E^{\prime})$ for $t>0$ small
enough, which contradicts our assumption. ∎

The same strategy above in fact implies the existence and uniqueness of the
Harder-Narasimhan filtration for nef divisors, generalizing the result for
ample ones.

######  Corollary 22.

Let $(D_{1},\ldots,D_{n-1})$ be a set of nef divisors of $X$. Then any
torsion-free coherent sheaf $E$ over $X$ admits a unique Harder-Narasimhan
filtration with respect to the polarization $(D_{1},\ldots,D_{n-1})$.

###### Proof.

Let $H$ be a fixed ample divisor over $X$. By Theorem 19, there exists a
unique filtration of saturated subsheaves $E_{i}$ of $E$, such that when $t>0$
is very small, $E_{i}$ is the Harder-Narasimhan filtration of $E$ with respect
to $(H_{1}(t),\ldots,H_{n-1}(t))$. Each $E_{i}$ is semistable with respect to
$(D_{1},\ldots,D_{n-1})$ by Lemma 21, and the slopes of $E_{i}/E_{i-1}$ is
non-strictly decreasing. Thus by picking the subsheaves in this filtration so
that the slopes of the factors become strictly decreasing, we get the Harder-
Narasimhan filtration of $E$ with respect to $(D_{1},\ldots,D_{n-1})$. At
last, the uniqueness of the Harder-Narasimhan filtration follows from the
uniqueness of the maximal destabilizing subsheaf with respect to
$(D_{1},\ldots,D_{n-1})$, where the latter can be checked via the continuity
of $\mu_{t}$. So we are done. ∎

At last, we give a simple example illustrating the degeneration of the Harder-
Narasimhan filtration. Example:

Let $X=\mathbb{P}^{1}\times\mathbb{P}^{1}$ be the product of two projective
lines over a field $k$. Let $D$ be the trivial divisor, $H$ be the ample
divisor of bidegree $(1,2)$ over $X$, and let $H(t):=D+tH$ be the sum. Denote
$L_{1}=(0,1)$ and $L_{2}=(1,0)$ to be the two line bundles over $X$, and let
$E$ be the the direct sum $L_{1}\oplus L_{2}$. Then for any $t>0$, the divisor
$H(t)$ is ample in $X$ as it can be written as a sum of the pullback of ample
divisors on two separate factors of $X$. Moreover, for each $t>0$, we have
$L_{1}\cdot H(t)=2t>t=L_{2}\cdot H(t)$, and thus

$0\subset L_{1}\subset E$

is the Harder-Narasimhan filtration of $E$ for the ample polarization $H(t)$,
for any $t>0$. However, when $t=0$, as $H(0)=D=0$ is the trivial divisor, any
subsheaf of $E$ has the same slope $0$. In particular, the Harder-Narasimhan
filtration of $E$ is the trivial filtration, and the filtration $(\ast)$
becomes a filtration of subsheaves whose graded pieces have the same slopes.

### Positive characteristic

The main difference in characteristic $p$ is that we need to work with the
notion of _strong semistability_ , as the Frobenius pullbacks of semistable
sheaves are not necessarily semistable.

First, we recall a few basic notions from algebraic geometry in positive
characteristic. Let $k$ be an algebraically closed field of characteristic
$p>0$. Let $X$ be a smooth projective $k$-variety. The _absolute Frobenius_
morphism $F_{X}:X\to X$ is the map on $X$ given locally on an open subset
$\mathrm{Spec}\ R\subset X$ by $a\mapsto a^{p}$. For simplicity of notation,
we just denote $F_{X}$ by $F$. Note that $F$ is not a map of $k$-schemes.

We also fix nef divisors $D_{1},\dots,D_{n-1}$ on $X$ and we will compute
slope with respect to the polarization $(D_{1},\ldots,D_{n-1})$.

######  Definition 23 (Strong semistability).

A coherent sheaf $E$ on $X$ is said to be _strongly semistable_ if
$(F^{e})^{*}E$ is a semistable sheaf on $X$ for all $e\geq 0$.

######  Remark 24.

A coherent sheaf that is semistable but not strongly semistable can be found
for example in [2, Corollary 2].

In the positive characteristic, instead of just keeping track of $\mu_{\min}$
and $\mu_{\max}$ of a coherent sheaf, we also keep track of a few other
invariants related to the Frobenius pullbacks. We note first that Frobenius
pullback alters the slope of a sheaf, indeed we have that
$\mu((F^{e})^{*}E)=p^{e}\mu(E)$. For this reason, we define the related
quantity

$L_{\max}(E):=\lim_{e\to\infty}\frac{\mu_{\max}({(F^{e})^{*}E})}{p^{e}}.$

Note that the sequence $\frac{\mu_{\max}({(F^{e})^{*}E})}{p^{e}}$ is
increasing in $e$, thus the limit $L_{\max}(E)$ exists in
$\mathbb{R}\cup\\{\infty\\}$. We will show later that this limit is indeed
finite. Similarly, we can define $L_{\min}$. By definition, we have
$L_{\max}(E)\geq\mu_{\max}(E)$ and $L_{\min}(E)\leq\mu_{\min}(E)$. It
immediately follows from the definition that if $E$ is strongly semistable,
then

$L_{\max}=\mu_{\max}=\mu(E)=\mu_{\min}(E)=L_{\min}(E).$

Let us set

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

We would like to find an upper estimate for $\alpha(E)$. To do this, we first
state a theorem on how to detect instability of the Frobenius pullback of a
semistable sheaf. For details, refer to [5, Section 2]. The idea is to make
use of the canonical connection $F^{*}E\to F^{*}E\otimes\Omega_{X}$ on the
Frobenius pullback.

######  Theorem 25.

Let $E$ be a semistable sheaf on $X$ such that $F^{*}E$ is not a semistable
sheaf on $X$. Let $0=E_{0}\subset E_{1}\subset\dots\subset E_{m}=F^{*}E$ be
the Harder-Narasimhan filtration of $F^{*}E$. Then, the natural
$\mathcal{O}_{X}$ homomorphisms $E_{i}\to(E/E_{i})\otimes\Omega_{X}$ induced
by the canonical connection are non-zero.

We now estimate the the minimum and maximum slopes of the Frobenius pullback
of a semistable sheaf. The idea is to use the non-zero maps above along with
Theorem 25 to get an estimate of how far apart the slopes can be.

######  Lemma 26.

Let $A$ be a nef divisor such that $T_{X}(A)$ is globally generated (or
equivalently if $\Omega\hookrightarrow\mathcal{O}_{X}(A)^{\oplus l}$ for some
$l$) and let $E$ be a torsion-free semistable sheaf on $X$. Then,

$\mu_{\max}(F^{*}E)-\mu_{\min}(F^{*}E)\leq(\mathrm{rk}(E)-1)A\cdot
D_{1}\cdot\dots\cdot D_{n-1}.$

###### Proof.

Let $0=E_{0}\subset E_{1}\subset\dots\subset E_{m}=F^{*}E$ be the Harder-
Narasimhan filtration of $F^{*}E$. Using Theorem 25, we have that the
$\mathcal{O}_{X}$-homomorphism $E_{i}\to F^{*}E/E_{i}\otimes\Omega_{X}$ is
non-zero. Thus, we get that

$\mu(E_{i}/E_{i-1})=\mu_{\min}(E_{i})\leq\mu_{\max}(F^{*}E/E_{i}\otimes\Omega_{X}).$

Since $\Omega\hookrightarrow\mathcal{O}_{X}(A)^{\oplus l}$, we get that
$\mu_{\max}(F^{*}E/E_{i}\otimes\Omega_{X})\leq\mu_{\max}(F^{*}E/E_{i}\otimes\mathcal{O}_{X}(A))=\mu(E_{i+1}/E_{i})+A\cdot
D_{1}\cdot\dots\cdot D_{n-1}$.

Thus, we get that

$\mu(E_{i}/E_{i-1})-\mu(E_{i+1}/E_{i})\leq A\cdot D_{1}\cdot\dots\cdot
D_{n-1}$

Summing this inequality, we get the result. ∎

######  Proposition 27.

If $A$ is a nef divisor such that $T_{X}(A)$ is globally generated and $E$ is
a torsion-free sheaf on $X$, then

$\frac{\mu_{\max}(F^{*}E)}{p}\leq\mu_{\max}(E)+\frac{(\mathrm{rk}(E)-1)}{p}A\cdot
D_{1}\dots D_{n-1}$

###### Proof.

Let $0=E_{0}\subset E_{1}\subset\dots\subset E_{m}=E$ be the Harder-Narasimhan
filtration of $E$. Applying the previous proposition to
$F^{*}(E_{i}/E_{i-1})$, we get that

$\frac{\mu_{\max}(F^{*}(E_{i}/E_{i-1}))}{p}\leq\mu(E_{i}/E_{i-1})+\frac{(\mathrm{rk}(E)-1)}{p}A\cdot
D_{1}\dots D_{n-1}.$

Note that $F^{*}E_{i}$ form a filtration of $F^{*}E$ and thus by Lemma 16
$\mu_{\max}(F^{*}E)\leq\max_{i}\\{\mu_{\max}(F^{*}(E_{i}/E_{i-1}))\\}$. Using
this, we get the required result. ∎

We have the following bound on $\alpha(E)$ (which in particular shows that
$L_{\max}(E)$ and $L_{\min}(E)$ are finite).

######  Proposition 28.

If $A$ is a nef divisor such that $T_{X}(A)$ is globally generated, then

$\alpha(E)\leq\frac{(\mathrm{rk}(E)-1)}{p-1}A\cdot D_{1}\dots D_{n-1}.$

###### Proof.

Applying induction to the previous proposition, we see that

$\frac{\mu_{\max}((F_{X}^{e})^{*}E)}{p^{e}}\leq\mu_{\max}(E)+(\mathrm{rk}(E)-1)\left(\frac{1}{p}+\frac{1}{p^{2}}+\dots+\frac{1}{p^{e}}\right)A\cdot
D_{1}\dots D_{n-1}.$

Letting $e\to\infty$, we get the required result. ∎

### Finite determinancy of the Harder-Narasimhan filtration.

We will now prove the following theorem.

######  Theorem 29.

For every torsion-free sheaf $E$, there exists a non-negative integer $e_{0}$
such that all the factors in the Harder-Narasimhan filtration of
$(F^{e_{0}})^{*}E$ are strongly semistable.

The proof uses the Harder-Narasimhan polygon (see Definition 17). For any
sheaf $G$ over a smooth projective variety $X$, let us define
$p(G):=(\mathrm{rk}G,\deg G)\in\mathbb{R}^{2}$. We also define

$\mathrm{HNP}_{e}(E):=\\{(x,y)\in\mathbb{R}^{2}\mid(x,p^{e}y)\in\mathrm{HNP}((F^{e})^{*}E)\\}.$

Note that $\mathrm{HNP}_{e}(E)$ forms an increasing sequence of convex subsets
of $\mathbb{R}^{2}$ and we define
$\mathrm{HNP}_{\infty}(E):=\overline{\bigcup_{e}\mathrm{HNP}_{e}(E)}$.

######  Proposition 30.

$HNP_{\infty}(E)$ is a bounded convex polygon.

###### Proof.

It is clear that $\mathrm{HNP}_{\infty}(E)$ is convex as each of the
$\mathrm{HNP}_{e}(E)$ are convex. To see that it is bounded, note that the
rank coordinates of $\mathrm{HNP}_{e}(E)$ lie in in the interval
$[0,\mathrm{rk}(E)]$. The fact that $\alpha(E)$ is finite also tells us that
there is a uniform bound on the degree coordinates of $\mathrm{HNP}_{e}(E)$.
Thus we see that $\mathrm{HNP}_{\infty}(E)$ is a bounded subset of
$\mathbb{R}^{2}$.

To show ${\mathrm{HNP}}_{\infty}(E)$ is a polygon, we claim that it is a
convex hull of the set $\\{(r,q_{\infty,r})\mid
r\in\\{0,\dots,\mathrm{rk}(E)\\}$, where
$q_{\infty,r}=\sup\\{d\mid(r,d)\in{\mathrm{HNP}}_{\infty}(E)\\}$. Note that
$q_{\infty,0}=p(0)=(0,0)$ and
$q_{\infty,\mathrm{rk}(E)}=p(E)=(\mathrm{rk}(E),\deg(E))$.

It is clear that the convex hull of these points is contained in
${\mathrm{HNP}}_{\infty}(E)$. To see the converse, first note that all
${\mathrm{HNP}}_{k}(E)$ and thus ${\mathrm{HNP}}_{\infty}(E)$ lie above the
line segment joining $p(E_{0})=(0,0)$ and $p(E)=(\mathrm{rk}(E),\deg(E))$. It
is thus enough to show that every vertex of ${\mathrm{HNP}}_{e}(E)$ lies in
the convex hull for all $e$. Pick any vertex $(r,d)$ of
${\mathrm{HNP}}_{e}(E)$ for some $e$ and some
$r\in\\{0,\dots,\mathrm{rk}(E)\\}$. Then, since $(r,d)$ lies above the line
segment $\overline{p(0)p(E)}$ and below the point $(r,q_{r})$, one can easily
see that $(r,d)$ lies in the convex hull of $p(0),p(E)$ and $(r,q_{r})$. ∎

Before proving the finite determinancy of the Harder-Narasimhan filtration,
let us introduce some notation. Let $0=E_{0,e}\subset\dots\subset E_{m_{e},e}$
denote the Harder-Narasimhan filtration of $(F^{e})^{*}E$. Let
$p_{i,e}=(\mathrm{rk}(E_{i,e}),\\\ \deg(E_{i,e})/p^{e})$ denote the vertices
of ${\mathrm{HNP}}_{e}(E)$.

Let $(0,0)=p_{0,\infty},\dots,p_{s,\infty}=p(E)$ denote the vertices of the
${\mathrm{HNP}}_{\infty}(E)$ and denote the coordinates of $p_{j,\infty}$ as
$p_{j,\infty}=(r_{j,\infty},d_{j,\infty})$. Let us denote the slope of the
line segment $\overline{p_{(j-1),\infty}p_{j,\infty}}$ as $\mu_{j,\infty}$.
For every $j$, there exists a sequence $p_{i_{j},e}\to p_{j,\infty}$ for some
sequence $i_{j}$ as $e\to\infty$.

###### Proof of Theorem 29.

We prove that there exists some $e_{0}$ such that
$HNP_{e_{0}}(E)=HNP_{\infty}(E)$. We show this by induction on rank. In the
rank one case, there is nothing to show.

Pick $0<\epsilon\ll 1$. Then we have the Euclidean distance
$\|p_{i_{j},e}-p_{j,\infty}\|<\epsilon$ for all $j$ and all $e\gg 0$.
Replacing $E$ by $(F^{e})^{*}E$ for some large $e$, we may assume that
$\|p_{i_{j},e}-p_{j,\infty}\|<\epsilon$ holds for all $e\geq 0$. Since the
rank coordinates of $p_{i_{j},e}$ and $p_{j,\infty}$ can only take integer
values, it must be the case that $r_{j,\infty}=\mathrm{rk}(E_{i_{j},e})$.

We first show that there exists an integer $e_{0}$ such that
$E_{i_{1},e}=(F^{e-e_{0}})^{*}E_{i_{1},e_{0}}$ for all $e\geq e_{0}$. Consider
the first line segment $s$ of the polygon
${\mathrm{HNP}}_{\infty}(E_{i_{1},0})$. Note that $s$ cannot lie above the
line segment $\overline{p_{0,\infty}p_{1,\infty}}$ (see Remark 18). We will
show that $s$ actually lies on the line segment
$\overline{p_{0,\infty}p_{1,\infty}}$. Assuming that is the case, we apply the
induction hypothesis to $E_{i_{1},0}$ to get: there exists a subsheaf $G$ of
$(F^{l})^{*}{E_{i_{1},0}}$ such that $(\mathrm{G},\frac{\deg{G}}{p^{l}})\in
s\subset\overline{p_{0,\infty}p_{1,\infty}}$ i.e. $G$ is strongly semistable.
Once again, we may replace $E$ by $(F^{l})^{*}E$ to assume that $l=0$. By
induction hypothesis, since the theorem holds for $E/G$ and since
$\mu(G)=\mu_{\max}(E)=\frac{d_{1,\infty}}{r_{1,\infty}}$, we also get the
theorem for $E$.

Now suppose that $s$ lies strictly below the line segment
$\overline{p_{0,\infty}p_{1,\infty}}$. We will deduce a contradiction from
this. Since $p_{i_{1},l}\to p_{i_{1},\infty}$ as $l\to\infty$, we can find an
integer $l$ such that $\overline{p_{0,l}p_{i_{1},l}}$ lies above $s$. Thus,
$\mu_{\max}((F^{l})^{*}E)>\mu_{\max}((F^{l})^{*}E_{i_{1},0})$ for such an
integer $l$.

Since
$\\{(F^{l})^{*}E_{i_{1},0},(F^{l})^{*}E_{i_{1}+1,0},\dots,(F^{l})^{*}E\\}$
form a filtration of $(F^{l})^{*}E$, using Lemma 16, we get that there exists
an integer $j>i_{1}$ such that

$\mu_{\max}((F^{l})^{*}(E_{j,0}/E_{j-1,0}))>\mu_{\max}((F^{l})^{*}E_{i_{1},0}).$

Consider a saturated subsheaf $G\subset(F^{l})^{*}(E_{j,0})$ such that
$\mu(G/(F^{l})^{*}(E_{j-1,0}))=\mu_{\max}((F^{l})^{*}(E_{j,0}/E_{j-1,0}))$.
Then, $(\mathrm{rk}(G),\frac{\deg(G)}{p^{l}})\in{\mathrm{HNP}}_{l}(E)$. Let us
try to estimate the difference in the areas of ${\mathrm{HNP}}_{l}(E)$ and
${\mathrm{HNP}}(E)$. There is a lower bound on the difference in areas by
considering the triangle $W$ joining the points $p_{j-1,0}$, $p_{j,0}$ and
$(\mathrm{rk}(G),\frac{\deg(G)}{p^{l}})$. Let
$\tilde{\mu}_{1}:=\frac{\mu((F^{l})^{*}(E_{j,0}/E_{j-1,0}))}{p^{l}}$ and
$\tilde{\mu}_{2}:=\frac{\mu(G/(F^{l})^{*}E_{j-1,0})}{p^{l}}$ denote the slopes
of the sides of $W$ that contain $p_{j-1,0}$. Using the fact that the
$\mathrm{rk}$-coordinates of these points can only take integer values, we can
get a get a lower bound on the area of the triangle $W$ by considering the
triangle $W^{\prime}$ joining $p_{j-1,0}$, $p_{j-1,0}+(1,\tilde{\mu}_{1})$,
$p_{j-1,0}+(1,\tilde{\mu}_{2})$ (see Figure 2).

$p_{j-1,0}$$p_{j,0}$$(\mathrm{rk}(G),\frac{\deg(G)}{p^{l}})$$p_{j-1}+(1,\tilde{\mu}_{1})$$p_{j-1}+(1,\tilde{\mu}_{2})$$W^{\prime}$
Figure 2. Triangles $W$ and $W^{\prime}$ appearing in the proof of Theorem 29.
The larger triangle is $W$ and the smaller triangle inside it is $W^{\prime}$.

Thus it follows:

$\displaystyle\mathrm{Area}({\mathrm{HNP}}_{l}(E))-\mathrm{Area}({\mathrm{HNP}}(E))$
$\displaystyle\geq\mathrm{Area}(W^{\prime})$
$\displaystyle\geq\frac{1}{2}(\tilde{\mu}_{2}-\tilde{\mu}_{1})$
$\displaystyle=\frac{1}{2}\left(\frac{\mu(G/(F_{X}^{l})^{*}(E_{j-1,0}))}{p^{l}}-\frac{\mu((F_{X}^{l})^{*}(E_{j,0}/E_{j-1,0}))}{p^{l}}\right)$
$\displaystyle\geq\frac{1}{2}\left(\mu({E_{i_{1},0}})-\mu(E_{j,0}/E_{j-1,0})\right)$
$\displaystyle\geq\frac{1}{2}((\mu_{1\infty}-\epsilon)-\mu(E_{j,0}/E_{j-1,0})).$

To finish the estimate, we need to get an upper bound on
$\mu(E_{j,0}/E_{j-1,0})$. To do this, let $u$ be such that $i_{u}<j\leq
i_{u+1}$. Then the line segment $\overline{p_{j-1,0}p_{j,0}}$ is sandwiched
between the line segments $\overline{p_{i_{u},0}p_{i_{u+1},0}}$ and
$\overline{p_{i_{u},\infty}p_{i_{u+1},\infty}}$ (see Figure 3). Using the fact
that $\|p_{i_{u},\infty}-p_{i_{u}0}\|\leq\epsilon$ and estimating the slope of
the line segment $\overline{p_{j-1,0}p_{j,0}}$, we get that
$|\mu(E_{j,0}/E_{j-1,0})-\mu_{u+1,\infty}|\leq 3\epsilon$. Thus, we have

$p_{i_{u},\infty}$$p_{i_{u+1},\infty}$$p_{i_{u},0}$$p_{i_{u+1},0}$$p_{j-1,0}$$p_{j,0}$$\leq\epsilon$$\leq\epsilon$
Figure 3. Estimating $\mu(E_{j,0}/E_{j-1,0})$. The red line shows the boundary
of ${\mathrm{HNP}}_{\infty}(E)$ and the blue line shows the boundary of
${\mathrm{HNP}}(E)$.

$\mathrm{Area}({\mathrm{HNP}}_{l}(E))-\mathrm{Area}({\mathrm{HNP}}(E))\geq\frac{1}{2}(\mu_{1\infty}-\mu_{2\infty}-4\epsilon).$

But the difference in areas between ${\mathrm{HNP}}(E)$ and
${\mathrm{HNP}}_{\infty}(E)$ is at most $r\epsilon$, which gives us a
contradiction for a small enough choice of $\epsilon$. ∎

##  3\. Bogomolov’s inequality

In this section, we give several equivalent statements on Bogomolov’s
inequality, about the positivity of $\Delta(E)\cdot D_{2}\cdots D_{n-1}$.

Fix a positive integer $r$. Let $X$ be a smooth projective variety of
dimension $n\geq 2$ over an algebraically closed field $k$ in any
characteristic. Note that in the characteristic zero case, we define strong
semistability to be the same as the semistability. We fix a nef divisor $A$
over $X$ such that $T_{X}(A)$ is globally generated. Define the constant
$\beta_{r}=\beta_{r}(A;D_{1},\ldots,D_{n-1})$ for a choice of divisors
$(D_{1},\ldots,D_{n-1})$ as below

$\beta_{r}(A;D_{1},\ldots,D_{n-1}):=\begin{cases}0,&\text{if}\leavevmode\nobreak\
\mathrm{char}(k)=0;\\\ \left(\frac{r(r-1)}{p-1}AD_{1}\cdots
D_{n-1}\right)^{2},&\text{if}\leavevmode\nobreak\
\mathrm{char}(k)=p.\end{cases}$

We also reference the _discriminant_ of a rank $r$ torsion-free sheaf $E$,
denoted as $\Delta(E)$, is defined as

$\Delta(E):=2r\cdot c_{2}(E)-(r-1)c_{1}(E)^{2}.$

When $G$ is a non-trivial subsheaf of rank $s$ in $E$, we set

$\xi_{G,E}=\frac{c_{1}(G)}{s}-\frac{c_{1}(E)}{r}.$

We also use $K^{+}$ to denote an open cone in $\mathrm{Num}(X)$, defined as in
the beginning of Subsection 3.

The main theorem is the following.

######  Theorem 31 (BI($r$)).

Let $(D_{1},\ldots,D_{n-1})$ be a collection of nef line bundles over $X$, and
let $d=D_{1}^{2}\cdot D_{2}\cdots D_{n-1}\geq 0$. Fix a positive integer $r$.
Then the following statements are equivalent.

  1. (i)

Assume each $D_{i}$ is ample, and $E$ is a strongly
$(D_{1},\ldots,D_{n-1})$-semistable torsion-free sheaf of rank $r^{\prime}\leq
r$. Then we have

$\Delta(E)D_{2}\ldots D_{n-1}\geq 0.$

  2. (ii)

Let $E$ be a $(D_{1},\ldots,D_{n-1})$-semistable torsion-free sheaf of rank
$r^{\prime}$ for some $r^{\prime}\leq r$. Then we have

$d\cdot\Delta(E)D_{2}\cdots D_{n-1}+\beta_{r^{\prime}}\geq 0.$

  3. (iii)

Let $E$ be a torsion-free sheaf of rank $r^{\prime}$ for some $r^{\prime}\leq
r$ over $X$, and assume $d\cdot\Delta(E)D_{2}\cdots
D_{n-1}+\beta_{r^{\prime}}<0$. Then there exists a saturated subsheaf
$E^{\prime}$ of $E$ such that $\xi_{E^{\prime},E}\in K^{+}$.

  4. (iv)

Let $E$ be a strongly $(D_{1},\ldots,D_{n-1})$-semistable torsion-free sheaf
of rank $r^{\prime}\leq r$ over $X$. Then

$\Delta(E)D_{2}\cdots D_{n-1}\geq 0.$

######  Remark 32.

In terms of the notations as in [5], the above corresponds to
$T^{5}(r)\Rightarrow T^{3}(r)\Rightarrow T^{4}(r)\Rightarrow
T^{2}(r)\Rightarrow T^{5}(r)$. Note that the last implication from $T^{2}(r)$
to $T^{5}(r)$ is trivial, as the statement (i) is the same as (iv) but with a
stronger assumption.

######  Remark 33.

In characteristic zero, since $\beta_{r}=0$ and the strong semistability is
the same as the semistability, the statement above can be more or less
combined into one single statement about the non-negativity of
$\Delta(E)D_{2}\cdots D_{n-1}\geq 0$. This was first proved by Bogomolov in
[1], thus the name.

### $(i)\Rightarrow(ii)$

In this subsection, our goal is to prove the implication $(i)\Rightarrow(ii)$
in Theorem  (BI($r$)). for torsion-free sheaf $E$ of rank $r$, and we follow
[5, $\S$3.6].

To prove the statement, we start with the following preparation. Let $H$ be an
ample line bundle over $X$, and let $(D_{1},\ldots,D_{n-1})$ be as in Theorem
(BI($r$)). (ii). Then
$(H_{1}(t),\ldots,H_{n-1}(t)):=(D_{1}+tH,\ldots,D_{n-1}+tH)$ is a polarization
of ample line bundles.

######  Lemma 34.

Assume Theorem  (BI($r$)). (i). Then for $t>0$ we have

$H_{1}(t)^{2}H_{2}(t)\cdots H_{n-1}(t)\cdot\Delta(E)H_{2}(t)\ldots
H_{n-1}(t)+\\\ r^{2}(L_{max,t}(E)-\mu_{t}(E))(\mu_{t}(E)-L_{min,t}(E))\geq 0.$

Here we follow the notations $L_{max}$ and $L_{min}$ as in Subsection 2, and
denote $L_{max,t}$, $L_{min,t}$ and $\mu_{t}$ as the ones defined using the
polarization $(H_{1}(t),\ldots,H_{n-1}(t))$.

######  Remark 35.

Before we prove the lemma, it is worth noting that the above inequality
together with its proof, which only uses Theorem  (BI($r$)). (i), applies also
to a general polarization $(D_{1},\ldots,D_{n-1})$ for ample $D_{i}$, in place
of $(H_{1}(t),\ldots,H_{n-1}(t))$. More explicitly, Theorem  (BI($r$))., (i)
will imply the inequality

$D_{1}^{2}D_{2}\cdots D_{n-1}\cdot\Delta(E)D_{2}\cdots
D_{n-1}+r^{2}(L_{\max}-\mu)(L_{\min}-\mu)\geq 0,$

where $D_{i}$ are all ample.

###### Proof.

We first notice that by Theorem 29, there exists $k\in\mathbb{N}$ such that
all of the factors in the Harder-Narasimhan filtration of $(F^{k})^{*}E$ are
strongly semistable. Denote $0=E_{0}\subset\cdots E_{m}=(F^{k})^{*}E$ to be
the corresponding filtration of $(F^{k})^{*}E$, and let $F_{i}$ be the
quotient $E_{i}/E_{i-1}$, $r_{i}=\mathrm{rk}(F_{i})$, and
$\mu_{i,t}=\mu_{t}(F_{i})$. Then by the Hodge index theorem, 222Here we are
using the form of the Hodge index theorem that $(D^{2}D_{2}\ldots
D_{n-1})\cdot(D_{1}^{2}D_{2}\ldots D_{n-1})\leq(DD_{1}D_{2}\ldots
D_{n-1})^{2}$, where $D_{i}$ are nef divisors satisfying $D_{1}^{2}D_{2}\cdots
D_{n-1}>0$. This can be proved using the standard Hodge index theorem for
ample divisors together with the approximation technique as in Subsection 2.
we have

$\displaystyle\frac{\Delta((F^{k})^{*}E)H_{2}(t)\cdots H_{n-1}(t)}{r}$
$\displaystyle=\sum_{i}\frac{\Delta(F_{i})H_{2}(t)\cdots
H_{n-1}(t)}{r_{i}}-\frac{1}{r}\sum_{i<j}r_{i}r_{j}(\frac{c_{1}(F_{i})}{r_{i}}-\frac{c_{1}(F_{j})}{r_{j}})^{2}H_{2}(t)\cdots
H_{n-1}(t)$ $\displaystyle\geq\sum_{i}\frac{\Delta(F_{i})H_{2}(t)\cdots
H_{n-1}(t)}{r_{i}}-\frac{1}{rd}\sum_{i<j}r_{i}r_{j}(\mu_{i,t}-\mu_{j,t})^{2}.$

The assumption of Theorem  (BI($r$)). (i) and the ampleness of $H_{i}(t)$
provides us with the inequality

$\Delta(F_{i})H_{2}(t)\cdots H_{n-1}(t)\geq 0.$

On the other hand, we have the following elementary inequality on $r_{i}$,
$\mu_{i}$ (with $r=\sum_{i}r_{i}$ and $r\mu=\sum_{i}r_{i}\mu_{i}$) that

(3.1) $\sum_{i<j}r_{i}r_{j}(\mu_{i}-\mu_{j})^{2}\leq
r^{2}(\mu_{1}-\mu)(\mu-\mu_{m}).$

Combine with the inequality above, we get the inequality in the statement of
lemma. ∎

With the above lemma together with the approximation technique as in
Subsection 2, we are ready to prove the aforementioned implication.

###### Proof of Theorem  (BI($r$))., $(i)\Rightarrow(ii)$.

Let $H$ be a fixed ample line bundle. By Theorem 19, we can find a filtration
of torsion-free coherent subsheaves $0=E_{0}\subset E_{1}\subset\cdots\subset
E_{m}=E$ of $E$ such that it is the Harder-Narasimhan filtration with respect
to the polarizations $(H_{1}(t),\ldots,H_{n-1}(t))$, where $t>0$ is very close
to $0$. As $E$ is $(D_{1},\ldots,D_{n-1})$-semistable, we have

$\displaystyle\mu(E)$ $\displaystyle\geq\mu(E_{1})$
$\displaystyle=\lim_{t\rightarrow 0^{+}}\mu_{t}(E_{1})$
$\displaystyle\geq\lim_{t\rightarrow 0^{+}}\mu_{t}(E)$ $\displaystyle=\mu(E).$

By the choice of $E_{1}$, the above implies the equality

$\mu(E)=\lim_{t\rightarrow 0^{+}}\mu_{{max},t}(E).$

Similarly, we have

$\mu(E)=\lim_{t\rightarrow 0^{+}}\mu_{{min},t}(E).$

On the other hand, Proposition 28 states that for a nef divisor $A$ such that
$T_{X}(A)$ is globally generated, we have

$\displaystyle L_{{max},t}(E)-\mu_{{max},t}(E),\leavevmode\nobreak\
\mu_{{min},t}(E)-L_{{min},t}(E)\leq\frac{r-1}{p-1}AH_{1}(t)\cdots H_{n-1}(t).$

In particular, the limit of the right hand side of the inequality above is
equal to $\frac{r-1}{p-1}AD_{1}\cdots D_{n-1}$, whose square is
$\frac{1}{r^{2}}\beta_{r}$. In this way, apply the inequality of Lemma 34 for
each $t>0$, we get

$\displaystyle D_{1}^{2}D_{2}\cdots D_{n-1}\cdot\Delta(E)D_{2}\ldots
D_{n-1}+\beta_{r}$ $\displaystyle=\lim_{t\rightarrow
0^{+}}\Bigg{(}D_{1}^{2}D_{2}\cdots D_{n-1}\cdot\Delta(E)D_{2}\ldots
D_{n-1}+{r^{2}}\left(\frac{r-1}{p-1}AH_{1}(t)\cdots
H_{n-1}(t)\right)^{2}\Bigg{)}$ $\displaystyle\geq\lim_{t\rightarrow
0^{+}}\left(H_{1}(t)^{2}H_{2}(t)\cdots H_{n-1}(t)\cdot\Delta(E)H_{2}(t)\cdots
H_{n-1}(t)\right)$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+r^{2}(L_{{max},t}(E)-\mu_{t}(E))(\mu_{t}(E)-L_{{min},t}(E))$
$\displaystyle\geq 0.$

∎

### $(ii)\Rightarrow(iii)$

In this subsection, we show the implication $(ii)\Rightarrow(iii)$, following
[5, $\S$3.7]

As a preparation, we define an open cone in $\mathrm{Num}(X)$ as follows.

######  Definition 36.

Let $K^{+}$ be the following open cone in $\mathrm{Num}(X)$:

$K^{+}=\\{D\in\text{Num}(X):D^{2}D_{2}\cdots D_{n-1}>0,\text{and}\
DD^{\prime}D_{2}\cdots D_{n-1}\geq 0\\\ \text{for all nef }D^{\prime}\\}.$

The following description is used in the proof.

######  Lemma 37.

A divisor $D\in K^{+}$ if and only if it satisfies the inequalities
$DLD_{2}\cdots D_{n}>0$ for all $L\in\bar{K}^{+}\setminus\\{0\\}$.

Before we start, we also mention several elementary computational results
whose proofs are left as exercises to the reader.

######  Lemma 38.

Let $0\to E^{\prime}\to E\to E^{\prime\prime}\to 0$ be a short exact sequence
of coherent sheaves. Let $r^{\prime}$ (resp. $r^{\prime\prime}$) be the rank
of $E^{\prime}$ (resp. $E^{\prime\prime}$), and assume they are positive. We
have

$\frac{\Delta(E)D_{2}\cdots
D_{n-1}}{r}+\frac{rr^{\prime}}{r^{\prime\prime}}\xi_{E^{\prime},E}^{2}D_{2}\cdots
D_{n-1}=\frac{\Delta(E^{\prime})D_{2}\cdots D_{n-1}}{r^{\prime}}+\\\
\frac{\Delta(E^{\prime\prime})D_{2}\cdots D_{n-1}}{r^{\prime\prime}}$

######  Lemma 39.

Let $0\to E^{\prime}\to E\to E^{\prime\prime}\to 0$ be a short exact sequence
of coherent sheaves. Let $r^{\prime}$ ($r^{\prime\prime}$) be the rank of
$E^{\prime}$ ($E^{\prime\prime}$) respectively.

  * •

If $G$ is a non-trivial subsheaf of $E^{\prime}$, then

$\xi_{G,E}=\xi_{E^{\prime},E}+\xi_{G,E^{\prime}}.$

  * •

If $G^{\prime\prime}\subset E^{\prime\prime}$ is a proper subsheaf of rank
$s$, and denote by $G$ the kernel of the map $E\to
E^{\prime\prime}/G^{\prime\prime}$, then

$\xi_{G,E}=\frac{r^{\prime}(r^{\prime\prime}-s)}{(r^{\prime}+s)r^{\prime\prime}}\xi_{E^{\prime},E}+\frac{s}{r^{\prime}+s}\xi_{G^{\prime\prime},E^{\prime\prime}}.$

###### Proof of Theorem  (BI($r$))., $(ii)\Rightarrow(iii)$.

We prove (iii) inductively on $\mathrm{rk}(E)$ $=r$, assuming the statement
(ii). As a starting point, we notice that when $E$ is of rank one, as
$\Delta(E)=0$ and $\beta_{1}\geq 0$, the statement (iii) is automatically
true. In general, the assumption of Theorem  (BI($r$)). (ii) tells us that $E$
is not semistable with respect to $(D_{1},\ldots,D_{n-1})$. Thus we have the
maximal destabilizing subsheaf $E^{\prime}\subset E$ with respect to the
polarization. Let $E^{\prime\prime}=E/E^{\prime}$, and
$r^{\prime},r^{\prime\prime}$ be the ranks of $E^{\prime},E^{\prime\prime}$
respectively. Then by Lemma 38 we have

$\frac{\Delta(E)D_{2}\cdots
D_{n-1}}{r}+\frac{rr^{\prime}}{r^{\prime\prime}}\xi_{E^{\prime},E}^{2}D_{2}\cdots
D_{n-1}=\\\ \frac{\Delta(E^{\prime})D_{2}\cdots
D_{n-1}}{r^{\prime}}+\frac{\Delta(E^{\prime\prime})D_{2}\cdots
D_{n-1}}{r^{\prime\prime}}.$

We also have

$\frac{\beta_{r}}{r}\geq\frac{\beta_{r^{\prime}}}{r^{\prime}}+\frac{\beta_{r^{\prime\prime}}}{r^{\prime\prime}}.$

So by multiplying $d=D_{1}^{2}D_{2}\cdots D_{n-1}$, either
$\xi_{E^{\prime},E}^{2}D_{2}\cdots D_{n-1}>0$, or one of
$d\Delta(E^{\prime})D_{2}\cdots D_{n-1}+\beta_{r^{\prime}}$ or
$d\Delta(E^{\prime\prime})D_{2}\cdots D_{n-1}+\beta_{r^{\prime\prime}}$ is
negative.

In the first case above, we claim that $\xi_{E^{\prime},E}\in K^{+}$. To see
this, by the definition of $K^{+}$ and the assumption on $\xi_{E^{\prime},E}$
it suffices to show that for any $L\in{K}^{+}\backslash\\{0\\}$ we have
$\xi_{E^{\prime},E}LD_{2}\cdots D_{n-1}>0$. Note first the assumption of
$E^{\prime}$ and $E$ implies that the inequality is true for $L=D_{1}$, and it
reduces to show the sign of the function
$L\mapsto\xi_{E^{\prime},E}LD_{2}\cdots D_{n-1}$ is not changing on $K^{+}$.
This then follows from the continuity of the function, the connectivity of
$K^{+}$, and the Hodge Index Theorem that

$(\xi_{E^{\prime},E}LD_{2}\cdots
D_{n-1})^{2}\geq\xi_{E^{\prime},E}^{2}D_{2}\cdots D_{n-1}\cdot
L^{2}D_{2}\cdots D_{n-1}>0,$

for $L\in K^{+}\backslash\\{0\\}$.

Suppose now $\xi_{E^{\prime},E}^{2}D_{2}\cdots D_{n-1}\leq 0$ and we are in
the second case above. Note that since both $E^{\prime}$ and
$E^{\prime\prime}$ are of smaller ranks, we can apply the induction hypotheses
of ranks to get the following dichotomy:

  * •

There is a saturated subsheaf $G\subset E^{\prime}$ such that
$\xi_{G,E^{\prime}}\in K^{+}$, or

  * •

There is a saturated subsheaf $G^{\prime\prime}\subset E^{\prime\prime}$ such
that $\xi_{G^{\prime\prime},E^{\prime\prime}}\in K^{+}$.

Applying Lemma 39, we get in either case that $\xi_{G,E}$ is a positive linear
combination of $\xi_{E^{\prime},E}$ and some element $L$ in $K^{+}$.

To proceed, define the open subcone
$\mathcal{C}(\xi):=\\{D\in\overline{K^{+}}\setminus\\{0\\}:\xi\cdot
DD_{2}\cdots D_{n-1}>0\\}\subset\overline{K^{+}}$ for a given
$\xi\in\text{Num}(X)$. We observe that by replacing $E^{\prime}$ with $G$ in
either case, we get strictly larger cones than before:
$\mathcal{C}(\xi_{E^{\prime},E})\subsetneq\mathcal{C}(\xi_{G,E})$. Here the
inclusion part is clear, as $\xi_{G,E}$ is the sum of $\xi_{E^{\prime},E}$
with some $L\in K^{+}$, where the second term only contributes positive values
in the product $\xi_{G,E}DD_{2}\cdots D_{n-1}$. To see the inclusion is
strict, as $\xi_{E^{\prime},E}$ is not in $K^{+}$, by Lemma 37 there exists
some $L^{\prime}\in\overline{K^{+}}\backslash\\{0\\}$ such that
$\xi_{E^{\prime},E}L^{\prime}D_{2}\cdots D_{n-1}\leq 0$ but
$LL^{\prime}D_{2}\cdots D_{n-1}>0$. On the other hand, as $E^{\prime}$ is the
maximal destabilizing subsheaf of $E$ with respect to the polarization
$(D_{1},\ldots,D_{n-1})$, we have $\xi_{E^{\prime},E}D_{1}\cdots D_{n-1}>0$
and $LD_{1}\cdots D_{n-1}>0$, where the latter follows as $L\in K^{+}$. In
this way, there exists some real number $t\in[0,1)$ such that
$\xi_{E^{\prime},E}(tD_{1}+(1-t)L^{\prime})D_{2}\cdots D_{n-1}=0$, and thus

$\displaystyle\xi_{G,E}$ $\displaystyle(tD_{1}+(1-t)L^{\prime})D_{2}\cdots
D_{n-1}$ $\displaystyle=\xi_{E^{\prime},E}(tD_{1}+(1-t)L^{\prime})D_{2}\cdots
D_{n-1}$ $\displaystyle+L(tD_{1}+(1-t)L^{\prime})D_{2}\cdots D_{n-1}$
$\displaystyle=0+L(tD_{1}+(1-t)L^{\prime})D_{2}\cdots D_{n-1}$
$\displaystyle>0=\xi_{E^{\prime},E}(tD_{1}+(1-t)L^{\prime})D_{2}\cdots
D_{n-1}.$

So the element $tD_{1}+(1-t)L^{\prime}$ is in $\mathcal{C}(\xi_{G,E})$ but not
in $\mathcal{C}(\xi_{E^{\prime},E})$. In this way, we get a sequence of
strictly increasing subcones of $\overline{K^{+}}$ by replacing $E^{\prime}$
by $G$, until we reach to the situation where
$\xi_{E^{\prime},E}^{2}D_{2}\cdots D_{n-1}>0$.

It remains to show that this process terminates within finite replacements.
First note that by choosing ample $\mathbb{R}$-basis $H_{1},\cdots,H_{\rho}$,
which are contained in $\mathcal{C}(\xi_{E^{\prime},E})$ for $\text{Num}(X)$,
333Note that since $\mathcal{C}(\xi_{E^{\prime}E})$ is an open subcone in
$K^{+}\subset\mathrm{Num}(X)$ that is non-empty (it contains $D_{1}$), such a
basis of $H_{i}$ exists. we have

$\xi_{G,E}\in\frac{1}{r!}(\mathbb{Z}H_{1}+\cdots+\mathbb{Z}H_{\rho}).$

Furthermore, we also have

$0<\xi_{G,E}H_{j}D_{2}\cdots D_{n-1}<\mu_{\max}^{j}(E)-\mu^{j}(E),$

for all $j=1,2,\cdots,\rho$, where $\mu^{j}$ denote the slopes with respect to
$(H_{j},D_{2},\cdots,D_{n-1})$. Thus $\xi_{G,E}$ is in fact contained in a
bounded discrete hence finite subset of $\text{Num}(X)$. ∎

### $(iii)\Rightarrow(iv)$

In this section, we follow [5, $\S$3.8] to show the implication
$(iii)\Rightarrow(iv)$.

We follow the definition of $\beta_{r}$ as in the beginning of this section,
and the definition of the cone $K^{+}$ with respect to the polarization
$(D_{1},\ldots,D_{n-1})$ as in the last subsection.

By the definition of $\xi_{E^{\prime},E}$ and $K^{+}$, the statement (iii)
implies that $E$ is not semistable with respect to $(D_{1},\ldots,D_{n-1})$.

######  Remark 40.

The implication $(iii)\implies(iv)$ is easy in $\mathrm{char}(k)=0$, by the
definition of $K^{+}$ applying at $\xi_{E^{\prime},E}D_{1}\cdots D_{n-1}$.

###### Proof of Theorem  (BI($r$))., $(iii)\Rightarrow(iv)$.

Assume $E$ is a torsion-free sheaf such that $\Delta(E)D_{2}\cdots D_{n-1}<0$.
We will deduce that $E$ is not strongly semistable. We would like to apply
Theorem  (BI($r$))., (iii) to the Frobenius pullback $(F^{l})^{*}E$ of $E$.
Since $\Delta((F^{l})^{*}E)=p^{2l}\Delta(E)$, we get that

$D_{1}^{2}D_{2}\cdots D_{n-1}\cdot\Delta((F^{l})^{*}E)D_{2}\cdots
D_{n-1}+\beta_{r}=\\\ D_{1}^{2}D_{2}\cdots D_{n-1}\cdot
p^{2l}\cdot\Delta(E)D_{2}\cdots D_{n-1}+\beta_{r}$

is negative when $l$ is large enough. By the assumption of Theorem
(BI($r$))., (iii) we can find a saturated subsheaf $E^{\prime}$ of
$(F^{l})^{*}E$ with $\xi_{E^{\prime},(F^{l})^{*}E}\in K^{+}$. By the
definition of $K^{+}$ as in Definition 36, we have

$\xi_{E^{\prime},(F^{l})^{*}E}D_{1}\cdots D_{n-1}>0,$

in particular, the pullback $(F^{l})^{*}E$ is not semistable. Hence the
torsion-free sheaf $E$ itself is not strongly semistable. ∎

##  4\. Restriction to hypersurfaces and Bogomolov’s inequality

In this section, we prove one of the main technical ingredients Theorem 41,
which controls the change of various numerical invariants as one passes to
hypersurfaces. Following this, we complete the induction schema sketched in
the introduction. We follow the proof as in [5, $\S$3.5, $\S$3.9] and [4].
Combining the section with the equivalent statements in Theorem  (BI($r$)).,
we finish the proof of all theorems in [5, Section 3].

Fix a positive integer $r$, and let $X$ be a smooth projective variety over
$k$. Consider the following statement.

######  Theorem 41 (Res($r$)).

Let $E$ be a torsion-free sheaf of rank $r$ over $X$, and let
$(D_{1},\ldots,D_{n-1})$ be a collection of nef divisors over $X$ with
$d:=D_{1}^{2}D_{2}\cdots D_{n-1}\geq 0$.

Assume that $D_{1}$ is very ample and the restriction of $E$ to a very general
divisor $D\in|D_{1}|$ is not semistable (with respect to
$(D_{2}|_{D},...,D_{n-1}|_{D})$). Let $\mu_{i}$ ($r_{i}$) denote the slopes
(ranks) of the Harder-Narasimhan filtration of $E|_{D}$. Then

$\sum_{i<j}r_{i}r_{j}(\mu_{i}-\mu_{j})^{2}\leq
d\Delta(E)D_{2}...D_{n-1}+2r^{2}(L_{max}-\mu)(\mu-L_{min}).$

Our goal this section is to show the following two implications

$Res(r)+BI(r-1)\Longrightarrow BI(r)\Longrightarrow Res(r+1).$

Note that since both Theorem  (Res($r$)). and Theorem  (BI($r$)). are empty
and thus automatically true for $r=1$, the above induction process proves that
both Res(r) and BI(r) hold true for all $r\geq 1$.

###  $Res(r)$ $+$ $BI(r-1)\Rightarrow$ $BI(r)$

In this subsection, we consider the implication $Res(r)+BI(r-1)\Rightarrow
BI(r)$, following [5, $\S$3.5] and [4]. This is achieved via an inductive
argument on the dimension of $X$. We first prove the case when $X$ is a
surface. Further, we do not need to assume BI($r-1$) for the base case.

######  Proposition 42.

Let $X$ be a surface. Assume Theorem  (Res($r$)). holds for all torsion-free
sheaves of rank $\leq r$, then Theorem  (BI($r$)). holds for all torsion-free
sheaves of rank $\leq r$ and all ample divisor $D_{1}$.

###### Proof.

By the equivalence of statements in Theorem  (BI($r$))., it suffices to show
the assumption implies the statement in Theorem  (BI($r$))., (i), i.e. the
inequality $\Delta(E)\geq 0$, for a strongly $D_{1}$-semistable torsion-free
sheaf $E$ of rank $\leq r$, where $D_{1}$ is ample.

Let $E$ be a rank $r$ vector bundle that is strongly $D_{1}$-semistable.
Suppose by way of contradiction that $\Delta(E)<0$. We may further assume that
$E$ is locally free, by the inequality

$\Delta(E^{**})=\Delta(E)-2r\operatorname{length}(E^{**}/E),$

where the reflection $E^{**}$ is locally free on $X$. We first claim that
under the assumption, the restriction to a general curve $C\in|D_{1}|$ is also
strongly semistable. To see this, since $E$ is strongly semistable,
$\mu=L_{max}=L_{\min}$, and since (by assumption) $\Delta(E)<0$, from the
assumption we would get a contradiction

$0\leq\sum_{i<j}r_{i}r_{j}(\mu_{i}-\mu_{j})^{2}<0,$

where we use the notation in  (Res($r$)).. Similarly, as $(F^{k})^{*}E$ is
semistable, its restriction to a general curve $C$ is also semistable. Thus,
the restriction of $E$ to a very general curve $C$ is strongly semistable. For
the remainder of the proof, fix such a very general $C$.

Now consider the symmetric power $S^{kr}E|_{C}$. The strategy is to compute
$\chi(S^{kr}E)$ in two different ways and use them to deduce a contradiction.
Since symmetric powers of strongly semistable sheaves on curves are strongly
semistable (see [3, Corollary 3.2.10] for the characteristic zero case and [7,
Sections 3 and 5] for the positive characteristic case), it follows that
$S^{kr}E|_{C}$ is strongly semistable. Consider the short exact sequence
arising from $C$:

$0\to S^{kr}E(-kc_{1}(E)-C)\to S^{kr}E(-kc_{1}(E))\to
S^{kr}E(-kc_{1}(E))|_{C}\to 0.$

This allows us to estimate

$h^{0}(S^{kr}E(-kc_{1}(E))\leq
h^{0}(S^{kr}E(-kc_{1}(E)-C))+h^{0}(S^{kr}E(-kc_{1}(E))|_{C}).$

Consider the first term on the right hand side. This quantity is equal to
$\operatorname{dim}\left(\operatorname{Hom}(\mathcal{O}_{X}(kc_{1}(E)+C),S^{kr}E)\right)$,
and we claim that this number is zero. Note that the slope of
$\mathcal{O}_{X}(kc_{1}(E)+C)$ is given by $kr\mu(E)+D_{1}^{2}$, and by the
splitting principle, the slope of the symmetric power can be seen to be

$\mu(S^{kr}(E))=\frac{\binom{r+kr-1}{kr}kc_{1}(E)D_{1}}{\binom{r+kr-1}{kr}}=kr\mu(E).$

Since $D_{1}$ is ample, $D_{1}^{2}>0$, so the claim follows as there are no
morphisms from semistable sheaves of higher slope to those of lower slope.
Thus we get $h^{0}(S^{kr}E(-kc_{1}(E))\leq h^{0}(S^{kr}E(-kc_{1}(E))|_{C})$.

Given any semistable vector bundle $G$ on a curve, we have the further
estimate $h^{0}(G)\leq\max\\{0,\operatorname{deg}G+\operatorname{rk}G\\}$.
Using the sequence

$0\longrightarrow G(-lP)\longrightarrow G\longrightarrow
G|_{lP}\longrightarrow 0,$

the above inequality can be seen from the estimation $h^{0}(G)\leq
h^{0}(G\otimes\mathcal{O}_{C}(-lP))+\mathrm{rk}(G)l=0+\mathrm{rk}(G)l\leq\max\\{0,\mathrm{rk}G+\mathrm{deg}G\\}$,
where $P$ is a point on $C$, $l=\max\\{0,\lceil\mu(G)\rceil\\}$, and
$h^{0}(G\otimes\mathcal{O}_{C}(-lP))=0$ as $\mu(\mathcal{O}_{C}(lP))>\mu(G)$.

Thus, we have $h^{0}(S^{kr}E(-kc_{1}(E))=O(k^{r})$. By Serre duality, we have
the same order of magnitude estimate for $h^{2}(S^{kr}E(-kc_{1}(E))$. On the
other hand, the splitting principle for Chern classes and Hirzebruch-Riemann-
Roch theorem can be used to get the following estimate (for example, see [1,
Section 10])

$\chi(X,S^{kr}E(-kc_{1}(E))=-\frac{r^{r}\Delta(E)}{2(r+1)!}k^{r+1}+O(k^{r}).$

Since $\Delta(E)<0$, this polynomial is eventually positive and is of order
$k^{r+1}$, but this is an obvious contradiction as $h^{0}$ and $h^{2}$ are of
order $k^{r}$. ∎

To show the implication for general $X$, we induct on the dimension of $X$,
using the above surface case as the base case. However, different from the
assumption in Proposition 42, for this to work we require the additional
assumption of $BI(r-1)$ (i.e. Theorem $BI(r-1)$). Precisely, we want to show
that assuming

  * •

$BI(r-1)$ and $Res(r)$ for $X$ of any dimension;

  * •

$BI(r)$ for varieties of dimension $<n$,

then we have $BI(r)$ holds for any variety $X$ of dimension $n$.

###### Proof of Theorem $BI(r)$.

Similar to the proposition above, we aim to prove Theorem  (BI($r$))., (i) for
$X$ of dimension $n$, namely the inequality $\Delta(E)D_{2}\cdots D_{n-1}\geq
0$ for a strongly $(D_{1},\ldots,D_{n-1})$-semistable torsion-free sheaf $E$
over $X$, where $D_{i}$ are all ample.

Assume to the contrary that $\Delta(E)D_{2}\dots D_{n-1}<0$. For a general
divisor in $|D_{2}|$ (which we also denote as $D_{2}$), we have
$\Delta(E|_{D_{2}})D_{3}\dots D_{n-1}=\Delta(E)D_{2}\dots D_{n-1}<0$. Consider
the polarization
$(D_{2}^{2},D_{3},\ldots,D_{n-1}):=(D_{2},D_{2},D_{3},\ldots,D_{n-1})$. We
first assume that $E$ is strongly $(D^{2}_{2},D_{3},\dots
D_{n-1})$-semistable. Applying $Res(r)$ to the Frobenius twists $(F^{k})^{*}E$
with respect to $(D_{2}^{2},\ldots,D_{n-1})$, then if $(F^{k})^{*}E|_{D_{2}}$
is not semistable we would get the inequality

$\sum_{i<j}r_{i}r_{j}(\mu_{i}-\mu_{j})^{2}\leq dp^{k}\Delta(E)D_{2}\cdots
D_{n-1}+0<0,$

which is impossible. Thus $E|_{D_{2}}$ is strongly semistable for ample
divisors $(D_{2},\ldots D_{n-1})$, and by the induction hypothesis, since
$E|_{D_{2}}$ is defined over a hypersurface of $X$ we have
$\Delta(E|_{D_{2}})D_{3}\dots D_{n-1}=\Delta(E)D_{2}\dots D_{n-1}\geq 0$, a
contradiction.

For the rest, we assume $E$ is not strongly
$(D^{2}_{2},D_{3},\ldots,D_{n-1})$-semistable. The trick is to interpolate
between the polarizations $(D_{1},D_{2},\dots,D_{n-1})$ and
$(D_{2}^{2},D_{3},\dots,D_{n-1})$. Let $B_{t}$ be the product
$((1-t)D_{1}+tD_{2})D_{2}\dots D_{n-1}.$ By assumption, $E$ is strongly
$B_{0}$-semistable while it is not strongly $B_{1}$-semistable. Since not
being strongly semistable is an open condition with respect to $t$, there
exists a number $t_{k}\in[0,1)$ such that $E$ is strongly
$B_{t_{k}}$-semistable but not strongly $B_{t}$-semistable for all
$t_{k}<t\leq 1$.

Let $k$ be a positive integer such that $(F^{k})^{*}E$ is not semistable for
$B_{1}$. Using Theorem 19, we see that the Harder-Narasimhan filtration of $E$
with respect to $B_{t}$ remains constant for all $t_{k}<t<t_{k}+\epsilon$ for
some $\epsilon>0$ small enough. We let $E^{\prime}$ denote the maximal
destabilizing sheaf of $(F^{k})^{*}E$ with respect to the polarization $B_{t}$
for $t_{k}<t<t_{k}+\epsilon$. By the continuity of the slope with respect to
$t$, it follows that $E^{\prime}$ and $(F^{k})^{*}E$ have the same
$B_{t_{k}}$-slope (see Lemma 21). Since $E^{\prime}$ is a subsheaf of a
semistable sheaf with the same slope, $E^{\prime}$ is also semistable with
respect to $B_{t_{k}}$. It also follows that the quotient
$E^{\prime\prime}=(F^{k})^{*}E/E^{\prime}$ is $B_{t_{k}}$-semistable as well.
Let $r$, $r^{\prime}$ and $r^{\prime\prime}$ denote the ranks of $E$,
$E^{\prime}$ and $E^{\prime\prime}$ respectively.

Now we apply Lemma 38 to the short exact sequence $E^{\prime}\to E\to
E^{\prime\prime}$ to get that

$\frac{\Delta((F^{k})^{*}E)D_{2}\dots
D_{n-1}}{r}=\frac{\Delta(E^{\prime})D_{2}\dots
D_{n-1}}{r^{\prime}}+\frac{\Delta(E^{\prime\prime})D_{2}\dots
D_{n-1}}{r^{\prime\prime}}-\\\
\frac{r^{\prime}r^{\prime\prime}}{r}\xi_{E^{\prime},E^{\prime\prime}}^{2}D_{2}\dots
D_{n-1}.$

Note that by the Hodge index theorem, we have

$\xi_{E^{\prime},E^{\prime\prime}}^{2}D_{2}\dots
D_{n-1}\cdot(t_{k}D_{1}+(1-t_{k})D_{2})^{2}D_{2}\dots
D_{n-1}\leq(\xi_{E^{\prime},E^{\prime\prime}}B_{t_{k}})^{2}.$

Since $E^{\prime},E^{\prime\prime}$ have the same $B_{t_{k}}$-slope,
$(\xi_{E^{\prime},E^{\prime\prime}}B_{t_{k}})^{2}=0$ and by ampleness of
$D_{1},\dots,D_{n-1}$, we have that

$d(t_{k}):=(t_{k}D_{1}+(1-t_{k})D_{2})^{2}D_{2}\dots D_{n-1}>0.$

Moreover by Theorem  (BI($r$))., (ii) but for $(r-1)$ in place of $r$, we get
that the inequalities as below

$\frac{\Delta((F^{k})^{*}E)D_{2}\dots
D_{n-1}}{r}\geq-\frac{1}{d(t_{k})}\left(\frac{\beta_{r^{\prime}}(t_{k})}{r^{\prime}}+\frac{\beta_{r^{\prime\prime}}(t_{k})}{r^{\prime\prime}}\right)\geq-\frac{\beta_{r}(t_{k})}{rd(t_{k})},$

where we denote $\beta_{r}(t)$ to be
$\beta_{r}((1-t)D_{1},+tD_{2},D_{2},\ldots,D_{n-1})$. In this way, we get

$\Delta(E)D_{2}\cdots D_{n-1}\geq-\frac{\beta_{r}(t_{k})}{d(t_{k})p^{2k}}.$

Notice that the function $\frac{\beta_{r}(t)}{d(t)}$ for $t\in[0,1]$ is
defined and continuous, thus bounded. Taking the limit as $k$ approaches
infinity, we get the inequality

$\Delta(E)D_{2}\cdots D_{n-1}\geq 0,$

which is a contradiction, so we are done. ∎

######  Remark 43.

Pointed out by the referee, the last paragraph of the proof above can be
argued as follows. As $E^{\prime}$ and $E^{\prime\prime}$ are of the same
$B_{t_{k}}$-slopes as the strongly $B_{t_{k}}$-semistable sheaf $E$, both of
them are strongly $B_{t_{k}}$-semistable. Thus applying the induction
hypothesis of Theorem  (BI($r$))., (i) to $E^{\prime}$, $E^{\prime\prime}$ for
the polarization $B_{t_{k}}$, we see

$\frac{\Delta(E^{\prime})D_{2}\dots
D_{n-1}}{r^{\prime}}+\frac{\Delta(E^{\prime\prime})D_{2}\dots
D_{n-1}}{r^{\prime\prime}}\geq 0.$

In this way, by the same vanishing of
$\xi_{E^{\prime},E^{\prime\prime}}D_{2}\cdots D_{n-1}$ proved above, we get
the positivity of $\Delta(E)D_{2}\cdots D_{n-1}$.

### $BI(r)\Rightarrow Res(r+1)$

We then consider the implication $BI(r)\Rightarrow Res(r+1)$, following [5,
$\S$3.9].

Let $r$ be a non-negative integer, and $(D_{1},\ldots,D_{n-1})$ be a nef
polarization over $X$ with $d=D_{1}^{2}D_{2}\cdots D_{n-1}\geq 0$.

###### Proof.

Let $\Pi$ be the projective space associated to the linear system $|D_{1}|$.
Let $Z\subset\Pi\times X$ be the incident scheme, defined as the locus
$\\{(D,x)\in\Pi\times X\leavevmode\nobreak\ |\leavevmode\nobreak\ x\in D\\}$,
where $p:Z\rightarrow\Pi$ and $q:Z\rightarrow X$ are the two natural maps
induced by the projections. For each $s\in\Pi$, we let $Z_{s}$ be its fiber
along the map $q$, which is a divisor in $X$. Let $0=E_{0}\subset
E_{1}\subset\cdots\subset E_{m}=q^{*}E$ be the Harder–Narasimhan filtration of
$q^{*}E$ associated to the polarizaion

$(p^{*}\mathcal{O}_{\Pi}(1)^{\operatorname{dim}\Pi},q^{*}D_{2},\ldots,q^{*}D_{n-1})=\\\
(p^{*}\mathcal{O}_{\Pi}(1),\ldots,p^{*}\mathcal{O}_{\Pi}(1),q^{*}D_{2},\ldots,q^{*}D_{n-1})$

over $Z$. Denote $F_{i}$ to be the subquotient $E_{i}/E_{i-1}$. Here we
observe that the pull back of the filtration $\\{E_{i}\\}$ at a general fiber
$Z_{s}$ for $s\in\Pi$ is the Harder–Narasimhan filtration of $E|_{Z_{s}}$ at
the hypersurface $Z_{s}\in|D|$. Indeed, the slope of $q^{*}E$ with respect to
$(p^{*}\mathcal{O}_{\Pi}(1)^{\operatorname{dim}\Pi},q^{*}D_{2},\ldots,q^{*}D_{n-1})$
is the same as the slope of $q^{*}E|_{Z_{s}}$. This shows that the filtration
$\\{E_{i}\\}$ is also the relative Harder-Narasimhan filtraion for $q^{*}E$
with respect to the map $p$. In particular, we have the equalities
$r_{i}=\mathrm{rk}F_{i}$ and $\mu_{i}=\mu(F_{i})$, where the latter is with
respect to the polarization given by
$(p^{*}\mathcal{O}_{\Pi}(1)^{\operatorname{dim}\Pi},q^{*}D_{2},\ldots,q^{*}D_{n-1})$.

Now let $\mathbb{P}^{1}\cong\Lambda\subset\Pi$ be a sufficiently general
pencil, corresponding to a linear subsystem of $|D|$ in $X$ parametrized by a
line $\mathbb{P}^{1}\subset\Pi$. Let $B\subset X$ be the base of $\Lambda$,
which is the intersection of $Z_{s}$ for any two (hence all) $s\in\Lambda$,
and is of codimension $2$ in $X$. Denote by $Y$ the incident scheme for
$\Lambda$, which is equal to the closed subscheme $p^{-1}\Lambda$ inside of
$Z$. Here by the construction of the incident scheme for a pencil, the
projection map $Y=p^{-1}\Lambda\rightarrow X$ coincides with the blowup of $X$
at the base locus $B$. Moreover, depending on the dimension of $X$, we can
write the first Chern class of the restriction $F_{i}|_{Y}$ in terms of the
following:

  * •

If $\operatorname{dim}(X)=2$, then $B$ consists of a union of a finite amount
of points, and we may write each $c_{1}(F_{i}|_{Y})$ as
$M_{i}+\sum_{j}b_{ij}N_{j}$, with $M_{i}\in\mathrm{Pic}(X)$ and $N_{j}$ being
the $j$-th exceptional divisor for $1\leq j\leq l$. We define $b_{i}$ to be
the number $\frac{\sum_{j}b_{ij}}{l}$.

  * •

If $\operatorname{dim}(X)\geq 3$, then by Bertini’s theorem the blowup center
$B$ is a smooth connected closed subscheme of codimension two in $X$, and we
may write each $c_{1}(F_{i}|_{Y})$ as $M_{i}+b_{i}N$, for
$M_{i}\in\mathrm{Pic}(X)$ and $N$ being the exceptional divisor.

In any of the above cases, with respect to the polarization

$(p^{*}\mathcal{O}_{\Lambda}(1),q^{*}D_{2},\ldots,q^{*}D_{n-1})$

over $Y$, we can write the slope $\mu_{i}$ as the following

$\mu_{i}=\frac{M_{i}D_{1}\cdots D_{n-1}+b_{i}d}{r_{i}}.$

Here as the collection of torsion-free sheaves $\\{F_{i}\\}$ forms the graded
pieces of a filtration of $q^{*}E$, we have

$\sum_{i}b_{i}=c_{1}(q^{*}E)p^{*}\mathcal{O}_{\Lambda}(1)q^{*}D_{2}\cdots
q^{*}D_{n-1}=0.$

Furthermore, since $(q|_{Y})_{*}(E_{i}|_{Y})\subset E$, we have the
inequalities

$\frac{\sum_{j\leq i}M_{j}D_{1}\cdots D_{n-1}}{\sum_{j\leq
i}r_{j}}\leq\mu_{\max},\text{ for all }i,$

which is equivalent to the following inequalities

$\sum_{j\leq i}b_{j}d\geq\sum_{j\leq i}r_{j}(\mu_{j}-\mu_{\max}).$

Finally, we are ready to prove the implication. As each $F_{i}$ is of rank
$\leq r-1$, by Theorem  (BI($r$))., (i) for the polarization
$(p^{*}\mathcal{O}_{\Pi}(1)^{\operatorname{dim}\Pi},q^{*}D_{2},\ldots,q^{*}D_{n-1})$
over $Z$, we have

$\Delta(F_{i})p^{*}\mathcal{O}_{\Pi}(1)^{\operatorname{dim}\Pi}q^{*}D_{2}\cdots
q^{*}D_{n-1}=\\\
\Delta(F_{i}|_{Y})p^{*}\mathcal{O}_{\Lambda}(1)q^{*}D_{2}\cdots
q^{*}D_{n-1}\geq 0,\text{ for all }i.$

Applying this, we get

$\displaystyle\frac{d\Delta(E)D_{2}\cdots D_{n-1}}{r}$
$\displaystyle=\sum_{i}\frac{d\Delta(F_{i}|_{Y})p^{*}\mathcal{O}_{\Lambda}(1)q^{*}D_{2}\cdots
q^{*}D_{n-1}}{r_{i}}$
$\displaystyle-\frac{d}{r}\sum_{i<j}r_{i}r_{j}\left(\frac{c_{1}(F_{i}|_{Y})}{r_{i}}-\frac{c_{1}(F_{j}|_{Y})}{r_{j}}\right)^{2}(q|_{Y})^{*}(D_{2}\cdots
D_{n-1})$
$\displaystyle\geq-\frac{d}{r}\sum_{i<j}r_{i}r_{j}\left(\frac{c_{1}(F_{i}|_{Y})}{r_{i}}-\frac{c_{1}(F_{j}|_{Y})}{r_{j}}\right)^{2}(q|_{Y})^{*}(D_{2}\cdots
D_{n-1})$
$\displaystyle=\frac{d}{r}\sum_{i<j}r_{i}r_{j}\left(d(\frac{b_{i}}{r_{i}}-\frac{b_{j}}{r_{j}})^{2}-(\frac{M_{i}}{r_{i}}-\frac{M_{j}}{r_{j}})^{2}D_{2}\cdots
D_{n-1}\right)$
$\displaystyle\geq\frac{1}{r}\sum_{i<j}r_{i}r_{j}\left(d^{2}(\frac{b_{i}}{r_{i}}-\frac{b_{j}}{r_{j}})^{2}-(\frac{M_{i}D_{1}\ldots
D_{n-1}}{r_{i}}-\frac{M_{j}D_{1}\cdots D_{n-1}}{r_{j}})^{2}\right),$

where the first inequality is the application of the statement (i) as above,
the equality after is a rearrangement using the formula (0), and the last
inequality follows from the Hodge index theorem applied to
$\frac{M_{i}}{r_{i}}-\frac{M_{j}}{r_{j}}$ and the polarization
$D_{1},\cdots,D_{n-1}$. Moreover, using the formula (0), we may rewrite the
last expression of the inequalities above as

$\frac{2d}{r}\sum_{i<j}(\mu_{i}-\mu_{j})(b_{i}r_{j}-b_{j}r_{i})-\frac{1}{r}\sum_{i<j}r_{i}r_{j}(\mu_{i}-\mu_{j})^{2}$

Using (1), one further simplifies the above to

$2\sum_{i}db_{i}\mu_{i}-\frac{1}{r}\sum_{i<j}r_{i}r_{j}(\mu_{i}-\mu_{j})^{2}.$

Here more concretely, the simplification above can be seen via an induction on
$m$. If $m=1$, this is obvious. Suppose this is true for $m-1$. Then

$\displaystyle\frac{2d}{r}\sum_{i<j}(\mu_{i}-\mu_{j})(b_{i}r_{j}-b_{j}r_{i})$
$\displaystyle=2\sum_{i<j\leq
m-1}(\mu_{i}-\mu_{j})((b_{i}+\frac{b_{m}}{m})r_{j}-(b_{j}+\frac{b_{m}}{m})r_{i})$
$\displaystyle+\frac{b_{m}}{m}\sum_{i<j\leq
m-1}(\mu_{i}-\mu_{j})(r_{i}-r_{j})+\frac{2d}{r}\sum_{i\leq
m-1}(\mu_{i}-\mu_{m})(b_{i}r_{m}-b_{m}r_{i})$
$\displaystyle=\frac{2(r-r_{m})}{r}\sum_{i\leq
m-1}d(b_{i}+\frac{b_{m}}{m})\mu_{i}$
$\displaystyle+\frac{2d}{r}\left(r_{m}\sum_{i\leq m-1}b_{i}\mu_{i}-b_{m}(\mu
r-\mu_{m}r_{m})+b_{m}\mu_{m}r_{m}+b_{m}\mu_{m}(r-r_{m})\right)$
$\displaystyle+\frac{b_{m}}{m}\sum_{i<j\leq
m-1}(\mu_{i}-\mu_{j})(r_{i}-r_{j})$ $\displaystyle=2\sum_{i}db_{i}\mu_{i}$

where the second equality follows from induction hypothesis. Furthermore,
using the formula (2) together with an elementary equality

$\sum_{i}a_{i}b_{i}=\sum_{i}(\sum_{j\leq i}a_{j})(b_{i}-b_{i+1}),$

we get

$\displaystyle\sum_{i}db_{i}\mu_{i}$ $\displaystyle=\sum_{i}(\sum_{j\leq
i}db_{j})(\mu_{i}-\mu_{j})$ $\displaystyle\geq\sum_{i}\left(\sum_{j\leq
i}r_{j}(\mu_{j}-\mu_{\max})(\mu_{i}-\mu_{i+1})\right)$
$\displaystyle=\sum_{i}(r_{i}\mu_{i}^{2}-r\mu\mu_{\max})$
$\displaystyle\geq\sum_{i}r_{i}\mu_{i}^{2}-r\mu^{2}+r(\mu-\mu_{\max})(\mu-\mu_{\min})$
$\displaystyle=\sum_{i<j}\frac{r_{i}r_{j}}{r}(\mu_{i}-\mu_{j})^{2}+r(\mu-\mu_{\max})(\mu-\mu_{\min}).$

As a consequence, we get

$\frac{d\Delta(E)D_{2}\cdots
D_{n-1}}{r}\geq\sum_{i<j}\frac{r_{i}r_{j}}{r}(\mu_{i}-\mu_{j})^{2}+2r(\mu-\mu_{\max})(\mu-\mu_{\min}).$

At last, by moving the second term in the right hand side above and the
inequalities $L_{\max}\geq\mu_{\max}$, $\mu_{\min}\geq L_{\min}$, we get the
one as in Theorem  (Res($r$)).. ∎

##  5\. Boundedness of torsion-free sheaves

At last, we use the ingredients in the last two sections to show the
boundedness, following [5, Section 4]

We start with the following result combining the inequalities in the last two
sections into a uniform one, following [5, Corollary 3.11].

######  Corollary 44.

Let $E$ be a torsion-free sheaf on $X$ and $D_{1}$ be very ample,
$D_{2},\dots,D_{n-1}$ be ample, and let $D\in|D_{1}|$ be a general divisor.
Then, we have that

$\frac{r}{2}(L_{\max}(E|_{D})-L_{\min}(E|_{D}))^{2}\leq d\Delta(E)D_{2}\dots
D_{n-1}+2r^{2}(L_{\max}-\mu)(\mu-L_{\min})$

###### Proof.

First consider the case when $E|_{D}$ is not strongly semistable. Then, the
inequality follows from Theorem  (Res($r$)). as well as an elementary
inequality [5, Lemma 1.3] as below:

$\sum_{i<j}r_{i}r_{j}(\mu_{i}-\mu_{j})^{2}\geq\frac{r_{1}r_{m}}{r_{1}+r_{m}}r(\mu_{1}-\mu_{m})^{2}\geq\frac{r}{2}(\mu_{1}-\mu_{m})^{2}$

where $r_{1},\dots,r_{m}$ are positive real numbers and
$\mu_{1},\dots,\mu_{m}$ are real numbers and $r=r_{1}+\dots+r_{m}$.

Now consider the case when $E|_{D}$ is strongly semistable. Then, the left-
hand side of the inequality is just zero and the result follows from the
inequality in Remark 35 (where the latter is proved using Theorem  (BI($r$)).,
(i)). ∎

The boundedness is deduced from the following result. A more precise version
is given in [5].

######  Theorem 45.

Let $H_{1},\dots,H_{n-1}$ be very ample divisors on $X$ and let
$X_{l}=|H_{1}|\cap\dots\cap|H_{l}|$, $1\leq l\leq n-1$ be very general
complete intersections. Pick a nef divisor $A$ such that $T_{X_{l}}(A)$ is
globally generated for all $0\leq l\leq n-1$. Set
$\beta_{r}=\beta(r;A,H_{1},\dots,H_{n-1})$. Let $\mu_{\max,l},\mu_{\min,l}$
denote the maximal and minimal slopes of the Harder-Narasimhan filtration of
$E|_{X_{l}}$. Then we have the following inequality,

$\mu_{\max,l}-\mu_{\min,l}\leq\\\
C(r,n)\left(\sqrt{\max\\{d\Delta(E)H_{2}\dots
H_{n-1},0\\}}+\sqrt{\beta_{r}}+(\mu_{\max}-\mu_{\min})\right),$

where $C(r,n)$ is a constant depending only on $r$ and $n$.

######  Remark 46.

Before we move on to the proof, it is worth mentioning that except for
$\mu_{\max}-\mu_{\min}$, the rest of terms in the right hand side together
with $\mu_{\min,l}$ can all be bounded using constants depending only on $X$
and the Hilbert polynomial of $E$.

###### Proof.

For $l=1$, we use Corollary 44, and get

$\displaystyle(L_{\max,1}-L_{\min,1})^{2}$
$\displaystyle\leq\frac{2}{r}d\Delta(E)H_{2}\dots
H_{n-1}+4r(L_{\max}-\mu)(\mu-L_{\min})$
$\displaystyle\leq\frac{2}{r}\max\\{d\Delta(E)H_{2}\dots
H_{n-1},0\\}+4r(L_{\max}-L_{\min})^{2}$ $\displaystyle\leq
4r\left(\max\\{d\Delta(E)H_{2}\dots
H_{n-1},0\\}+(L_{\max}-L_{\min})^{2}\right)$

Now using the definition and Proposition 28 that $\mu_{\max}-\mu_{\min}\leq
L_{\max}-L_{\min}\leq\mu_{\max}-\mu_{\min}+\frac{2\sqrt{\beta_{r}}}{r}$, we
get

$(\mu_{\max,1}-\mu_{\min,1})^{2}\\\ \leq 4r\left(\max\\{d\Delta(E)H_{2}\dots
H_{n-1},0\\}+(\mu_{\max}-\mu_{\min}+2\sqrt{\beta_{r}})^{2}\right)$

Using $\sqrt{a+b}\leq\sqrt{a}+\sqrt{b}$ for $a,b\geq 0$, we get

$\mu_{\max,1}-\mu_{\min,1}\\\ \leq
4\sqrt{r}\left(\sqrt{\max\\{d\Delta(E)H_{2}\dots
H_{n-1},0\\}}+(\mu_{\max}-\mu_{\min})+\sqrt{\beta_{r}}\right),$

which gives the required inequality for $l=1$. For higher $l$, we use
induction. ∎

Now we are ready to prove the main theorem of the article, which states that
the moduli space, parametrizing torsion-free sheaves with a fixed Hilbert
polynomial, and with a upper-bound for the slopes of their Harder-Narasimhan
filtrations, is bounded.

######  Theorem 47.

Let $X$ be a projective variety over an algebraically closed field with an
ample line bundle $\mathcal{O}_{X}(1)$. Let $P$ be a polynomial of degree $d$
and let $\mu_{0}\in\mathbb{R}$. Then, the family of torsion-free sheaves whose
Hilbert polynomial are $P$ and whose $\mu_{\max}$ are at most $\mu_{0}$ is
bounded.

###### Proof.

We use the same notation as in Theorem 45. According to Kleiman’s criteria
(Theorem 14) and Lemma 15, to prove boundedness, it is enough to give an upper
bound of $\mu_{\max,l}$ (or more specifically for $l=1$). Note that
$\mu_{l}=\mu(E|_{X_{l}})$ is independent of $E$ and depends only on $P$ for
all $0\leq l\leq n$. To get such an upper bound, we use Theorem 45 to get

$\displaystyle\mu_{\max,l}(E)$
$\displaystyle\leq\mu_{l}+(\mu_{\max,l}(E)-\mu_{\min,l}(E))$
$\displaystyle\leq C_{1}+C_{2}(\mu_{\max}(E)-\mu_{\min}(E))$
$\displaystyle\leq C_{1}+C_{2}r(\mu_{0}-\mu),$

where $C_{1},C_{2}$ are constants independent of $E$. Here the last inequality
follows from $\mu_{\max}\leq\mu_{0}$ and the observation that
$r\mu\leq\mu_{\min}(E)+(r-1)\mu_{\max}(E)$. ∎

A more general version of the above theorem for pure-dimensional sheaves can
be proved by imitating the proof in [8, Theorem 1.1].

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