# Positivity vs. slope semistability for bundles with vanishing discriminant

Mihai Fulger  Department of Mathematics, University of Connecticut, Storrs, CT
06269-1009, USA  Institute of Mathematics of the Romanian Academy, P. O. Box
1-764, RO-014700, Bucharest, Romania<EMAIL_ADDRESS>and  Adrian
Langer  Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097
Warszawa, Poland<EMAIL_ADDRESS>
###### Abstract.

It is known that a strongly slope semistable bundle with vanishing
discriminant is nef if and only if its determinant is nef. We give an
algebraic proof of this result in all characteristics and generalize it to
arbitrary proper schemes. We also address a question of S. Misra.

###### 2010 Mathematics Subject Classification:

Primary 14C17, 14C20, 14F06, 14H60, 14N05

The first author was partially supported by the Simons Foundation
Collaboration Grant 579353. The second author was partially supported by
Polish National Centre (NCN) contract numbers 2018/29/B/ST1/01232 and
2021/41/B/ST1/03741.

##  1\. Introduction

Let $\mathcal{E}$ be a vector bundle of rank $r$ on a smooth projective
variety $X$ defined over an algebraically closed field. Inspired by the case
of line bundles, one might hope that the positivity of $\mathcal{E}$ is
determined by the positivity of its characteristic classes. The bundle
$\mathcal{O}_{\mathbb{P}^{1}}(n)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-n)$ is an
easy counterexample. To rectify this, one adds stability assumptions on
$\mathcal{E}$.

Let $H$ be an ample polarization on $X$. In characteristic zero, say that
$\mathcal{E}$ is _strongly slope semistable with respect to $H$_ if it is
slope semistable in the usual sense. In positive characteristic, say that
$\mathcal{E}$ is _strongly slope semistable with respect to $H$_ if
$\mathcal{E}$ and all its iterated Frobenius pullbacks are slope semistable.

On curves the polarization is irrelevant and we just say that $\mathcal{E}$ is
_strongly semistable_. Here the connection between positivity and strong
semistability is well known by work of Hartshorne [Har71], Barton [Bar71], and
Miyaoka [Miy87]. A strongly semistable bundle $\mathcal{E}$ on a curve is
ample (or just nef) if and only if $\int_{X}c_{1}(\mathcal{E})>0$ (resp. $\geq
0$). On surfaces, even on $\mathbb{P}^{2}$, it is not sufficient. See Example
2.1. Furthermore, on curves $\mathcal{E}$ is strongly semistable if and only
if the twisted normalized bundle
$\mathcal{E}\langle-\frac{1}{r}\det\mathcal{E}\rangle$ is nef (equivalently
$\mathcal{E}{\rm nd}\,\mathcal{E}$ is nef).

A link in codimension two between semistability and positivity comes from the
famous Bogomolov inequality. The discriminant of $\mathcal{E}$ is
$\Delta(\mathcal{E})=2rc_{2}(\mathcal{E})-(r-1)c_{1}^{2}(\mathcal{E})$. Assume
that $\mathcal{E}$ is strongly slope semistable with respect to some ample
polarization. The classical form of the inequality states that if $X$ is a
surface, then the degree of the discriminant of $\mathcal{E}$ is non-negative.
When $X$ has arbitrary dimension, the Mehta–Ramanathan theorem [MR82] implies
that $\Delta(\mathcal{E})$ has nonnegative degree with respect to any
polarization.

Note that $\Delta(\mathcal{E})=0$ on curves. This suggests a close connection
between (strong) semistability and positivity for vector bundles that are
extremal with respect to the Bogomolov inequality, meaning that
$\Delta(\mathcal{E})$ is numerically trivial. We have the following known
results:

######  Theorem 1.1.

Let $X$ be a smooth projective variety of dimension $n$, and let $H$ be an
ample class on $X$. Let $\mathcal{E}$ be a reflexive sheaf of rank $r$ on $X$.
The following are equivalent:

  1. (1)

$\mathcal{E}$ is strongly slope semistable with respect to $H$, and
$\Delta(\mathcal{E})\cdot H^{n-2}=0$.

  2. (2)

$\mathcal{E}$ is locally free and $\mathcal{E}{\rm nd}\,\mathcal{E}$ is nef.

  3. (3)

$\mathcal{E}$ is universally semistable (see below).

In particular, if $\mathcal{E}$ is strongly slope semistable with respect to
$H$, and $\Delta(\mathcal{E})\cdot H^{n-2}=0$, then $\mathcal{E}$ is a nef
(resp. ample) vector bundle if and only if $\det\mathcal{E}$ is nef (resp.
ample).

In the above theorem we say that $\mathcal{E}$ is _universally semistable_ if
for every morphism $f:Y\to X$ from any projective manifold $Y$, the pullback
$f^{*}\mathcal{E}$ is slope semistable with respect to any ample polarization
on $Y$. It is sufficient to check this condition for morphisms from curves.

When $\det\mathcal{E}$ is numerically trivial, condition $(2)$ is simply that
the vector bundle $\mathcal{E}$ is nef, equivalently $\mathcal{E}$ is
_numerically flat_ ($\mathcal{E}$ and $\mathcal{E}^{\vee}$ are nef). Theorem
1.1 takes the following form:

######  Theorem 1.2.

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically
closed field, and let $H$ be an ample polarization. Let $\mathcal{E}$ be a
reflexive sheaf on $X$. The following are equivalent:

  1. (1)

$\mathcal{E}$ is strongly slope semistable with respect to $H$, and
$c_{1}(\mathcal{E})\cdot H^{n-1}=\operatorname{ch}_{2}(\mathcal{E})\cdot
H^{n-2}=0$.

  2. (2)

$\mathcal{E}$ is universally semistable and all the Chern classes
$c_{i}(\mathcal{E})$ are numerically trivial for $i\geq 1$.

  3. (3)

$\mathcal{E}$ is locally free and numerically flat.

In particular, if $\mathcal{E}$ is a strongly slope semistable with respect to
$H$ with $c_{1}(\mathcal{E})$ numerically trivial, then $\mathcal{E}$ is nef
if and only if $\Delta(\mathcal{E})$ is numerically trivial.

The result fails if $\mathcal{E}$ is only assumed to be torsion free. It also
fails if the condition $\operatorname{ch}_{2}(\mathcal{E})\cdot H^{n-2}=0$ in
(1) is replaced by $c_{2}(\mathcal{E})\cdot H^{n-2}=0$.

The two theorems are essentially equivalent. As mentioned before, they are not
new. In characteristic zero, Theorem 1.1 is proved by [Nak99]. It is proved by
[GKP16] for more general polarizations by movable curves on varieties with
mild singularities. It is proved by [BB08] for principal $G$-bundles, and by
[BHR06] as part of their study of Higgs bundles. Note that for Higgs bundles
with nontrivial Higgs field it is not known whether an analogue of
$(3)\Rightarrow(1)$ holds (this is known as Bruzzo’s conjecture). See [BBG19].
The proofs given in the references above have important transcendental
components coming from [Sim92] or [DPS94]. The last statement of Theorem 1.1
has also been observed by [MR21]. In positive characteristic, a version of the
result is proved algebraically by the second named author in [Lan11] making
crucial use of the Frobenius morphism. In characteristic zero, [Lan19,
Corollary 4.10] gives an algebraic proof of the implication
$(1)\Rightarrow(3)$ for the more general case of Higgs bundles by reduction to
positive characteristic. See also [Lan15, Theorem 12]. In characteristic zero,
Theorem 1.2 is proved in [Sim92, Theorem 2]. A positive characteristic version
appears in [Lan11, Proposition 5.1].

A conjecture of Bloch predicts that the Chern classes of numerically flat
bundles vanish in $B^{*}(X)\otimes\mathbb{Q}$, where $B^{m}(X)$ is the group
of codimension $m$ cycles modulo algebraic equivalence (see [Lan21, Conjecture
3.2]). This would strengthen Theorem 1.2. However, this is known only if $k$
has positive characteristic (see [Lan21, Corollary 3.7]). In this case the
result holds also for general proper schemes (see Corollary 3.10).
Analogously, one can formulate a similar conjecture and a result for bundles
in Theorem 1.1 (see [Lan21, Proposition 3.6]). If $k=\mathbb{C}$ one can prove
only a weaker version of the vanishing of Chern classes in the rational
cohomology $H^{2*}(X^{\rm an},\mathbb{Q})$, where $X^{\rm an}$ is the complex
manifold underlying variety $X$. This result would follow from Bloch’s
conjecture by using the cycle map $B^{*}(X)\otimes\mathbb{Q}\to H^{2*}(X^{\rm
an},\mathbb{Q})$.

###  1.1. Main results

We give algebraic proofs for Theorems 1.1 and 1.2 that are characteristic
free. We avoid the application of the non-abelian Hodge theorem [Sim92,
Corollary 1.3] or the reduction to positive characteristic techniques of
[Lan11, Lan19]. Moreover, we generalize both theorems to general proper
schemes over an algebraically closed field (see Theorem 3.13 and Corollary
3.14).

###  1.2. On a question of S. Misra

On curves $C$ it follows from [Miy87] that $\mathcal{E}$ is strongly
semistable if and only if every effective divisor on
$\mathbb{P}_{C}(\mathcal{E})$ is nef. The first named author proved in [Ful11]
a generalization of this for cycles of arbitrary (co)dimension in
$\mathbb{P}_{C}(\mathcal{E})$. It is interesting to see to what extent does
Miyaoka’s result carry over to a projective manifold $X$ of arbitrary
dimension. The equality
$\operatorname{\overline{Eff}}(X)=\operatorname{{Nef}}(X)$ of cones of
divisors is a necessary condition which held trivially on curves. Here one has
the following result:

######  Theorem 1.3.

Let $X$ be a smooth projective variety such that
$\operatorname{\overline{Eff}}(X)=\operatorname{{Nef}}(X)$. Let $\mathcal{E}$
be a strongly slope semistable bundle with respect to some ample polarization
of $X$ and assume that $\Delta(\mathcal{E})\equiv 0$. Then
$\operatorname{\overline{Eff}}(\mathbb{P}(\mathcal{E}))=\operatorname{{Nef}}(\mathbb{P}(\mathcal{E}))$.

In the characteristic zero case this result was proven by S. Misra in [Mis21,
Theorem 1.2] as an application of Theorem 1.1. A similar proof works also in
an arbitrary characteristic. Misra [Mis21, Question 3.11] also asks about a
possible converse to this result.

######  Question 1.4.

Let $X$ be a smooth projective variety. Let $\mathcal{E}$ be a vector bundle
on $X$ such that every effective divisor on $\mathbb{P}(\mathcal{E})$ is nef.
Is it true that $\mathcal{E}$ is slope semistable with respect to some (any)
polarization of $X$ and $\Delta(\mathcal{E})\equiv 0$? Or equivalently, is
$\operatorname{{\mathcal{E}}nd\,}\mathcal{E}$ numerically flat?

For every $n\geq 2$, the tangent bundle $T_{\mathbb{P}^{n}}$ is a
counterexample to the current phrasing of the question. In Example 5.7 we even
give a slope unstable counterexample. However [Mis21] also observes that under
the hypothesis of Theorem 1.3 the effective and nef cones of divisors coincide
on $\mathbb{P}(\operatorname{Sym}^{m}\mathcal{E})$ for all $m\geq 0$. This
motivates the following positive answer to a version of Question 1.4 in
arbitrary characteristic.

######  Theorem 1.5.

Let $X$ be a smooth projective variety. Let $\mathcal{E}$ be a vector bundle
on $X$ such that every effective divisor on
$\mathbb{P}(\operatorname{Sym}^{m}\mathcal{E})$ is nef for all $m\geq 0$. Then
$\mathcal{E}$ is strongly slope semistable with respect to any polarization
$H$ and $\Delta(\mathcal{E})\equiv 0$.

Our proof shows that in fact existence of only one positive _even_ value $2m$
for which the cones
$\operatorname{\overline{Eff}}(\mathbb{P}(\operatorname{Sym}^{2m}\mathcal{E}))$
and $\operatorname{{Nef}}(\mathbb{P}(\operatorname{Sym}^{2m}\mathcal{E}))$
coincide is sufficient. The key idea is a result on plethysms which guarantees
that the line bundle $(\det\mathcal{E})^{\otimes 2m}$ is a subbundle of
$\operatorname{Sym}^{r}\operatorname{Sym}^{2m}\mathcal{E}$. In characteristic
zero, this also follows from [BCI11]. One can also see that equality
$\operatorname{\overline{Eff}}(\mathbb{P}(\operatorname{{\mathcal{E}}nd\,}\mathcal{E}))=\operatorname{{Nef}}(\mathbb{P}(\operatorname{{\mathcal{E}}nd\,}\mathcal{E}))$
implies that $\operatorname{{\mathcal{E}}nd\,}\mathcal{E}$ is numerically
flat, which provides a satisfactory answer to the original question.

###  1.3. Acknowledgments

The authors would like to thank S. Misra, D. S. Nagaraj and J. Weyman for
useful comments, suggestions, and references.

##  2\. Preliminaries

Let $X$ be a connected proper scheme over an algebraically closed field $k$
and $\mathcal{E}$ be a vector bundle on $X$. Note that terms vector bundle and
locally free sheaf are used interchangeably (since $X$ is connected, a locally
free sheaf has the same rank at every point of $X$).

###  2.1. Positive bundles

Let $\mathbb{P}(\mathcal{E})=\mathbb{P}_{X}(\mathcal{E})={\rm
Proj}_{\mathcal{O}_{X}}\operatorname{Sym}^{\bullet}\mathcal{E}$ with natural
bundle map $\pi:\mathbb{P}(\mathcal{E})\to X$, and let
$\xi=c_{1}(\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1))$. We say that
$\mathcal{E}$ is _ample_ (resp.  _nef_) on $X$, if $\xi$ is ample (resp. nef)
on $\mathbb{P}(\mathcal{E})$. The definition also makes sense for coherent
sheaves.

###  2.2. Numerically trivial Chern classes

In the following we write $A_{*}(X)$ for the Chow group of rational
equivalence classes on $X$ and $A^{*}(X)=A^{*}(X\stackrel{{\scriptstyle\rm
id}}{{\longrightarrow}}X)$ for the operational Chow ring, i.e., the group of
bivariant rational equivalence classes (see [Ful98, Chapter 17]). The Chern
classes of $\mathcal{E}$ are operations on the Chow group (see [Ful98, Chapter
3]) so they are elements of $A^{*}(X)$. Following [Ful98, Chapter 19], we say
that $c_{j}(\mathcal{E})$ is _numerically trivial_ if for every proper closed
subscheme $Y\subseteq X$ of dimension $j$ we have
$\int_{Y}c_{j}(\mathcal{E})=\int_{X}c_{j}(\mathcal{E})\cap[Y]=0$, where
$\int_{X}:A_{0}(X)\to\mathbb{Z}$ denotes the natural degree map. Then we write
$c_{j}(\mathcal{E})\equiv 0$. By additivity it is sufficient to check the
condition $\int_{X}c_{j}(\mathcal{E})\cap[Y]=0$ for all $Y$ that are
irreducible and reduced. Similarly, we can define numerical triviality for any
polynomial in Chern classes of $\mathcal{E}$, or even in Chern classes of
finitely many bundles.

###  2.3. Positive polynomials

Consider $n\geq 1$ and grade $\mathbb{Q}[c_{1},\ldots,c_{n}]$ so that $\deg
c_{i}=i$. Let $P(c_{1},\ldots,c_{n})$ be a weighted homogeneous polynomial of
degree $n$. Fulton and Lazarsfeld proved in [FL83] that
$\int_{X}P(\mathcal{E})\geq 0$ for every $n$-dimensional variety $X$ and every
nef vector bundle $\mathcal{E}$ on $X$ if and only if $P$ is a linear
combination with nonnegative coefficients of Schur polynomials of degree $n$.
We call such polynomials _positive_. For example, the degree 1 positive
polynomials are spanned over $\mathbb{Q}_{\geq 0}$ by $c_{1}$, while the
degree 2 ones are spanned by $c_{2}$ and by $c_{1}^{2}-c_{2}$. In particular,
$c_{1}^{2}=c_{2}+(c_{1}^{2}-c_{2})$ is positive, but $c_{1}^{2}-2c_{2}$ is
not.

Let $X$ be a smooth projective surface and let $\mathcal{E}$ be a nef vector
bundle of rank $r$ on it. Then $c_{1}(\mathcal{E})$ is nef, and
$\int_{X}c_{2}(\mathcal{E})$ and
$\int_{X}(c_{1}^{2}(\mathcal{E})-c_{2}(\mathcal{E}))$ are nonnegative
integers. If $\mathcal{E}$ is strongly slope semistable with respect to some
polarization, Bogomolov’s inequality gives
$\int_{X}(2rc_{2}(\mathcal{E})-(r-1)c_{1}^{2}(\mathcal{E}))\geq 0$. However,
there exist examples of strongly slope semistable vector bundles $\mathcal{E}$
on surfaces such that all the above positivity conditions hold for the
characteristic classes of $\mathcal{E}$ without $\mathcal{E}$ being nef.

######  Example 2.1.

Let
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{2}}(-1)\otimes\bigotimes^{3}(T_{\mathbb{P}^{2}}(-1))$.
This is strongly slope semistable since $T_{\mathbb{P}^{2}}$ is strongly slope
semistable. Let $h=c_{1}(\mathcal{O}_{\mathbb{P}^{2}}(1))$. Then $r={\rm
rk}\,\mathcal{E}=8$, $c_{1}(\mathcal{E})=4h$, and
$c_{2}(\mathcal{E})=16h^{2}$. We compute
$c_{1}^{2}(\mathcal{E})-c_{2}(\mathcal{E})=0$. So the characteristic classes
of $\mathcal{E}$ suggest that $\mathcal{E}$ might be nef. However, the
restriction of $\mathcal{E}$ to every line in $\mathbb{P}^{2}$ is
$\mathcal{O}_{\mathbb{P}^{1}}(-1)\otimes\bigotimes^{3}(\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus\mathcal{O}_{\mathbb{P}^{1}})$
and this has $\mathcal{O}_{\mathbb{P}^{1}}(-1)$ as a summand, hence it is not
nef. ∎

See also [BHP14, Section 5] for a related example.

######  Convention 2.2.

For the rest of this section we assume that $X$ is a smooth projective variety
of dimension $n$.

###  2.4. Positive cones of cycles

Denote by $N^{c}(X)$ the space of numerical classes of cycles of codimension
$c$ with real coefficients. For example $N^{1}(X)$ is the real Néron–Severi
space spanned by Cartier divisors modulo numerical equivalence. The space
$N^{c}(X)$ is finite dimensional. It contains important convex cones. For
instance it contains the _pseudoeffective cone_
$\operatorname{\overline{Eff}}^{c}(X)$, the closure of the cone spanned by
classes of closed subsets of codimension $c$. It also contains the _nef cone_
$\operatorname{{Nef}}^{c}(X)$, the set of classes that intersect every
$c$-dimensional subvariety nonnegatively, the dual of the pseudoeffective cone
in the complementary codimension. The nef cone and the pseudoeffective cone in
codimension $c$ each span $N^{c}(X)$ and do not contain linear subspaces of
$N^{c}(X)$. See [FL17a].

When $c=1$, we put
$\operatorname{\overline{Eff}}(X)=\operatorname{\overline{Eff}}^{1}(X)$ and
$\operatorname{{Nef}}(X)=\operatorname{{Nef}}^{1}(X)$. In this case,
$\operatorname{{Nef}}(X)\subseteq\operatorname{\overline{Eff}}(X)$. The
interior of the nef cone is the ample cone, and the interior of the
pseudoeffective cone is the big cone of numerical classes with positive volume
[Laz04a, Chapter 2.2].

###  2.5. Determinant and discriminant

Let $\mathcal{F}$ be a torsion free sheaf on the smooth projective $X$. Then
the _singular_ (non-locally free) locus of $\mathcal{F}$ has codimension $\geq
2$. If $\mathcal{F}$ is reflexive, then the singular locus has codimension
$\geq 3$.

One can abstractly compute Chern classes of coherent sheaves in $A^{*}(X)$ by
taking locally free resolutions and using the additivity of the Chern
character ${\rm ch}$. The _determinant_ line bundle $\det\mathcal{F}$ of a
torsion free sheaf can be defined by extending from the locally free locus, or
considering the _reflexive hull_ $(\bigwedge^{\rm
rk\,\mathcal{F}}\mathcal{F})^{\vee\vee}$. This gives a concrete definition of
$c_{1}$. The second Chern class of a reflexive sheaf can be similarly
concretely defined.

######  Remark 2.3.

Let $f:Y\to X$ be a morphism of nonsingular projective varieties and let
$\mathcal{E}$ be a coherent sheaf on $X$. Then $\operatorname{ch}({\bf
L}f^{*}\mathcal{E})=f^{*}\operatorname{ch}(\mathcal{E})$ in the Chow ring of
$Y$, where ${\bf L}f^{*}\mathcal{E}$ is the derived pullback. In particular
$c_{i}(f^{*}\mathcal{E})=f^{*}c_{i}(\mathcal{E})$ for all $i$ in each of the
following cases:

  1. (1)

$\mathcal{E}$ is locally free.

  2. (2)

$f$ is flat.

  3. (3)

$f$ is the inclusion of a Cartier divisor and $\mathcal{E}$ is torsion free.

If $\imath:D\hookrightarrow X$ is the inclusion of a Cartier divisor and
$\mathcal{E}$ is reflexive, then $\imath^{*}\mathcal{E}$ is torsion free. If
$|H|$ is a basepoint free linear system and $\mathcal{E}$ is torsion free
(resp. reflexive), then $\imath^{*}\mathcal{E}$ is again torsion free (resp.
reflexive) for $D$ _general_ in $|H|$. See [HL10, Corollary 1.1.14].

The _discriminant_ of $\mathcal{E}$ is

$\Delta(\mathcal{E})=2rc_{2}(\mathcal{E})-(r-1)c_{1}^{2}(\mathcal{E}),$

where $r={\rm rk}\,\mathcal{E}$. It is an element of $A^{2}(X)$, but we use
the same notation for its image in $N^{2}(X)$. The class

$\log
r+\frac{c_{1}(\mathcal{E})}{r}-\frac{\Delta(\mathcal{E})}{2r^{2}}+\ldots$

in $A^{*}(X)_{\mathbb{R}}$ is the formal logarithm of the Chern character
${\rm ch}(\mathcal{E})$. It follows that $\frac{\Delta(\mathcal{E})}{2{\rm
rk}^{2}\,\mathcal{E}}$ is additive for tensor products if one of the factors
is locally free, just like the slope. In particular,
$\Delta(\mathcal{E})=\Delta(\mathcal{E}\otimes\mathcal{O}_{X}(D))$ for every
divisor $D$ on $X$. Furthermore, note that if $c_{1}(\mathcal{E})\equiv 0$,
then $\Delta(\mathcal{E})\equiv 0$ if and only if $c_{2}(\mathcal{E})\equiv
0$. The vanishing $\Delta(\mathcal{E})\equiv 0$ is equivalent to the numerical
vanishing of the second Chern class $c_{2}$ of
$\mathcal{E}\langle-\frac{1}{r}\det\mathcal{E}\rangle$, the _formal twist_ of
$\mathcal{E}$ in the sense of [Laz04b, Chapter 6.2]. If $\mathcal{E}$ is
locally free, then $\Delta(\mathcal{E})\equiv 0$ is also equivalent to the
numerical vanishing of $c_{2}$ of $\mathcal{E}^{\otimes
r}\otimes\det\mathcal{E}^{\vee}$ or ${\mathcal{E}}{\rm nd}\,\mathcal{E}$.

###  2.6. Semistability

Let $H$ be an ample (or just nef) divisor on $X$. For a nonzero torsion free
sheaf $\mathcal{E}$ on $X$, we define the _slope_ by
$\mu_{H}(\mathcal{E})=\frac{c_{1}(\mathcal{E})\cdot H^{n-1}}{{\rm
rk}\,\mathcal{E}}$. We say that $\mathcal{E}$ is $\mu_{H}$-_semistable_ (or
_slope semistable_ with respect to $H$) if no proper subsheaf
$0\neq\mathcal{F}\subsetneq\mathcal{E}$ verifies
$\mu_{H}(\mathcal{F})>\mu_{H}(\mathcal{E})$. We say that $\mathcal{E}$ is
$\mu_{H}$-_stable_ (or _slope stable_ with respect to $H$) if no proper
subsheaf $0\neq\mathcal{F}\subsetneq\mathcal{E}$ with ${\rm
rk}\,\mathcal{F}<{\rm rk}\,\mathcal{E}$ has
$\mu_{H}(\mathcal{F})\geq\mu_{H}(\mathcal{E})$. When $X$ is a curve and $H$ is
ample, slope semistability is independent of $H$. In this case we simply say
that $\mathcal{E}$ is semistable. In positive characteristic it is useful to
also consider Frobenius pullbacks.

######  Definition 2.4.

A torsion free sheaf $\mathcal{E}$ is called _strongly $\mu_{H}$-(semi)stable_
if $(F_{X}^{m})^{*}\mathcal{E}$ is $\mu_{H}$-(semi)stable for all $m\geq 0$.
Here $F_{X}$ denotes the Frobenius morphism in positive characteristic, and
the identity morphism in characteristic zero.

Examples show that these notions are indeed stronger that
$\mu_{H}$-(semi)stability in positive characteristic. The first example
showing this is due to J.-P. Serre and it was published in [Gie71]. More
precisely, the example shows that there exists a stable bundle of rank $2$ and
degree $1$ on a genus $3$ curve in characteristic $3$, whose Frobenius
pullback splits as direct sum of line bundles of different degrees. Nowadays
there are many more examples of stable bundles that are not strongly
semistable in an arbitrary positive characteristic. See, e.g., [JP15, Theorem
1.1.3] for recent examples that appear on any smooth projective curve in large
characteristic.

###  2.7. Semistability on curves

In the sequel we will use the following well–known lemma. It goes back to R.
Hartshorne [Har70] in the characteristic zero case, with a subsequent
algebraic proof in any characteristic due to C. M. Barton [Bar71, Theorem
2.1]. See also [Mor98, Proposition 7.1].

######  Lemma 2.5.

Let $C$ be a smooth projective curve defined over some algebraically closed
field $k$ and let $\mathcal{E}$ be a degree $0$ vector bundle on $C$. Then
$\mathcal{E}$ is strongly semistable if and only if $\mathcal{E}$ is nef.

We will also use the following standard lemma (see, e.g., [LP97, Lemma
7.1.2]).

######  Lemma 2.6.

Let $\mathcal{E}$ be a semistable vector bundle on a smooth projective curve
$C$ defined over some algebraically closed field $k$. Then

$h^{0}(C,\mathcal{E})\leq{\rm rk}\,\mathcal{E}+\deg\mathcal{E}.$

##  3\. Numerically flat vector bundles on proper schemes

Let $X$ be a proper scheme over a field $k$.

######  Definition 3.1.

A vector bundle $\mathcal{E}$ on $X$ is called _numerically flat_ if both
$\mathcal{E}$ and $\mathcal{E}^{\vee}$ are nef.

######  Remark 3.2.

A bundle $\mathcal{E}$ is numerically flat if and only if it is nef with
$c_{1}(\mathcal{E})\equiv 0$. Indeed if $\mathcal{E}$ and $\mathcal{E}^{\vee}$
are nef, then $c_{1}(\det\mathcal{E})=c_{1}(\mathcal{E})$ is numerically
trivial. Conversely, if $\mathcal{E}$ is nef with $c_{1}(\mathcal{E})\equiv 0$
then by [Laz04b, Theorem 6.2.12] any exterior power of $\mathcal{E}$ is nef
and hence
$\mathcal{E}^{\vee}\simeq\det\mathcal{E}^{\vee}\otimes\bigwedge^{r-1}\mathcal{E}$
is also nef.

It follows that if $\mathcal{E}$ is numerically flat, then all tensor functors
of $\mathcal{E}$ and their duals are numerically flat, e.g., tensor powers,
symmetric powers, divided powers, exterior powers, and all other Schur and co-
Schur (Weyl) functors associated to $\mathcal{E}$.

###  3.1. Numerical triviality of Chern classes of numerically flat bundles

######  Proposition 3.3.

Let $X$ be a proper scheme over a field $k$ and let $\mathcal{E}$ be a
numerically flat vector bundle on $X$. Then $c_{j}(\mathcal{E})$ is
numerically trivial for all $j>0$.

###### Proof.

The proof is by induction on $j$. For $j=1$ the assertion is clear since by
assumption for any proper curve $Y\subseteq X$ we have
$\int_{X}c_{1}(\mathcal{E})\cap[Y]\geq 0$ and
$\int_{X}c_{1}(\mathcal{E}^{\vee})\cap[Y]=-\int_{X}c_{1}(\mathcal{E})\cap[Y]\geq
0$.

Let $Y\subseteq X$ be a $j$-dimensional subvariety of $X$. Let
$s_{j}(\mathcal{E}^{\vee})=\pi_{*}(\xi^{j+r-1})$ be the $j$-th Segre class of
the dual of $\mathcal{E}$. By [Ful98, Chapter 3.2], we have
$s_{j}(\mathcal{E}^{\vee})=(-1)^{j+1}c_{j}(\mathcal{E})-\sum_{i=1}^{j-1}(-1)^{i}s_{i}(\mathcal{E}^{\vee})c_{j-i}(\mathcal{E})$.
By our induction hypothesis, we deduce
$\int_{Y}s_{j}(\mathcal{E}_{Y}^{\vee})=(-1)^{j+1}\cdot\int_{Y}c_{j}(\mathcal{E}_{Y})$,
so is suffices to prove that $\int_{Y}s_{j}(\mathcal{E}^{\vee}_{Y})=0$. Set
$Z\coloneqq{\mathbb{P}(\mathcal{E}_{Y})}$ and
$\mathcal{L}\coloneqq\mathcal{O}_{\mathbb{P}(\mathcal{E}_{Y})}(1)$. Since
$\mathcal{E}$ is nef, $\mathcal{E}_{Y}$ and $\mathcal{L}$ are also nef. By the
asymptotic Riemann–Roch theorems (see [Kol96, Chapter VI, Corollary 2.14 and
Theorem 2.15]) we have

$h^{0}(Z,\mathcal{L}^{\otimes m})=\chi(Z,\mathcal{L}^{\otimes
m})+O(m^{j+r-2})=\frac{\int_{Z}c_{1}(\mathcal{L})^{j+r-1}}{(j+r-1)!}m^{j+r-1}+O(m^{j+r-2}).$

Since $\mathcal{E}_{Y}$ is numerically flat, ${\rm Sym}^{m}\mathcal{E}_{Y}$ is
also numerically flat. Hence for any ample divisor $H$ on $Y$ we have
$h^{0}(Y,({\rm Sym}^{m}\mathcal{E}_{Y})(-H))=0$ (otherwise ${\rm
Sym}^{m}\mathcal{E}_{Y}$ contains $\mathcal{O}(H)$ contradicting the nefness
of $(\operatorname{Sym}^{m}\mathcal{E}_{Y})^{\vee}$). Again using the
asymptotic Riemann–Roch theorem we get

$\displaystyle h^{0}(Z,\mathcal{L}^{\otimes m})=h^{0}(Y,{\rm
Sym}^{m}\mathcal{E}_{Y})$ $\displaystyle\leq h^{0}(Y,({\rm
Sym}^{m}\mathcal{E}_{Y})(-H))+h^{0}(H,{\rm Sym}^{m}\mathcal{E}_{H})$
$\displaystyle=h^{0}(\mathbb{P}(\mathcal{E}_{H}),\mathcal{O}_{\mathbb{P}(\mathcal{E}_{H})}(m))=O(m^{j+r-2}).$

Summing up, we have
$\int_{Y}s_{j}(\mathcal{E}^{\vee}_{Y})=\int_{Z}c_{1}(\mathcal{L})^{j+r-1}=0$
as required. ∎

######  Remark 3.4.

The idea of proof of vanishing of highest Segre classes comes from the proof
of [Ful20, Proposition 5.1], in turn inspired by the proof of the Bogomolov
inequality. We avoid using this result and semistability and give a proof
working in an arbitrary characteristic. The proof of an analogue of [Ful20,
Proposition 5.1] would require small rewriting and the use of deep results of
Ramanan and Ramanathan [RR84] on the behaviour of strong slope semistability.

######  Remark 3.5.

If $X$ is a smooth complex manifold the above proposition was proven in
[DPS94, Corollary 1.19] using earlier deep analytic results. If $X$ is a
smooth variety and $k$ has positive characteristic the above proposition
follows from [Lan11, Proposition 5.1 and Theorem 4.1]. The proof of [Lan11,
Proposition 5.1] cites rather deep results from [FL83] although it uses only a
much weaker and easier result of Kleiman [Kle69]. However, it also depends on
[Lan04] and the proof above is much more elementary.

######  Remark 3.6.

From [BG71] or [FL83], the Chern class $c_{k}(\mathcal{E})$ is nef if
$\mathcal{E}$ is a nef bundle. Thus for $\mathcal{E}$ numerically flat the
classes $c_{k}(\mathcal{E})$, $c_{k}(\mathcal{E}^{\vee})$, and
$s_{k}(\mathcal{E}^{\vee})$ are nef. Using induction and the fact that
$\operatorname{{Nef}}^{k}(X)$ does not contain linear subspaces (cf. [FL17a]),
this gives another argument than the one above. Note though that [BG71] and
consequently also [FL83] use the hard Lefschetz theorem on cohomology so this
proof is much harder.

Together with the main result of [Lan04] the above propostion implies the
following corollary:

######  Corollary 3.7.

Let $f:X\to S$ be a flat projective morphism of noetherian schemes. Then the
set of numerically flat vector bundles of fixed rank $r$ on the fibers of $f$
is bounded.

###### Proof.

Let $\mathcal{O}_{X}(1)$ be an $f$-very ample line bundle on $X$ and let
$\mathcal{E}$ be a rank $r$ numerically flat vector bundle on a geometric
fiber $X_{s}$ for some geometric point $s$ of $S$. The singular
Grothendieck–Riemann–Roch theorem (see [Ful98, Corollary 18.3.1]) and
Proposition 3.3 imply that

$\chi(X_{s},\mathcal{E}(m))=\int_{X_{s}}\operatorname{ch}(\mathcal{E}(m))\cap{\mathrm{Td}}({X_{s}})=r\int_{X_{s}}\operatorname{ch}(\mathcal{O}_{X_{s}}(m))\cap{\mathrm{Td}}({X_{s}})=r\chi({X_{s}},\mathcal{O}_{X_{s}}(m)).$

Since $f$ is flat, for every connected component $S_{0}$ of $S$ the Hilbert
polynomial $P_{s}(m)=\chi({X_{s}},\mathcal{O}_{X_{s}}(m))$ is independent of
the geometric point $s$ of $S_{0}$. Moreover, any numerically flat vector
bundle on $X_{s}$ is slope $\mathcal{O}_{X_{s}}(1)$-semistable (the general
definition of slope semistability in case of singular projective schemes can
be found in [HL10, Definition and Corollary 1.6.9]). Therefore the required
assertion follows from [Lan04, Theorem 4.4]. ∎

######  Remark 3.8.

In case $X$ is a normal variety and $S=\operatorname{Spec}k$, the above
corollary was proved in [Lan12, Theorem 1.1]. If $S=\operatorname{Spec}k$ and
$k$ is a finite field the above corollary was proved in [DW20, Theorem 2.4].

######  Theorem 3.9.

Let $X$ be a projective scheme over a perfect field $k$ of positive
characteristic. Let $\mathcal{E}$ be a rank $r$ vector bundle on $X$. Then the
following conditions are equivalent:

  1. (1)

$\mathcal{E}$ is numerically flat.

  2. (2)

The set $\\{(F_{X}^{m})^{*}\mathcal{E}\\}_{m\in\mathbb{Z}_{\geq 0}}$ is
bounded.

  3. (3)

There exist $m_{1}>m_{2}\geq 0$ such that $(F_{X}^{m_{1}})^{*}\mathcal{E}$ and
$(F_{X}^{m_{2}})^{*}\mathcal{E}$ are algebraically equivalent.

###### Proof.

If $\mathcal{E}$ is numerically flat then all
$\mathcal{E}_{m}=(F_{X}^{m})^{*}\mathcal{E}$ are numerically flat, so
$(1)\Rightarrow(2)$ follows from Corollary 3.7. Assume $(2)$. Then by
definition there exists a $k$-scheme $S$ of finite type and an $S$-flat
coherent sheaf $\mathcal{F}$ on $X_{S}:=X\times_{k}S$ such that for every
$m\in\mathbb{Z}$ there exists a geometric $k$-point $s_{m}$ in $S$ such that
$\mathcal{F}_{X_{s_{m}}}\simeq\mathcal{E}_{m}$. Now $(3)$ follows by the
pigeonhole principle applied to the finitely many connected components of $S$.
Assume $(3)$. Then for all $m\geq 0$ the bundles
$(F_{X}^{m_{1}+m})^{*}\mathcal{E}$ and $(F_{X}^{m_{2}+m})^{*}\mathcal{E}$ are
algebraically equivalent. This implies that the family
$\\{(F_{X}^{m_{2}+m(m_{1}-m_{2})})^{*}\mathcal{E}\\}_{m\in\mathbb{Z}_{\geq
0}}$ is bounded. The implication $(3)\Rightarrow(1)$ follows as in the first
part of proof of [Lan11, Proposition 5.1]. ∎

The following corollary can be thought of as a generalization of [DW20,
Theorem 2.3] from finite fields to arbitrary perfect fields of positive
characteristic. In case $X$ is smooth the result is contained in [Lan21,
Corollary 3.7].

######  Corollary 3.10.

Let $X$ be a projective scheme over a perfect field $k$ of positive
characteristic. Let $\mathcal{E}$ be a numerically flat vector bundle on $X$.
Then for all $i>0$ the Chern classes $c_{i}(\mathcal{E})$ are, up to torsion,
algebraically equivalent to $0$.

###### Proof.

By the above theorem we know that there exist $m_{1}>m_{2}\geq 0$ such that
$(F_{X}^{m_{1}})^{*}\mathcal{E}$ and $(F_{X}^{m_{2}})^{*}\mathcal{E}$ are
algebraically equivalent. Since
$c_{i}((F_{X}^{m})^{*}\mathcal{E})=p^{im}c_{i}(\mathcal{E})$ in $A^{*}(X)$, we
get

$0=c_{i}((F_{X}^{m_{1}})^{*}\mathcal{E})-c_{i}((F_{X}^{m_{2}})^{*}\mathcal{E})=(p^{im_{1}}-p^{im_{2}})c_{i}(\mathcal{E})$

in $B^{*}(X)$, so $c_{i}(\mathcal{E})=0$ in $B^{*}(X)_{\mathbb{Q}}$. ∎

###  3.2. Characterizations of numerically flat bundles

Let $X$ be a proper scheme over an algebraically closed field $k$.

######  Definition 3.11.

A vector bundle $\mathcal{E}$ on $X$ is called _universally semistable_ if for
all $k$-morphisms $f:C\to X$ from smooth connected projective curves $C$ over
$k$ the pullback $f^{*}\mathcal{E}$ is semistable. We say that $\mathcal{E}$
is _Nori semistable_ if it is universally semistable and $c_{1}(\mathcal{E})$
is numerically trivial.

To better justify the terminology, note that if $\mathcal{E}$ is universally
semistable, and $f:Y\to X$ is a morphism from a projective manifold $Y$ over
$k$, then for every polarization $H$ on $Y$ the pullback $f^{*}\mathcal{E}$ is
$\mu_{H}$-semistable.

If $f:Y\to X$ is a proper generically finite morphism between smooth
projective varieties and $\mathcal{E}$ is a strongly slope $H$-semistable
bundle on $X$ then $f^{*}\mathcal{E}$ is slope $f^{*}H$-semistable bundle.
This motivates the following definition:

######  Definition 3.12.

If $X$ is irreducible then we say that a vector bundle $\mathcal{E}$ on $X$ is
_strongly semistable_ if there exist a proper generically finite $k$-morphism
$f:Y\to X$ from a smooth projective $k$-variety $Y$ to $X$ and an ample
divisor $H$ on $Y$ such that the bundle $f^{*}\mathcal{E}$ is strongly slope
$H$-semistable. In general, we say that a vector bundle $\mathcal{E}$ on $X$
is _strongly semistable_ if its restriction to every irreducible component of
$X$ is strongly semistable.

A line bundle $\mathcal{L}$ is said to be $\tau$-trivial, if
$\mathcal{L}^{\otimes m}$ is algebraically equivalent to $\mathcal{O}_{X}$ for
some $m\geq 1$. This notion is equivalent to $\mathcal{L}$ being numerically
trivial by [Kle05, Theorem 6.3]. Numerically flat bundles can be seen as a
higher rank version of $\tau$-trivial bundles. In the proof we use Theorem
4.8.

######  Theorem 3.13.

Let $X$ be a proper scheme over an algebraically closed field $k$. Let
$\mathcal{E}$ be a rank $r$ vector bundle on $X$. Then the following
conditions are equivalent:

  1. (1)

$\mathcal{E}$ is numerically flat.

  2. (2)

$\mathcal{E}$ is Nori semistable.

  3. (3)

$\mathcal{E}$ is strongly semistable and $c_{j}(\mathcal{E})$ is numerically
trivial for all $j>0$.

  4. (4)

$\mathcal{E}$ is strongly semistable and both $c_{1}(\mathcal{E})$ and
$c_{2}(\mathcal{E})$ are numerically trivial.

  5. (5)

$\mathcal{E}$ is strongly semistable and for every coherent sheaf
$\mathcal{F}$ on $X$ we have
$\chi(X,\mathcal{E}\otimes\mathcal{F})=r\cdot\chi(X,\mathcal{F})$.

###### Proof.

The equivalence of $(1)$ and $(2)$ is well known and it follows from Lemma 2.5
(see, e.g., [Lan11, 1.2]). Assume that $\mathcal{E}$ is numerically flat and
let $X_{0}$ be some irreducible component of $X$. By [dJ96] there exists a
proper generically finite $k$-morphism $f:Y\to X_{0}$ from a smooth projective
$k$-variety $Y$. Then $f^{*}\mathcal{E}$ is numerically flat, so it is
strongly $\mu_{H}$-semistable for every ample divisor $H$ on $X$. In
particular, $\mathcal{E}$ is strongly semistable. Now implication
$(1)\Rightarrow(3)$ follows from Proposition 3.3.

In proof of the implication $(3)\Rightarrow(5)$ we use singular Riemann–Roch
[Ful98, Chapter 18]. Let $f:X\to Y=\operatorname{Spec}k$ be the structural
morphism. We denote by $[\mathcal{E}]$ the class of $\mathcal{E}$ in
$K^{0}(X)$ and by $[\mathcal{F}]$ the class of $\mathcal{F}$ in $K_{0}(X)$. By
[Ful98, Theorem 18.3] we have canonical maps $\tau_{X}:K_{0}(X)\to
A_{*}(X)_{\mathbb{Q}}$ and $\tau_{Y}:K_{0}(Y)\to
A_{*}(Y)_{\mathbb{Q}}=\mathbb{Q}$, which satisfy the following equalities:

$\chi(X,\mathcal{E}\otimes\mathcal{F})=\tau_{Y}f_{*}([\mathcal{E}]\otimes[\mathcal{F}])=f_{*}\tau_{X}([\mathcal{E}]\otimes[\mathcal{F}])=f_{*}(\operatorname{ch}([\mathcal{E}])\cap\tau_{X}([\mathcal{F}]))=r\tau_{Y}f_{*}([\mathcal{F}])=r\chi(X,\mathcal{F}).$

To prove that $(5)$ implies $(1)$ we can assume that $X$ is irreducible. Then
by assumption there exist a proper generically finite $k$-morphism $f:Y\to X$
from a smooth projective $k$-scheme $Y$ to $X$ and an ample divisor $H$ on $Y$
such that the bundle $f^{*}\mathcal{E}$ is strongly slope $H$-semistable. By
the Leray spectral sequence and the projection formula we have

$\displaystyle\chi(Y,f^{*}\mathcal{E}(mH))$
$\displaystyle=\sum_{i}(-1)^{i}\chi(X,R^{i}f_{*}(f^{*}\mathcal{E}(mH)))=\sum_{i}(-1)^{i}\chi(X,\mathcal{E}\otimes
R^{i}f_{*}\mathcal{O}_{Y}(mH))$
$\displaystyle=r\sum_{i}(-1)^{i}\chi(X,R^{i}f_{*}\mathcal{O}_{Y}(mH))=r\chi(Y,\mathcal{O}_{Y}(mH)).$

So by the implication $(4)\Rightarrow(2)$ of Theorem 4.8 we see that
$f^{*}\mathcal{E}$ is numerically flat. Since $f$ is surjective, this implies
that $\mathcal{E}$ is also numerically flat.

The implication $(3)\Rightarrow(4)$ is obvious, so it is sufficient to prove
that $(4)\Rightarrow(1)$. Without loss of generality we can assume that $X$ is
irreducible and there exist a proper generically finite $f:Y\to X$ from a
smooth projective $k$-variety $Y$ and an ample divisor $H$ on $Y$ such that
the bundle $f^{*}\mathcal{E}$ is strongly $\mu_{H}$-semistable. Then
$c_{1}(f^{*}\mathcal{E})$ and $c_{2}(f^{*}\mathcal{E})$ are numerically
trivial. So by the implication $(1)\Rightarrow(2)$ of Theorem 4.8 we see that
$f^{*}\mathcal{E}$ is numerically flat. As before this implies that
$\mathcal{E}$ is also numerically flat. ∎

######  Corollary 3.14.

Let $\mathcal{E}$ be a vector bundle on $X$. Then the following conditions are
equivalent:

  1. (1)

$\mathcal{E}$ is strongly semistable and $\Delta(\mathcal{E})$ is numerically
trivial.

  2. (2)

$\mathcal{E}{\rm nd}\,\mathcal{E}$ is nef.

  3. (3)

$\mathcal{E}$ is universally semistable.

###### Proof.

If $\mathcal{E}$ is strongly semistable and $\Delta(\mathcal{E})\equiv 0$ then
$\mathcal{E}{\rm nd}\,\mathcal{E}$ is strongly semistable and both
$c_{1}(\mathcal{E}{\rm nd}\,\mathcal{E})$ and $c_{2}(\mathcal{E}{\rm
nd}\,\mathcal{E})$ are numerically trivial. So $\mathcal{E}{\rm
nd}\,\mathcal{E}$ is numerically flat, which proves $(1)\Rightarrow(2)$. If
$\mathcal{E}{\rm nd}\,\mathcal{E}$ is nef then it is also numerically flat (as
it isomorphic to its dual) and hence it is Nori semistable. This implies
$(3)$. To prove that $(3)\Rightarrow(1)$ it is sufficient to prove that
$\Delta(\mathcal{E})\equiv 0$. But $\mathcal{E}{\rm nd}\,\mathcal{E}$ is
universally semistable with trivial determinant, so it is Nori semistable.
Hence it is numerically flat and $\Delta(\mathcal{E})=c_{2}(\mathcal{E}{\rm
nd}\,\mathcal{E})\equiv 0$. ∎

##  4\. Algebraic proofs of Theorems 1.1 and 1.2

###  4.1. Restriction theorems

In the sequel we frequently use the following strengthening of the
Mehta–Ramanathan theorem for sheaves with vanishing discriminant. The result
follows from [Lan04, Theorem 5.2] with a different proof from the
Mehta–Ramanathan theorem.

######  Lemma 4.1.

Let $X$ be a smooth projective variety defined over an algebraically closed
field $k$ and let $H$ be an ample divisor class on $X$. Let $\mathcal{E}$ be a
torsion free sheaf with $\Delta(\mathcal{E})\equiv 0$. Then there exists
$m_{0}=m_{0}(X,H,{\rm rk}\,\mathcal{E})\geq 1$ such that for all $m\geq m_{0}$

  1. (1)

If $\mathcal{E}$ is strongly $\mu_{H}$-stable and $\mathcal{E}_{D}$ is torsion
free for some normal divisor $D\in|mH|$, then $\mathcal{E}_{D}$ is strongly
$\mu_{H_{D}}$-stable.

  2. (2)

If $\mathcal{E}$ is strongly $\mu_{H}$-semistable, then for general $D\in|mH|$
the restriction $\mathcal{E}_{D}$ is strongly $\mu_{H_{D}}$-semistable.

###### Proof.

In characteristic zero pick $m_{0}$ such that $|mH|$ is basepoint free for all
$m\geq m_{0}$. In positive characteristic we also need to exceed a constant
$\beta_{r}$ depending on $X$, $H$, and ${\rm rk}\,\mathcal{E}$. See the
inequality in [Lan04, Theorem 5.2]. (1) follows immediately from [Lan04,
Theorem 5.2]. We obtain (2) as a consequence of (1) as in [Lan04, Corollary
5.4]. The factors (successive quotients) in any Jordan–Hölder filtration of
$\mathcal{E}$ are $\mu_{H}$-stable. As in [Lan04, Theorem 5.4] we observe that
they also have numerically trivial discriminant. Their restriction to a
general $D$ in a basepointfree $|mH|$ is again torsion free. In characteristic
zero then (2) follows from (1). In positive characteristic strong
$\mu_{H}$-semistability also takes into account the countably many Frobenius
pullbacks. We remark that there exists some $s_{0}$ such that the factors in a
Jordan–Hölder filtration of $(F_{X}^{s_{0}})^{*}\mathcal{E}$ are strongly
$\mu_{H}$-stable. Then for a general divisor $D\in|mH|$ the restrictions of
the factors in a Jordan–Hölder filtration of the sheaves
$\\{(F_{X}^{s})^{*}\mathcal{E}\\}_{s\leq s_{0}}$ to $D$ are torsion free. The
restrictions of the factors in a Jordan–Hölder filtration of
$(F_{X}^{s})^{*}\mathcal{E}$ for all $s\geq s_{0}$ to such a divisor are also
torsion free since they are pullbacks of those of
$(F_{X}^{s_{0}})^{*}\mathcal{E}$. ∎

###  4.2. The surface case

######  Proposition 4.2.

Let $(X,H)$ be an amply polarized smooth projective surface defined over an
algebraically closed field $k$. Let $\mathcal{E}$ be a strongly
$\mu_{H}$-semistable locally free sheaf of rank $r$ with
$c_{i}(\mathcal{E})\equiv 0$ for $i=1,2$. Then

  1. (1)

For any line bundle $L$ on $X$ and any $i\in\\{0,1,2\\}$ we have
$h^{i}(X,L\otimes\operatorname{Sym}^{m}\mathcal{E})=O(m^{r-1})$.

  2. (2)

$\mathcal{E}$ is numerically flat.

###### Proof.

Without loss of generality we can assume that $H$ is very ample. Since
$\dim\mathbb{P}(\mathcal{E})=r+1$, the claimed growth rate in $(1)$ is two
degrees lower than expected. The proof is similar to the proof of the
Bogomolov inequality in [HL10, Theorem 7.3.1].

By [RR84, Theorem 3.23 and the remark at the end of Section 4] the bundle
$\operatorname{Sym}^{m}\mathcal{E}$ is strongly $\mu_{H}$-semistable. Since
$\Delta(\operatorname{Sym}^{m}\mathcal{E})\equiv 0$ Lemma 4.1 implies that if
$D\in|H|$ is a general divisor then $(\operatorname{Sym}^{m}\mathcal{E})_{D}$
is strongly semistable of degree $0$. By Lemma 2.6 we have

$h^{0}(D,(L\otimes\operatorname{Sym}^{m}\mathcal{E})_{D})\leq(1+\deg
L_{D})\cdot{\rm rk}\,\operatorname{Sym}^{m}\mathcal{E}=O(m^{r-1})$

From the short exact sequence

$0\to(L\otimes\operatorname{Sym}^{m}\mathcal{E})(-H)\to
L\otimes\operatorname{Sym}^{m}\mathcal{E}\to(L\otimes\operatorname{Sym}^{m}\mathcal{E})_{D}\to
0$

we have

$h^{0}(X,L\otimes\operatorname{Sym}^{m}\mathcal{E})\leq
h^{0}(D,(L\otimes\operatorname{Sym}^{m}\mathcal{E})_{D})+h^{0}(X,(L\otimes\operatorname{Sym}^{m}\mathcal{E})(-H)).$

Changing $H$ by its multiple (which does not depend on $m$) if necessary, we
can assume that $L(-H)$ has negative degree with respect to $H$ (e.g., we can
assume that $L^{\vee}(H)$ is effective). Then the bundle
$(L\otimes\operatorname{Sym}^{m}\mathcal{E})(-H)$ is $\mu_{H}$-semistable with
negative slope so it does not have any nonzero sections. Therefore
$h^{0}(X,L\otimes\operatorname{Sym}^{m}\mathcal{E})=O(m^{r-1})$.

By Serre’s duality,
$h^{2}(X,L\otimes\operatorname{Sym}^{m}\mathcal{E})=h^{0}(X,(\operatorname{Sym}^{m}\mathcal{E})^{\vee}\otimes\omega_{X}\otimes
L^{\vee})$. The bundle $(\operatorname{Sym}^{m}\mathcal{E})^{\vee}$ is also
strongly semistable with numerically trivial Chern classes. An analogous proof
to the case $i=0$ gives
$h^{2}(X,L\otimes\operatorname{Sym}^{m}\mathcal{E})=O(m^{r-1})$. Note that in
positive characteristic this equality does not follow formally from the
previous case applied to $\mathcal{E}^{\vee}$.

Finally, to prove that $h^{1}(X,L\otimes\operatorname{Sym}^{m}\mathcal{E})$
grows at most like $O(m^{r-1})$, it is sufficient to prove that
$\chi(X,L\otimes\operatorname{Sym}^{m}\mathcal{E})=O(m^{r-1})$. Let
$\pi:\mathbb{P}(\mathcal{E})\to X$ be the bundle map and let us set
$\xi\coloneqq c_{1}(\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1))$. Since the
Chern classes $c_{1}(\mathcal{E})$ and $c_{2}(\mathcal{E})$ are both
numerically trivial, we have $\xi^{r}=0$. By the Riemann–Roch theorem

$\chi(X,L\otimes\operatorname{Sym}^{m}\mathcal{E})=\chi(\mathbb{P}(\mathcal{E}),\mathcal{O}_{\mathbb{P}(\mathcal{E})}(m)\otimes\pi^{*}L)=\int_{\mathbb{P}(\mathcal{E})}\exp(m\xi+\pi^{*}c_{1}(L))\cap\operatorname{Td\,}\mathbb{P}(\mathcal{E}),$

so the coefficients of $m^{r+1}$ and $m^{r}$ in the expression above are $0$.
The claim is proved.

$(2)$. Let $f:C\to X$ be a morphism from a smooth projective curve and let
$C^{\prime}\subset X$ be the (possibly singular) image of $f$. Consider the
restriction sequence

$0\to(\operatorname{Sym}^{m}\mathcal{E})(-C^{\prime})\to\operatorname{Sym}^{m}\mathcal{E}\to\operatorname{Sym}^{m}\mathcal{E}_{C^{\prime}}\to
0.$

Let $\xi_{C}=c_{1}(\mathcal{O}_{\mathbb{P}(f^{*}\mathcal{E})}(1))$. If
$f^{*}\mathcal{E}$ is not strongly semistable, then since it has degree 0, it
follows from Lemma 2.5 that $\xi_{C}$ is not nef. It is however big since some
Frobenius pullback of $f^{*}\mathcal{E}$ has a strongly semistable subbundle
of positive degree, so an ample subbundle. Bigness is invariant under
birational pullback (cf. [Laz04a, Chapter 2.2]) and even under dominant
generically finite pullback, hence
$h^{0}(C^{\prime},\operatorname{Sym}^{m}\mathcal{E}_{C^{\prime}})=h^{0}(\mathbb{P}(\mathcal{E}_{C^{\prime}}),\mathcal{O}_{\mathbb{P}(\mathcal{E}_{C^{\prime}})}(m))$
grows like $O(m^{r})$. However, we have

$h^{0}(C^{\prime},\operatorname{Sym}^{m}\mathcal{E}_{C^{\prime}})\leq
h^{0}(X,\operatorname{Sym}^{m}\mathcal{E})+h^{1}(X,(\operatorname{Sym}^{m}\mathcal{E})(-C^{\prime})).$

We get a contradiction from part $(1)$. ∎

###  4.3. Local freeness via the vanishing of the discriminant

######  Lemma 4.3.

Let $(X,H)$ and $(Y,A)$ be amply polarized smooth projective varieties defined
over an algebraically closed field $k$. Let $\mathcal{E}$ be a
$\mu_{H}$-semistable ($\mu_{H}$-stable, strongly $\mu_{H}$-semistable or
strongly $\mu_{H}$-stable) torsion free sheaf on $X$. Then ${\rm
pr}_{X}^{*}\mathcal{E}$ is $\mu_{L}$-semistable (respectively
$\mu_{L}$-stable, strongly $\mu_{L}$-semistable or strongly $\mu_{L}$-stable)
for $L={\rm pr}_{X}^{*}H+{\rm pr}_{Y}^{*}A$.

###### Proof.

Let $r={\rm rk}\,\mathcal{E}$ and denote $n=\dim X$ and $m=\dim Y$. Let
$\mathcal{F}\subseteq{\rm pr}_{X}^{*}\mathcal{E}$ be a subsheaf of rank less
than $r$. For $x\in X$ and $y\in Y$, let
$\mathcal{F}_{x}\coloneqq\mathcal{F}_{\\{x\\}\times Y}$ and
$\mathcal{F}_{y}\coloneqq\mathcal{F}_{X\times\\{y\\}}$. For general points
$x\in X(k)$ and $y\in Y(k)$ we have
$\mathcal{F}_{x}\subseteq\mathcal{O}_{Y}^{\oplus r}$ and
$\mathcal{F}_{y}\subseteq\mathcal{E}$. Since $\mathcal{E}$ is
$\mu_{H}$-semistable and $\mathcal{O}_{Y}^{\oplus r}$ is $\mu_{A}$-semistable,
we deduce that $\frac{c_{1}(\mathcal{F})\cdot{\rm pr}_{X}^{*}H^{n-1}{\rm
pr}_{Y}^{*}A^{m}}{(A^{m})\cdot{\rm
rk}\,\mathcal{F}}=\frac{c_{1}(\mathcal{F}_{y})\cdot H^{n-1}}{{\rm
rk}\,\mathcal{F}}\leq\mu_{H}(\mathcal{E})$ and
$\frac{c_{1}(\mathcal{F})\cdot{\rm pr}_{X}^{*}H^{n}{\rm
pr}_{Y}^{*}A^{m-1}}{(H^{n})\cdot{\rm
rk}\,\mathcal{F}}=\frac{c_{1}(\mathcal{F}_{x})\cdot A^{m-1}}{{\rm
rk}\,\mathcal{F}}\leq 0$. Then

$\mu_{L}(\mathcal{F})=\frac{c_{1}(\mathcal{F})\cdot L^{n+m-1}}{{\rm
rk}\,\mathcal{F}}=\frac{c_{1}(\mathcal{F})\cdot\left({{n+m-1}\choose{n-1}}{\rm
pr}_{X}^{*}H^{n-1}{\rm pr}_{Y}^{*}A^{m}+{{n+m-1}\choose n}{\rm
pr}_{X}^{*}H^{n}{\rm pr}_{Y}^{*}A^{m-1}\right)}{{\rm rk}\,\mathcal{F}}\\\
\leq{{n+m-1}\choose{n-1}}(A^{m})\cdot\mu_{H}(\mathcal{E})=\frac{{{n+m-1}\choose{n-1}}\cdot(c_{1}(\mathcal{E})\cdot
H^{n-1})(A^{m})}{{\rm rk}\,\mathcal{E}}=\mu_{L}({\rm pr}_{X}^{*}\mathcal{E}).$

If $\mathcal{E}$ is $\mu_{H}$-stable then $\frac{c_{1}(\mathcal{F})\cdot{\rm
pr}_{X}^{*}H^{n-1}{\rm pr}_{Y}^{*}A^{m}}{(A^{m})\cdot{\rm
rk}\,\mathcal{F}}=\frac{c_{1}(\mathcal{F}_{y})\cdot H^{n-1}}{{\rm
rk}\,\mathcal{F}}<\mu_{H}(\mathcal{E})$ and we get
$\mu_{L}(\mathcal{F})<\mu_{L}({\rm pr}_{X}^{*}\mathcal{E})$ as required.

Applying the above assertions for slope semistability and slope stability to
all Frobenius pull-backs gives immediately the assertions for strong slope
semistability and strong slope stability. ∎

######  Proposition 4.4.

Let $X$ be a smooth projective variety of dimension $n$ defined over an
algebraically closed field $k$ and let $H$ be an ample polarization on $X$.
Let $\mathcal{E}$ be a strongly $\mu_{H}$-stable torsion free sheaf on $X$
with $c_{1}(\mathcal{E})\equiv 0$. Then the following conditions are
equivalent:

  1. (1)

$\mathcal{E}$ is reflexive and $c_{2}(\mathcal{E})\cdot H^{n-2}=0$.

  2. (2)

$\mathcal{E}$ is locally free and numerically flat.

  3. (3)

$c_{j}(\mathcal{E})\equiv 0$ for all $j\geq 1$.

  4. (4)

The normalized Hilbert polynomial $\frac{1}{{\rm
rk}\,\mathcal{E}}\cdot\chi(X,\mathcal{E}(mH))$ of $\mathcal{E}$ is equal to
the Hilbert polynomial of $\mathcal{O}_{X}$.

###### Proof.

We argue by induction on $n$. If $n=1$, the equivalence of all four conditions
is tautological, with $(2)$ being implied by Lemma 2.5. Let us assume that
$n=2$. Since every reflexive sheaf on a smooth surface is locally free, the
implication $(1)\Rightarrow(2)$ follows from Proposition 4.2. The implication
$(2)\Rightarrow(3)$ follows from Proposition 3.3 and $(3)\Rightarrow(4)$
follows from the Hirzebruch–Riemann–Roch theorem. To prove
$(4)\Rightarrow(1)$, consider the exact sequence
$0\to\mathcal{E}\to\mathcal{E}^{\vee\vee}\to Q\to 0,$ where $Q$ has a finite
support. Since $c_{1}(\mathcal{E}^{\vee\vee})=c_{1}(\mathcal{E})\equiv 0$ and
$\mathcal{E}^{\vee\vee}$ is strongly $\mu_{H}$-stable, the Bogomolov type
inequality (see [Lan04, Theorem 3.2]) gives
$\int_{X}\Delta(\mathcal{E}^{\vee\vee})\geq 0$. Our assumption on the Hilbert
polynomial of $\mathcal{E}$ and the Hirzebruch–Riemann–Roch theorem imply that
$\int_{X}c_{2}(E)=0$. Therefore with $r={\rm rk}\,\mathcal{E}$

$\int_{X}\Delta(\mathcal{E}^{\vee\vee})=2r\int_{X}c_{2}(\mathcal{E}^{\vee\vee})=2r\int_{X}(c_{2}(\mathcal{E})+c_{2}(Q))=2r\int_{X}c_{2}(Q)\leq
0,$

which gives $\int_{X}c_{2}(Q)=0$. But then $Q=0$ and $\mathcal{E}$ is
reflexive.

Let us now assume that the result holds for varieties of dimension $<n$, where
$n\geq 3$.

$(1)\Rightarrow(2)$

Let $\mathcal{E}$ be reflexive, strongly $\mu_{H}$-stable, with
$c_{1}(\mathcal{E})$ numerically trivial and $c_{2}(\mathcal{E})\cdot
H^{n-2}=0$. Let $\imath:D\to X$ be the inclusion of a general member of $|mH|$
for sufficiently large $m$. Then $D$ is smooth of dimension $n-1$. The
restriction $\imath^{*}\mathcal{E}$ is still reflexive by the general choice
of $D$ and it is strongly $\mu_{\imath^{*}H}$-stable by Lemma 4.1. We also
have
$c_{2}(\imath^{*}\mathcal{E})\cdot\imath^{*}H^{n-3}=c_{2}(\mathcal{E})\cdot
H^{n-2}$ by Remark 2.3. Using the implication $(1)\Rightarrow(4)$ on $D$ we
see that

$\displaystyle\chi(X,\mathcal{E}(mH))-\chi(\mathcal{E}((m-1)H))$
$\displaystyle=\chi(D,\imath^{*}\mathcal{E}(mH))=r\chi(D,\mathcal{O}_{D}(mH))$
$\displaystyle=r(\chi(X,\mathcal{O}_{X}(mH))-\chi(\mathcal{O}_{X}((m-1)H))).$

Now assume that $\mathcal{E}$ is not locally free. For sufficiently large $m$,
let $\imath:D\to X$ be an embedding of a member of $|mH|$ that is general
among those that pass through one of the points where $\mathcal{E}$ is not
locally free. By [DH91] we know that $D$ is smooth. We also know that the
restriction $\imath^{*}\mathcal{E}$ is torsion-free. So Lemma 4.1 implies that
$\imath^{*}\mathcal{E}$ is strongly $\mu_{\imath^{*}H}$-stable. By the above,
we also know that

$\displaystyle\chi(D,\imath^{*}\mathcal{E}(mH))$
$\displaystyle=\chi(X,\mathcal{E}(mH))-\chi(\mathcal{E}((m-1)H))$
$\displaystyle=r(\chi(X,\mathcal{O}_{X}(mH))-\chi(\mathcal{O}_{X}((m-1)H)))=r\chi(D,\mathcal{O}_{D}(mH)).$

Then the induction assumption implies that $\imath^{*}\mathcal{E}$ is locally
free. By [Lan19, Lemma 1.14] we deduce that $\mathcal{E}$ is locally free
around $D$, a contradiction. Thus $\mathcal{E}$ is locally free and we need to
prove that it is numerically flat. Let $f:C\to X$ be a morphism from a smooth
projective curve. Let $A$ be any ample polarization on $C$ and let
$\widetilde{C}\subset X\times C$ be the embedding of the graph of $C$. Denote
by ${\rm pr}_{X}:X\times C\to X$ and ${\rm pr}_{C}:X\times C\to C$ the
projections onto the two factors. Let us also set $L\coloneqq{\rm
pr}_{X}^{*}H+{\rm pr}_{C}^{*}A$ and $\widetilde{\mathcal{E}}\coloneqq{\rm
pr}_{X}^{*}\mathcal{E}$. Then $\widetilde{C}\simeq C$ and
$f^{*}\mathcal{E}\simeq\widetilde{\mathcal{E}}_{\widetilde{C}}$, so it is
sufficient to check that $\widetilde{\mathcal{E}}_{\widetilde{C}}$ is
semistable of degree $0$. We have $c_{2}(\widetilde{\mathcal{E}})\cdot
L^{n-1}=\sum_{j=0}^{n-1}{{n-1}\choose j}{\rm
pr}_{X}^{*}(c_{2}(\widetilde{\mathcal{E}})\cdot H^{n-1-j})\cdot{\rm
pr}_{C}^{*}A^{j}$. For dimension reasons, all terms except possibly $j=1$
vanish. The term $j=1$ also vanishes from the assumption
$c_{2}(\mathcal{E})\cdot H^{n-2}=0$. Therefore we have
$c_{1}(\widetilde{\mathcal{E}})\equiv 0$ and
$c_{2}(\widetilde{\mathcal{E}})\cdot L^{n-1}=0$. By Lemma 4.3 we also know
that $\widetilde{\mathcal{E}}$ is strongly $\mu_{L}$-stable. By [DH91, Theorem
3.1] for large $m$ there exists a chain of smooth varieties $S=X_{2}\subset
X_{3}\subset...\subset X_{n}=X\times C$ containing $\widetilde{C}$ such that
all $X_{i}\in|mL_{X_{i+1}}|$ are smooth. Then Lemma 4.1 implies that
$\widetilde{\mathcal{E}}_{S}$ is strongly $\mu_{L_{S}}$-stable with
$c_{1}(\widetilde{\mathcal{E}}_{S})\equiv 0$ and
$\int_{S}c_{2}(\widetilde{\mathcal{E}}_{S})=0$. So the required assertion
follows from Proposition 4.2.

$(2)\Rightarrow(3)$ follows from Proposition 3.3.

$(3)\Rightarrow(4)$ follows from the Hirzebruch–Riemann–Roch theorem.

$(4)\Rightarrow(1)$ Let $\mathcal{E}$ be a rank $r$ torsion free, strongly
$\mu_{H}$-stable sheaf with $c_{1}(\mathcal{E})\equiv 0$ and
$\chi(X,\mathcal{E}(mH))=r\chi(X,\mathcal{O}_{X}(mH))$ for all
$m\in\mathbb{Z}$. Comparing coefficients of these polynomials at $m^{n-2}$ we
see that $c_{2}(\mathcal{E})\cdot H^{n-2}=0$. Consider the exact sequence

$0\to\mathcal{E}\to\mathcal{E}^{\vee\vee}\to Q\to 0,$

where $Q$ is a torsion sheaf on $X$. Then $\mathcal{E}^{\vee\vee}$ is
reflexive, strongly $\mu_{H}$-stable sheaf with
$c_{1}(\mathcal{E}^{\vee\vee})\equiv 0$. As in the surface case, the Bogomolov
type inequality (see [Lan04, Theorem 3.2]) gives

$\Delta(\mathcal{E}^{\vee\vee})\cdot H^{n-2}=2rc_{2}(Q)\cdot H^{n-2}\geq 0,$

which implies $\Delta(\mathcal{E}^{\vee\vee})\cdot H^{n-2}=c_{2}(Q)\cdot
H^{n-2}=0$. Using already proven implication $(1)\Rightarrow(4)$ we see that
Hilbert polynomials of $\mathcal{E}$ and $\mathcal{E}^{\vee\vee}$ coincide.
Therefore the Hilbert polynomial of $Q$ vanishes. This implies that $Q=0$ and
hence $\mathcal{E}$ is reflexive with $c_{2}(\mathcal{E})\cdot H^{n-2}=0$. ∎

The following known example shows that reflexivity assumption is necessary in
condition $(1)$ of Proposition 4.4.

######  Example 4.5.

Let $X$ be a smooth projective variety of dimension $n\geq 3$. Let
$\mathcal{I}\subset\mathcal{O}_{X}$ be the ideal sheaf of a nonempty closed
subset of codimension $j\geq 3$. Then $\mathcal{I}$ is torsion-free, strongly
slope semistable with respect to any polarization, and $c_{1}(\mathcal{I})$
and $c_{2}(\mathcal{I})$ are numerically trivial. However $c_{j}(\mathcal{I})$
is not numerically trivial and of course $\mathcal{I}$ is not locally free.

######  Corollary 4.6.

Let $\mathcal{E}$ be a reflexive strongly $\mu_{H}$-semistable sheaf with
$c_{1}(\mathcal{E})\equiv 0$ and $c_{2}(\mathcal{E})\cdot H^{n-2}=0$. Then
$\mathcal{E}$ is locally free and numerically flat. Moreover, every factor in
a Jordan–Hölder filtration of $\mathcal{E}$ is also locally free and
numerically flat.

This proof is analogous to [Nak99, Proposition 2.5] and [Lan19, Section 2.2].

###### Proof.

The statement is clear if $n=1$ so we assume that $n\geq 2$. We perform
induction on the rank $r$ of $\mathcal{E}$. The case $r=1$ or more generally
$\mathcal{E}$ is strongly $\mu_{H}$-stable is Proposition 4.4. Thus we may
assume that $\mathcal{E}$ is not strongly $\mu_{H}$-stable. First, let us
assume that there exists an exact sequence

$0\to\mathcal{E}_{1}\to\mathcal{E}\to\mathcal{E}_{2}\to 0$

where $\mathcal{E}_{1}$ is $\mu_{H}$-stable, strongly $\mu_{H}$-semistable and
reflexive and $\mathcal{E}_{2}$ is strongly $\mu_{H}$-semistable and torsion
free, both $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ are nonzero sheaves and
$\mu_{H}(\mathcal{E}_{1})=\mu_{H}(\mathcal{E}_{2})=0$. By the Hodge index
theorem we have

$\displaystyle 0$ $\displaystyle=\frac{\Delta(\mathcal{E})\cdot
H^{n-2}}{r}=\frac{\Delta(\mathcal{E}_{1})\cdot
H^{n-2}}{r_{1}}+\frac{\Delta(\mathcal{E}_{2})\cdot
H^{n-2}}{r_{2}}-\frac{r_{1}r_{2}}{r}\left(\frac{c_{1}(\mathcal{E}_{1})}{r_{1}}-\frac{c_{1}(\mathcal{E}_{2})}{r_{2}}\right)^{2}\cdot
H^{n-2}$ $\displaystyle\geq\frac{\Delta(\mathcal{E}_{1})\cdot
H^{n-2}}{r_{1}}+\frac{\Delta(\mathcal{E}_{2})\cdot H^{n-2}}{r_{2}}.$

Using the Bogomolov type inequality [Lan04, Theorem 3.2] for $\mathcal{E}_{1}$
and $\mathcal{E}_{2}$, we see that $\Delta(\mathcal{E}_{1})\cdot
H^{n-2}=\Delta(\mathcal{E}_{2})\cdot H^{n-2}=0$. Equality in the Hodge index
inequality implies also that $c_{1}(\mathcal{E}_{1})$ and
$c_{1}(\mathcal{E}_{2})$ are numerically trivial. The induction hypothesis
directly applies only to the reflexive sheaf $\mathcal{E}_{1}$ (not the
torsion free $\mathcal{E}_{2}$) and we deduce that it is locally free and
numerically flat. But we also have an exact sequence

$0\to\mathcal{E}_{2}\to{\mathcal{E}_{2}}^{\vee\vee}\to Q\to 0,$

where $Q$ is supported in codimension at least $2$. Since
${\mathcal{E}_{2}}^{\vee\vee}$ is also strongly $\mu_{H}$-semistable,
$\Delta({\mathcal{E}_{2}}^{\vee\vee})\cdot H^{n-2}\geq 0$. But
$\Delta({\mathcal{E}_{2}}^{\vee\vee})\cdot H^{n-2}=2rc_{2}(Q)\cdot H^{n-2}\leq
0$, so $\Delta({\mathcal{E}_{2}}^{\vee\vee})\cdot H^{n-2}=0$ and $Q$ is
supported in codimension at least $3$. By the induction hypothesis,
${\mathcal{E}_{2}}^{\vee\vee}$ is locally free and every factor in a
Jordan–Hölder filtration of ${\mathcal{E}_{2}}^{\vee\vee}$ is also locally
free and numerically flat. An Ext computation using that $Q$ has codimension
at least $3$ shows that we have a commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}_{2}}^{\vee\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

for some sheaf $\mathcal{E}^{\prime}$ that is then necessarily locally free.
See [Nak99, Proposition 2.5] or [Lan19, Lemma 1.12] for details. The middle
vertical arrow is an isomorphism since $\mathcal{E}$ and
$\mathcal{E}^{\prime}$ are both reflexive, and isomorphic on the locally free
locus of $\mathcal{E}_{2}$. This implies that $\mathcal{E}$ is locally free
and $\mathcal{E}_{2}$ is reflexive, so we can apply the induction assumption
also to $\mathcal{E}_{2}$.

To finish the proof one needs to deal with the case when $\mathcal{E}$ is
$\mu_{H}$-stable but not strongly $\mu_{H}$-stable. Then we can apply the
above arguments for some Frobenius pull-back $(F_{X}^{m})^{*}\mathcal{E}$.
Since local freeness and numerical flatness of $(F_{X}^{m})^{*}\mathcal{E}$
implies local freeness and numerical flatness of $\mathcal{E}$, we get the
required assertion. ∎

######  Corollary 4.7.

Let $\mathcal{E}$ be a reflexive strongly $\mu_{H}$-semistable sheaf of rank
$r$ with $\Delta(\mathcal{E})\cdot H^{n-2}=0$. Then $\mathcal{E}$ is locally
free and $\operatorname{{\mathcal{E}}nd\,}\mathcal{E}$ is numerically flat.
Moreover, every factor in a Jordan–Hölder filtration of $\mathcal{E}$ is also
locally free and its endomorphism bundle is numerically flat.

###### Proof.

We use a finite cover to extract an $r$-th root of $\det\mathcal{E}$ and
reduce to the case $c_{1}(\mathcal{E})\equiv 0$ where Corollary 4.6 applies.
Let $\varphi:Y\to X$ be a Bloch–Gieseker cover (see [BG71, Lemma 2.1]), i.e.,
a finite surjective map from a smooth projective variety $Y$ such that
$\varphi^{*}\det\mathcal{E}=\mathcal{L}^{\otimes r}$ for some line bundle
$\mathcal{L}$ on $Y$. The map $\varphi$ is flat so $\varphi^{*}\mathcal{E}$ is
reflexive. By Remark 2.3 we also have
$c_{i}(\varphi^{*}\mathcal{E})=\varphi^{*}c_{i}(\mathcal{E})$ for all $i$. In
particular, we have
$\det(\varphi^{*}\mathcal{E}\otimes\mathcal{L}^{\vee})=\mathcal{O}_{Y}$ and
$\Delta(\varphi^{*}\mathcal{E})\cdot\varphi^{*}H^{n-2}=0$. Furthermore,
$\varphi^{*}H$ is ample, and $\varphi^{*}\mathcal{E}$ is strongly
$\mu_{\varphi^{*}H}$-semistable. Then
$\varphi^{*}\mathcal{E}\otimes\mathcal{L}^{-1}$ is reflexive, strongly
$\mu_{\phi^{*}H}$-semistable, with trivial determinant, and
$\Delta(\varphi^{*}\mathcal{E}\otimes\mathcal{L}^{-1})\cdot\varphi^{*}H^{n-2}=\Delta(\varphi^{*}\mathcal{E})\cdot\varphi^{*}H^{n-2}=0$.
From the results above we deduce that
$\varphi^{*}\mathcal{E}\otimes\mathcal{L}^{-1}$ is locally free and
numerically flat. Hence
$\varphi^{*}\operatorname{{\mathcal{E}}nd\,}\mathcal{E}=\operatorname{{\mathcal{E}}nd\,}(\varphi^{*}\mathcal{E}\otimes\mathcal{L}^{-1})$
is also numerically flat. This implies that $\mathcal{E}$ is locally free and
$\operatorname{{\mathcal{E}}nd\,}\mathcal{E}$ is numerically flat. The last
part follows analogously. The only difference is that the pull-back of
$\mu_{H}$-stable sheaf need not be $\mu_{\varphi^{*}H}$-stable and it is only
$\mu_{\varphi^{*}H}$-semistable. So one needs to take a refinement of the
pull-back of a Jordan–Hölder filtration of $\mathcal{E}$ to a Jordan–Hölder
filtration of $\varphi^{*}\mathcal{E}$ and then use Corollary 4.6. ∎

###  4.4. Main theorems in the smooth case

######  Theorem 4.8.

Let $X$ be a smooth projective variety of dimension $n$ defined over an
algebraically closed field $k$ and let $H$ be an ample polarization on $X$.
Let $\mathcal{E}$ be a torsion free sheaf on $X$. Then the following
conditions are equivalent:

  1. (1)

$\mathcal{E}$ is reflexive, strongly $\mu_{H}$-semistable and
$\operatorname{ch}_{1}(\mathcal{E})\cdot
H^{n-1}=\operatorname{ch}_{2}(\mathcal{E})\cdot H^{n-2}=0$.

  2. (2)

$\mathcal{E}$ is locally free and numerically flat.

  3. (3)

$\mathcal{E}$ is strongly $\mu_{H}$-semistable and $c_{j}(\mathcal{E})\equiv
0$ for all $j\geq 1$.

  4. (4)

$\mathcal{E}$ is strongly $\mu_{H}$-semistable and the normalized Hilbert
polynomial of $\mathcal{E}$ equals to the Hilbert polynomial of
$\mathcal{O}_{X}$.

In particular, if $\mathcal{E}$ is a strongly slope semistable vector bundle
on $X$ with $c_{1}(\mathcal{E})\equiv 0$, then $\mathcal{E}$ is nef if and
only if $\Delta(\mathcal{E})\equiv 0$.

###### Proof.

The conditions $\operatorname{ch}_{1}(\mathcal{E})\cdot H^{n-1}=0$ and
$\operatorname{ch}_{2}(\mathcal{E})\cdot H^{n-2}=0$ imply that
$c_{1}(\mathcal{E})\cdot H^{n-2}$ is numerically trivial and
$\Delta(\mathcal{E})\cdot H^{n-2}=0$ by the Bogomolov inequality and by the
Hodge index theorem on surfaces (see [Lan11, Lemma 4.2]). The condition
$c_{1}(\mathcal{E})\cdot H^{n-2}=0$ implies that $c_{1}(\mathcal{E})$ is
numerically trivial by [Ful98, Example 19.3.3]. Therefore $(1)\Rightarrow(2)$
follows from Corollary 4.6. The implication $(2)\Rightarrow(3)$ follows from
Proposition 3.3. The proofs of implications $(3)\Rightarrow(4)$ and
$(4)\Rightarrow(1)$ are analogous to the proofs of the corresponding
implications in Proposition 4.4.

If $\mathcal{E}$ is a nef vector bundle with $c_{1}(\mathcal{E})\equiv 0$,
then $\mathcal{E}$ is numerically flat and in particular
$\Delta(\mathcal{E})\equiv 0$. Conversely, if $\mathcal{E}$ is a strongly
semistable vector bundle with $c_{1}(\mathcal{E})\equiv 0$, and
$\Delta(\mathcal{E})\equiv 0$, then $\operatorname{ch}_{2}(\mathcal{E})\equiv
0$ and so $\mathcal{E}$ is numerically flat. ∎

Even in the locally free case, one cannot replace condition $(3)$ in Theorem
4.8 with $c_{j}(\mathcal{E})\cdot H^{n-j}=0$ for all $j\geq 1$, or the
condition $\operatorname{ch}_{2}(\mathcal{E})\cdot H^{n-2}=0$ in (1) with
$c_{2}(\mathcal{E})\cdot H^{n-2}=0$.

######  Example 4.9.

Let $X$ be a smooth projective surface of Picard rank at least 3. Let
$H,L,L^{\prime}$ be divisors on $X$ with $H$ ample such that the intersection
pairing on ${\rm span}(H,L,L^{\prime})\subseteq N^{1}(X)$ has diagonal matrix
with respect to the basis $(H,L,L^{\prime})$. Let
$\mathcal{E}=\mathcal{O}_{X}(L)\oplus\mathcal{O}_{X}(L^{\prime})$. It is
strongly $\mu_{H}$-semistable. Furthermore, we have $c_{1}(\mathcal{E})\cdot
H=0$ and $\int_{X}c_{2}(\mathcal{E})=0$. However, $\mathcal{E}$ is not
numerically flat since $c_{1}(\mathcal{E})=L+L^{\prime}$ is not numerically
trivial. Note that in this case
$\int_{X}\operatorname{ch}_{2}(\mathcal{E})<0$.

######  Theorem 4.10.

Let $X$ be a smooth projective variety of dimension $n$ defined over an
algebraically closed field $k$ and let $H$ be an ample polarization on $X$.
Let $\mathcal{E}$ be a reflexive sheaf of rank $r$ on $X$. Then the following
conditions are equivalent:

  1. (1)

$\mathcal{E}$ is strongly $\mu_{H}$-semistable and $\Delta(\mathcal{E})$ is
numerically trivial.

  2. (2)

$\mathcal{E}$ is strongly $\mu_{H}$-semistable and $\Delta(\mathcal{E})\cdot
H^{n-2}=0$.

  3. (3)

$\mathcal{E}$ is locally free and the twisted normalized bundle
$\mathcal{E}\langle-\frac{1}{r}\det\mathcal{E}\rangle$ is nef.

  4. (4)

$\mathcal{E}$ is locally free and
$\operatorname{{\mathcal{E}}nd\,}\mathcal{E}$ is nef.

  5. (5)

For every morphism $f:C\to X$ from a smooth projective curve,
$f^{*}\mathcal{E}$ is semistable.

  6. (6)

For every morphism $f:Y\to X$, where $Y$ is a smooth projective variety,
$f^{*}\mathcal{E}$ is strongly slope semistable with respect to any ample
polarization on $Y$.

In particular, if $\mathcal{E}$ is strongly $\mu_{H}$-semistable and
$\Delta(\mathcal{E})\cdot H^{n-2}=0$, then $\mathcal{E}$ is locally free.
Furthermore, it is nef (resp. ample) if and only if $\det\mathcal{E}$ is nef
(resp. ample).

###### Proof.

$(1)\Rightarrow(2)$ is trivial. For locally free $\mathcal{E}$, the nefness of
$\mathcal{E}\langle-\frac{1}{r}\det\mathcal{E}\rangle$ is equivalent to the
nefness of $\varphi^{*}\mathcal{E}\otimes\mathcal{L}^{\vee}$, where
$\varphi:Y\to X$ is a finite cover as in Corollary 4.7. Since this bundle has
trivial determinant, its nefness is equivalent to the nefness (equivalently
numerical flatness) of $\mathcal{E}{\rm nd}\,\varphi^{*}\mathcal{E}$ and then
to that of $\mathcal{E}{\rm nd}\,\mathcal{E}$. We get the implications
$(2)\Rightarrow(3)\Leftrightarrow(4)$ by Corollary 4.7. We also have
$(4)\Rightarrow(1)$ by Proposition 3.3.

Numerically flat bundles are universally slope semistable, in fact universally
strongly slope semistable. We obtain $(4)\Rightarrow(6)\Rightarrow(5)$. Then
non-torsion semistable sheaves on smooth projective curves are torsion free,
in particular locally free. General complete intersection curves of high
degree passing through a given point $x\in X$ are smooth by [DH91]. Assuming
$(5)$, we obtain that $\mathcal{E}$ is locally free. By precomposing $f:C\to
X$ with iterates of the Frobenius $F_{C}$, we see that $(5)$ is equivalent to
the analogous statement for strong semistability. On $C$, the strong
semistability of $f^{*}\mathcal{E}$ is equivalent to the nefness of
$f^{*}\mathcal{E}{\rm nd}\,\mathcal{E}$. We deduce that $(5)\Rightarrow(4)$.

For the last statements, if $\mathcal{E}$ is strongly $\mu_{H}$-semistable
with $\Delta(\mathcal{E})\cdot H^{n-2}=0$, then $\mathcal{E}$ is locally free
by Corollary 4.7. Clearly if $\mathcal{E}$ is nef (resp. ample), then
$\det\mathcal{E}$ is nef (resp. ample). The implication $(2)\Rightarrow(3)$
and the identity
$\mathcal{E}=\bigl{(}\mathcal{E}\langle-\frac{1}{r}\det\mathcal{E}\rangle\bigr{)}\langle\frac{1}{r}\det\mathcal{E}\rangle$
give the converse. ∎

######  Remark 4.11.

In the above theorem one can also give another condition analogous to $(3)$ of
Theorem 4.8. See [Lan19, Theorem 2.2] for the precise formulation. We leave an
easy proof of this result along the above lines to the interested reader.

######  Remark 4.12.

If $X$ is a complex projective manifold then homological equivalence over
$\mathbb{Q}$ implies numerical equivalence. Classically they are known to
agree for divisors. Lieberman [Lie68] also proved it for codimension 2 cycles
using the hard Lefschetz theorem. So in this case proving that
$\Delta(\mathcal{E})$ is 0 in $H^{4}(X,\mathbb{Q})$ is equivalent to proving
that it is numerically trivial.

##  5\. On Misra’s question

######  Definition 5.1.

Let $X$ be a projective variety defined over an algebraically closed field
$k$. We say that $X$ is _$1$ -homogeneous_ if
$\operatorname{{Nef}}(X)=\operatorname{\overline{Eff}}(X)$.

Curves, or more generally varieties of Picard rank 1, and homogeneous spaces
are $1$-homogeneous. By [Miy87, Theorem 3.1 and remark on p. 464] if
$\mathcal{E}$ is a vector bundle on a smooth projective curve then
$\mathbb{P}(\mathcal{E})$ is $1$-homogeneous if and only if $\mathcal{E}$ is
strongly semistable.

######  Remark 5.2.

  1. (i)

If $X$ is a projective variety with Picard number 2 and $A$ and $B$ are
globally generated line bundles, but not big, and not proportional in
$N^{1}(X)$, then $X$ is 1-homogeneous and $A$ and $B$ span the boundary rays
of $\operatorname{{Nef}}(X)=\operatorname{\overline{Eff}}(X)$.

  2. (ii)

If $X$ is projective with Picard number 1 and dimension $n$, and $\mathcal{E}$
is a vector bundle of rank $r$ on $X$ such that $\mathcal{E}$ can be generated
by fewer than $n+r$ global sections, then $\mathbb{P}(\mathcal{E})$ is
$1$-homogeneous. (We get an induced morphism
$f:\mathbb{P}(\mathcal{E})\to\mathbb{P}^{N}$ for some
$N<\dim\mathbb{P}(\mathcal{E})$. In particular $r\geq 2$ and
$\mathbb{P}(\mathcal{E})$ has Picard rank 2. The fibers of
$\pi:\mathbb{P}(\mathcal{E})\to X$ are embedded by $f$. Thus if $H$ is a very
ample line bundle on $X$, then $\pi^{*}H$ and
$f^{*}\mathcal{O}_{\mathbb{P}^{N}}(1)$ satisfy the requirements of (i).)

###  5.1. Misra’s theorem in an arbitrary characteristic

The following theorem generalizes [Mis21, Theorem 1.2] to an arbitrary
characteristic:

######  Theorem 5.3.

Let $X$ be a smooth projective variety defined over an algebraically closed
field $k$ and let $\mathcal{E}$ be a strongly slope semistable bundle with
respect to some ample polarization of $X$. Let us also assume that
$\Delta(\mathcal{E})\equiv 0$. Then the following conditions are equivalent:

  1. (1)

$X$ is $1$-homogeneous,

  2. (2)

$c_{1}(\pi_{*}{\mathcal{O}}_{\mathbb{P}(E)}(D))$ is nef for every effective
divisor $D$ on $\mathbb{P}(\mathcal{E})$,

  3. (3)

$\mathbb{P}(\mathcal{E})$ is $1$-homogeneous.

###### Proof.

We specify the adjustments needed to port the proof of [Mis21, Theorem 1.2] to
positive characteristic. Once can use Theorem 4.10 instead of [Mis21, Theorem
2.1]. Apart from that the implication $(1)\Rightarrow(2)$ uses the fact that
symmetric powers of $\mathcal{E}$ are semistable. This follows either from the
Ramanan–Ramanathan theorem (see the proof of Proposition 4.2) or one can use
Theorem 4.10 and the fact that symmetric powers of nef bundles are nef.

Finally, the proof of $(1)\Rightarrow(2)$ uses the duality between the cone of
strongly movable curves and pseudo-effective divisors. This fact also holds in
positive characteristic by [FL17b, Theorem 2.22] (see also [Das20, Theorem
1.4]). The rest of the proof is the same as in [Mis21]. ∎

###  5.2. Syzygy bundle counterexamples to Question 1.4

######  Definition 5.4.

Let $X$ be a projective variety. Let $\mathcal{V}$ be a globally generated
vector bundle on $X$. The associated _syzygy bundle_ $M_{\mathcal{V}}$ is the
kernel of the natural evaluation morphism
$H^{0}(X,\mathcal{V})\otimes\mathcal{O}_{X}\overset{{\rm
ev}}{\twoheadrightarrow}\mathcal{V}$. Put
$E_{\mathcal{V}}=M_{\mathcal{V}}^{\vee}$. This is a globally generated vector
bundle.

For example the Euler sequence on $\mathbb{P}^{n}$ gives
$M_{\mathcal{O}_{\mathbb{P}^{n}}(1)}=\Omega_{\mathbb{P}^{n}}(1)$. If
$\mathcal{L}$ is a globally generated line bundle, and
$\varphi:X\to\mathbb{P}^{N}$ is the induced morphism with
$\varphi^{*}\mathcal{O}_{\mathbb{P}^{N}}(1)=\mathcal{L}$, then
$M_{\mathcal{L}}=\varphi^{*}\Omega_{\mathbb{P}^{N}}(1)$ and
$E_{\mathcal{L}}=\varphi^{*}T_{\mathbb{P}^{N}}(-1)$.

######  Remark 5.5.

If $\mathcal{V}$ is a globally generated vector bundle on $X$ with $\dim
X>{\rm rk}\,\mathcal{V}$, then $r={\rm
rk}\,E_{\mathcal{V}}=h^{0}(X,\mathcal{V})-{\rm rk}\,\mathcal{V}$ and
$E_{\mathcal{V}}$ is generated by $h^{0}(X,\mathcal{V})=r+{\rm
rk}\,\mathcal{V}<\dim X+r$ global sections. In particular, if $X$ has Picard
rank 1 then $\mathbb{P}(E_{\mathcal{V}})$ is $1$-homogeneous by Remark 5.2.

######  Proposition 5.6.

For $n\geq 2$ we have that

  1. (i)

$\mathbb{P}(T_{\mathbb{P}^{n}})$ is $1$-homogeneous.

  2. (ii)

$T_{\mathbb{P}^{n}}$ is strongly slope semistable with respect to the
hyperplane class.

  3. (iii)

$\Delta(T_{\mathbb{P}^{n}})\neq 0$.

  4. (iv)

The restriction of $T_{\mathbb{P}^{n}}$ to every line in $\mathbb{P}^{n}$ is
unstable.

###### Proof.

$(i)$. Apply Remark 5.5 to $\mathcal{V}=\mathcal{O}_{\mathbb{P}^{n}}(1)$.
$(ii)$ is classical and it follows, e.g., from the Bott vanishing. $(iii)$. By
direct computation,
$\Delta(T_{\mathbb{P}^{n}})=\Delta(T_{\mathbb{P}^{n}}(-1))=n+1$. $(iv)$.
$T_{\mathbb{P}^{n}}$ restricts as $\mathcal{O}(2)\oplus\mathcal{O}(1)^{\oplus
n-1}$ on every line. ∎

The easy counterexample above is slope semistable. We also give a slope
unstable example inspired in part by suggestions of S. Misra and D. S.
Nagaraj.

######  Example 5.7.

On $\mathbb{P}^{3}$ consider the globally generated bundle
$\mathcal{V}=\mathcal{O}_{\mathbb{P}^{3}}(1)\oplus\mathcal{O}_{\mathbb{P}^{3}}(2)$.
Consider the associated syzygy bundle $M_{\mathcal{V}}$ and let
$\mathcal{E}=E_{\mathcal{V}}=M_{\mathcal{V}}^{\vee}$. Then $\mathcal{E}$ is
slope unstable, has positive discriminant, and $\mathbb{P}(\mathcal{E})$ is
1-homogeneous. The bundle $\mathcal{E}$ has rank 12, $c_{1}(\mathcal{E})=3$
and $c_{2}(\mathcal{E})=7$. It is an immediate computation that
$\Delta(\mathcal{E})>0$. We have that
$M_{\mathcal{V}}=M_{\mathcal{O}_{\mathbb{P}^{3}}(1)}\oplus
M_{\mathcal{O}_{\mathbb{P}^{3}}(2)}$. The summands have slopes $-1/3$ and
respectively $-2/9$. Thus $M_{\mathcal{V}}$ and its dual $\mathcal{E}$ are
unstable. Since ${\rm rk}\,\mathcal{V}=2<\dim\mathbb{P}^{3}$, we get that
$\mathbb{P}(\mathcal{E})$ is 1-homogeneous by Remark 5.5.∎

We list related problems asking if our counterexamples are the
simplest/smallest possible.

######  Question 5.8.

  1. (1)

Does there exist a complex projective manifold $X$ of Picard rank 1 and
dimension at least 2 supporting an ample and globally generated line bundle
$L$ such that the syzygy bundle $M_{L}$ is $\mu_{L}$-unstable, but
$\mathbb{P}(M_{L}^{\vee})$ is $1$-homogeneous? 111[Sch05] constructs an
example on curves. The semistability of syzygy bundles is an active topic of
research. We refer to [EL92, ELM13, BPMGNO19] and the references therein for a
history of the problem.

  2. (2)

Does there exist a complex projective surface $X$ supporting a slope unstable
$\mathcal{E}$ such that $\mathbb{P}(\mathcal{E})$ is $1$-homogeneous?

  3. (3)

Are there any $\mu_{H}$-unstable bundles $\mathcal{E}$ with
$\Delta(\mathcal{E})\cdot H^{n-2}=0$ such that $\mathbb{P}(\mathcal{E})$ is
$1$-homogeneous?

###  5.3. A positive result

######  Lemma 5.9.

Let $V$ be a free module of rank $r$ over a commutative ring $k$. Then for any
$a,b\geq 1$ there exist

  1. (1)

a surjection of ${\rm GL}\,(V)$-modules

$\operatorname{Sym}^{a}(\operatorname{Sym}^{b}V)\twoheadrightarrow\operatorname{Sym}^{ab}V,$

  2. (2)

an inclusion of ${\rm GL}\,(V)$-modules

$(\bigwedge^{r}V)^{\otimes
2a}\hookrightarrow\operatorname{Sym}^{r}(\operatorname{Sym}^{2a}V).$

Moreover, the composition

$(\bigwedge^{r}V)^{\otimes
2a}\hookrightarrow\operatorname{Sym}^{r}(\operatorname{Sym}^{2a}V)\to\operatorname{Sym}^{2ra}V$

is zero.

Over $\mathbb{C}$, assertion $(2)$ is a particular case of the main result of
[BCI11] which also applies to other even partitions than $\lambda=((2a)^{r})$.

###### Proof.

We have a canonical surjection
$\bigotimes_{i=1}^{a}\operatorname{Sym}^{b}V\twoheadrightarrow\operatorname{Sym}^{ab}V$
coming from the symmetric multiplication. By definition we also have a
canonical surjection
$\bigotimes_{i=1}^{a}\operatorname{Sym}^{b}V\twoheadrightarrow\operatorname{Sym}^{a}(\operatorname{Sym}^{b}V)$.
Using the universal property of the symmetric product we get an induced map
$\operatorname{Sym}^{a}(\operatorname{Sym}^{b}V)\to\operatorname{Sym}^{ab}V$,
which is also surjective. This gives the first assertion.

To prove the second assertion we reduce to the case $k=\mathbb{Z}$. For
$k=\mathbb{Z}$ we construct an explicit non-zero map of ${\rm
GL}\,(V)$-modules that has an associated matrix with an entry equal to $1$.
The map is then a split inclusion as a morphism of $\mathbb{Z}$-modules, hence
base changing to any commutative ring $k$ is still injective.

Let $\lambda=((2a)^{r})=(2a,...,2a)$ be a partition of $2ar$, i.e., we have a
rectangle of size $(r\times 2a)$. Let $\Sigma$ be the set of all tableaux $T$
of shape $\lambda$ with the entries in $[1,r]=\\{1,...,r\\}$ so that in each
column we have a permutation of the set $[1,r]$ and the first column
corresponds to an even permutation. We set
$\operatorname{sgn\,}T=\prod_{i=1}^{2a}\operatorname{sgn\,}\sigma_{i}$, where
$\sigma_{i}$ is the permutation of $[1,r]$ corresponding to the $i$-th column
of $T$ and $\operatorname{sgn\,}\sigma$ is the sign of permutation $\sigma$.
Now we define the map

$\varphi:\prod_{i=1}^{2a}\left(\prod_{j=1}^{r}V\right)\to\operatorname{Sym}^{r}(\operatorname{Sym}^{2a}V)$

by setting

$\varphi((v_{11},...,v_{r1}),...,(v_{1,2a},...,v_{r,2a}))=\sum_{T\in\Sigma}{\operatorname{sgn\,}T}\prod_{j=1}^{r}\left(\prod_{i=1}^{2a}v_{T(j,i),i}\right).$

Since this map is multilinear it factors to the map

$\varphi:(V^{\otimes r})^{\otimes
2a}=\bigotimes_{i=1}^{2a}\left(\bigotimes_{j=1}^{r}V\right)\to\operatorname{Sym}^{r}(\operatorname{Sym}^{2a}V)$

Note that this last map is alternating in each set of variables
$(v_{1m},...,v_{rm})$, where $m\in[1,2a]$. Since we work over $\mathbb{Z}$, it
is sufficient to check that the corresponding multilinear form is
antisymmetric. This is clear for $m>1$ as exchanging $v_{im}$ with $v_{jm}$
defines a bijection on the set $\Sigma$ that replaces the tableau $T$ with
another tableau with exchanged entries between $(i,m)$ and $(j,m)$ places. For
$m=1$ it follows from the fact that exchanging $v_{i1}$ and $v_{j1}$ defines a
bijection on the set $\Sigma$ that replaces the tableau $T$ with another
tableau with the same first column but exchanged $i$-th and $j$-th entries on
all of the remaining $2a-1$ columns. This changes the sign with which the
corresponding product is taken. Therefore $\varphi$ is antisymmetric also in
the variables $(v_{11},...,v_{r1})$. This implies that the formula

$\bigotimes_{i=1}^{2a}(v_{1i}\wedge...\wedge
v_{ri})\to\sum_{T\in\Sigma}{\operatorname{sgn\,}T}\prod_{j=1}^{r}\left(\prod_{i=1}^{2a}v_{T(j,i),i}\right)$

defines a map of ${\rm GL}\,(V)$-modules

$(\bigwedge^{r}V)^{\otimes
2a}\to\operatorname{Sym}^{r}(\operatorname{Sym}^{2a}V).$

If $(e_{1},...,e_{r})$ is a basis of $V$, the element
$\bigotimes_{i=1}^{2a}(e_{1}\wedge...\wedge e_{r})$ is mapped to

$W=\sum_{T\in\Sigma}{\operatorname{sgn\,}T}\prod_{j=1}^{r}\left(\prod_{i=1}^{2a}e_{T(j,i)}\right).$

Note that $\operatorname{Sym}^{r}(\operatorname{Sym}^{2a}V)$ has a standard
basis corresponding to

$\prod_{j=1}^{r}\left(\prod_{i=1}^{r}e_{i}^{n_{ij}}\right),$

where $\sum_{i}n_{ij}=2a$ for $j=1,...,r$. If we write $W$ in this basis, the
coefficient at the element $\prod_{j=1}^{r}e_{j}^{2a}$ is equal to $1$, so the
corresponding map is non-zero.

To see the last part of the lemma, it is sufficient to remark that we have

$\sum_{T\in\Sigma}{\operatorname{sgn\,}T}\prod_{j=1}^{r}\prod_{i=1}^{2a}e_{T(j,i)}=\sum_{T\in\Sigma}{\operatorname{sgn\,}T}\prod_{i=1}^{2a}\prod_{j=1}^{r}e_{T(j,i)}=(\sum_{T\in\Sigma}{\operatorname{sgn\,}T})\prod_{i=1}^{2a}\prod_{j=1}^{r}e_{j}=0$

in $\operatorname{Sym}^{2ra}V$. ∎

######  Remark 5.10.

[Wei90, Example 1.9] shows that the plethysm
$\operatorname{Sym}^{5}(\operatorname{Sym}^{3}\mathbb{C}^{5})$ does not
contain $(\bigwedge^{5}\mathbb{C}^{5})^{\otimes 3}$ as a ${\rm
GL}\,(\mathbb{C},5)$-submodule. Thus the parity condition in the above lemma
is necessary.

######  Corollary 5.11.

Let $\mathcal{E}$ be a rank $r$ vector bundle on some scheme $X$ defined over
some commutative ring $k$. Then for any $a,b\geq 1$ we have

  1. (1)

a canonical surjection

$\operatorname{Sym}^{a}(\operatorname{Sym}^{b}\mathcal{E})\twoheadrightarrow\operatorname{Sym}^{ab}\mathcal{E},$

  2. (2)

a canonical inclusion

$(\det\mathcal{E})^{\otimes
2a}\hookrightarrow\operatorname{Sym}^{r}(\operatorname{Sym}^{2a}\mathcal{E})$

onto a subbundle.

Moreover, the composition

$(\det\mathcal{E})^{\otimes
2a}\hookrightarrow\operatorname{Sym}^{r}(\operatorname{Sym}^{2a}\mathcal{E})\twoheadrightarrow\operatorname{Sym}^{2ra}\mathcal{E}$

is zero.

###### Proof.

The corollary follows immediately from the previous lemma. For the convenience
of the reader we recall the idea of proof. Let $V$ be a free $k$-module of
rank $r$ and let $P\to X$ be the principal ${\rm GL\,}(V)$-bundle associated
to $\mathcal{E}$. Then for any ${\rm GL}\,(V)$-module $W$ we have the
associated vector bundle $P(W)$ and maps of ${\rm GL}\,(V)$-modules induce the
corresponding maps of vector bundles. Applying this construction to maps from
Lemma 5.9, we get the corresponding maps from the corollary. ∎

When $\mathcal{E}$ is a $\mu_{H}$-semistable bundle on $(X,H)$ such that
$\Delta(\mathcal{E})\equiv 0$, [Mis21, Lemma 2.3] observes that
$\operatorname{Sym}^{m}\mathcal{E}$ is also $\mu_{H}$-semistable and
$\Delta(\operatorname{Sym}^{m}\mathcal{E})\equiv 0$ for all $m\geq 0$. If
furthermore $X$ is $1$-homogeneous, then it follows from [Mis21, Theorem 1.2]
that $\mathbb{P}(\operatorname{Sym}^{m}\mathcal{E})$ is $1$-homogeneous for
all $m\geq 0$. Question 1.4 should also consider $m\geq 1$.

######  Example 5.12.

On $\mathbb{P}^{2}$ consider
$\mathcal{E}=\operatorname{Sym}^{2}(T_{\mathbb{P}^{2}}(-1))$. Then
$\mathbb{P}(\mathcal{E})\simeq{\rm Hilb}^{2}\mathbb{P}^{2}$. The divisor $E$
of nonreduced length 2 subschemes of $\mathbb{P}^{2}$ is contracted by the
birational Hilbert–Chow morphism. In particular, it is effective, but not a
nef divisor. If $L=\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ and $H$ is the
pullback of the class of a line in $\mathbb{P}^{2}$ then $E$ is linearly
equivalent to $2(L-H)$. See [FL17b, Section 7.2] for details.

Another perspective at this example is as follows. Since $\mathcal{E}$ has
rank $2$, Corollary 5.11 and comparison of ranks imply that we have a short
exact sequence of vector bundles

$0\to(\det\mathcal{E})^{\otimes
2}\to\operatorname{Sym}^{2}(\operatorname{Sym}^{2}\mathcal{E})\to\operatorname{Sym}^{4}\mathcal{E}\to
0.$

This gives a short exact sequence

$0\to\mathcal{O}_{\mathbb{P}^{2}}\to\bigl{(}\operatorname{Sym}^{2}(\operatorname{Sym}^{2}T_{\mathbb{P}^{2}}(-1))\bigr{)}(-2)\to\bigl{(}\operatorname{Sym}^{4}(T_{\mathbb{P}^{2}}(-1))\bigr{)}(-2)\to
0.$

In particular, we see that
$\bigl{(}\operatorname{Sym}^{2}(\operatorname{Sym}^{2}T_{\mathbb{P}^{2}}(-1))\bigr{)}(-2)$
is effective. However, it is not nef, e.g., because its quotient
$\bigl{(}\operatorname{Sym}^{4}(T_{\mathbb{P}^{2}}(-1))\bigr{)}(-2)$ restricts
to
$\mathcal{O}_{L}(-2)\oplus\mathcal{O}_{L}(-1)\oplus\mathcal{O}_{L}\oplus\mathcal{O}_{L}(1)\oplus\mathcal{O}_{L}(2)$
on every line $L$ in $\mathbb{P}^{2}$.

######  Remark 5.13.

Let $\mathcal{E}$ is a vector bundle on $X$. Then one can easily see that the
following conditions are equivalent:

  1. (1)

$\mathbb{P}(\mathcal{E})$ is $1$-homogeneous

  2. (2)

If $D$ is a divisor on $X$ such that
$\bigl{(}\operatorname{Sym}^{m}\mathcal{E}\bigr{)}(D)$ is effective for some
$m\geq 0$ then $\bigl{(}\operatorname{Sym}^{m}\mathcal{E}\bigr{)}(D)$ is nef.

######  Theorem 5.14.

Let $X$ be a smooth projective variety defined over an algebraically closed
field $k$. Let $\mathcal{E}$ be a vector bundle of rank $r$ on $X$. Then
$\mathcal{E}$ is strongly slope semistable with respect to any ample
polarization and $\Delta(\mathcal{E})\equiv 0$ if any of the following
conditions hold.

  1. (1)

For every $m\geq 0$, we have that
$\mathbb{P}(\operatorname{Sym}^{m}\mathcal{E})$ is $1$-homogeneous.

  2. (2)

For every divisor $D$ on $X$ and every $m,l\geq 0$, we have that
$(\operatorname{Sym}^{l}(\operatorname{Sym}^{m}\mathcal{E}))(D)$ is effective
if and only if it is nef.

  3. (3)

There exists $m\geq 1$ such that
$\operatorname{Sym}^{r}(\operatorname{Sym}^{2m}\mathcal{E})\otimes(\det\mathcal{E}^{\vee})^{\otimes
2m}$ is nef.

  4. (4)

$\mathbb{P}(\mathcal{E}^{\otimes r})$ is $1$-homogeneous.

  5. (5)

$\mathbb{P}(\operatorname{{\mathcal{E}}nd\,}\mathcal{E})$ is $1$-homogeneous.

###### Proof.

The equivalence of $(1)$ and $(2)$ follows from Remark 5.13. We focus on
$(2)$. By Corollary 5.11 for all $m\geq 1$, the bundle
$\operatorname{Sym}^{r}(\operatorname{Sym}^{2m}\mathcal{E})$ contains
$(\det\mathcal{E})^{\otimes 2m}$. Thus
$\operatorname{Sym}^{r}(\operatorname{Sym}^{2m}\mathcal{E})\otimes(\det\mathcal{E}^{\vee})^{\otimes
2m}$ is effective and hence it is also nef. Since Corollary 5.11 implies that
$\operatorname{Sym}^{2mr}\mathcal{E}\otimes(\det\mathcal{E}^{\vee})^{\otimes
2m}$ is a quotient of
$\operatorname{Sym}^{r}(\operatorname{Sym}^{2m}\mathcal{E})\otimes(\det\mathcal{E}^{\vee})^{\otimes
2m}$, it is also nef. Since nefness for (twisted) vector bundles is
homogeneous (cf. [Laz04b, Theorem 6.2.12], or [FM21, Lemma 3.24 and Remark
3.10]), we deduce that
$\mathcal{E}\langle-\frac{1}{r}\det\mathcal{E}^{\vee}\rangle$ is nef. Conclude
by Theorem 1.1. This argument also handled $(3)$.

$(4)$ We have a natural inclusion
$\det\mathcal{E}=\bigwedge^{r}\mathcal{E}\hookrightarrow\mathcal{E}^{\otimes
r}$. It is obtained by dualizing the natural surjection
$(\mathcal{E}^{\vee})^{\otimes
r}\twoheadrightarrow\bigwedge^{r}(\mathcal{E}^{\vee})$. It shows that
$\mathcal{E}^{\otimes r}\otimes\det\mathcal{E}^{\vee}$ is effective. By the
assumption on the positive cones, it is then also nef. Hence so is its
quotient $\operatorname{Sym}^{r}\mathcal{E}\otimes\det\mathcal{E}^{\vee}$.
Argue as above.

$(5)$ We have a natural inclusion
$\mathcal{O}_{X}\hookrightarrow{\mathcal{E}}{\rm nd}\,\mathcal{E}$ induced by
sending $1\in\mathcal{O}_{X}(U)$ to ${\rm id}_{\mathcal{E}(U)}$. Therefore
${\mathcal{E}}{\rm nd}\,\mathcal{E}$ is effective and hence our assumption
implies that it is nef. Now Theorem 4.10 implies the required assertion. ∎

Together with Theorem 5.3 this implies the following result:

######  Corollary 5.15.

Let $X$ be a smooth projective variety defined over an algebraically closed
field $k$ and let $\mathcal{E}$ be a vector bundle of rank $r$ on $X$. Then
the following conditions are equivalent:

  1. (1)

$\mathbb{P}(\operatorname{Sym}^{m}\mathcal{E})$ is $1$-homogeneous for every
$m\geq 0$.

  2. (2)

$\mathbb{P}(\operatorname{Sym}^{2m}\mathcal{E})$ is $1$-homogeneous for some
$m\geq 1$.

  3. (3)

$\mathbb{P}(\mathcal{E}^{\otimes r})$ is $1$-homogeneous.

  4. (4)

$\mathbb{P}(\operatorname{{\mathcal{E}}nd\,}\mathcal{E})$ is $1$-homogeneous.

  5. (5)

The bundle $\operatorname{{\mathcal{E}}nd\,}\mathcal{E}$ is nef and $X$ is
$1$-homogeneous.

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