destabilizing $j^{[*]}({\mathcal{E}},\theta)$ would destabilize it on $H\cap
U$. This would show that one of the restrictions
$j^{*}({\mathcal{E}}_{i},\theta_{i})|_{H\cap U}$ is not slope
$(j^{*}L_{2},\dots,j^{*}L_{n-1})$-semistable. But this contradicts the fact
that the reflexivization of its extension to $H$ (which is equal to
$j^{[*]}({\mathcal{F}}_{i},\tilde{\theta}_{i})$) is slope
$(j^{*}L_{2},\dots,j^{*}L_{n-1})$-stable. ∎

For $r=2$ the assumptions of this theorem can be slightly relaxed (cf. [La5,
Theorem 10]). Note that unlike in [La5] we do not have any assumptions on
lifting on $X$. These assumptions were added in [La5] only to avoid the term
containing $\beta_{r}$ so that the results could hold uniformly in all
characteristics (including $0$). The above result is restricted to the
positive characteristic and it was not known in the characteristic zero case
even if one assumes that $D=0$ and $X$ has klt singularities (cf. [GKPT1,
Theorem 5.22] for a non-effective restriction theorem for general
hypersurfaces). The above theorem will be used to obtain a strong restriction
theorem for Higgs sheaves in characteristic zero in Section 7.

###  5.3 Deformations to systems of Hodge sheaves

Let $X$ be a normal projective variety defined over an algebraically closed
field $k$ and let $L$ be a Lie algebroid on $X$, which is coherent as an
${\mathcal{O}}_{X}$-module.

It is convenient to consider $L$-modules as modules over the universal
enveloping algebra $\Lambda_{L}$ of differential operators associated to $L$
(see [La4, Section 2.2]). So we consider an $L$-module as a pair
$({\mathcal{E}},\nabla)$, where ${\mathcal{E}}$ is a quasi-coherent
${\mathcal{O}}_{X}$-module and
$\nabla:\Lambda_{L}\otimes_{{\mathcal{O}}_{X}}{\mathcal{E}}\to{\mathcal{E}}$
is a $\Lambda_{L}$-module structure. If the underlying sheaf of an $L$-module
is coherent as an ${\mathcal{O}}_{X}$-module, we say (at the risk of abusing
the notation) that $({\mathcal{E}},\nabla)$ is a _coherent $L$-module_. If the
underlying sheaf of an $L$-module is coherent and torsion free as an
${\mathcal{O}}_{X}$-module, we say (again abusing the notation) that
$({\mathcal{E}},\nabla)$ is a _torsion free $L$-module_.

If $({\mathcal{E}},\nabla)$ is a coherent $L$-module then we say that a
filtration ${\mathcal{E}}=N^{0}\supset N^{1}\supset...\supset N^{m}=0$
satisfies _Griffiths transversality_ if it is a filtration of ${\mathcal{E}}$
by coherent ${\mathcal{O}}_{X}$-submodules and
$\nabla(\Lambda_{L}\otimes_{{\mathcal{O}}_{X}}N^{i})\subset N^{i-1}$. For
every such filtration the associated graded object
${\mathop{Gr}}_{N}({\mathcal{E}}):=\bigoplus_{i}N^{i}/N^{i+1}$ carries a
canonical coherent $L$-Higgs module structure
$\theta:L\otimes_{{\mathcal{O}}_{X}}{\mathop{Gr}}_{N}({\mathcal{E}})\to{\mathop{Gr}}_{N}({\mathcal{E}})$
defined by $\nabla$. This can be seen by considering the following commutative
diagram:

$\textstyle{\Lambda_{0}\otimes_{{\mathcal{O}}_{X}}N^{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda_{0}\otimes_{{\mathcal{O}}_{X}}N^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda_{1}\otimes_{{\mathcal{O}}_{X}}N^{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda_{1}\otimes_{{\mathcal{O}}_{X}}N^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N^{i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda_{1}/\Lambda_{0}\otimes_{{\mathcal{O}}_{X}}N^{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda_{1}/\Lambda_{0}\otimes_{{\mathcal{O}}_{X}}N^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N^{i-1}/N^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

where $\Lambda_{0}\subset\Lambda_{1}\subset...\subset\Lambda_{L}$ is the
standard filtration on $\Lambda_{L}$. It follows from the above diagram that
the map $\Lambda_{1}\otimes_{{\mathcal{O}}_{X}}N^{i+1}\to N^{i-1}/N^{i}$ is
zero and hence the map
$\Lambda_{1}/\Lambda_{0}\otimes_{{\mathcal{O}}_{X}}N^{i+1}\to N^{i-1}/N^{i}$
is also zero. So we have an induced map
$\Lambda_{1}/\Lambda_{0}\otimes_{{\mathcal{O}}_{X}}N^{i}/N^{i+1}\to
N^{i-1}/N^{i}$. But $\Lambda_{1}/\Lambda_{0}=L$ and one can easily check that
the obtained map gives an $L$-Higgs module structure on
${\mathop{Gr}}_{N}({\mathcal{E}})$. Note also that by construction the
obtained pair $({\mathop{Gr}}_{N}({\mathcal{E}}),\theta)$ is a system of
$L$-Hodge sheaves on $X$.

In the remainder of this section to define semistability we fix a collection
$(L_{1},...,L_{n-1})$ of nef line bundles on $X$ such that
$L_{1}L_{2}....L_{n-1}$ is numerically non-trivial.

We say that a Griffiths transverse filtration $N^{\bullet}$ on
$({\mathcal{E}},\nabla)$ is _slope gr-semistable_ if the associated $L$-Higgs
sheaf $({\mathop{Gr}}_{N}({\mathcal{E}}),\theta)$ is (torsion free and) slope
semistable. A _partial $L$-oper_ is a triple
$({\mathcal{E}},\nabla,N^{\bullet})$ consisting of a torsion free coherent
${\mathcal{O}}_{X}$-module ${\mathcal{E}}$ with a ${\Lambda_{L}}$-module
structure $\nabla$ and a Griffiths transverse filtration $N^{\bullet}$, which
is slope gr-semistable.

######  Remark 5.5.

Note that analogous definitions in [La4, Section 5.2] work only for smooth Lie
algebroids. The above definitions allow us to deal with general Lie algebroids
and they are equivalent to those in [La4, Section 5.2] in case of smooth Lie
algebroids.

The following theorem can be proven in the same way as [La6, Theorem 5.5]. The
only difference is that in the proof one needs to consider $L$-modules as
$\Lambda_{L}$-modules.

######  Theorem 5.6.

If $({\mathcal{E}},\nabla)$ is slope semistable then there exists a
canonically defined slope gr-semistable Griffiths transverse filtration
$S^{\bullet}$ on $({\mathcal{E}},\nabla)$ providing it with a partial $L$-oper
structure. This filtration is preserved by the automorphisms of
$({\mathcal{E}},\nabla)$.

The above filtration $S^{\bullet}$ is called _Simpson’s filtration_. Even in
the case of a trivial Lie algebroid structure on $L$ the above theorem gives a
non-trivial corollary:

######  Corollary 5.7.

Let
$({\mathcal{E}},\theta:L\otimes_{{\mathcal{O}}_{X}}{\mathcal{E}}\to{\mathcal{E}})$
be a slope semistable $L$-Higgs sheaf. Then there exists a decreasing
filtration ${\mathcal{E}}=N^{0}\supset N^{1}\supset...\supset N^{m}=0$ such
that $\theta(L\otimes_{{\mathcal{O}}_{X}}N^{i})\subset N^{i-1}$ and the
associated graded is a slope semistable system of $L$-Hodge sheaves.

###  5.4 Higgs–de Rham sequences on normal varieties

Let $X$ be a normal projective variety defined over an algebraically closed
field $k$ of positive characteristic $p$. Let $D$ be an effective reduced Weil
divisor on $X$.

Let us assume that $(X,D)$ is almost liftable to $W_{2}(k)$. Then we can find
a big open subset $U\subset X$ such that the pair $(U,D_{U}=D\cap U)$ is log
smooth and liftable to $W_{2}(k)$. Let us fix a lifting
$(\tilde{U},\tilde{D}_{U})$ of $(U,D_{U})$.

Let $({\mathcal{E}},\theta:T_{X}(\log D)\otimes{\mathcal{E}}\to{\mathcal{E}})$
be a reflexive logarithmic Higgs ${\mathcal{O}}_{X}$-module of rank $r\leq p$.
Let us assume that $({\mathcal{E}},\theta)$ is slope semistable. Then by
Corollary 5.7 there exists a canonical filtration $N^{\bullet}$ on
${\mathcal{E}}$ such that the associated graded
$(\bar{\mathcal{E}}_{0},\bar{\theta}_{0})$ is a slope semistable system of
logarithmic Hodge sheaves. Let $({\mathcal{E}}_{0},\theta_{0})$ be the
reflexive hull of $(\bar{\mathcal{E}}_{0},\bar{\theta}_{0})$. By construction,
it is a slope semistable reflexive logarithmic system of Hodge sheaves. In
particular, since its rank $r$ is $\leq p$, it is also a reflexive logarithmic
Higgs ${\mathcal{O}}_{X}$-module with a nilpotent Higgs field of level less or
equal to $p-1$. So we can define
$(V_{0},\nabla_{0}):=C_{(\tilde{U},\tilde{D}_{U})}^{[-1]}({\mathcal{E}}_{0},\theta_{0})$.
Let $S_{0}^{\bullet}$ be (decreasing) Simpson’s filtration on
$(V_{0},\nabla_{0})$ and let
$(\bar{\mathcal{E}}_{1}={\mathop{Gr}}_{S_{0}}(V_{0}),\bar{\theta}_{1})$ be the
associated system of Hodge sheaves. Then we set
$({\mathcal{E}}_{1},\theta_{1}):=((\bar{\mathcal{E}}_{1})^{**},{\bar{\theta}_{1}}^{**})$
and repeat the procedure. In this way we get the following sequence

$\textstyle{({\mathcal{E}},\theta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathop{Gr}}_{N})^{**}}$$\textstyle{(V_{0},\nabla_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathop{Gr}}_{S_{0}})^{**}}$$\textstyle{(V_{1},\nabla_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathop{Gr}}_{S_{1}})^{**}}$$\textstyle{({\mathcal{E}}_{0},\theta_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C^{[-1]}}$$\textstyle{({\mathcal{E}}_{1},\theta_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C^{[-1]}}$$\textstyle{...}$

in which each logarithmic Higgs sheaf $({\mathcal{E}}_{j},\theta_{j})$ is
reflexive rank $r\leq p$ and slope semistable. We call this sequence a
_canonical Higgs–de Rham sequence_ of $({\mathcal{E}},\theta)$.

######  Remark 5.8.

Higgs de Rham sequences were invented by G. Lan. M. Sheng and K. Zuo in [LSZ2]
and their existence was proven in [LSZ2] and [La4]. Canonical Higgs–de Rham
sequences in the above sense first appeared in the proof of [La7, Lemma 3.10].
They are better suited to dealing with normal varieties as one cannot define
suitable Chern classes for torsion free sheaves on normal varieties.

######  Remark 5.9.

Although the above construction is very general, it does not seem easy to
compare numerical invariants of the sheaves ${\mathcal{E}}_{i}$ without some
further assumptions on the singularities of the pair $(X,D)$.

###  5.5 Inverse Cartier transform on log varieties with locally $F$-liftable
singularities.

Let $X$ be a normal variety defined over an algebraically closed field $k$ of
positive characteristic $p$. We define the _Grothendieck group
$K^{{\mathop{\operatorname{\rm ref}}}}(X)$ of reflexive sheaves on $X$_ as the
free abelian group on the isomorphism classes $[{\mathcal{E}}]$ of coherent
reflexive ${\mathcal{O}}_{X}$-modules modulo the relations
$[{\mathcal{E}}_{2}]=[{\mathcal{E}}_{1}]+[{\mathcal{E}}_{3}]$ for each locally
split short exact sequence

$0\to{\mathcal{E}}_{1}\to{\mathcal{E}}_{2}\to{\mathcal{E}}_{3}\to 0$

of coherent reflexive ${\mathcal{O}}_{X}$-modules.

For a coherent ${\mathcal{O}}_{X}$-module ${\mathcal{F}}$ we denote by
$\nabla_{can}^{{\mathcal{F}}}$ the canonical connection on
$F_{X}^{*}{\mathcal{F}}$ given by differentiating along the fibers of the
Frobenius morphism.

######  Lemma 5.10.

Let $D$ be an effective reduced Weil divisor on $X$ such that $(X,D)$ is
liftable to $W_{2}(k)$ and it is locally $F$-liftable. If
$({\mathcal{E}},\theta)=(\bigoplus{\mathcal{E}}^{i},\theta)$ is a reflexive
system of logarithmic Hodge sheaves on $(X,D)$ and we set
${\mathcal{E}}_{j}=\bigoplus_{j\leq i}{\mathcal{E}}^{i}$ with induced
$\theta_{j}$ then for every $j$ we have a short exact sequence

$0\to C^{[-1]}({\mathcal{E}}_{j},\theta_{j})\to
C^{[-1]}({\mathcal{E}}_{j+1},\theta_{j+1})\to(F_{X}^{[*]}{\mathcal{E}}^{j+1},\nabla_{{\mathop{\rm
can}}}^{{\mathcal{E}}_{j+1}})\to 0$

of reflexive ${\mathcal{O}}_{X}$-modules with a logarithmic connection, which
is locally split as a sequence of ${\mathcal{O}}_{X}$-modules. In particular,
we have $[C^{[-1]}{\mathcal{E}}]=[F_{X}^{[*]}{\mathcal{E}}]$ in
$K^{{\mathop{\operatorname{\rm ref}}}}(X)$.

###### Proof.

By construction we have a short exact sequence of Higgs sheaves

$0\to({\mathcal{E}}_{j},\theta_{j})\to({\mathcal{E}}_{j+1},\theta_{j+1})\to({\mathcal{E}}^{j+1},0)\to
0,$

which is split as a sequence of ${\mathcal{O}}_{X}$-modules. Applying
$C^{[-1]}$ to this sequence we get

$0\to C^{[-1]}({\mathcal{E}}_{j},\theta_{j})\to
C^{[-1]}({\mathcal{E}}_{j+1},\theta_{j+1})\to(F_{X}^{[*]}{\mathcal{E}}^{j+1},\nabla_{{\mathop{\rm
can}}}^{{\mathcal{E}}_{j+1}})\to 0,$

because
$C^{[-1]}({\mathcal{E}}^{j+1},0)=(F_{X}^{[*]}{\mathcal{E}}^{j+1},\nabla_{{\mathop{\rm
can}}}^{{\mathcal{E}}_{j+1}})$. So it is sufficient to show that this sequence
is locally split. To do so we fix a point $x\in X$ and an open neighbourhood
$x\in U\subset X$, which is $F$-liftable. Let $V$ be a big open subset of $U$,
which is contained in the log smooth locus of $(X,D)$. The pair $(V,D\cap V)$
has an $F$-lifting $\tilde{F}_{V}:\tilde{V}\to\tilde{V}$ compatible with the
$W_{2}(k)$-lifting $(\tilde{V},\tilde{D})$ induced from the given
$W_{2}(k)$-lifting of $(X,D)$. On $V$ we have a short exact sequence of
modules with integrable connections

$0\to(F_{V}^{*}{\mathcal{E}}_{j},\nabla_{{\mathop{\rm
can}}}^{{\mathcal{E}}_{j}})\to(F_{V}^{*}{\mathcal{E}}_{j+1},\nabla_{{\mathop{\rm
can}}}^{{\mathcal{E}}_{j+1}})\to(F_{V}^{*}{\mathcal{E}}^{j+1},\nabla_{{\mathop{\rm
can}}}^{{\mathcal{E}}^{j+1}})\to 0,$

which is split as a sequence of ${\mathcal{O}}_{V}$-modules. Extending the
above sequence to $U$, we get a short exact sequence of reflexive
$T_{U}$-modules

$0\to F_{U}^{[*]}{\mathcal{E}}_{j}\to F_{U}^{[*]}{\mathcal{E}}_{j+1}\to
F_{U}^{[*]}{\mathcal{E}}^{j+1}\to 0,$

which is split as a sequence of ${\mathcal{O}}_{U}$-modules. By construction
(see Section 8) $C^{[-1]}({\mathcal{E}}_{j})|_{U}\simeq
F_{U}^{[*]}{\mathcal{E}}_{j}$ and $C^{[-1]}(\theta_{j})|_{U}$ is obtained by
extension of the logarithmic connection $\nabla_{{\mathop{\rm
can}}}^{{\mathcal{E}}_{j}}+\zeta_{V}(F^{*}_{V}\theta_{j})$, where
$\zeta_{V}:=p^{-1}\tilde{F}_{V}:F_{V}^{*}\Omega_{V}\to\Omega_{V}$ (see
Subsection 8.1). Since the above isomorphisms are compatible with restrictions
to $V$, we see that the sequence

$0\to C^{[-1]}({\mathcal{E}}_{j},\theta_{j})\to
C^{[-1]}({\mathcal{E}}_{j+1},\theta_{j+1})\to(F_{X}^{[*]}{\mathcal{E}}^{j+1},\nabla_{{\mathop{\rm
can}}}^{{\mathcal{E}}_{j+1}})\to 0$

is split as a sequence of ${\mathcal{O}}_{U}$-modules. ∎

######  Corollary 5.11.

Let $X$ be a normal projective variety with a collection $(L_{1},...,L_{n-2})$
of nef line bundles. Let $D$ be an effective reduced Weil divisor on $X$ such
that $(X,D)$ is almost liftable to $W_{2}(k)$ and it has $F$-liftable
singularities in codimension $2$. Then we have

$\int_{X}{\mathop{\rm
ch\,}}_{2}(C^{[-1]}{\mathcal{E}})L_{1}...L_{n-2}=p^{2}\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}.$

###### Proof.

By Theorem 1.18 we can reduce the assertion to the surface case. Then Theorem
1.13 says that $(X,D)$ satisfies assumptions of Lemma 2.1 and hence we get

$\displaystyle\int_{X}{\mathop{\rm
ch\,}}_{2}(C^{[-1]}{\mathcal{E}})L_{1}...L_{n-2}$
$\displaystyle=\sum_{j}\int_{X}{\mathop{\rm
ch\,}}_{2}(F_{X}^{[*]}{\mathcal{E}}^{j})L_{1}...L_{n-2}=p^{2}\sum_{j}\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}}^{j})L_{1}...L_{n-2}$
$\displaystyle=p^{2}\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}.$

∎

###  5.6 Bogomolov’s inequality for Higgs sheaves

In this subsection we give the first version of Bogomolov’s inequality for
logarithmic Higgs sheaves on singular varieties. The following theorem
generalizes Bogomolov’s inequality for logarithmic Higgs sheaves to singular
varieties (see [La5, Theorem 8] in case $X$ is smooth and [La6, Theorem 3.3]
for the log smooth case).

######  Theorem 5.12.

Let $(L_{1},...,L_{n-1})$ be a collection of nef line bundles on $X$ such that
$L_{1}L_{2}....L_{n-1}$ is numerically non-trivial. Assume that the pair
$(X,D)$ is almost liftable to $W_{2}(k)$ and and it has $F$-liftable
singularities in codimension $2$. Then for any slope
$(L_{1},...,L_{n-1})$-semistable logarithmic reflexive Higgs sheaf
$({\mathcal{E}},\theta)$ of rank $r\leq p$ we have

$\int_{X}\Delta({\mathcal{E}})L_{2}...L_{n-1}\geq 0.$

###### Proof.

Let $({\mathcal{E}},\theta:T_{X}(\log
D)\otimes_{{\mathcal{O}}_{X}}{\mathcal{E}}\to{\mathcal{E}})$ be a reflexive
logarithmic Higgs ${\mathcal{O}}_{X}$-module of rank $r\leq p$. Let us assume
that $({\mathcal{E}},\theta)$ is slope semistable. Let

$\textstyle{({\mathcal{E}},\theta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathop{Gr}}_{N})^{**}}$$\textstyle{(V_{0},\nabla_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathop{Gr}}_{S_{0}})^{**}}$$\textstyle{(V_{1},\nabla_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathop{Gr}}_{S_{1}})^{**}}$$\textstyle{({\mathcal{E}}_{0},\theta_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C^{[-1]}}$$\textstyle{({\mathcal{E}}_{1},\theta_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C^{[-1]}}$$\textstyle{...}$

be the canonical Higgs–de Rham sequence of $({\mathcal{E}},\theta)$.

By Lemma 4.15 there exists $\alpha$ such that
$\mu_{\max,L}({\mathcal{E}}_{m})-\mu_{L}({\mathcal{E}}_{m})\leq\alpha$ for all
$m\geq 0$. So by Corollary 3.8 there exists some constant $C$ such that for
every non-negative integer $m$ we have

$\int_{X}\Delta({\mathcal{E}}_{m})L_{2}...L_{n-1}\geq C.$

Lemma 2.4 implies that

$\int_{X}\Delta({\mathcal{E}})L_{2}...L_{n-1}\geq\int_{X}\Delta({\mathcal{E}}_{0})L_{2}...L_{n-1}$

and

$\int_{X}\Delta(V_{m})L_{2}...L_{n-1}\geq\int_{X}\Delta({\mathcal{E}}_{m+1})L_{2}...L_{n-1}.$

By Corollary 5.11 we have

$\int_{X}\Delta(V_{m})L_{2}...L_{n-1}=p^{2}\int_{X}\Delta({\mathcal{E}}_{m})L_{2}...L_{n-1}.$

Therefore

$C\leq\int_{X}\Delta({\mathcal{E}}_{m})L_{2}...L_{n-1}\leq
p^{2m}\int_{X}\Delta({\mathcal{E}})L_{2}...L_{n-1}.$

Dividing by $p^{2m}$ and passing with $m$ to infinity, we get
$\int_{X}\Delta({\mathcal{E}})L_{2}...L_{n-1}\geq 0$. ∎

######  Remark 5.13.

The above theorem holds also for reflexive sheaves with an integrable
logarithmic connection. Indeed, if $({\mathcal{E}},\nabla)$ is a rank $r\leq
p$ slope $L$-semistable reflexive sheaf with an integrable logarithmic
connection and $S^{\bullet}$ is its Simpson’s filtration then by the above
theorem and Lemma 2.4 we have

$\int_{X}\Delta({\mathcal{E}})L_{2}...L_{n-1}\geq\int_{X}\Delta(({\mathop{Gr}}_{S}{\mathcal{E}})^{**})L_{2}...L_{n-1}\geq
0.$

##  6 The Miyaoka–Yau inequality on singular varieties in positive
characteristic

In this section we prove the Miyaoka–Yau inequality on some mildly singular
varieties in positive characteristic. The ideas are similar to that from [Si]
and [La6] but we show full proofs to show where they need additional facts
related to the use of our Chern classes.

We fix a log pair $(X,D)$ defined over an algebraically closed field of
characteristic $p>0$. We assume that $(X,D)$ is almost liftable to $W_{2}(k)$
and it has $F$-liftable singularities in codimension $2$. Let $n=\dim X$ and
let us fix a collection $L=(L_{1},...,L_{n-1})$ of nef line bundles on $X$
such that $L_{1}^{2}L_{2}....L_{n-1}>0$. As in Subsection 3.2 we consider a
positive open cone $K^{+}_{L}\subset N_{L}(X)$.

The proof of the following proposition is essentially the same as that of
[La6, Proposition 4.1].

######  Proposition 6.1.

Let ${\mathcal{L}}$ be a rank $1$ reflexive sheaf contained in
$\Omega_{X}^{[1]}(\log D)$. Then $c_{1}({\mathcal{L}})\not\in K^{+}_{L}$.

###### Proof.

Assume that $c_{1}({\mathcal{L}})\in K^{+}_{L}$ and consider a system of
logarithmic Hodge sheaves
$({\mathcal{E}}:={\mathcal{E}}^{1}\oplus{\mathcal{E}}^{0},\theta)$ with
${\mathcal{E}}^{1}={\mathcal{L}}$, ${\mathcal{E}}^{0}={\mathcal{O}}_{X}$ and
$\theta:{\mathcal{E}}^{1}\to{\mathcal{E}}^{0}\hat{\otimes}\Omega_{X}^{[1]}(\log
D)=\Omega_{X}^{[1]}(\log D)$ given by the inclusion. Then
$({\mathcal{E}},\theta)$ is slope $L$-stable since the only rank $1$
logarithmic subsystem of Hodge sheaves of $({\mathcal{E}},\theta)$ is of the
form $({\mathcal{O}}_{X},0)$. Therefore by Lemma 2.1 and Theorem 5.12 we have

$0=4\int_{X}c_{2}({\mathcal{E}})L_{2}...L_{n-1}\geq\int_{X}c_{1}^{2}({\mathcal{E}})L_{2}...L_{n-1}=c_{1}({\mathcal{L}})^{2}.L_{2}...L_{n-1},$

a contradiction. ∎

Similarly as [La6, Theorem 4.4] one can also get the following theorem
generalizing the Miyaoka–Yau inequality in the surface case:

######  Theorem 6.2.

Let us assume that $p\geq 3$ and let
${\mathcal{F}}\subset\Omega_{X}^{[1]}(\log D)$ be a rank $2$ reflexive
subsheaf with $c_{1}({\mathcal{F}})\in\overline{K^{+}_{L}}$. Then

$3\int_{X}c_{2}({\mathcal{F}})L_{2}...L_{n-1}\geq\int_{X}c_{1}({\mathcal{F}})^{2}L_{2}...L_{n-1}.$

###### Proof.

Let us consider the system of logarithmic Hodge sheaves
$({\mathcal{E}}:={\mathcal{E}}^{1}\oplus{\mathcal{E}}^{0},\theta)$ given by
${\mathcal{E}}^{1}={\mathcal{F}}$, ${\mathcal{E}}^{0}={\mathcal{O}}_{X}$ and
$\theta:{\mathcal{E}}^{1}={\mathcal{E}}\hookrightarrow\Omega_{X}^{[1]}(\log
D)={\mathcal{E}}^{0}\hat{\otimes}\Omega_{X}^{[1]}(\log D)$. If
$({\mathcal{E}},\theta)$ is slope $L$-semistable, then by Lemma 2.1 and
Theorem 5.12 we have

$3\int_{X}c_{2}({\mathcal{F}})L_{2}...L_{n-1}=3\int_{X}c_{2}({\mathcal{E}})L_{2}...L_{n-1}\geq\int_{X}c_{1}({\mathcal{E}})^{2}L_{2}...L_{n-1}=\int_{X}c_{1}({\mathcal{F}})^{2}L_{2}...L_{n-1}.$

So we can assume that $({\mathcal{E}},\theta)$ is not slope $L$-semistable.
Let $({\mathcal{E}}^{\prime},\theta^{\prime})$ be its maximal destabilizing
subsystem of logarithmic Hodge subsheaves. $({\mathcal{E}},\theta)$ contains
only one saturated rank $1$ system of logarithmic Hodge subsheaves, namely
$({\mathcal{O}}_{X},0)$. Since this subsystem does not destabilize
$({\mathcal{E}},\theta)$, the sheaf ${\mathcal{E}}^{\prime}$ has rank $2$.
Note that $({\mathcal{E}}^{\prime},\theta^{\prime})$ is slope $L$-stable so
$c_{1}({\mathcal{M}}).L_{1}...L_{n-1}>0$. We can decompose
${\mathcal{E}}^{\prime}$ into a direct sum
${\mathcal{O}}_{X}\oplus{\mathcal{M}}$, where ${\mathcal{M}}$ is a saturated
rank $1$ reflexive sheaf contained in ${\mathcal{F}}$. By assumption
$({\mathcal{E}}^{\prime},\theta^{\prime})$ destabilizes
$({\mathcal{E}},\theta)$ so

$\mu_{L}({\mathcal{E}}^{\prime})=\frac{c_{1}({\mathcal{M}}).L_{1}...L_{n-1}}{2}>\mu_{L}({\mathcal{E}})=\frac{c_{1}({\mathcal{F}}).L_{1}...L_{n-1}}{3}.$

Therefore $(3c_{1}({\mathcal{M}})-2c_{1}({\mathcal{F}})).L_{1}...L_{n-1}>0$.
If $(3c_{1}({\mathcal{M}})-2c_{1}({\mathcal{F}}))\in K_{L}^{+}$ then
$3c_{1}({\mathcal{M}})=(3c_{1}({\mathcal{M}})-2c_{1}({\mathcal{F}}))+2c_{1}({\mathcal{F}})\in
K_{L}^{+}$, which contradicts Proposition 6.1. This shows that

$(3c_{1}({\mathcal{M}})-2c_{1}({\mathcal{F}}))^{2}.L_{2}...L_{n-1}\leq 0.$

Let us set ${\mathcal{L}}:=({\mathcal{F}}/{\mathcal{M}})^{**}$. Then the
sequence

$0\to{\mathcal{M}}\to{\mathcal{F}}\to{\mathcal{L}}$

satisfies assumptions of Lemma 2.1 and hence

$\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{F}})L_{2}...L_{n-1}\leq\frac{1}{2}c_{1}({\mathcal{M}})^{2}.L_{2}...L_{n-1}+\frac{1}{2}c_{1}({\mathcal{L}})^{2}.L_{2}...L_{n-1},$

which after rewriting gives

$\displaystyle\int_{X}(3c_{2}({\mathcal{F}})-c_{1}({\mathcal{F}})^{2})L_{2}...L_{n-1}$
$\displaystyle+\frac{3}{4}c_{1}({\mathcal{M}})^{2}.L_{2}...L_{n-1}$
$\displaystyle\geq-\frac{1}{4}(3c_{1}({\mathcal{M}})-2c_{1}({\mathcal{F}}))^{2}.L_{2}...L_{n-1}\geq
0.$

Since by Proposition 6.1 we have $c_{1}({\mathcal{M}})^{2}.L_{2}...L_{n-1}\leq
0$, this implies the required inequality. ∎

##  7 Applications to characteristic zero

Here we show a few applications of our results to study varieties defined in
characteristic zero. In particular, we prove Theorems 0.5, 0.6 and 0.7.

First we recall the following lemma that follows from Lemma 3.19 in the
preprint version of [GKPT1] (note that any normal surface with quotient
singularities is klt).

######  Lemma 7.1.

Let $X$ be a normal projective surface with at most quotient singularities
defined over an algebraically closed field $k$ of characteristic $0$. Let
${\mathcal{E}}$ be a coherent reflexive ${\mathcal{O}}_{X}$-module. Then there
exists a normal projective surface $Y$ and a finite morphism $\pi:Y\to X$ such
that $\pi^{[*]}{\mathcal{E}}$ is locally free. In this case we have

$\int_{X}\Delta({\mathcal{E}})=\frac{1}{\deg\pi}\int_{Y}\Delta(\pi^{[*]}{\mathcal{E}})$

and

$\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})=\frac{1}{\deg\pi}\int_{Y}{\mathop{\rm
ch\,}}_{2}(\pi^{[*]}{\mathcal{E}}).$

From now on we fix the following notation in this section. Let $X$ be a normal
projective variety of dimension $n$ defined over an algebraically closed field
$k$ of characteristic $0$. We assume that $X$ has quotient singularities in
codimension $2$ and we fix a reduced divisor $D\subset X$ such that the pair
$(X,D)$ is log canonical in codimension $2$. For sheaves on such a variety we
use Chern classes defined in [La9, 5.3]. They coincide with classical
Mumford’s ${\mathbb{Q}}$-Chern classes considered in [Ko, Chapter 10] and in
[GKPT1, Theorem 3.13] (see [La9, Remark 5.9]).

###  7.1 Strong restriction theorems

Let us fix a collection $(L_{1},...,L_{n-1})$ of ample line bundles and let us
set $d=L_{1}^{2}L_{2}...L_{n-1}$. The proof of the following theorem is based
on a standard spreading out argument.

######  Theorem 7.2.

Let $({\mathcal{E}},\theta)$ be a reflexive logarithmic Higgs sheaf of rank
$r\geq 2$ on $(X,D)$. Let $m_{0}$ be a non-negative integer such that
$T_{X}(\log\,D)\otimes L_{1}^{\otimes m_{0}}$ is globally generated. Let $m$
be an integer such that

$m>\max\left(\left\lfloor\frac{r-1}{r}\int_{X}\Delta({\mathcal{E}})L_{2}\dots
L_{n-1}+\frac{1}{r(r-1)d}\right\rfloor,2(r-1)m_{0}^{2}\right).$

Let $H\in|L_{1}^{\otimes m}|$ be good for $(X,D)$.

  1. 1.

If $({\mathcal{E}},\theta)$ is slope $(L_{1},...,L_{n-1})$-stable then the
logarithmic Higgs sheaf $({\mathcal{E}},\theta)|_{H}$ on $(H,D\cap H)$ is
slope $(L_{2}|_{H},\dots,L_{n-1}|_{H})$-stable.

  2. 2.

If $({\mathcal{E}},\theta)$ is slope $(L_{1},...,L_{n-1})$-semistable and
restrictions of all quotients of a Jordan–Hölder filtration of
$({\mathcal{E}},\theta)$ to $H$ are torsion free then
$({\mathcal{E}},\theta)|_{H}$ is slope
$(L_{2}|_{H},\dots,L_{n-1}|_{H})$-semistable.

###### Proof.

We can find a subring $R\subset k$, which is finitely generated over
${\mathbb{Z}}$ and there exists a flat projective morphism ${\mathcal{X}}\to
S={\mathop{{\rm Spec\,}}}R$ with a relative reduced Weil divisor
${\mathcal{D}}$ on ${\mathcal{X}}/S$ such that
$(X,D)\simeq({\mathcal{X}}\times_{S}{\mathop{{\rm
Spec\,}}}k,{\mathcal{D}}\times_{S}{\mathop{{\rm Spec\,}}}k)$. We can assume
that there exist line bundles ${\mathcal{L}}_{1},...,{\mathcal{L}}_{n}$ on
${\mathcal{X}}$ lifting $L_{1},...,L_{n-1}$, a relative logarithmic Higgs
sheaf
$(\tilde{\mathcal{E}},\tilde{\theta}:T_{{\mathcal{X}}/S}(\log{\mathcal{D}})\otimes_{{\mathcal{O}}_{X}}\tilde{\mathcal{E}}\to\tilde{\mathcal{E}})$
lifting $({\mathcal{E}},\theta)$ and a relative effective Cartier divisor
${\mathcal{H}}\in|{\mathcal{L}}_{1}^{\otimes m}|$ lifting $H$ (in particular,
${\mathcal{H}}\to S$ is flat). Shrinking $S$ if necessary we can assume that
the following conditions are satisfied:

  1. 1.

$S$ is regular,

  2. 2.

all fibers of ${\mathcal{X}}\to S$ and ${\mathcal{H}}\to S$ are geometrically
integral and geometrically normal,

  3. 3.

${\mathcal{L}}_{1},...,{\mathcal{L}}_{n}$ are relatively ample,

  4. 4.

$T_{{\mathcal{X}}/S}(\log\,{\mathcal{D}})\otimes{\mathcal{L}}_{1}^{\otimes
m_{0}}$ is relatively globally generated,

  5. 5.

for all closed points $s\in S$ the fiber ${\mathcal{X}}_{s}$ is liftable
modulo $W_{2}(\kappa(s))$ (see the proof of [La5, Theorem 7]),

  6. 6.

for all geometric points $\bar{s}$ of $S$, ${\mathcal{H}}_{\bar{s}}$ is good
for $({\mathcal{X}}_{\bar{s}},{\mathcal{D}}_{\bar{s}})$,

  7. 7.

for all geometric points $\bar{s}$ of $S$ the sheaf
$\tilde{\mathcal{E}}_{\bar{s}}$ is a coherent reflexive
${\mathcal{O}}_{{\mathcal{X}}_{\bar{s}}}$-module,

  8. 8.

a fixed Jordan–Hölder filtration ${\mathcal{E}}_{\bullet}$ of
$({\mathcal{E}},\theta)$ extends to a filtration
$\tilde{\mathcal{E}}_{\bullet}$ of $(\tilde{\mathcal{E}},\tilde{\theta})$.

Following the proof of [Ma, Theorem 4.2] (see also [HL, Proposition 2.3.1]),
one can see that geometric slope (semi)stability of logarithmic Higgs sheaves
is an open condition in flat families. It is sufficient to prove that there
exists some geometric point $\bar{s}$ of $S$, the logarithmic Higgs sheaf
$(\tilde{\mathcal{E}},\tilde{\theta})|_{{\mathcal{H}}_{\bar{s}}}$ on
$({\mathcal{X}}_{\bar{s}},{\mathcal{H}}_{\bar{s}})$ is slope
$({\mathcal{L}}_{2}|_{{\mathcal{H}}_{\bar{s}}},\dots,{\mathcal{L}}_{n-1}|_{{\mathcal{H}}_{\bar{s}}})$-(semi)stable.
Then the restriction of
$(\tilde{\mathcal{E}},\tilde{\theta})|_{{\mathcal{H}}}$ to the fiber of
${\mathcal{H}}\to S$ over the generic geometric point of $S$ is also slope
(semi)stable, proving the theorem.

By the above openness of semistability, we can assume that for all geometric
points $\bar{s}$ of $S$ the logarithmic Higgs sheaf
$(\tilde{\mathcal{E}}_{\bar{s}},\tilde{\theta}_{\bar{s}})$ is slope
(semi)stable. By the same argument the restriction of the filtration
$\tilde{\mathcal{E}}_{\bullet}$ to ${\mathcal{X}}_{\bar{s}}$ gives a
Jordan–H”older filtration of
$(\tilde{\mathcal{E}}_{\bar{s}},\tilde{\theta}_{\bar{s}})$. Note also that
restrictions of quotients of this filtration to ${\mathcal{H}}_{\bar{s}}$ are
torsion free for $\bar{s}$ over an open subset of $S$. We need to check that
there exists some non-empty open subset $U\subset S$ such that for all
geometric points $\bar{s}$ over closed points of $U$ we have

$\int_{{\mathcal{X}}_{\bar{s}}}\Delta(\tilde{\mathcal{E}}_{{\bar{s}}}){\mathcal{L}}_{2}|_{{\mathcal{X}}_{\bar{s}}}\dots\mathcal{L}_{n-1}|_{{\mathcal{X}}_{\bar{s}}}=\int_{X}\Delta({\mathcal{E}})L_{2}\dots
L_{n-1}.$

This is not obvious as $\tilde{\mathcal{E}}$ is not locally free and Chern
numbers of reflexive sheaves do not remain constant in flat families (see
Example 2.2). Using Theorems 1.17, 1.18 and [La9, Theorem 5.8] we can reduce
to the surface case. By Lemma 7.1 we can find a normal projective surface $Y$
and a finite covering $\pi:Y\to X$ such such that $\pi^{[*]}{\mathcal{E}}$ is
locally free. Then, shrinking $S$ if necessary, we can find a flat projective
morphism $\tilde{\pi}:{\mathcal{Y}}\to S$ and a morphism
${\mathcal{Y}}\to{\mathcal{X}}$ lifting $\pi:Y\to X$. We can also assume that
all fibers of $g:{\mathcal{Y}}\to S$ are geometrically integral and
geometrically normal. Since $S$ is normal, the schemes ${\mathcal{X}}$ and
${\mathcal{Y}}$ are also normal. So we can consider
$\tilde{\pi}^{[*]}\tilde{\mathcal{E}}$, which is reflexive on ${\mathcal{Y}}$.
This sheaf is locally free outside of a closed subscheme
$Z\subset{\mathcal{Y}}$ of codimension $\geq 2$. Since $Z$ does not intersect
the generic fiber of ${\mathcal{Y}}\to S$,
$\tilde{\pi}^{[*]}\tilde{\mathcal{E}}$ is locally free over a non-empty open
subset $S^{\prime}=S\backslash g(Z)\subset S$. Now let us consider a
commutative diagram

$\textstyle{{\mathcal{Y}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{s}}$$\scriptstyle{\tilde{\pi}_{s}}$$\textstyle{{\mathcal{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\pi}}$$\textstyle{{\mathcal{X}}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{s}}$$\textstyle{{\mathcal{X}}.}$

Since $j_{s}^{*}(\tilde{\pi}^{[*]}\tilde{\mathcal{E}})$ is locally free for
$s\in S^{\prime}$ we have an induced map $\varphi_{s}$ that fits into a
commutative diagram

$\textstyle{\tilde{\pi}_{s}^{*}i_{s}^{*}(\tilde{\mathcal{E}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\tilde{\pi}_{s}^{[*]}(i_{s}^{*}(\tilde{\mathcal{E}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{s}}$$\textstyle{j_{s}^{*}(\tilde{\pi}^{*}\tilde{\mathcal{E}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j_{s}^{*}(\tilde{\pi}^{[*]}\tilde{\mathcal{E}}).}$

Let us set $U:=\\{x\in X:\tilde{\mathcal{E}}_{x}\hbox{ is a free
${\mathcal{O}}_{X,x}$-module}\\}$. Since $\varphi_{s}$ is an isomorphism over
a big open subset ${\mathcal{Y}}_{s}\cap\tilde{\pi}^{-1}(U)$ of
${\mathcal{Y}}_{S}$ and
$\tilde{\pi}_{s}^{[*]}(i_{s}^{*}(\tilde{\mathcal{E}}))$ is reflexive,
$\varphi_{s}$ is an isomorphism. So by Theorem 1.18 and Lemma 7.1 we have

$\displaystyle\int_{{\mathcal{X}}_{s}}\Delta({\mathcal{E}}_{s})=\frac{1}{\deg\tilde{\pi}_{s}}\int_{{\mathcal{Y}}_{s}}\Delta(\tilde{\pi}_{s}^{[*]}{\mathcal{E}}_{s})=\frac{1}{\deg\pi}\int_{Y}\Delta(\pi^{[*]}{\mathcal{E}})=\int_{X}\Delta({\mathcal{E}})$

as claimed. Now the required assertion follows by applying Theorem 5.4 to
fibers over geometric points $\bar{s}$ of $S$ with large characteristic of the
residue field (then $\beta_{r}(\bar{s})\to 0$). ∎

The same argument as above show also that Theorem 0.1 implies the following
strong restriction theorem of Bogomolov’s type:

######  Theorem 7.3.

Let ${\mathcal{E}}$ be a coherent reflexive ${\mathcal{O}}_{X}$-module of rank
$r\geq 2$. Let $m$ be an integer such that

$m>\left\lfloor\frac{r-1}{r}\int_{X}\Delta({\mathcal{E}})L_{2}\dots
L_{n-1}+\frac{1}{r(r-1)d}\right\rfloor$

and let $H\in|L_{1}^{\otimes m}|$ be a normal hypersurface.

  1. 1.

If ${\mathcal{E}}$ is slope $(L_{1},...,L_{n-1})$-stable then
${\mathcal{E}}|_{H}$ is slope $(L_{2}|_{H},\dots,L_{n-1}|_{H})$-stable.

  2. 2.

If ${\mathcal{E}}$ is slope $(L_{1},...,L_{n-1})$-semistable and restrictions
of all quotients of a Jordan–Hölder filtration of ${\mathcal{E}}$ to $H$ are
torsion free then ${\mathcal{E}}|_{H}$ is slope
$(L_{2}|_{H},\dots,L_{n-1}|_{H})$-semistable.

######  Remark 7.4.

Although Theorem 0.1 works for any normal varieties in positive
characteristic, it does not seem easy to use a similar spreading out argument
to obtain even the usual Mehta–Ramanathan theorem for ample multipolarizations
on a general normal projective variety in characteristic zero. The problem is
that the choice of spreading out depends on $m$ as we need to spread out
divisors $H\in|L_{1}^{\otimes m}|$. But since Chern numbers of reflexive
sheaves are in general not well behaved in families of normal varieties, we
cannot choose one $m$ so that Theorem 0.1 works for this fixed $m$ on even one
geometric fiber ${\mathcal{X}}_{\bar{s}}$. However, using Corollary 3.11 one
can show bounds on the maximal destabilizing slope of ${\mathcal{E}}|_{H}$ on
any normal variety in terms of numerical invariants of reductions of
${\mathcal{E}}$.

###  7.2 Bogomolov’s inequality for logarithmic Higgs sheaves

We will need an analogue of the first part of Lemma 2.1 in the characteristic
zero case (the analogue of the second part also holds but we will not need
it):

######  Lemma 7.5.

Let $L=(L_{1},...,L_{n-2})$ be a collection of nef line bundles on $X$. If

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

is a left exact sequence of reflexive sheaves on $X$, which is also right
exact on some big open subset of $X$ then

$\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}})L_{1}...L_{n-2}\leq\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}}_{1})L_{1}...L_{n-2}+\int_{X}{\mathop{\rm
ch\,}}_{2}({\mathcal{E}}_{2})L_{1}...L_{n-2}.$

###### Proof.

As in the proof of Lemma 2.1 we first reduce to the surface case. Then by
Lemma 7.1 there exists a normal projective surface $Y$ and a finite morphism
$\pi:Y\to X$ such that $\pi^{[*]}{\mathcal{E}}$, $\pi^{[*]}{\mathcal{E}}_{1}$
and $\pi^{[*]}{\mathcal{E}}_{2}$ are locally free. As in the proof of Lemma
2.1 we see that the sequence

$0\to\pi^{[*]}{\mathcal{E}}_{1}\to\pi^{[*]}{\mathcal{E}}\to\pi^{[*]}{\mathcal{E}}_{2}\to
0$

is left exact on $Y$ and right exact on some big open subset of $Y$. Therefore

$\chi(Y,\pi^{[*]}{\mathcal{E}})\leq\chi(Y,\pi^{[*]}{\mathcal{E}}_{1})+\chi(Y,\pi^{[*]}{\mathcal{E}}_{2}).$

Using the Riemann–Roch theorem for locally free sheaves on normal projective
surfaces this can be rewritten as

$\int_{Y}{\mathop{\rm
ch\,}}_{2}(\pi^{[*]}{\mathcal{E}})\leq\int_{Y}{\mathop{\rm
ch\,}}_{2}(\pi^{[*]}{\mathcal{E}}_{1})+\int_{Y}{\mathop{\rm
ch\,}}_{2}(\pi^{[*]}{\mathcal{E}}_{2}).$

Dividing by the degree of $\pi$, we get the required inequality from the
second part of Lemma 7.1. ∎

######  Theorem 7.6.

Let $L=(L_{1},...,L_{n-1})$ be a collection of nef line bundles on $X$ such
that $L_{1}^{2}L_{2}....L_{n-1}>0$. For any slope
$(L_{1},...,L_{n-1})$-semistable logarithmic reflexive Higgs sheaf
$({\mathcal{E}},\theta)$ we have

$\int_{X}\Delta({\mathcal{E}})L_{2}...L_{n-1}\geq 0.$

###### Proof.

First assume that $L_{1},...,L_{n-1}$ are ample. Then by the above theorem we
can restrict to the surface case. Since finite quotients of smooth affine log
surface pairs are $F$-liftable in large characteristics (cf. [Zd, Lemma
4.21]), we can apply Theorem 5.12 and an easy spreading out argument.

In general, we reduce to the above case by an argument analogous to that from
the proof of [La2, 3.6]. Namely, we fix an ample line bundle $A$ and consider
the classes $L_{i}(t)=c_{1}(L_{i})+tc_{1}(A)$ in $N_{L}(X)\otimes{\mathbb{Q}}$
for $t\in{\mathbb{Q}}_{>0}$. These classes are ample and the Harder–Narasimhan
filtration of $({\mathcal{E}},\theta)$ with respect to
$(L_{1}(t),...,L_{n-1}(t))$ is independent of $t$ for small
$t\in{\mathbb{Q}}_{>0}$. We have an analogue of Lemma 2.5 for normal
projective varieties in characteristic zero that have quotient singularities
in codimension $2$ (this follows from Lemma 7.5 in the same way as Lemma 2.5
follows from Lemma 2.1). Applying this result to the above filtration, using
the inequality for ample collections of line bundles and taking the limit as
$t\to 0$ gives the required inequality. ∎

##  8 Appendix: inverse Cartier transform after Lan–Sheng–Zuo

In this section we recall the construction of inverse Cartier transform from
Ogus–Vologodsky’s correspondence [OV], following [LSZ]. For simplicity of
notation we consider only the non-logarithmic case. The logarithmic case is
essentially the same.

###  8.1 Results of Deligne and Illusie

Below we recall the construction from [DI] of a canonical splitting of the
Cartier operator that is associated to a fixed lifting of Frobenius of a
smooth $F$-liftable variety.

Let $k$ be a perfect field of characteristic $p>0$ and let us set
$S={\mathop{{\rm Spec\,}}}k$ and $\tilde{S}={\mathop{{\rm Spec\,}}}W_{2}(k)$.
Let $X$ be a smooth $k$-variety with a fixed lifting $\tilde{X}/\tilde{S}$.
Let $X^{\prime}$ be the fiber product of $X$ over the absolute Frobenius
morphism of $S$. Then we have an induced relative Frobenius morphism
$F_{X/S}:X\to X^{\prime}$. Note that $X^{\prime}$ has a natural lifting
$\tilde{X}^{\prime}$ to $\tilde{S}$, which is defined as the base change of
$\tilde{X}\to\tilde{S}$ via $\tilde{S}\to\tilde{S}$ coming from
$\sigma_{2}:W_{2}(k)\to W_{2}(k)$. Let us assume that $F_{X/S}:X\to
X^{\prime}$ has a lifting $\tilde{F}_{X/S}:\tilde{X}\to\tilde{X}^{\prime}$ so
that we have a commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{X/S}}$$\textstyle{\tilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{F}_{X/S}}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{X}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{S}}$

Since the map
$\tilde{F}^{*}:\tilde{F}^{*}_{X/S}\Omega_{\tilde{X}^{\prime}/\tilde{S}}^{1}\to\Omega_{\tilde{X}/\tilde{S}}^{1}$
vanishes after pulling back to $X$, we can define

$\zeta=p^{-1}\tilde{F}^{*}:F^{*}_{X/S}\Omega_{X^{\prime}/S}^{1}\to\Omega_{X/S}^{1}$

Since $d\zeta=0$ we can consider $\zeta$ as the map of sheaves of abelian
groups $F^{*}_{X/S}\Omega_{X^{\prime}/S}^{1}\to Z_{X/S}^{1}$, where
$Z^{1}_{X/S}$ is the kernel of $d:\Omega^{1}_{X/S}\to\Omega^{2}_{X/S}$. Its
adjoint $\zeta^{{\mathop{\rm ad\,}}}:\Omega_{X^{\prime}/S}^{1}\to
F_{X/S,*}Z_{X/S}^{1}$ is ${\mathcal{O}}_{X^{\prime}}$-linear and it splits the
composition

$F_{X/S,*}Z_{X/S}^{1}\longrightarrow{\mathcal{H}}^{1}(F_{X/S,*}\Omega_{X/S}^{\bullet})\mathop{\longrightarrow}^{C_{X/S}}\Omega_{X^{\prime}/S}^{1}$

of the Cartier operator with the canonical projection.

###  8.2 Inverse Cartier transform

Assume that $X$ is smooth and there exists a global lifting $\tilde{X}$ of $X$
to $W_{2}(k)$.

Let $({\mathcal{E}},\theta)$ be a Higgs ${\mathcal{O}}_{X^{\prime}}$-modules
with a nilpotent Higgs field of level $\leq(p-1)$. We want to construct an
${\mathcal{O}}_{X}$-module $V$ with an integrable connection $\nabla$, whose
$p$-curvature is nilpotent of level $\leq(p-1)$. The pair $(V,\nabla)$ will be
denoted by $C^{-1}_{\tilde{X}/W_{2}(k)}({\mathcal{E}},\theta)$ and called the
_inverse Cartier transform_ of $(E,\theta)$. Let us fix a $W_{2}(k)$-lifting
$\tilde{X}^{\prime}$ of $X^{\prime}/k$ and take a covering
$\\{\tilde{U}_{\alpha}\\}_{\alpha\in I}$ of $\tilde{X}$ such that for each
$\alpha\in I$ there exists
$\tilde{F}_{\alpha}:\tilde{U}_{\alpha}\to\tilde{U}_{\alpha}^{\prime}$ lifting
the relative Frobenius morphism $F_{\alpha}:U_{\alpha}\to
U_{\alpha}^{\prime}$. By previous subsection, the lifting $\tilde{F}_{\alpha}$
allows us to construct
$\zeta_{\alpha}=p^{-1}\tilde{F}_{\alpha}^{*}:F^{*}_{\alpha}\Omega_{U^{\prime}_{\alpha}/k}^{1}\to\Omega_{U_{\alpha}/k}^{1}.$
Therefore over each $U_{\alpha}$ we can define $(V_{\alpha},\nabla_{\alpha})$
by setting $V_{\alpha}:=F^{*}_{\alpha}({\mathcal{E}}|_{U_{\alpha}})$ and
$\nabla_{\alpha}:=\nabla_{can}+\zeta_{\alpha}(F^{*}_{\alpha}\theta|_{U_{\alpha}^{\prime}}).$
To glue $(V_{\alpha},\nabla_{\alpha})$ and $(V_{\beta},\nabla_{\beta})$ over
$U_{\alpha\beta}=U_{\alpha}\cap U_{\beta}$ one uses the following lemma due to
Deligne and Illusie (see [DI]):

######  Lemma 8.1.

There exist ${\mathcal{O}}_{U_{\alpha\beta}^{\prime}}$-linear maps
$h_{\alpha\beta}:\Omega_{U^{\prime}_{\alpha\beta}}\to(F_{U^{\prime}_{\alpha\beta}})_{*}{\mathcal{O}}_{\alpha\beta}$
such that

  1. 1.

for all $\alpha,\beta$ we have

$\zeta_{\alpha}^{{\mathop{\rm ad\,}}}-\zeta_{\beta}^{{\mathop{\rm
ad\,}}}=dh_{\alpha\beta},$

  2. 2.

for all $\alpha,\beta,\gamma$ we have over
$U_{\alpha\beta\gamma}=U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$

$h_{\alpha\beta}+h_{\beta\gamma}=h_{\alpha\gamma}.$

Let
$h_{\alpha\beta}^{\prime}:F_{U^{\prime}_{\alpha\beta}}^{*}\Omega_{U^{\prime}_{\alpha\beta}}\to{\mathcal{O}}_{\alpha\beta}$
be adjoint to $h_{\alpha\beta}$. Now we define gluing maps
$g_{\alpha\beta}:V_{\alpha}|_{U_{\alpha\beta}}\to
V_{\beta}|_{U_{\alpha\beta}}$ using
$h_{\alpha\beta}^{\prime}(F^{*}\theta|_{U_{\alpha\beta}}):F^{*}{\mathcal{E}}|_{U_{\alpha\beta}}\to
F^{*}{\mathcal{E}}|_{U_{\alpha\beta}}\otimes
F^{*}\Omega_{U^{\prime}_{\alpha\beta}}\to
F^{*}{\mathcal{E}}|_{U_{\alpha\beta}}$ by setting

$g_{\alpha\beta}:=\exp(h_{\alpha\beta}(F^{*}\theta|_{U_{\alpha\beta}}))=\sum_{i=0}^{p-1}\frac{(h_{\alpha\beta}(F^{*}\theta|_{U_{\alpha\beta}}))^{i}}{i!}.$

The maps $g_{\alpha\beta}$ allow us to glue $(V_{\alpha},\nabla_{\alpha})$ and
$(V_{\beta},\nabla_{\beta})$ over $U_{\alpha\beta}$ to a global object
$(V,\nabla)\in{\mathop{\operatorname{MIC}\,}}_{p-1}(X/k)$.

## Acknowledgements

The author would like to thank P. Achinger and T. Kawakami for useful
conversations. A part of the paper was written while the author was an
External Senior Fellow at Freiburg Institute for Advanced Studies (FRIAS),
University of Freiburg, Germany. The author would like to thank Stefan Kebekus
for his hospitality during the author’s stay in FRIAS.

The author was partially supported by Polish National Centre (NCN) contract
numbers 2018/29/B/ST1/01232 and 2021/41/B/ST1/03741. The research leading to
these results has received funding from the European Union’s Horizon 2020
research and innovation programme under the Maria Skłodowska-Curie grant
agreement No 754340.

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