Bogomolov’s inequality and Higgs sheaves on normal varieties in positive characteristic

. We prove Bogomolov’s inequality on a normal projective variety in positive characteristic and we use it to show some new restriction theorems and a new boundedness result. We also redefine Higgs sheaves on normal varieties and we prove restriction theorems and Bogomolov-type inequalities for semistable logarithmic Higgs sheaves on some normal varieties in an arbitrary characteristic.


Introduction
Let X be a smooth projective variety of dimension n defined over an algebraically closed field k and let H be an ample line bundle on X . Let E be a coherent torsion free O X -module of rank r and let us set ∆(E ) := 2rc 2 (E ) − (r − 1)c 1 (E ) 2 .
A well known Bogomolov's theorem says that if k has characteristic zero and E is slope H-semistable then X ∆(E )H n−2 ≥ 0.
This theorem was first proven by F. Bogomolov in the surface case. The higher dimensional case follows from restriction theorems for semistability (e.g., one can use the Mehta-Ramanathan restriction theorem). An analogue of this theorem for slope H-stable Higgs bundles was proven by C. Simpson in [Si] using analytic methods. Simpson's paper contains also applications of this result to uniformization and the Miyaoka -Yau inequality in higher dimensions (although only in the non-logarithmic case). Later, T. Mochizuki in [Mo] generalized this inequality to the logarithmic case (he also used analytic methods). An algebraic proof of Bogomolov's inequality for Higgs sheaves appeared in [La5] and in the logarithmic case in [La6]. These papers contained also generalization of these results to positive characteristic.
More recently, in characteristic zero the above results have been generalized in [GKPT1] to projective varieties with klt singularities (but not in the logarithmic case). In the mildly singular logarithmic case one also knows the Miyaoka-Yau inequality (see [Ko,Chapter 10] and [La1] for the 2-dimensional case and [GT] for higher dimensions).
One of the main motivations behind this paper is generalization of the above results to positive characteristic and strengthening of the results known in the characteristic zero. We also deal with semistability defined by a collection of nef line bundles instead of one ample line bundle. An importance of considering this generalized situation was first recognized by Y. Miyaoka in [Mi], who proved Bogomolov's inequality for torsion free sheaves on normal varieties smooth in codimension 2 in case of collections of ample and one nef line bundles. However, it is not completely clear to the author if the original proof of Mehta-Ramanathan's restriction theorem works so easily for multipolarizations on normal varieties as claimed in [Mi,Corollary 3.13]. In case of one ample line bundle on a normal projective variety defined over an algebraically closed field of characteristic zero, restriction theorem for semistable sheaves has been proven by H. Flenner in [Fl]. However, it seems that his proof cannot be generalized to multipolarizations.
In case of smooth projective varieties the corresponding Bogomolov's inequality in any characteristic was proven in [La2] (however, the proof uses a different, new restriction theorem). Using resolution of singularities one can use this to obtain Mehta-Ramanathan's restriction theorem for multipolarizations on normal varieties in characteristic zero. In case of Higgs sheaves on smooth projective varieties, restriction theorem and Bogomolov's inequality for multipolarizations has been proven in [La5] and in the logarithmic case in [La6].
Our first main result is a strong restriction theorem for multipolarized normal varieties in positive characteristic, analogous to [La2,Theorem 5.2 and Corollary 5.4]. One of the problems here is with the definition of Chern classes. Below we use Chern classes of reflexive sheaves defined in [La9] (see Subsection 1.3 for a few basic properties).
Let X be a normal projective variety of dimension n defined over an algebraically closed field k of characteristic p > 0. Let us fix a collection (L 1 , ..., L n−1 ) of ample line bundles on X (in fact, we usually need weaker assumptions on this collection). Let us set d = L 2 1 L 2 ...L n−1 . Then we have the following result (see Subsection 3.1 for the definition of β r ). and let H ∈ |L ⊗m 1 | be a normal hypersurface. If E is slope (L 1 , ..., L n−1 )-stable then E | H is slope (L 2 | H , . . . , L n−1 | H )-stable.
The above theorem implies the following boundedness result. THEOREM 0.2. Let us fix some positive integer r, integer ch 1 and some real numbers ch 2 and µ max . Then the set of reflexive coherent O X -modules E of rank r with X ch 1 (E )L 1 ...L n−1 = ch 1 , X ch 2 (E )L 1 ... L i ...L n−1 ≥ ch 2 for i = 1, ..., n − 1, and µ max (E ) ≤ µ max is bounded.
In the statement above it is not even clear that the number of Hilbert polynomials of sheaves in the considered family is finite. In case L 1 = ... = L n−1 the above theorem follows from [La2,Theorem 4.4]. If X is smooth then the above theorem follows from [La2,Corollary 5.4]. But it is no longer the case if we consider multipolarizations on normal varieties.
We also prove an analogue of Theorem 0.1 for (semi)stable Higgs sheaves on normal varieties (see Theorem 5.4 for a more precise version). This theorem generalizes [La5,Theorem 10] that works for smooth varieties liftable to W 2 (k).
Finally, we use the above results to prove the following Bogomolov's inequality for reflexive Higgs sheaves on mildly singular normal varieties. Note that, unlike previously known results for singular varieties in characteristic zero, our theorem holds for log pairs. THEOREM 0.4. Let D ⊂ X be an effective reduced Weil divisor such that the pair (X , D) is almost liftable to W 2 (k) and it has F-liftable singularities in codimension 2. Then for any slope (L 1 , ..., L n−1 )-semistable logarithmic reflexive Higgs sheaf (E , θ ) of rank r ≤ p we have X ∆(E )L 2 ...L n−1 ≥ 0.
For the meaning of almost liftable log pair and F-liftable singularities we refer the reader to Definitions 1.3 and 1.5. If X is liftable to W 2 (k) then it is almost liftable to W 2 (k) and almost all reductions of varieties from characteristic zero satisfy this condition. To understand the second notion we note that a reduction of quotient surface singularity is F-liftable in large characteristics (see Subsection 1.2). In fact, for a dense set of primes, reductions of surfaces with log canonical singularities have F-liftable singularities (we do not prove this non-trivial fact as we will not need it in the following).
Now let X be a normal projective variety of dimension n defined over an algebraically closed field k of characteristic 0. Assume that X has at most quotient singularities in codimension 2. Let us fix a collection (L 1 , ..., L n−1 ) of ample line bundles on X and set d = L 2 1 L 2 ...L n−1 . Then Theorem 0.1 implies the following restriction theorem: THEOREM 0.5. Let E be a coherent reflexive O X -module of rank r ≥ 2. Let m be an integer such that m > r − 1 r X ∆(E )L 2 . . . L n−1 + 1 dr(r − 1) and let H ∈ |L ⊗m 1 | be a normal hypersurface. If E is slope (L 1 , ..., L n−1 )-stable then E | H is slope (L 2 | H , . . . , L n−1 | H )-stable. Now let us also fix an effective reduced Weil divisor D ⊂ X such that the pair (X , D) is log canonical in codimension 2. Theorem 0.3 implies the following result: THEOREM 0.6. Let (E , θ ) be a reflexive logarithmic Higgs sheaf of rank r ≥ 2 on (X , D). Let m 0 be a non-negative integer such that T X (log D) ⊗ L ⊗m 0 1 is globally generated. Let m be an integer such that m > max r − 1 r X ∆(E )L 2 . . . L n−1 + 1 dr(r − 1) , 2(r − 1)m 2 0 .
Theorem 0.6 generalizes [GKPT1,Theorem 5.22] and [GKPT2,Theorem 6.1], which are non-effective. In fact, we prove a stronger version of Theorem 0.6 (see Theorem 7.2) that works for all normal divisors H for which restriction of (E , θ ) to H gives a logarithmic Higgs sheaf on (H, D ∩ H).
Similarly, Theorem 0.4 can be used to prove the following inequality generalizing [GKPT1,Theorem 6.1].
Note that Bogomolov's inequality for logarithmic Higgs sheaves has not been known so far even on klt pairs. As in [Si] and [GKPT1] the above theorem implies Miyaoka-Yau inequalities for singular log pairs. Here we show only some simple applications of Theorem 0.4 to general Miyaoka-Yau inequalities in positive characteristic (see Section 6), leaving statement of general results in characteristic zero to the interested reader. Let us remark that unlike previous works on Chern number inequalities in higher dimensions (e.g., [GKPT1] and [GT]) our method should work in much more general situations in characteristic zero, the only obstacle being unknown behaviour of Chern classes of reflexive sheaves under tensor operations on normal surfaces (see [La9]). In particular, an analogue of Theorems 0.1, 0.2 and 0.3 should hold on any normal variety in characteristic zero and an analogue of Theorem 0.4 should hold for any pair (X , D) which is log canonical in codimension 2. Appropriate versions are also expected if D is an arbitrary effective Weil Q-divisor.
Here we should warn the reader that our definition of a reflexive Higgs sheaf is weaker than the one used in [GKPT1] and [GKPT2]. More precisely, a logarithmic reflexive Higgs sheaf (E , θ ) in our sense is a pair consisting of a coherent reflexive O X -module E and an O X -linear map T X (log D) ⊗ O X E → E satisfying additional integrability condition. In the situation of [GKPT1] this would corre- This is why in Section 4 we carefully explain differences between our approach and [GKPT1]. D. Greb et al. use a different definition as they need to pullback Higgs sheaves by birational morphisms to pass to a resolution of singularities. On the other hand, they cannot take duals or pushforward Higgs sheaves by open embeddings as is allowed in our approach. In this paper we do not use Kebekus's pullback functor for reflexive differentials on klt pairs (see [Ke]) and we do not pullback Higgs sheaves by birational morphisms (cf. Subsection 4.8). This allows us to obtain stronger results, e.g., Bogomolov's inequality for reflexive extensions of semistable Higgs sheaves on the regular locus (cf. [GKPT1,Theorem 6.1]).
Further applications of the obtained results to non-abelian Hodge theory and Simpson's correspondence are postponed to [La10].
The structure of the paper is as follows. In the first section we gather some auxiliary results and introduce some notation. In Section 2 we prove a few results on Chern classes of reflexive sheaves on normal varieties in positive characteristic. These resuts are used in Section 3 to prove generalized versions of Theorems 0.1 and 0.2. In Section 4 we study modules over Lie algebroids and generalized Higgs sheaves on normal varieties. In Section 5 we prove generalized versions of Theorems 0.3 and 0.4. In Section 6 we apply these results to obtain the Miyaoka-Yau inequality for some normal varieties in positive characteristic. In Section 7 we show some applications of the obtained results in characteristic zero, proving Theorems 0.5, 0.6 and 0.7. Section 8 contains an appendix in which we recall construction of the inverse Cartier transform used in Section 5.

Notation
If f : X → Y is a morphism between normal schemes and E is a coherent reflexive O Y -module then we set If X is a normal scheme of characteristic p then we denote by F X its absolute Frobenius morphism. If E is a coherent reflexive O X -module then for any positive integer m we set F [m] comes with an associative and symmetric tensor product⊗ given by An open subset U ⊂ X is called big if its complement X \U has codimension ≥ 2 in X .
Proof. We have a natural map If E is torsion free then there exists a big open subset V ⊂ Y such that E V is finite locally free. Then the above map is an isomorphism on If E is not torsion free, then the pullback of the quotient map E →Ẽ := E /Torsion is surjective. Hence the dual map ( f * Ẽ ) * → ( f * E ) * is an isomorphism. But if we apply the lemma toẼ then we get f [ * ] (E * ) ≃ ( f * Ẽ ) * , which proves the required assertion.

F-liftable schemes
A Weil divisor on a locally Noetherian integral scheme X is a formal sum ∑ a i D i , where a i ∈ Z and D i are prime divisors. However, if all nonzero a i are equal to 1 then we can consider an effective reduced Weil divisor as a reduced induced scheme structure on D := {i:a i =0} D i ⊂ X . If f : X → S is a morphism of schemes and X is an integral locally Noetherian normal scheme then we say that a subscheme D ⊂ X is a relative effective reduced Weil divisor on X /S if D is an effective reduced Weil divisor and D → S is a flat morphism.
A log pair (X , D) is a pair consisting of a normal variety X defined over a perfect field k and an effective reduced Weil divisor D on X (we allow D = 0). We say that (X , D) is log smooth if X is smooth and D is a normal crossing divisor. In this subsection we assume that k has positive characteristic p. We also set S = Spec k andS = SpecW 2 (k).
if there exists a flat morphismX →S and a relative effective is called a lifting of (X , D) to W 2 (k).
2. F-liftable if there exists a lifting (X,D) of (X , D) to W 2 (k) and a morphism F X :X →X restricting to F X modulo p such that for each D i the image of . In this case we say thatF X is compatible withD and we callF X an F-lifting of (X , D) (compatible with (X,D)).

almost liftable to W
. The corresponding lifting of (U, D U ) is called an almost lifting of (X , D).

almost F-liftable if there exists a big open subset
The corresponding lifting is called an almost F-lifting of (X , D). (3)-(4) exists then we can always find a big open subset V ⊂ X such that (V, D V ) is log smooth and the corresponding condition is satisfied.
This observation follows from the corresponding fact for F-liftable log smooth pairs (see [AWZ2,Lemma 3.2] for a simple proof).
We also need to introduce some notions of singularities in presence of liftings: is liftable to W 2 (k) then we say that it is locally F-liftable if there exists a lifting (X,D) of (X , D) such that every x ∈ X has an open neighbourhood V ⊂ X for which there exists an F-lifting of (V, D V ) compatible with the lifting induced from (X,D).

If
Remark 1.6. 1. If (X , D) is log smooth and (almost) liftable to W 2 (k) then it is also locally F-liftable.
2. Note that if there exists a big open subset U ⊂ X such that the pair (U, D U = D ∩ U ) is liftable to W 2 (k) and locally F-liftable then (X , D) is almost liftable to W 2 (k) but it does not need to be locally F-liftable.
..,Y s are closed subschemes of X then we say that they are compatibly F-split by ϕ if ϕ((F X ) * I Y j ) ⊂ I Y j for all j. For the basic facts about these notions we refer the reader to [BK]. In the proof of the next proposition we need the following generalization of the second part of [BK,Proposition 1.3.11]. LEMMA 1.7. Let X be a smooth variety defined over an algebraically closed field of characteristic p > 0. Let us assume that ϕ ∈ H 0 (X , Proof. Let Y be an irreducible component of D and let x be a smooth point of the support of Z(ϕ) that belongs to Y . Then we can choose a system of local coordinates (t 1 , ...,t n ) at x such that the local equation of Y is given by t 1 = 0. Note that by assumption the local expansion of ϕ at x is given by where g(t 1 , ...,t n ) is not divisible by t 1 and m ≥ 1 is the multiplicity of Y in D. Since ϕ splits X by [BK,Theorem 1.3.8 is nonzero. Hence m = 1 and the splitting ϕ is compatible with Y at x. It follows that ϕ is compatible with Y at smooth points of the support of Z(ϕ). So the required assertion follows from [BK,Lemma 1.1.7,(ii)]. PROPOSITION 1.8. Let (X , D) be a log pair.

If X is F-split compatibly with all irreducible components of D then (X , D)
is liftable to W 2 (k).

If (X , D) is almost F-liftable then X is F-split compatibly with all irreducible components of D.
Proof. In case D = 0 the first part is contained in [La5,Proposition 4] and the second one follows from [BTLM,Theorem 2] (see also [AWZ2,Section 2]). In general, the first part follows from [AZ,Lemma 5.2.1]. By [BK,Lemma 1.1.7,(ii) and (iii)] to prove the second part it is sufficient to prove that if (X , D) is log smooth and F-liftable then irreducible components of D are compatibly Fsplit. Note that the F-splitting induced by a liftingF X that is compatible withD vanishes to order (p − 1) along D (see the proof of [AWZ2, Lemma 3.1]). So we can conclude by Lemma 1.7. Remark 1.9. If (X , D) is log smooth then the fact that D is compatibly F-split is claimed in [AWZ2, Lemma 3.1] but the proof there contains a gap. The problem is that the Frobenius splitting coming from the lifting of the Frobenius morphism to W 2 (k) does not need to come from (p −1)-th power of a section of H 0 (X , ω −1 X ). See below for an explicit example.
where k is a perfect field of characteristic p > 2. LetX := SpecW 2 (k)[x 1 , x 2 ] be a lifting of X to W 2 (k) and letD := (x 1 = 0) ⊂X be a lifting of D ⊂ X . Let us consider a liftingF X of F X given by x 1 → x p 1 and x 2 → x p 2 + px 2 2 . This lifting is compatible withD. However, it is easy to see that the Frobenius splitting associated toF X is given by In fact, in this case one cannot find any open subset U ⊂ X such that ϕ| U is a (p − 1)-th power of a section of H 0 (U, ω −1 U ). On U = {x 2 (x p−2 2 + 2) = 0} one can multiply ϕ by an invertible u ∈ Γ(U, O * U ) so that u · ϕ| U = ψ p−1 for some ψ ∈ H 0 (U, ω −1 U ) and apply [BK,Proposition 1.3.11] to this new splitting. This shows that u · ϕ| U splits U compatibly with D ∩ U . However, this is not sufficient to apply [BK,Lemma 1.1.7,(ii)] to conclude that ϕ splits X compatibly with D.
Example 1.11. The following example is motivated by [Zd,Example 5.1] (note that the argument showing F-liftability works for p > 2; for p = 2 F-liftability needs to be proven using [Zd,Corollary 4.12]).
Let us consider divisor D : where k is a perfect field of characteristic p > 0. ThenX := SpecW 2 (k)[x 1 , x 2 ] is a lifting of X to W 2 (k) and it has a natural liftingF X of F X given by x i → x p i for i = 1, 2. LetD := (x 1 x 2 (x 1 + x 2 ) = 0) ⊂X be a lifting of D ⊂ X . If p > 2 thenF X induces a compatible liftingF X |D :D →D of F D . However,F X is not compatible withD . In fact, an explicit computation shows that (X , D) is not F-liftable. Note however that there exist splittings of X that are compatible with D. For example, one can take splitting of X corresponding to We need also the following logarithmic version of [AWZ,Theorem 3.3.6 (a), (iii)]. The proof is analogous to the one from [AWZ] and we leave it to the reader. LEMMA 1.12. Let (X , D) be a log scheme and let U ⊂ X be a big open subset of X . Let (X,D) be a W 2 (k)-lifting of (X , D) and let FŨ be an F-lifting of (Ũ,D U ), wherẽ U = (U, OX | U ) andD U = (D U , OD| D U ). Then there exists an F-liftingF X :X →X compatible withD.
The following theorem shows that an almost liftable log pair, which is locally almost F-liftable is already liftable to W 2 and locally F-liftable.
. By Proposition 1.8 we know that V is F-split compatibly with irreducible components of D V and hence we have a canonical lifting of (V, D V ) to W 2 . Moreover, this lifting extends lifting ( V ∩U,D U∩V ). So again using Lemma 1.12 we can extend F V ∩U to an F-lifting of (V, D V ). This shows (2). Now let us remark that for all x we can glue the obtained canonical liftings (Ṽ ,D V ) to (Ũ,D U ), obtaining a lifting of (X , D) to W 2 (k), which is locally Fliftable. One can do that since an F-lifting is uniquely determined up to a canonical isomorphism (this is a log version of [AWZ2, Theorem 2.7]).
The above theorem immediately implies the following corollary: COROLLARY 1.14. If (X , D) is almost liftable to W 2 (k) and it has F-liftable singularities in codimension 2 then there exists a closed subset Z ⊂ X of codimension ≥ 3 such that (X \Z, D\Z) is liftable to W 2 (k) and it is locally F-liftable. Remark 1.15. Note that it is usually much easier to lift to W 2 (k) a big open subset of X than the whole X . For example, if X is a smooth projective surface then any open subset U X is liftable to W 2 (k). This follows from the fact that the obstruction to lifting of U to W 2 (k) lies in H 2 (T U ), which vanishes by Lichtenbaum's theorem (see [SP, Theorem 0G5F]). Theorem 1.13 says that if X is not liftable to W 2 (k) then it is not locally (almost) F-liftable with respect to any lifting of U . Remark 1.16. If X is F-liftable then it does not need to have rational singularities. In fact, [Zd,Example 5.2] shows that the cone over an ordinary elliptic curve is F-liftable. This singularity is log canonical but not klt. Let us also recall that by [AWZ2,Theorem 2.10,(c)] if X is F-liftable and G is a linearly reductive group acting on X then the quotient X / / G is also F-liftable.
Finally, note that by [Zd,Theorem 4.15] ordinary double points of the form (x 2 1 + ... + x 2 n = 0) ⊂ A n k for n ≥ 5 in characteristic p ≥ 3 are F-split but they are not (locally) F-liftable. These singularities are not only log canonical but even terminal. The hypersurface (x 2 1 + ... + x 2 n = 0) ⊂ A n k is almost liftable to W 2 (k) and it has F-liftable singularities in codimension 2 (since for n ≥ 4 it is regular in codimension 2).

Intersection theory on normal varieties
Let X be a normal projective variety of dimension n defined over an algebraically closed field k. In the following we write A 1 (X ) for the class group of X , i.e., the group of Weil divisors modulo rational equivalence on X . If E is a coherent O Xmodule of rank r ≥ 1 then the sheaf det E = ( r E ) * * is reflexive of rank 1 and we can consider the associated class c 1 (E ) ∈ A 1 (X ) of Weil divisors on X .
More generally, we write A m (X ) for the group of m-cycles modulo rational equivalence on X . Chern classes of vector bundles on X are considered as in [Fu] as operations on A * (X ).
We say that two line bundles L and M on X are numerically equivalent if for every proper curve C ⊂ X we have If L is numerically equivalent to O X then we say that L is numerically trivial. The group of line bundles modulo numerical equivalence is denoted by N 1 (X ). This is a torsion free quotient of the Néron-Severi group of X . So by theorem of the base, N 1 (X ) is a free Z-module of finite rank.

If D 2 is a Cartier divisor then
3. If L 1 , ..., L n−2 are very ample then for a general complete intersection surface S ∈ |L 1 | ∩ ... ∩ |L n−2 | we have where on the right hand side we have Mumford's intersection of Weil divisors on a normal surface. THEOREM 1.18. Assume that k has positive characteristic. For any normal projective variety X /k and for any coherent reflexive O X -module E on X there exists a Z-multilinear symmetric form X ch 2 (E ) : N 1 (X ) ×(n−2) → R such that: 2. If k ⊂ K is an algebraically closed field extension then 3. If n > 2 and L 1 is very ample then for a very general hypersurface H ∈ |L 1 | we have Once we have the above theorems we can define some other Chern numbers as follows.
Definition 1.19. For any reflexive coherent O X -module E of rank r and any line bundles L 1 , ..., L n−2 on X we define the following Chern numbers: By linearity we can also extend obtained forms to Q-line bundles. In this way we get symmetric Q-multilinear forms

Numerical groups of divisors
Let X be an irreducible normal scheme defined over an algebraically closed field k. We say that a Weil divisor D is algebraically equivalent to zero if there exists a smooth variety T , a Weil divisor G on X × k T and k-points t 1 ,t 2 ∈ T such that [Fu,10.3]). Then we write D ∼ alg 0. The group of algebraic equivalence classes of Weil divisors on X is denoted by B 1 (X ). Bȳ B 1 (X ) we denote the quotient of B 1 (X ) by torsion.
Let us recall that if X is proper then N 1 (X ) ≃ Pic X /Pic τ X . If X is also smooth then N 1 (X ) ≃B 1 (X ) (see [Kl,Theorem 9.6.3]). We will need the following variant of theorem of the base. It is a special case of [Ka, Théorème 3] but we provide a different simple proof in the case used in the paper. LEMMA 1.20. Let X be an irreducible, normal, proper scheme defined over an algebraically closed field k. ThenB 1 (X ) is a free Z-module of finite rank.
Proof. By definitionB 1 (X ) is torsion free. So it is sufficient to prove thatB 1 (X ) is finitely generated as a Z-module. By [dJ] there exists an alterationỸ → X from a smooth projective varietyỸ . Taking Stein's factorization we get a proper birational map g :Ỹ → Y to normal variety and a finite surjective morphism π : Y → X .
Let E be the exceptional locus of g. Then using the localization sequence (see [Fu,Example 10.3.4]) we get a surjective map showing thatB 1 (Y ) is finitely generated.
There exists a big open subset U ⊂ X such that π : V := π −1 (U ) → U is flat. Then using flat pullback and the localization sequence (see [Fu,Proposition 10.3 and Example 10.3.4]) we have a well defined map induced by pullback of Weil divisors. Since π * π * is multiplication by the degree of π on A 1 (U ) (and hence also on B 1 (U )), the induced mapB 1 (X ) →B 1 (Y ) is injective. This implies thatB 1 (X ) is also finitely generated.
From now one in this subsection X is a normal projective variety of dimension n defined over an algebraically closed field k. LEMMA 1.21. If a Weil divisor D 1 is algebraically equivalent to zero then for every Weil divisor D 2 and all line bundles L 1 , ..., L n−2 we have Proof. Let us first assume that X is a surface and let f :X → X be a resolution of singularities. By assumption there exists a smooth variety T , a Weil divisor G on X × k T and k-points t 1 In general, by linearity of the intersection product it is sufficient to prove that D 1 .D 2 .L 1 ...L n−2 = 0 assuming that L 1 , ..., L n−2 are very ample. Let S ∈ |L 1 | ∩ ... ∩ |L n−2 | be a general complete intersection surface. Since cycles algebraically equivalent to zero are preserved by Gysin homomorphisms (see [Fu,Proposition 10.3]) the restriction D 1 | S is algebraically equivalent to zero. So by Theorem 1.17 we have The above lemma shows that the intersection pairing Let us fix a collection L = (L 2 , ..., L n−1 ) of nef line bundles on X . Assume that there exists a nef line bundle L 1 such that L 1 L 2 ....L n−1 is numerically non-trivial, i.e., there exists some Weil divisor D such that D.L 1 ...L n−1 = 0. Let us consider a Q-valued intersection pairing ·, · L : Let us write N L (X ) for the quotient of B 1 (X ) modulo the radical of this intersection pairing. Then we have an induced non-degenerate intersection pairing Proof. Let us fix some ample line bundle A. Then the Q-line bundles L i + εA are ample for ε ∈ Q >0 . So by [La9, Lemma 2.6] we have inequalities Taking the limit when ε → 0, we get D 2 1 L 2 ...L n−1 ≤ 0. Now let us assume that which gives a contradiction with some t ∈ Z.
The following lemma generalizes [La9,Lemma 2.6]. LEMMA 1.23. N L (X ) is a free Z-module of finite rank. If rk Z N L (X ) = s then the intersection pairing ·, · L has signature (1, s − 1).
Proof. By definition N L (X ) is torsion free and it is a quotient of B 1 (X ). So by Lemma 1.20 N L (X ) is also a free Z-module of finite rank. If L 2 1 L 2 ...L n−1 > 0 then the second assertion follows from Lemma 1.22. In general, there exists some Weil divisor D such that D.L 1 ...L n−1 = 0. Without loss of generality we can assume that D.L 1 ...L n−1 > 0. Then for every ample Cartier divisor H we have does not depend on L 1 , we get the required assertion from the previous case.
As in [La8, Lemma 2.1] the above lemma implies the following result (in fact, the first part follows from the proof of Lemma 1.23).

Several auxiliary results on Chern classes
In this section we prove several results on Chern classes of reflexive sheaves that will be needed throughout the paper. We assume that X is a normal projective variety of dimension n defined over an algebraically closed field k of characteristic p > 0. We also fix a collection (L 1 , ..., L n−1 ) of nef line bundles on X .
be a left exact sequence of reflexive sheaves on X , which is also right exact on some big open subset of X . Then we have Moreover, if the above sequence is exact on X and locally split in codimension 2 then we have equality Proof. Since numerical equivalence classes of nef line bundles are limits of classes of ample Q-line bundles, we can by continuity assume that all L i are ample Q-line bundles. Passing to their multiples we can also assume that all L i are very ample line bundles. By Theorem 1.18, (2) and (3), we can assume that the base field k is uncountable and then by restricting to a very general complete intersection surface S ∈ |L 1 | ∩ ... ∩ |L n−2 | we can assume that X is a surface. Let U be a big open subset on which all E , E 1 and E 2 are locally free and let j : U ֒→ X denote the open embedding. Since X is normal, we can also assume that U is contained in the regular locus X reg of X . Since F * U is exact, the sequences X E 2 are exact and the cokernel of the last map is supported on the closed subset X \U of codimension ≥ 2. So we get inequalities Dividing by p 2m , passing to the limit and using Theorem 1.18, (3), we get the required inequality. Equality follows from the fact that the above mentioned left exact sequence becomes right exact if the sequence Example 2.2. If the short exact sequence in the above lemma is not locally split, then it is well known that the inequality can be strict. For example, if X ⊂ P 3 is the cone over a smooth quadric curve in P 2 and D is its generator, then we have a short exact sequence This example shows also that Chern classes of reflexive sheaves on a normal projective surface are not deformation invariant, i.e., they can change in flat families.
Remark 2.3. Equality in Lemma 2.1 is one of the results that is not known for general normal surfaces defined over an algebraically closed field of characteristic 0 (see [La9]).
. Then the following conditions are satisfied: Proof. The first condition is clear as We prove the second condition by induction on the length s of the filtration. If s > 1 then N 1 is reflexive as N 0 /N 1 is torsion free. So by Lemma 2.1 we have Applying the induction assumption to the filtration N 1 ⊃ N 2 ⊃ ... ⊃ N s = 0 of N 1 , we get the required inequality. The last condition follows from (1) and (2).
LEMMA 2.5. Let E be a rank r reflexive coherent O X -module and let E = N 0 ⊃ N 1 ⊃ ... ⊃ N s = 0 be a filtration such that all N i /N i+1 are torsion free. Let us assume that L is numerically nontrivial and let us set Proof. Passing to an algebraically closed and uncountable field extension of k we can assume that k is uncountable. If we set F := F i then Lemma 2.1 gives After rewriting this gives But by the Hodge index theorem (see Lemma 1.23) we have So (1) follows from Lemma 2.4, (3). Under assumption (2), Corollary 1.24 implies that so again the inequality follows from Lemma 2.4, (3).
LEMMA 2.6. Let H ∈ |L 1 | be a normal variety and let T be a rank τ torsion free O D -module and let i : H ֒→ X be the closed embedding. Let Proof. Note that G is a coherent reflexive O X -module. Since both sides of our inequality depend continuously on L 2 , ..., L n−1 when considered as functions on N 1 (X ) Q , and the inequality does not change when we pass to multiples, we can assume that L 2 , ..., L n−1 are very ample. By Theorems 1.17 and 1.18 we can assume that the base field k is uncountable and then we can restrict to a very general complete intersection surface in |L 2 | ∩ ... ∩ |L n−1 | to reduce the assertion to the surface case. An exact sequence which is an isomorphism on the set where F X is flat, i.e., on X reg . But X is a surface and H is a smooth curve, which is also a Cartier divisor.
To compute the last limit, let us consider a resolution of singularities f :X → X , which is an isomorphism on X reg . So we have a closed embeddingĩ : H ֒→X such that f •ĩ = i. Then we have a short exact sequence whereG is a rank r vector bundle. Then by the same arguments as above we have To compute X ch 2 (ĩ * T ) one can use the Riemann-Roch theorem onX and on H to get Summing up, we get After rewriting, using 2r

Boundedness on normal varieties in positive characteristic
In this section we prove strong restriction theorems for semistable sheaves and we show some boundedness results. In particular we prove Theorems 0.1 and 0.2. Let X be a normal projective variety of dimension n defined over an algebraically closed field k of characteristic p > 0 and let L = (L 1 , ..., L n−1 ) be a collection of nef line bundles on X .

Slope semistability and its behaviour under pullbacks
For a coherent O X -module E of positive rank r we define its slope with respect to the collection L by In this section we usually consider slope semistability with respect to our fixed collection L (unless explicitly stated). So for simplicity of notation we usually ignore dependence of slopes on L.
Every coherent O X -module E of positive rank admits the Harder-Narasimhan It is a unique filtration by coherent O Xsubmodules such that E 0 is torsion, quotients E i := E i /E i−1 are torsion free and slope L-semistable for i = 1, ..., s and we have µ 1 = µ(E 1 ) > ... > µ s = µ(E s ). In the following we write µ max (E ) for µ 1 and µ min (E ) for µ s .
The proof of [La2, Theorem 2.7] works on normal varieties and it gives the following result: By the above theorem these numbers are well defined rational numbers.

Now choose a nef Cartier divisor
A on X such that T X (A) is globally generated. Then [La2, Corollary 2.5] still holds and it implies the following lemma.
As in [La2] we also set

Restriction theorem and Bogomolov's inequality
We define an open cone in N L (X ) As in the smooth case, by Lemma 1.23 this cone is "self-dual" in the following sense: Let us fix a coherent reflexive O X -module E . Using Lemma 2.5 one can follow the proofs of [La2,Theorems 3.1,3.2,3.3 and 3.4] and get the following results: THEOREM 3.3. Assume that L 1 is very ample and the restriction of E to a general divisor H ∈ |L 1 | is not slope (L 2 | H , . . . , L n−1 | H )-semistable. Let r i and µ i denote ranks and slopes (with respect to The only difference in proofs with respect to [La2] is that one should consider F Also in the surface case one needs to use [La9,Corollary 6.6] instead of using the arguments of [La2] that do not work for normal surfaces.
This immediately implies the following corollary.
As in [La2] we can use Theorem 3.7 to prove strong restriction theorems for reflexive sheaves on normal varieties. We take this opportunity to show proof of a stronger result that combines [La2,Theorem 5.2] with [La8,Theorem 3.7].
Since T is a torsion free O H -module, E ′ is a coherent reflexive O X -module and we have two short exact sequences: . .
So Theorem 3.7 gives which implies the required assertions.
Remark 3.10. The above proof works also for an arbitrary irreducible normal divisor D ⊂ X , which is nef and Cartier. In this way one gets restriction theorems taking into account the difference of directions of lines given by classes of D and L 1 in N L (X ). We leave the details of proof to the interested reader.
As in [La2,Corollary 5.4] the above theorem together with Lemma 2.5 implies the following result: COROLLARY 3.11. Let E be a coherent reflexive O X -module of rank r ≥ 2. Assume that E is slope (L 1 , ..., L n−1 )-semistable and let H ∈ |L ⊗m 1 | be an irreducible normal divisor. Let 0 = E 0 ⊂ E 1 ⊂ ... ⊂ E s = E be a Jordan-Hölder filtration of E and let us assume that all (E i /E i−1 )| H are torsion free.
Proof. Note that all E i are reflexive as they are saturated in E . Let us set F i := (E i /E i−1 ) * * and r i = rk F i . By Lemma 2.5 we have In the first case either r i = 1 or r i ≥ 2 and Note that in the above inequality we need to worry about the term 1 dr i (r i −1) , which can be larger than 1 dr(r− 1) . However, the difference is compensated by the other terms unless both of them are 0 in which case ⌊ 1 dr i (r i −1) ⌋ = 0. Applying Theorem 3.9 we see that F i | H is stable with the same slope as E | H . Since (E i /E i−1 )| H are torsion free, the sequences The second case is completely analogous. We just need to use the fact that Remark 3.12. The above results give also restriction theorems for torsion free sheaves. More precisely, let E be a coherent torsion free O X -module, which is slope (L 1 , ..., L n−1 )-(semi)stable. Then its reflexive hull E * * is also slope (L 1 , ..., L n−1 )-(semi)stable, so we can apply Theorem 3.9 and Corollary 3.11 to E * * . If E | H is torsion free then it is slope (

Boundedness
In this subsection we assume that n ≥ 2 and all line bundles L 1 , ..., L n−1 are ample.  Proof. The first assertion follows from the definition and Lemma 1.20. By [Fu,Proposition 10.3] we have a Gysin homomorphism  Corollary 3.11 can be used to prove the following boundedness result, which does not immediately follow from Kleiman's criterion. Part of the proof follows the idea of proof of [La3,Theorem 3.4]. THEOREM 3.14. Let us fix some classes c 1 ∈ C 1 (X ; L 1 , ..., L n−1 ), a positive integer r and some real numbers c 2 and µ max . Let A be the set of reflexive coherent O Xmodules E such that Then the set A is bounded.
Proof. For n = 2 the assertion is well-known (see, e.g., [La2,Theorem 4.4]), so we can assume that n ≥ 3. Without loss of generality we can also assume that all L i are very ample. Note that if k ⊂ K is an algebraically closed field extension then the set A is bounded if and only if the set A K := {E K : E ∈ A } of sheaves on X K is bounded. This follows from [HL,Lemma 1.7.6] and the fact that the Castelnuovo-Mumford regularity of E (with respect to some fixed very ample line bundle O X (1)) coincides with the Castelnuovo-Mumford regularity of E (here we use the fact that H i (X K , E ( j) K K). So by Theorem 1.18, (2) and an analogue of [HL,Theorem 1.3.7] for slope semistability, we can assume that the base field k is uncountable.
For fixed E ∈ A and for a very general divisor H ∈ |L 1 |, the following conditions are satisfied: 1. E | H is reflexive as an O H -module (by [HL,Corollary 1.1.14]), Let us fix a normal hypersurface H ∈ |L 1 | such that the Gysin homomorphism Lemma 3.13).
Let us consider the set A H of all sheaves E ∈ A that satisfy the above conditions (1)-(4). By Corollary 3.8 there exists C such that for every slope (L 1 , ..., Let us set r i = rk F i and µ i := µ L (F i ). By Lemma 2.5 and [La2,Lemma 1.4] we have Corollary 3.11 and condition (4) imply that for all i we have µ max (F i | H ) ≤ C 1 for some C 1 that depends only on r, c 1 , c 2 and µ max . Condition (4) implies also that the sequences For any E ∈ A the class of The second condition above follows from the well-known Enriques-Severi-Zariski lemma for reflexive sheaves on normal varieties (see [SP, Lemma 0FD8]). Let us fix E ∈ A H . For all m ∈ Z we have short exact sequences Let us consider the embedding X ֒→ P N given by the linear system |O X (1)| and letH ⊂ P N be the hyperplane defining H. For any m ∈ Z we have a commutative diagram Assume that h 1 (X , E (m)) = h 1 (X , E (m − 1)) for some m ≥ a + n − 1. Then the map β 1 in the above diagram is surjective. Since m ≥ a + n − 1, we also know that H i (H, E | H (m − i)) = 0 for i > 0 so by the Castelnuovo-Mumford theorem the map α 2 is also surjective. It follows that β 2 is surjective. This implies that h 1 (X , E (m + 1)) = h 1 (X , E (m)). So by Serre's vanishing theorem we see that h 1 (X , E (m)) = 0. This shows that for m ≥ a + n − 2 the sequence {h 1 (X , E (m))} is strictly decreasing until it reaches 0. So h 1 (X , E (l)) = 0 for l ≥ h 1 (X , E (a + n − 2)) +a +n −2. Since E is reflexive and X is normal we know that h 1 (X , E (−l)) = 0 for l ≫ 0 (here we again use [SP, Lemma 0FD8]). So for all m ∈ Z we have In particular, h 1 (X , E (a + n − 2)) ≤ (a + n − 2 + b)c. Therefore h 1 (X , E (m)) = 0 for m ≥ (a + n − 2)(c + 1) + bc. This shows that there exists a constant m 0 such that all E ∈ A H are m-regular for all m ≥ m 0 and hence A H is a bounded family (see [HL,Lemma 1.7.6]). Since the family of divisors H ∈ |L 1 | is bounded, this also gives boundedness of the family A .
Remark 3.15. If L 1 = ... = L n−1 then the above theorem follows from [La2,Theorem 4.4] (see [La9,Theorem 6.4]). In general, the problem is that we need restriction theorems for multipolarizations on normal varieties and the method of proof of [La2,Theorem 4.4] for singular varieties depends on the projection method that works only if we have one polarization. In case of characteristic zero, restriction theorems needed for multipolarizations follow easily from the results of [La2] by passing to the resolution of singularities and using Bertini's theorem. Unfortunately, this method also does not work for varieties defined over a field of positive characteristic. However, even in this case Theorem 3.14 is new.
As in [La9, Corollary 6.7], Corollary 3.8 and the above theorem imply the following result.  (X ). Since the canonical map

But by the Hodge index theorem (see Lemma 1.22) the intersection form is nega-
is injective, there are also only finitely many possibilities for the classes [c 1 (E )] ∈ C 1 (X ; L 1 , ..., L n−1 ). Now the assertion follows from Theorem 3.14.
The above corollary has some nontrivial implications even in the rank one case: COROLLARY 3.17. The canonical mapB 1 (X ) → C 1 (X ; L 1 , ..., L n−1 ) is an isomorphism.
Proof. Let D be a Weil divisor representing the class in the kernel of B 1 (X ) → C 1 (X ; L 1 , ..., L n−1 ). Corollary 3.16 implies that the set {O X (mD)} m∈Z is bounded.

So the set {[mD]} m∈Z of the corresponding classes in
Remark 3.18. The above corollary implies that some multiple of a Weil divisor D on X is algebraically equivalent to 0 if and only if D.L n−1 = D 2 .L n−2 = 0 for some ample line bundle L. This allows to give generalization of [Kl,Theorem 9.6.3] to rank 1 reflexive sheaves on normal projective varieties.

Modules over Lie algebroids and Higgs sheaves
In this section we show various definitions and simple results on modules over Lie algebroids and on generalized Higgs sheaves. We also compare our notion with the one used in [GKPT1]. We finish the section with definition of semistability and with a restriction theorem for generalized Higgs sheaves. In the whole section X is a scheme over some fixed field k.

Basic definitions
Let us recall that a tangent sheaf T X/k is defined as H om O X (Ω X/k , O X ). For every open subset U ⊂ X we have a canonical isomorphism The above definitions allow us to talk about the category L-Mod (X ) of Lmodules.

Extensions of modules over Lie algebroids
Let X be an integral normal locally Noetherian scheme over some field k and let L be a Lie algebroid on X /k, whose underlying O X -module is coherent and reflexive. By abuse we will call such Lie algebroid reflexive.
Let j : U ֒→ X be a big open subscheme X . Let (E , ∇ : L U → E nd k E ) be an L U -module. By assumption L = j * L U , so we can set j * (E , ∇) := ( j * E , j * ∇), where j * ∇ acts as ∇ on the sections of j * E (which are the same as sections of E ). In this way we can define the functor j * : L U -Mod (U ) → L-Mod (X ).
We say that an L-module (E , ∇) is reflexive if E is coherent and reflexive as an O X -module. By L-Mod ref (X ) we denote the full subcategory of L-Mod (X ), whose objects are reflexive L-modules. Note that after restricting to reflexive modules j * and j * define adjoint equivalences of categories L U -Mod ref (U ) and L-Mod ref (X ).

Tensor operations on modules over Lie algebroids
If E 1 and E 2 are L-modules then we can define natural L-module structures on E 1 ⊗ O X E 2 and on H om O X (E 1 , E 2 ). In particular, if E is an L-module then E * has a canonical L-module structure.
If ∇ 1 : L → E nd k E 1 and ∇ 2 : L → E nd k E 2 are L-module structures on E 1 and E 2 then we define an L-module structure ∇ : Similarly, we define the L-module structure ∇ : This shows that for any L-module E we can define a natural L-module struc- X is not locally free one cannot define the dual of a Higgs sheaf as a Higgs sheaf. So we cannot also take a reflexivization of a (torsion free) Higgs sheaf in the sense of [GKPT1]. This is one of the main reasons why we need to use a different definition of a Higgs sheaf.

Generalized Higgs sheaves
Let X be a scheme and let L be a quasi-coherent O X -module. We can equip L with a trival Lie algebroid structure with zero Lie bracket and zero anchor map.
On any scheme X , if E , F , G are sheaves of O X -modules then we have a functorial isomorphism of Γ(X , O X )-modules In particular, we have an isomorphism This shows that we can replace θ by an O X -linear map L⊗ O X E → E that by abuse of notation will also be denoted by θ .
Note that we have a map L ⊗ O X L → L ⊗ O X L given by sending x ⊗ y to x ⊗ y − y ⊗ x. Since it maps x ⊗ x to 0, this map factors through the canonical projection L ⊗ O X L → 2 L. Hence we get the map ι : 2 L → L ⊗ O X L fitting into an exact sequence where the second map is the canonical projection.
The following lemma explains how to check when an O X -linear map L ⊗ O X E → E gives rise to an L-Higgs sheaf.
is a homomorphism of sheaves of Lie rings.
3. The map θ extends to a Sym • L-module structure on E , i.e., there exists an O X -linear mapθ : Proof. Let x, y be local sections of L and e a local section of E . Then the first conditions means that which can be rewritten asθ (x)θ (y) =θ (y)θ (x), i.e., [θ (x),θ (y)] = 0 =θ ([x, y]). This shows equivalence of the first two conditions. If these conditions are satisfied then there exists a homomorphism of sheaves of O X -algebras Sym • L → E nd O X (E ) extending α(θ ). This follows from the definition of Sym • L as the quotient of the tensor algebra of L by the two-sided ideal generated by local sections of the form x ⊗ y − y ⊗ x.
This map provides E with a Sym • L-module structure. Clearly, if we have such a structure then also the second condition is satisfied.
Interpretation of a Higgs field as an O X -linear map θ : L ⊗ O X E → E is sometimes more convenient. For example, we can use it to introduce the following definition that will play an important role in the paper.

Morphisms of generalized Higgs sheaves
Let X be a scheme over a field k and let L be a quasi-coherent O X -module. Lemma 4.6 shows that we can treat an L-Higgs sheaf as a pair (E , θ ), where θ : L⊗ O X E → E is an O X -linear map satisfying certain additional conditions (e.g., condition 1 from Lemma 4. 6). This point of view is convenient when one wants to consider morphisms between L-Higgs sheaves, because giving a morphism of L-Higgs sheaves ϕ : (E 1 , θ 1 ) → (E 2 , θ 2 ) is equivalent to giving an O X -linear map ϕ : E 1 → E 2 such that the diagram is commutative. In the following we denote the category of L-Higgs sheaves on X by HIG L (O X ). By HIG re f L (O X ) we denote the category of reflexive L-Higgs sheaves on X .

Reflexive Higgs sheaves
In this subsection we assume that X is integral locally Noetherian and L is a coherent O X -module. We set Ω [m] L = ( m L) * for m ≥ 1. So in particular we have Ω [1] L := L * . We also fix a reflexive coherent O X -module E . By we have a canonical isomorphism

linear maps corresponding to each other under the above isomorphism then (E , θ ) is an L-Higgs sheaf if and only if the composition
L⊗ Ω

vanishes.
Proof. Note that we have a canonical isomorphism So the required assertion follows from Lemma 4.6 and the fact that the above defined composition E → E⊗Ω [2] L corresponds to the composition We need to change the sign in the last map to make it compatible with de Rham sequences for modules over Lie algebroids.
The above lemma and Lemma 1.1 give a different proof of the following corollary (cf. Subsection 4.2). COROLLARY 4.9. Assume that X is normal and L is reflexive. If j : U ֒→ X is a big open subscheme X then j * and j * define adjoint equivalences of categories HIG  Proof. The assertion follows immediately from the previous lemma and the remark that Ω [1] L * * = Ω [1] L and Ω COROLLARY 4.11. Let E be a reflexive coherent O X -module on a normal kvariety X . Letθ : E → E ⊗Ω [1] X be an O X -linear map such that the composition

vanishes. Then after composingθ with the reflexivization map E ⊗Ω
Proof. Let U ⊂ X be the maximal open subset of the regular locus of X on which E is locally free. Note that this open subset is big. By assumption the composition / / E ⊗Ω Remark 4.12. Note that not all reflexive Higgs sheaves in our sense come from reflexive Higgs sheaves as defined in [GKPT1]. More precisely, if (E , θ ) is a reflexive Higgs sheaf then we get the corresponding mapθ : E → E⊗Ω [1] X . However, this map does not need to factor through E → E ⊗Ω [1] X . Even if the aboveθ factors through E → E ⊗Ω [1] X then the composition / / E ⊗Ω

Pullback of generalized Higgs sheaves
If f : X → Y is a morphism of schemes and L is a quasi-coherent O Y -module then we can easily define the pullback of L-Higgs sheaves. Namely, if (E , θ : is an L-Higgs sheaf then it is easy to see that is an f * L-Higgs sheaf (for example one can check condition 1 from Lemma 4. 6). This defines the pullback functor on the corresponding categories of generalized Higgs sheaves, which is functorial with respect to morphisms between schemes.

Pullback of Higgs sheaves in characteristic zero
Let (X , D) be a klt pair in the characteristic zero case. Then [Ke,Theorems 1.3 and 5.2] constructs pullback functor for reflexive differentials on klt pairs that is compatible with the usual pullback of Kähler differentials. More precisely, if f : Y → X is a morphism from a normal variety Y then we get an O Y -linear map d re f f : f * Ω [1] X → Ω [1] Y . This gives rise to the dual map If (E , θ ) is a Higgs sheaf then the above construction gives a structure of f * T X/k -Higgs sheaf on f * E . By Corollary 4.10 this induces a f [ * ] T X/k -Higgs sheaf structure on f * E . Unfortunately, the canonical map f [ * ] T X/k → ( f * Ω [1] X ) * is not an isomorphism in general and we do not have any canonical map T Y → f [ * ] T X/k . So we cannot pullback general Higgs sheaves on X to Higgs sheaves on Y . However, if (E , θ ) is a Higgs bundle then we can define its pullback by taking the composition This construction should be compared to [GKPT1,5.3] and the last sentence in [GKPT1,5.2].
Remark 4.13. T X/k has a canonical Lie algebroid structure with the standard Lie bracket and identity anchor map. A module with an integrable connection is a T X/k -module for the above Lie algebroid structure. Note that one cannot define reflexive pullback for reflexive modules with an integrable connection even if Y is smooth. The problem is that the pullback would give a reflexive module with an integrable connection. Such modules are locally free and have vanishing Chern classes. However, one can show explicit examples where the reflexive pullback of a reflexive sheaf underlying a module with an integrable connection does not have vanishing Chern classes.

Reflexive pullback for Higgs sheaves under finite morphisms
Let f : X → Y be a finite dominant morphism of integral normal schemes, locally of finite type over k.

Pullback in the separable case
Since this map is non-interesting for purely inseparable morphisms, from now on we assume that the induced field extension k(Y ) ֒→ k(X ) is separable. In this case the map T X/k → f [ * ] T Y /k is injective and it uniquely extends to a homomorphism of sheaves of O X -algebras If (E , θ ) be a Higgs sheaf on Y then ( f * E , f * θ ) is an f * T X/k -Higgs sheaf. Assume that E is reflexive. Taking reflexivization we get an f * T X/k -Higgs module structure on f [ * ] E . By Corollary 4.10 we also have an induced f [ * ] T X/k -Higgs module structure on f [ * ] E . Then the homomorphism Sym E with a canonical Higgs module structure. This Higgs module will be denoted by f [ * ] One can also describe the above construction explicitly in the following way (this will be useful in the next construction). Namely, let (E , θ : E → E⊗Ω [1] Y ) be a reflexive Higgs sheaf on Y (see Lemma 4.8)

. Then there exists a big open subset
is a big open subset of X . Let j : U ֒→ X be the corresponding open embedding. Then we can define the map

This gives the map
is a reflexive Higgs sheaf on X .

Pullback in the inseparable case
Unfortunately, the above construction is rather useless in case f is purely inseparable as then d f = 0 and f [ * ] θ always vanishes. However, if f = F X and the big open subset U is F-liftable (see Definition 1.3) then we have an induced map ξ : F * U Ω U → Ω U . Now if in the above construction we replace d f by ξ , then for any reflexive Higgs sheaf (E , θ ) on X we can define This construction is used in the proof of Lemma 5.10. Note that this map depends on the choice of the lifting. Similar constructions as above work also for log pairs.

Semistability for L-modules
Let X be a normal projective variety of dimension n defined over an algebraically closed field k. Let (L 1 , ..., L n−1 ) be a collection of nef line bundles on X and let L be a Lie algebroid on X /k such that L is coherent as an O X -module. Definition 4.14. Let (E , ∇) be an L-module such that E is coherent and torsion free as an O X -module. We say that (E , ∇) is slope (L 1 , ..., In further part of this subsection we consider slope semistability with respect to a fixed collection of nef line bundles and we omit it from the notation. If and similarly for slope stability). This should be compared with [GKPT1,Definition 4.13] that considers semistability using so called generically θ -invariant subsheaves. It is easy to see that for reflexive Higgs sheaves in the sense of [GKPT1] the obtained notions of semistability coincide.
Let us also remark that if (E , θ ) is a system of L-Hodge sheaves then we can define notion of semistability using only subsystems of L-Hodge sheaves. It is easy to see that this is equivalent to semistability of (E , θ ) as an L-Higgs sheaf. We will use this fact in Section 6.
We have the following general lemma allowing to bound instability of slope semistable L-modules. It is a weak form of [La5,Lemma 5] but it works also for nef polarizations LEMMA 4.15

. Let A be an ample Cartier divisor A such that L(A) is globally generated. Let (E , ∇) be an L-module such that E is coherent and torsion free of rank r as an
this map vanishes then F has a natural structure of an L-submodule of (E , ∇).
Let E 0 = 0 ⊂ E 1 ⊂ ... ⊂ E s = E be the Harder-Narasimhan filtration of E and let us set E i := E i /E i−1 for i = 1, ..., s. Then we have non-zero O X -linear maps

Strong restriction theorem for generalized Higgs sheaves
We keep the notation from the previous subsection but we assume that L has trivial Lie algebroid structure. The same proofs as that of Theorem 3.9 and Corollary 3.11 give the following theorem (cf. [La5,Theorem 9] in the smooth case). See Subsection 4.7 for the definition of pullback used in the statement. THEOREM 4.16. Let (E , θ ) be a reflexive L-Higgs sheaf of rank r ≥ 2. Let us assume that d = L 2 1 L 2 ...L n−1 > 0 and let m be an integer such that Let H ∈ |L ⊗m 1 | be an irreducible normal divisor and let i : H ֒→ X denote the corresponding embedding.
.., L n−1 )-semistable and restrictions of all quotients of a Jordan-Hölder filtration of (E , θ ) to H are torsion free then the i * L-Higgs Note that the above theorem should be thought of as a restriction theorem for sheaves with operators and not a genuine restriction theorem for Higgs sheaves (cf. Theorem 5. 4).

Bogomolov's inequality for logarithmic Higgs sheaves on singular varieties
This section contains proofs of Theorems 0.3 and 0.4. The main idea is to use Ogus-Vologodsky's correspondence and suitably generalized Higgs-de Rham sequences.

Ogus-Vologodsky's correspondence on normal varieties
Let X be a normal variety defined over an algebraically closed field k of positive characteristic p. Let D be an effective reduced Weil divisor on X .
Let us consider a (reflexive) Lie algebroid L, whose underlying O X -module is T X (log D) with the anchor map T X (log D) ֒→ T X/k and the Lie bracket induced from T X/k . An L-module for this Lie algebroid is called an O X -module with an integrable logarithmic connection on (X , D). In fact, L carries a restricted Lie algebroid structure (see [La4,Section 4]) given by raising logarithmic derivations to the p-th power. This allows us to talk about logarithmic p-curvature F * X L → E nd O X E of an O X -module with an integrable logarithmic connection.
If (E , ∇ : T X (log D) → E nd k E ) is a reflexive O X -module with an integrable logarithmic connection then we can also define its residues in the following way. For every open subset U ⊂ X an element δ ∈ (T X (log D))(U ) can be considered as a logarithmic k-derivation of O U . Let J be the ideal subsheaf of O U generated by the image of δ . Then induces an endomorphism ρ δ of (E | U )/(JE | U ) called the residue associated to δ . We say that the residues of (E , ∇) are nilpotent of order ≤ p if for every U ⊂ X and δ ∈ (T X (log D))(U ) we have ρ p δ = 0. Similarly, one can consider T X (log D) with the trivial Lie bracket and zero anchor map. Modules over this reflexive Lie algebroid are called logarithmic Higgs sheaves on (X , D). We say that a logarithmic Higgs sheaf (E , θ : The following theorem generalizes Ogus-Vologodsky's correspondence to normal varieties: THEOREM 5.1. Let us assume that there exists a big open subset U ⊂ X such that the pair (U, D U = D ∩ U ) is log smooth and liftable to W 2 (k). Let us fix a lifting (Ũ,D U ) of (U, D U ). Then there exists a Cartier transform C (Ũ,D U ) , which defines an equivalence of categories of reflexive O X -modules with an integrable logarithmic connection whose logarithmic p-curvature is nilpotent of level less or equal to p − 1 and the residues are nilpotent of order less than or equal to p on U , and reflexive logarithmic Higgs O X -modules with a nilpotent Higgs field of level less or equal to p − 1.
Proof. As remarked in Subsection 4.2, for any reflexive Lie algebroid L and any big open subset U ⊂ X , we have an equivalence of categories of reflexive Lmodules on X and reflexive L U -modules on U . So the results of Ogus and Vologodsky in the usual case (see [OV]) and Schepler in the logarithmic one (see [Sc]; see also [La5,Theorem 2.5] and [LSYZ, Appendix]) give the above correspondence on U . One needs only to check that extension to X preserves the remaining conditions. For Higgs modules it is clear that having a nilpotent Higgs field of level ≤ (p − 1) on U gives the same condition on X . Similarly, for modules with logarithmic connections checking nilpotency of the logarithmic p-curvature on U implies the one on X .
A quasi-inverse to C (Ũ,D U ) is denoted by C [−1] (Ũ,D U ) (or simply C [−1] ) and it is called the reflexivized inverse Cartier transform.

Strong restriction theorem for logarithmic Higgs sheaves
We keep the notation from the previous subsection.
Definition 5.2. Let j : H ֒→ X be a locally principal closed subscheme of X (i.e., a scheme associated to an effective Cartier divisor). We say that H is good for the pair (X , D) if the following conditions are satisfied: In particular, if H is good for (X , D) then Corollary 4.10 shows that any logarithmic Higgs sheaf (E , θ ) on (X , D) gives rise to a reflexive logarithmic Higgs sheaf structure Remark 5.3. If L is a very ample line bundle then Bertini's theorem implies that for all m ≥ 1 a general hypersurface H ∈ |L ⊗m | is good for (X , D).
THEOREM 5.4. Let (E , θ ) be a reflexive logarithmic Higgs sheaf of rank r ≥ 2 on (X , D). Let us assume that L 1 is ample and let m 0 be a non-negative integer such that T X (log D) ⊗ L ⊗m 0 1 is globally generated. Assume also that d = L 2 1 L 2 ...L n−1 > 0 and let m be an integer such that Let H ∈ |L ⊗m 1 | be good for (X , D) with closed embedding j : H ֒→ X .
.., L n−1 )-semistable and restrictions of all quotients of a Jordan-Hölder filtration of (E , θ ) to H are torsion free then j [ * ] Proof. Using Theorem 4.16, one can follow the proof of [La5,Theorem 10] to obtain the first part of the theorem. Now let us remark that if 0 show that one of the restrictions j * (E i , θ i )| H∩U is not slope ( j * L 2 , . . ., j * L n−1 )semistable. But this contradicts the fact that the reflexivization of its extension to H (which is equal to j [ * ] (F i ,θ i )) is slope ( j * L 2 , . . . , 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 β 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.

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 O X -module.
It is convenient to consider L-modules as modules over the universal enveloping algebra Λ L of differential operators associated to L (see [La4,Section 2.2]). So we consider an L-module as a pair (E , ∇), where E is a quasi-coherent O Xmodule and ∇ : Λ L ⊗ O X E → E is a Λ L -module structure. If the underlying sheaf of an L-module is coherent as an O X -module, we say (at the risk of abusing the notation) that (E , ∇) is a coherent L-module. If the underlying sheaf of an L-module is coherent and torsion free as an O X -module, we say (again abusing the notation) that (E , ∇) is a torsion free L-module.
If (E , ∇) is a coherent L-module then we say that a filtration For every such filtration the associated graded object Gr N (E ) := i N i /N i+1 carries a canonical coherent L-Higgs module structure θ : L ⊗ O X Gr N (E ) → Gr N (E ) defined by ∇. This can be seen by considering the following commutative diagram: and one can easily check that the obtained map gives an L-Higgs module structure on Gr N (E ). Note also that by construction the obtained pair (Gr N (E ), θ ) 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 nontrivial.
We say that a Griffiths transverse filtration N • on (E , ∇) is slope gr-semistable if the associated L-Higgs sheaf (Gr N (E ), θ ) is (torsion free and) slope semistable. A partial L-oper is a triple (E , ∇, N • ) consisting of a torsion free coherent O Xmodule E with a Λ L -module structure ∇ and a Griffiths transverse filtration N • , 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 Λ Lmodules.
THEOREM 5.6. If (E , ∇) is slope semistable then there exists a canonically defined slope gr-semistable Griffiths transverse filtration S • on (E , ∇) providing it with a partial L-oper structure. This filtration is preserved by the automorphisms of (E , ∇).
The above filtration S • 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:

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 ⊂ X such that the pair (U, D U = D ∩U ) is log smooth and liftable to W 2 (k). Let us fix a lifting (Ũ,D U ) of (U, D U ).
Let (E , θ : T X (log D) ⊗ E → E ) be a reflexive logarithmic Higgs O X -module of rank r ≤ p. Let us assume that (E , θ ) is slope semistable. Then by Corollary 5.7 there exists a canonical filtration N • on E such that the associated graded (Ē 0 ,θ 0 ) is a slope semistable system of logarithmic Hodge sheaves. Let (E 0 , θ 0 ) be the reflexive hull of (Ē 0 ,θ 0 ). By construction, it is a slope semistable reflexive logarithmic system of Hodge sheaves. In particular, since its rank r is ≤ p, it is also a reflexive logarithmic Higgs O X -module with a nilpotent Higgs field of level less or equal to p − 1. So we can define Simpson's filtration on (V 0 , ∇ 0 ) and let (Ē 1 = Gr S 0 (V 0 ),θ 1 ) be the associated system of Hodge sheaves. Then we set (E 1 , θ 1 ) := ((Ē 1 ) * * ,θ * * 1 ) and repeat the procedure. In this way we get the following sequence in which each logarithmic Higgs sheaf (E j , θ j ) is reflexive rank r ≤ p and slope semistable. We call this sequence a canonical Higgs-de Rham sequence of (E , θ ).
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 E i without some further assumptions on the singularities of the pair (X , D).

Inverse Cartier transform on log varieties with locally Fliftable singularities.
Let X be a normal variety defined over an algebraically closed field k of positive characteristic p. We define the can ) → 0 of reflexive O X -modules with a logarithmic connection, which is locally split as a sequence of O X -modules. In particular, we have [C [−1] Proof. By construction we have a short exact sequence of Higgs sheaves which is split as a sequence of O X -modules. Applying C [−1] to this sequence we get So it is sufficient to show that this sequence is locally split. To do so we fix a point x ∈ X and an open neighbourhood x ∈ U ⊂ 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 ∩V ) has an F-liftingF V :Ṽ →Ṽ compatible with the W 2 (k)-lifting (Ṽ ,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 can ) → 0, which is split as a sequence of O V -modules. Extending the above sequence to U , we get a short exact sequence of reflexive T U -modules U E j+1 → 0, which is split as a sequence of O U -modules. By construction (see Section 8) Since the above isomorphisms are compatible with restrictions to V , we see that the sequence can ) → 0 is split as a sequence of 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 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

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 (E , θ ) of rank r ≤ p we have be the canonical Higgs-de Rham sequence of (E , θ ).
By Lemma 4.15 there exists α such that µ max,L (E m ) − µ L (E m ) ≤ α for all m ≥ 0. So by Corollary 3.8 there exists some constant C such that for every nonnegative integer m we have By Corollary 5.11 we have Dividing by p 2m and passing with m to infinity, we get X ∆(E )L 2 ...L n−1 ≥ 0.
Remark 5.13. The above theorem holds also for reflexive sheaves with an integrable logarithmic connection. Indeed, if (E , ∇) is a rank r ≤ p slope L-semistable reflexive sheaf with an integrable logarithmic connection and S • is its Simpson's filtration then by the above theorem and Lemma 2.4 we have

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 2 1 L 2 ....L n−1 > 0. As in Subsection 3.2 we consider a positive open cone K + L ⊂ N L (X ). The proof of the following proposition is essentially the same as that of [La6,Proposition 4.1]. PROPOSITION 6.1. Let L be a rank 1 reflexive sheaf contained in Ω Proof. Assume that c 1 (L ) ∈ K + L and consider a system of logarithmic Hodge X (log D) given by the inclusion. Then (E , θ ) is slope L-stable since the only rank 1 logarithmic subsystem of Hodge sheaves of (E , θ ) is of the form (O X , 0). Therefore by Lemma 2.1 and Theorem 5.12 we have 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 ≥ 3 and let F ⊂ Ω [1] X (log D) be a rank 2 reflexive subsheaf with c 1 (F ) ∈ K + L . Then Proof. Let us consider the system of logarithmic Hodge sheaves (E : So we can assume that (E , θ ) is not slope L-semistable. Let (E ′ , θ ′ ) be its maximal destabilizing subsystem of logarithmic Hodge subsheaves. (E , θ ) contains only one saturated rank 1 system of logarithmic Hodge subsheaves, namely (O X , 0). Since this subsystem does not destabilize (E , θ ), the sheaf E ′ has rank 2. Note Since by Proposition 6.1 we have c 1 (M ) 2 .L 2 ...L n−1 ≤ 0, this implies the required inequality.

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 E be a coherent reflexive O X -module. Then there exists a normal projective surface Y and a finite morphism π : Y → X such that π [ * ] E is locally free. In this case we have .
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 ⊂ 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 Q-Chern classes considered in [Ko,Chapter 10] and in [GKPT1,Theorem 3.13] (see [La9,Remark 5.9]).

Strong restriction theorems
Let us fix a collection (L 1 , ..., L n−1 ) of ample line bundles and let us set d = L 2 1 L 2 ...L n−1 . The proof of the following theorem is based on a standard spreading out argument. THEOREM 7.2. Let (E , θ ) be a reflexive logarithmic Higgs sheaf of rank r ≥ 2 on (X , D). Let m 0 be a non-negative integer such that T X (log D) ⊗ L ⊗m 0 1 is globally generated. Let m be an integer such that 2. If (E , θ ) is slope (L 1 , ..., L n−1 )-semistable and restrictions of all quotients of a Jordan-Hölder filtration of (E , θ ) to H are torsion free then (E , θ )| H is slope (L 2 | H , . . ., L n−1 | H )-semistable.
Proof. We can find a subring R ⊂ k, which is finitely generated over Z and there exists a flat projective morphism X → S = Spec R with a relative reduced Weil divisor D on X /S such that (X , D) ≃ (X × S Spec k, D × S Spec k). We can assume that there exist line bundles L 1 , ..., L n on X lifting L 1 , ..., L n−1 , a relative logarithmic Higgs sheaf (Ẽ ,θ : T X /S (log D) ⊗ O XẼ →Ẽ ) lifting (E , θ ) and a relative effective Cartier divisor H ∈ |L ⊗m 1 | lifting H (in particular, H → S is flat). Shrinking S if necessary we can assume that the following conditions are satisfied: exists some non-empty open subset U ⊂ S such that for all geometric pointss over closed points of U we have This is not obvious asẼ 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 π : Y → X such such that π [ * ] E is locally free. Then, shrinking S if necessary, we can find a flat projective morphismπ : Y → S and a morphism Y → X lifting π : Y → X . We can also assume that all fibers of g : Y → S are geometrically integral and geometrically normal. Since S is normal, the schemes X and Y are also normal. So we can considerπ [ * ]Ẽ , which is reflexive on Y . This sheaf is locally free outside of a closed subscheme Z ⊂ Y of codimension ≥ 2. Since Z does not intersect the generic fiber of Y → S,π [ * ]Ẽ is locally free over a non-empty open subset S ′ = S\g(Z) ⊂ S. Now let us consider a commutative diagram Y s j s / / π s Yπ X s i s / / X .
Let us set U := {x ∈ X :Ẽ x is a free O X,x -module}. Since ϕ s is an isomorphism over a big open subset Y s ∩π −1 (U ) of Y S andπ [ * ] s (i * s (Ẽ )) is reflexive, ϕ s is an isomorphism. So by Theorem 1.18 and Lemma 7.1 we have as claimed. Now the required assertion follows by applying Theorem 5.4 to fibers over geometric pointss of S with large characteristic of the residue field (then β r (s) → 0).
The same argument as above show also that Theorem 0.1 implies the following strong restriction theorem of Bogomolov's type: 1. If E is slope (L 1 , ..., L n−1 )-stable then E | H is slope (L 2 | H , . . ., L n−1 | H )-stable.
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 ∈ |L ⊗m 1 |. 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 Xs. However, using Corollary 3.11 one can show bounds on the maximal destabilizing slope of E | H on any normal variety in terms of numerical invariants of reductions of E .

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 is a left exact sequence of reflexive sheaves on X , which is also right exact on some big open subset of X then X ch 2 (E )L 1 ...L n−2 ≤ X ch 2 (E 1 )L 1 ...L n−2 + X ch 2 (E 2 )L 1 ...L n−2 .
Using the Riemann-Roch theorem for locally free sheaves on normal projective surfaces this can be rewritten as Dividing by the degree of π, 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 2 1 L 2 ....L n−1 > 0. For any slope (L 1 , ..., L n−1 )-semistable logarithmic reflexive Higgs sheaf (E , θ ) we have X ∆(E )L 2 ...L n−1 ≥ 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 ) ⊗ Q for t ∈ Q >0 . These classes are ample and the Harder-Narasimhan filtration of (E , θ ) with respect to (L 1 (t), ..., L n−1 (t)) is independent of t for small t ∈ 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 → 0 gives the required inequality.

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.

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 = Spec k and S = SpecW 2 (k). Let X be a smooth k-variety with a fixed liftingX/S. Let X ′ 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 → X ′ . Note that X ′ has a natural liftingX ′ toS, which is defined as the base change ofX →S viaS →S coming from σ 2 : W 2 (k) → W 2 (k). Let us assume that F X/S : X → X ′ has a lifting F X/S :X →X ′ so that we have a commutative diagram Since the mapF * :F * X/S Ω 1X ′ /S → Ω 1X /S vanishes after pulling back to X , we can define ζ = p −1F * : F * X/S Ω 1 X ′ /S → Ω 1

X/S
Since dζ = 0 we can consider ζ as the map of sheaves of abelian groups F * X/S Ω 1 X ′ /S → Z 1 X/S , where Z 1 X/S is the kernel of d : Ω 1 X/S → Ω 2 X/S . Its adjoint ζ ad : Ω 1 X ′ /S → F X/S, * Z 1 X/S is O X ′ -linear and it splits the composition F X/S, * Z 1 X/S −→ H 1 (F X/S, * Ω • X/S ) of the Cartier operator with the canonical projection.

Inverse Cartier transform
Assume that X is smooth and there exists a global liftingX of X to W 2 (k).