Nearby cycles and semipositivity in positive characteristic

We study restriction of logarithmic Higgs bundles to the boundary divisor and we construct the corresponding nearby-cycles functor in positive characteristic. As applications we prove some strong semipositivity theorems for analogs of complex polarized variations of Hodge structures and their generalizations. This implies, e.g., semipositivity for the relative canonical divisor of a semistable reduction in positive characteristic and it gives some new strong results generalizing semipositivity even for complex varieties.


Introduction
Let X be a smooth projective variety defined over an algebraically closed field k of characteristic p and let D be a simple normal crossing divisor on X . In this introduction we assume that (X , D) lifts to the ring W 2 (k) of Witt vectors of length at most 2.
A logarithmic Higgs sheaf on (X , D) is a pair (E, θ ) consisting of a coherent O X -module and an O X -linear map θ : E → E ⊗ Ω X (log D) such that θ ∧ θ = 0.
Equivalently, replacing θ byθ : T X (log D) ⊗ E → E one can consider a logarithmic Higgs sheaf on (X , D) as a Sym • T X (log D)-module, which is coherent when considered as an O X -module.
Let MinHIG 0 (X , D) be the category of locally free logarithmic Higgs sheaves of rank r ≤ p on (X , D), which have vanishing Chern classes in H 2 * et (X , Q l ) for some l = p and are semistable. In this case semistable means slope H-semistable with respect to some ample divisor H, but one can show that the category MinHIG 0 (X , D) does not depend on the choice of H. One can also replace slope semistability by Gieseker semistability and the category remains the same.
Let Y be an irreducible component of D and let ı : Y → X be the corresponding embedding. One of the main aims of this paper is to prove the following theorem: THEOREM 0.1. Let (E, θ ) be an object of MinHIG 0 (X , D). The restriction of (E,θ ) to Y defines a semistable Sym • ı * T X (log D)-module. Moreover, this restriction can be deformed to an element of MinHIG 0 (Y, D Y ), where D Y is the restriction of the divisor D −Y to Y .
The precise statement of this theorem is contained in Theorem 3.9 and Corollary 3.11. In fact, we prove a more general version that works also for Higgs sheaves (or modules with an integrable connection) with non-vanishing Chern classes.
Together with the restriction theorem for curves not contained in the boundary divisor D (see Theorem 2.17) this gives an inductive procedure for studying restriction of elements of MinHIG 0 (X , D) to curves. In particular, it implies the following theorem (see Definition 4.1 for the definition of a strongly liftable morphism).
THEOREM 0.2. Let (E, θ ) be an object of MinHIG 0 (X , D). Let C be a smooth projective curve and let ν : C → (X , D) be a separable morphism that is strongly liftable to W 2 (k). Then the induced Sym • ν * T X (log D)-module ν * E is semistable. In particular, if G is a subsheaf of the kernel of ν * θ : ν * E → ν * E ⊗ ν * Ω X (log D) then deg G ≤ 0. This theorem has an obvious analogue in characteristic zero (see Theorem 4.4). But even the last part of this theorem was not known in characteristic zero. Already this part implies essentially all known semipositivity results (see below) for Higgs bundles or complex polarized variations of Hodge structures due to Fujita [Fu], Kawamata [Kw], Zuo [Zu], Fujino-Fujisawa [FF,Theorem 5.21], Brunebarbe [Br1,Theorems 1.8 and 4.5], [Br2,Theorem 1.2] and many others.
Note that almost all the proofs of such results are analytic and use Hodge theory. A notable exception is Arapura's proof of [Ar,Theorem 2] that uses reduction to positive characteristic. However, his proof uses vanishing theorems and it does not give any semipositivity results in positive characteristic.
We say that a sheaf E on (X , D) is W 2 -nef if for any smooth projective curve C and any morphism ν : C → (X , D) that is strongly liftable to W 2 (k) (see Definition 4.1), we know that all quotients of ν * E have a non-negative degree.
The following corollary is a direct analogue of [Br2,Theorem 1.2] in positive characteristic. In fact, it implies its generalization from polystable to the semistable case. COROLLARY 0.3. Let (E, θ ) be an object of MinHIG 0 (X , D). If E ′ is a locally split subsheaf of E contained in the kernel of θ then its dual (E ′ ) * is W 2 -nef on (X , D).
Over complex numbers a typical example of application of such a result is to semipositivity of direct images of relative canonical sheaves. This happens also in positive characteristic and we prove the following result (see Corollary 4.10 for a more precise version).
COROLLARY 0.4. Let X and Y be smooth projective varieties and let B be a normal crossing divisor on Y . Let f : X → Y be a smooth surjective morphism of relative dimension d, which has semi-stable reduction along B. Let us set D = f −1 (B). Assume that there exists a liftingf : (X,D) → (Ỹ ,B) of f to W 2 (k) withf a semi-stable reduction alongB. Assume that p > d +dimY . Then R j f * (ω X/Y (D)) is a W 2 -nef locally free sheaf on (Y, B) for all integers j ≥ 0. This is a positive characteristic analogue of various semipositivity results due to Griffiths [Gr], Fujita [Fu], Kawamata [Kw], Fujino-Fujisawa [FF] and others.
In positive characteristic p there are well-known examples due to L. Moret-Bailly (see [Sz,Exposé 8]), who showed for any integer n ≥ 1 and any p a family of smooth abelian surfaces f : X → P 1 such that f * ω X/P 1 = O P 1 (−n) ⊕ O P 1 (pn). In particular, one needs to add some additional assumptions to be able to get semipositivity results. The only known results on semipositivity in positive characteristic concern either f * (ω m X/Y (mD)) for m ≫ 0 (see [Pa1] in case ω X/Y (D) is f -nef, or [Ej] in case of relative dimension 1 or 2) or they deal with f * ω X/Y adding very strong assumptions on the fibers (see [Sz2] for the case dim X = 2 and dimY = 1, and [Pa2,Theorem 6.4] for a rather complicated statement).
One of the important results that we prove is the following theorem that is a special case of Theorem 2.1. THEOREM 0.5. Let E be a rank r reflexive sheaf with c 1 (E) = 0 (in H 2 et (X , Q l ) for some l = p) and c 2 (E)H n−2 = 0. Assume that E has a filtration M • such that all factors of the filtration are torsion free of rank ≤ p with µ H (Gr M j E) = µ H (E). Let us also assume that each factor has a structure of a slope H-semistable sheaf with an integrable logarithmic connection on (X , D). Then E is locally free and it has vanishing Chern classes in H 2 * et (X , Q l ) for any l = p. Moreover, every quotient Gr M j E is locally free and has vanishing Chern classes in H 2 * et (X , Q l ).
This result can be thought of as an analogue of a graded version of Schmid's nilpotent orbit theorem (see Remark 2.15). In the case D = 0 Theorem 0.5 gives [La3,Theorem 11] and fills in a gap in its proof. The stronger version, Theorem 2.2, generalizes Theorem 0.5 to the case of Higgs sheaves with possibly nontrivial Chern classes and in the case D = 0 it is indispensable for the proofs of [SYZ,Theorem 3.6,. In this last case Theorem 2.2 allows to compute higher Chern classes of twisted preperiodic Higgs bundles.
The structure of the paper is as follows. In Section 1 we recall some results and prove a few auxiliary results used in the sequel. In Section 2 we prove Theorem 0.5 and we show several applications. In Section 3 we construct a nearby-cycles functor and we check that it preserves some semistability conditions. We also study semistability of factors of the monodromy filtration associated to residue endomorphisms of logarithmic Higgs sheaves. Section 3 is devoted to applications of these results to semistability and semipositivity of restriction of Higgs bundles to curves. We also give some geometric applications to semipositivity of direct images of relative canonical sheaves. The appendix contains a proof of the functoriality of the inverse Cartier transform in the logarithmic case.

Notation
Let X be a smooth variety defined over an algebraically closed field k and let D be a normal crossing divisor on X . We often view D as a closed subscheme of X given locally by one equation but by abuse of notation we also identify D with the corresponding Weil divisor and write D = 0 instead of D = / 0. All normal crossing divisors in the paper are reduced simple normal crossing divisors. Sometimes we add "simple" to stress the place, where we need to use this assumption.
Let us recall that a logarithmic Higgs sheaf is a pair (E, θ ) consisting of a coherent O X -module and an O X -linear map θ : E → E ⊗ Ω X (log D) such that θ ∧ θ = 0. A system of logarithmic Hodge sheaves is a Higgs sheaf (E, θ ) with a decomposition E = E p,q such that θ maps E p,q into E p−1,q+1 ⊗ Ω X (log D).
In this paper if X is projective and we say that a logarithmic Higgs sheaf (E, θ ) is slope H-semistable for some ample H then we always implicitly assume that E is torsion free. Let us recall that a system of logarithmic Hodge sheaves is slope H-semistable as a system of logarithmic Hodge sheaves if and only if it is slope H-semistable as a logarithmic Higgs sheaf (see [La2,Corollary 3.5]). Now let S be any scheme. We say that (X , D) is a smooth log pair over S if X is a smooth S-scheme and D is a relatively simple normal crossing divisor over S.
We say that f : (Y, B) → (X , D) is a morphism of smooth log pairs if f : Y → X is a morphism and the support of B contains the support of f −1 (D).
If E is a coherent sheaf of rank r on a smooth projective variety X then we denote by ch (E) the Chern character of E. This is defined as an element of the rational Chow ring CH * (X ) ⊗ Q but in this paper we abuse notation and denote by ch (E) the image of this class by the cycle map and we treat it as an element of thé etale cohomology ring H * et (X , Q l ), where l is different from the characteristic of the base field (or an element of H * (X , Q) in case of complex manifolds). By ∆(E) we denote the discriminant of E defined as 2rc 2 (E) − (r − 1)c 2 1 (E). In case of surfaces we use the degree map X to identify the cohomology group H 4 et (X , Q l ) (or H 4 (X , Q)) with Q l (respectively, Q) and we think of ∆(E) as an integer. Similarly, in higher dimensions the top degree intersections like ∆(E)H dim X−2 denote the degree of the cycle ∆(E)H dimX−2 .

Logarithmic Higgs sheaves
In this subsection we recall a few results on semistable logarithmic Higgs sheaves. Throughout this subsection we fix the following notation.
Let X be a smooth variety of dimension n defined over an algebraically closed field k of characteristic p. Let D be a normal crossing divisor on X .
Let us recall the following theorem due to Ogus and Vologodsky in the usual case (see [OV]) and Schepler in the logarithmic one (see [Sc]; see also [La4,Theorem 2.5] and [LSYZ, Appendix]): THEOREM 1.1. Let us assume that (X , D) is liftable to W 2 (k) and let us fix such a lifting (X,D). There exists a Cartier transform C (X,D) , which defines an equivalence of categories of torsion free 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, and torsion free logarithmic Higgs O X -modules with a nilpotent Higgs field of level less or equal to p − 1.
From now on in this subsection we assume that X is projective and we fix an ample divisor H on X . Let us recall the following boundedness result for logarithmic Higgs sheaves. THEOREM 1.2. Let us fix some number ∆ and a class c ∈ H 2 et (X , Q l ) for some l = p. The family of slope H-semistable logarithmic Higgs sheaves (E, θ ) such that E is reflexive with fixed rank r, c 1 (E) = c and ∆(E)H n−2 ≤ ∆ is bounded.
Let us note that in the above theorem it is not sufficient to fix r, c 1 (E)H n−1 and ∆(E)H n−2 . We will also need the following theorem, which is a special case of [La2,Theorem 5.5]. THEOREM 1.3. Let (E, θ ) be a slope H-semistable logarithmic Higgs sheaf. Then there exists a decreasing filtration E = N 0 ⊃ N 1 ⊃ ... ⊃ N m = 0 such that θ (N i ) ⊂ N i−1 ⊗ Ω X (log D) and the associated graded is a slope H-semistable system of logarithmic Hodge sheaves. [La2,Theorem 5.5] (see also [LSZ,Theorem A.4] in case of flat torsion free sheaves) implies also the following result: THEOREM 1.4. If (E, ∇) is a slope H-semistable sheaf with an integrable logarithmic connection then there exists a canonical Griffiths transverse filtration E = S 0 ⊃ S 1 ⊃ ... ⊃ S m = 0 such that the associated graded system of logarithmic Hodge sheaves is slope H-semistable.
The canonical filtration S • from Theorem 1.4 is called Simpson's filtration of (E, ∇). This notion is used in the following generalization of [La2,Theorem 5.12] and [LSZ2,Theorem 2.2] (see [La4,Theorem 3.1]). THEOREM 1.5. Assume the pair (X , D) admits a lifting (X,D) to W 2 (k). If (E, θ ) is a slope H-semistable system of logarithmic Hodge sheaves of rank r ≤ p then there exists a canonically defined Higgs-de Rham sequence The following theorem is a generalization of [La3,Theorem 10] to the logarithmic case. We skip its proof as it is the same as in the non-logarithmic case.
Moreover, if r > 2 let us also assume that d > 2(r − 1) 2 d 0 . Let Y ∈ |dH| be a smooth divisor such that E Y has no torsion and D ∩ Y is a normal crossing divisor on Y . Then the logarithmic Higgs sheaf COROLLARY 1.7. Assume the pair (X , D) is liftable to W 2 (k) and let d 0 be as in the previous theorem. Let (E, θ ) be a slope H-semistable logarithmic Higgs sheaf of rank r ≤ p and let d be an integer satisfying the same inequalites as in the previous theorem. Then for a general divisor Y ∈ |dH| the restriction ( Proof. Let M • be a Jordan-Hölder filtration of (E, θ ). By definition this means that all the quotients Gr M i E are slope H-stable logarithmic Higgs sheaves with slopes µ H (Gr M i E) equal to µ H (E). Existence of such a filtration for logarithmic slope H-semistable logarithmic Higgs sheaves is standard and follows by the same arguments as for the usual slope H-stable sheaves (see, e.g., [HL,1.5 and 1.6]).
To simplify notation let us set E i = Gr M i E (we consider it as a logarithmic Higgs sheaf and not only a sheaf) and r i = rk E i . Then the Hodge index theorem implies that Therefore our assumptions on d imply that we can apply Theorem 1.6 to each quotient E i . So if we choose a smooth divisor Y ∈ |mH| such that D ∩ Y is a normal crossing divisor on Y and (E i ) Y has no torsion for every i then the restricted logarithmic Higgs sheaf Note that general Y ∈ |mH| satisfies the above assumptions by Bertini's theorem and Lemma 1.13.
Remark 1.8. In Theorem 1.6 and Corollary 1.7 we can replace a logarithmic Higgs sheaf with a sheaf with an integrable logarithmic connection. The proofs of the results remain the same.
Let us also recall Bogomolov's inequality for logarithmic Higgs sheaves (see [La4,Theorem 3.3] for a more general version). THEOREM 1.9. Assume that (X , D) admits a lifting to W 2 (k). Then for any slope H-semistable logarithmic Higgs sheaf (E, θ ) of rank r ≤ p we have Remark 1.10. The above theorem holds also for sheaves with an integrable logarithmic connection. Indeed, if (E, ∇) is a rank r ≤ p slope H-semistable sheaf with an integrable logarithmic connection and S • is its Simpson's filtration then by the above theorem

Higher discriminants
Let us fix a smooth projective variety X defined over an arbitrary algebraically closed field k.
Let E be a rank r > 0 coherent sheaf on X . Let us fix a prime l non-equal to the characteristic of the base field k and let us write for some classes ∆ i (E) ∈ H 2i et (X , Q l ) that we call higher discriminants of E (we can also use ∆ i (E) ∈ H * (X , Q) in case of complex manifolds). These discriminants are polynomials in Chern classes of E with integral coefficients. They are variants of Drezet's logarithmic invariants (with somewhat different normalization to get integral coefficients and ∆ 2 (E) = ∆(E)). Note that for any line bundle L we have ∆ i (E ⊗ L) = ∆ i (E) for i ≥ 2. This follows immediately from the fact that log(ch (E ⊗ L)) = log(ch (E) · ch (L)) = log(ch (E)) + c 1 (L).
In the following we often use this property of discriminants without further notice.
LEMMA 1.11. The following conditions in H * et (X , Q l ) (or in H * (X , Q) in case of complex manifolds) are equivalent: Proof. Equivalence of 2 and 3 is clear as ∆ 1 (E) = c 1 (E). For simplicity of notation let us assume that E is locally free. Proof in the general case is the same except that we need to replace E by its class in K 0 (X ) and do all the computations in Grothendieck's K-group. By the Bloch-Gieseker covering trick (see [BG,Lemma 2.1]) there exists a smooth projective varietyX and a finite flat surjective covering f :X → X together with a line bundle L such that f * (det E) −1 = L ⊗r . Let us setẼ := f * E ⊗ L. Note that c 1 (Ẽ) = 0 and , the second condition is equivalent to the vanishing of ∆ i (Ẽ) for all i ≥ 1, i.e., to the equality log ch (Ẽ) = log r. Clearly, this is equivalent to c i (Ẽ) = 0 for all i ≥ 1.
For all i ≥ 0 we have Using c 1 (L) = − 1 r c 1 ( f * E) and the fact that the map H * et (X , Q l ) → H * et (X, Q l ) is injective, we see that the second condition is equivalent to the equalities for all i = 1, ..., r. This follows from the fact that the equalities (1) for i ≤ m are equivalent to the equalities for i = 1, ..., m. We prove this by induction on m. For m = 1 it is clear, so let us assume it holds for 1, ..., m − 1. We can assume that (2) holds for i < m. Then we have This proves that under our assumptions, (1) for i = m is equivalent to (2) for i = m.

Criterion for local freeness and restriction to divisors
Let X be an integral noetherian scheme and let E be a coherent sheaf of O Xmodules. Let S(E) be the set of points x ∈ X such that E x is not a free O X,xmodule. We call S(E) the singular set of E. Let us define the function ϕ : ). Let η be the generic point of X . For a point x ∈ X , by [Ha,Chapter II,Lemma 8.9], E x is a free O X,x -module if and only if ϕ(x) = ϕ(η). On the other hand, by Nakayama's lemma the function ϕ is upper semicontinuous (see [Ha,Chapter III,Example 12.7 In the following we say that E is locally free outside a finite number of points if S(E) is a finite set of points. Now let X be a smooth projective variety of dimension n defined over an arbitrary algebraically closed field k. In the following we will use several times the following criterion for local freeness of graded sheaves associated to filtrations. LEMMA 1.12. Let us assume that n ≥ 3 and let V be a reflexive sheaf on X with a filtration N m = 0 ⊂ N m−1 ⊂ ... ⊂ N 0 = V such that each N i is saturated in V . Let W = N i /N i−1 be the associated graded and let us assume that 1. the reflexivization W * * of W is locally free, and 2. W is locally free outside a finite number of points.
Then both V and W are locally free.
Proof. It is sufficient to prove the lemma for m = 2. The general case follows easily from this special one by induction on the length m of the filtration.
Assuming m = 2 our assumptions imply that N 1 is locally free and we have a short exact sequence for some sheaf T supported on a finite number of points. By assumption we also know that (N 0 /N 1 ) * * is locally free. Let us note that by Serre's duality Ext 2 (T, N 1 ) is dual to Ext n−2 (N 1 , T ⊗ω X ) = H n−2 (T ⊗ω X ⊗(N 1 ) * ) = 0 as n ≥ 3. Hence by the long Ext exact sequence, the canonical map Ext 1 ((N 0 /N 1 ) * * , N 1 ) → Ext 1 (N 0 /N 1 , N 1 ) is surjective. Therefore there exists a coherent sheafṼ such that the following diagram is commutative: But since V is reflexive andṼ is locally free, the map V →Ṽ is an isomorphism (as it is an isomorphism outside of the support of T ). Hence T = 0 and W * * = W . This immediately implies the required assertion.
We will also need the following lemmas allowing us to keep track of singularities of sheaves when restricting to divisors. LEMMA 1.13. Let Λ be a base point free linear system on X and let E be a coherent O X -module.

If E is reflexive and Y
2. If E is torsion free (reflexive) and Y ∈ Λ is general then E Y is also torsion free (reflexive, respectively) as an O Y -module.
The above lemma follows from [HL, Lemma 1.1.12 and Corollary 1.1.14].
LEMMA 1.14. Let E be a rank r torsion free sheaf on X and let Y be an integral divisor on X such that E Y is locally free. Then S(E) ∩Y = / 0, i.e., E is locally free at all points of Y . Moreover, if Y is ample then E is locally free outside a finite number of points.
Proof. Since every torsion free sheaf on a smooth curve is locally free, we can assume that the dimension n of X is greater than 1. Since S(E) has codimension ≥ 2 in X , there exists a codimension 1 point y ∈ Y − S(E). Let η be the generic point of X and η ′ the generic point of Y . Since E Y is locally free, E Y,y is a free O Y,y -module and hence By the choice of y the O X,y -module E y is free and hence Then [Ha,Chapter II,Lemma 8.9] implies that E z is a free O X,z -module, which proves the first assertion. Now let us assume that Y is ample. The singular set S(E) is a closed subset of X and S(E) ∩Y = / 0, so it does not have any irreducible components of dimension ≥ 1. So S(E) is zero-dimensional.

Local freeness
In this section we fix the following notation. Let X be a smooth projective variety of dimension n ≥ 2 defined over an algebraically closed field k of characteristic p and let D be a simple normal crossing divisor on X . We assume that D ⊂ X admits a lifting to W 2 (k). We also fix an ample divisor H on X .
The main aim of this section is to prove the following generalization of Theorem 0.5:  (Gr M j This theorem is a strong version of the following theorem to which we will reduce its proof. THEOREM 2.2. Let (E, θ ) ((E, ∇)) be a rank r ≤ p slope H-semistable logarithmic Higgs sheaf (a rank r ≤ p slope H-semistable sheaf with a logarithmic connection, respectively). Then the following conditions are equivalent: for all m ≥ 1 and any l = p.
Remark 2.3. To simplify notation in Theorems 2.1 and 2.2 we deal with only one polarization although one can also replace H by a collection of ample divisors as in, e.g., [La3,Theorem 10].
Theorem 2.2 generalizes [La3,Theorem 11] to the case of logarithmic Higgs sheaves with possibly non-trivial Chern classes. It also generalizes [SYZ, Theorems 3.6 and 3.10], which deal with systems of Hodge sheaves of rank r < p on X defined over k =F p . In this last case Theorem 2.2 allows to compute higher Chern classes of twisted preperiodic Higgs bundles. Let us also remark that a special case of the above result was implicitly used in proof of [Ar,Theorem 3] (see Remark 2.12).
The strategy of our proof of Theorem 2.2 in the case D = 0 is modelled on the proof of [La3,Theorem 11]. Unfortunately, the proof of [La3,Theorem 11] contains a serious gap: it is not clear that the family of Higgs sheaves {(E i , θ i )} considered in the proof is bounded as a priori the sheaves E i need not be reflexive. However, if one assumes that in [La3,Theorem 11] all Chern classes vanish, then the arguments there show that E is locally free. This is already sufficient for almost all the applications mentioned in [La3] (except for Corollary 6 that also needs an additional assumption on vanishing Chern classes; one also needs to slightly adjust the proof of [La3,Corollary 5]).
In general, one can easily find examples of Higgs-de Rham sequences starting with a locally free sheaf for which other sheaves in the sequence are not reflexive. This causes several complications that we need to overcome. The same error appeared independently in the first version of [SYZ,Theorem 3.10], where the authors claimed existence of a certain map on the open subset of the moduli space of semistable sheaves, parameterizing reflexive sheaves. However, in case of [SYZ,Theorem 3.10], it is not so easy to adjust the arguments adding additional assumptions (this would require at least Lemma 1.11 and repeating the proof of [La3,Theorem 11]). So Theorem 2.2 offers in this case the only available proof.
A new idea appearing in the general proof of Theorem 2.2, when compared to the case D = 0, is that we need to use a nearby cycles functor to prove local freeness of the restriction of E to the irreducible components of D.

Reduction from filtrations to sheaves
In this subsection we show how to reduce the proof of Theorem 2.1 to Theorem 2.2. Before we do that let us prove a few independent lemmas: LEMMA 2.4. Let E be a rank r reflexive sheaf with ∆(E)H n−2 = 0. Assume that E has a filtration M • such that all factors of the filtration are torsion free of rank ≤ p with µ H (Gr M j Let us also assume that each factor has a structure of a slope H-semistable logarithmic Higgs sheaf (or of a slope H-semistable integrable logarithmic connection) on (X , D). Then for all j we have , the Hodge index theorem and Theorem 1.9 imply that H n−2 = 0 for all i and j. By assumption we also have . Now let us recall that by [Kl,Theorem 9.6.3] if B is a divisor on X such that BH n−1 = B 2 H n−2 = 0, then the class of B in the group of divisors on X modulo algebraic equivalence is torsion. Using the cycle map we obtain equality we get the required equalities.
Proof. Equality θ ∧ θ = 0 implies θ * * ∧ θ * * = 0, soθ := θ * * is a logarithmic Higgs field. In the second case we can extend ∇, e.g., in the following way. Let us set U := X − S(E) and let j : U ֒→ X be the corresponding embedding. Then E * * = j * ( j * E) and we can define∇ by∇ = j * ∇ j * . It is easy to see that this is a logarithmic connection. It is integrable, because∇ ∧∇ : Note that for any torsion free sheaf G the line bundles det(G * * ) and det(G) are isomorphic on X − S(G) and S(G) has codimension ≥ 2. So det(G * * ) ≃ det(G) and c 1 (G * * ) = c 1 (G). Now for any subsheaf G ⊂ E * * we have (E ∩ G) * * = G * * as both sheaves are reflexive and equal outside of codimension ≥ 2. So the sheaf E contains subsheaf E ∩ G of the same slope as G. This shows that passing to the reflexivization preserves slope H-semistability (and also slope H-stability).
To prove the second part note that the canonical map E → E * * is injective as by assumption E is torsion free. Let T be the cokernel of this map. Without any loss of generality we can assume that H is very ample. After restricting to a general complete intersection surface Y ∈ |H| ∩ ... ∩ |H|, we get a short exact Since T is supported in codimension ≥ 2, T Y is supported on a finite number of points. We have Since (E * * , θ * * ) is slope H-semistable, by Bogomolov's inequality for logarithmic Higgs bundles (see Theorem 1.9 and Remark 1.
LEMMA 2.6. Replacing H by some its multiple we can assume that any Y ∈ |H| is liftable toỸ ⊂X. Moreover, for any closed point x ∈ U := X − Supp D a general divisor Y ∈ |H| passing through x is smooth and the divisor D + Y is a normal crossing divisor. Then D Y = D ∩Y is a normal crossing divisor on Y and the pair Proof. The first part follows from the proof of [La3,Theorem 11]. Replacing H by its multiple we can also assume that H is very ample. As in the proof of [DH,Theorem 3.1] we can also assume that the subsystem Λ ⊂ |H| consisting of all divisors containing x has x as its scheme-theoretic base locus. If π : X ′ → X is the blow up of x then, replacing H if necessary by its multiple, we can also assume that π * H − E is very ample (see [Ha,Chapter II,Proposition 7.10]). Let {D i } i∈I be the irreducible components of D viewed as reduced closed subschemes of X . Let us set D ′ i = π −1 (D i ) and D ′ = π −1 (D). By Bertini's theorem for any J ⊂ I, general Y ′ ∈ |π * H − E| intersects all irreducible components of j∈J D ′ j along smooth divisors. Then D ′ +Y ′ is a normal crossing divisor on X ′ . By [DH, Theorem 2.1] the image of a general divisor Y ′ ∈ |π * H − E| is smooth and it is a general divisor in Λ. Hence for general Y ∈ Λ, D +Y is a normal crossing divisor on X . Moreover,D +Ỹ ⊂X is its lifting to W 2 (k). This implies that also (Ỹ ,Ỹ ∩D) Proof. The proof is by induction on the dimension n of X . If n = 1 then the assertion follows from the fact that torsion free sheaves on a smooth curve are locally free. Assume that the implication holds for varieties of dimension less than n and let X be of dimension n.
First we consider the case in which each factor of the filtration from Theorem 2.1 has a structure of a slope H-semistable logarithmic Higgs sheaf. Let us write E j for Gr M j E and r j for its rank. Replacing H by its multiple we can assume that is log smooth and liftable to W 2 (k). By Corollary 1.7 applied to each quotient of the filtration M • , for large d and for a general section Y ∈ |dH|, the restriction of each quotient E j := Gr M j E to Y is a slope H Y -semistable logarithmic Higgs sheaf and the restriction E Y is reflexive (here we use Lemma 1.13). Hence by the induction assumption each (E j ) Y is locally free. So by Lemma 1.14 each E j is locally free outside a finite number of points of X .
Since by Lemma 2.4 we have ∆(E j )H n−2 = 0, Theorem 2.2 applied to X implies that all E * * j are locally free. Hence the assumptions of Lemma 1.12 are satisfied and we conclude that E and all quotients E j are locally free. By Theorem 2.2 this implies that which with equality c 1 (E j ) = r j r c 1 (E) finishes the proof of the second part of Theorem 2.1. Now a simple computation of Chern classes shows that we also have Now let us consider the case in which each factor of the filtration M • from Theorem 2.1 has a structure of a slope H-semistable sheaf with an integrable logarithmic connection. The same arguments as above allow us to prove that for general Y as above the restriction (Gr M E) Y is locally free (here we use Remark 1.8 instead of Corollary 1.7). So Gr M E is locally free outside a finite number of points and by Lemma 1.12 it is sufficient to prove that (Gr M E) * * is locally free. Then Gr M E is locally free and we can finish as in the case of logarithmic Higgs sheaves. Let For n = 2 such equalities are clear as we need to check them only for m = 1. Let us recall that by Lemma 2.4 we have ∆(E j )H n−2 = 0. So by Lemma 2.5 E j := (E j ) * * is a slope H-semistable sheaf with an integrable logarithmic connection and we have ∆(Ē j )H n−2 = 0. Theorem 1.4 allows us to construct a filtration S j,• ofĒ j such that the associated graded Gr S jĒ j is a slope H-semistable Higgs sheaf with ∆(Gr S jĒ j )H n−2 = 0. Again using Lemma 2.5, we see that (Gr S jĒ j ) * * satisfies condition (1) of Theorem 2.1 and hence it is locally free.
Note that for general Y as above we have Therefore the Chern classes of (Gr S jĒ j ) Y satisfy condition (3) of Theorem 2.2 and thus (Gr S jĒ j ) Y is locally free. So by Lemma 1.14 the sheaf Gr S jĒ j is locally free outside a finite number of points. Now we can use Lemma 1.12 to conclude that E j is locally free. This proves that (Gr M E) * * = jĒ j is locally free as required.

Local freeness for sheaves
In this subsection we show the proof of Theorem 2.2.
Proof of Theorem 2.2. We prove the required assertion by induction on the dimension n of X . Let us assume that n = 2. Then equivalence of (1) and (2) is obvious since every reflexive sheaf on a smooth surface is locally free. The fact that (2) implies (3) is also obvious as equality in (3) for m = 1 is trivial and for m = 2 it is equivalent to ∆(E) = 0. The fact that (3) implies (1) follows from Lemma 2.5. Now let us assume that n ≥ 3 and equivalence of conditions (1), (2) and (3) holds for varieties of dimension less than n. Replacing H by its multiple we can assume T X (− log D) ⊗ O X (d 0 H) is globally generated and by Lemma 2.6 we can also assume that a general divisor Y ∈ |H| the pair (Y, D Y = D ∩Y ) is log smooth and liftable to W 2 (k).
Proof. For large d and for a general hyperplane section Y ∈ |dH|, by Corollary 1.7 (or Remark 1. Proof. Since E is reflexive for any smooth hypersurface Y ∈ |H| the restriction E Y is torsion free (see Lemma 1.13). Then, possibly replacing H with some its multiple, Lemma 2.6 implies that for every closed point x ∈ U := X − Supp D we can find Y ∈ |H| passing through x such that the pair (Y, D Y = D ∩Y ) is log smooth and liftable to W 2 (k). We can use Theorem 1.6 (or Remark 1.8) to conclude that for any such Y the restriction (3) and our induction assumption implies that E Y is locally free. Then Lemma 1.14 implies that E is locally free at all points of Y . This shows that E is locally free on U . Now let Y be an irreducible component of D. To finish the proof it is sufficient to show that E Y is locally free as then E is locally free along Y by Lemma 1.14. This part requires the results of Subsections 3.1-3.4 (that do not depend on Theorems 2.1 and 2.2). The proof is similar but more complicated than that of Theorem 3.9. As far as possible we will keep the notation from that proof and show the necessary adjustments.
We construct a certain sequence analogous to the canonical Higgs-de Rham sequence of (E, θ ) ((E, ∇)) in the following way. By Theorem 1.3 there exists a decreasing Griffiths transverse filtration N • of E such that the associated graded is a slope H-semistable system of logarithmic Hodge sheaves (in particular,θ 0 is nilpotent). In case of logarithmic connections we use Simpson's filtration S • instead of N • . Then using Lemma 2.5 we define 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 case of logarithmic connections the sequence is the same except that we replace (E, θ ) by (E, ∇). An easy induction shows also that c 1 (E j ) = p j c 1 (E) for all j ≥ 0. Now let us write p m = rs m + q m for some non-negative integers s m and 0 ≤ q m < r. Let us can take only finitely many values. So Theorem 1.2 implies that the family of reflexive slope H-semistable logarithmic Higgs sheaves {(G m , θ G m )} m≥0 is bounded. It follows that the family of sheaves Note that E m is locally free outside a finite number of points and by Lemma 2.5Ē m is isomorphic to E m outside of a closed subset of codimension ≥ 3. So allĒ m are locally free outside of a closed subset of codimension ≥ 3. In particular, (Ē m ) Y is locally free outside of a closed subset of codimension ≥ 2. This, similarly as in proof of Theorem 3.9, allows us to construct an L 0 all sheaves are reflexive and isomorphic on the set where (Ē 1 ) Y is locally free, i.e., outside of a closed subset of codimension ≥ 2 in Y . Now V ′ 0 ⊂ (V 0 ) Y has a filtration induced fromS • Y and the reflexivization of the associated graded is a subsheaf of ((E 1 ) Y ) * * that after intersecting with (E 1 ) Y gives the required submodule.
Repeating the above procedure allows us to construct a sequence Then as in the proof of Theorem 3.9 the boundedness of the family so Lemma 1.11 and our induction assumption show that (E 0 ) Y is locally free. This implies that E Y is locally free as required.
Now we can prove that E is always locally free. Let M • be a Jordan-Hölder filtration of (E, θ ) (or (E, ∇)) and let us set E i = Gr M i (E) and r i = rk E i . Then by Lemma 2.4 we know that ∆(E i )H n−2 = 0 and for all i we have c 1 (E i ) = r i r c 1 (E) in H 2 et (X , Q l ) for l = p. By Theorem 1.6 (Remark 1.8, respectively) for large d and a general smooth hypersurface Y ∈ |dH| the restriction (E i ) Y is a slope H Y -stable logarithmic sheaf. Since E is reflexive by Lemma 1.13 the restriction E Y is also reflexive for general Y . Therefore by the induction assumption E Y is locally free. Moreover, our induction assumption and Lemma 2.7 imply that all the factors (E i ) Y are also locally free. So by Lemma 1.14 all E i are locally free outside a finite number of points.
However, we also know that the logarithmic Higgs sheaf (respectively, the sheaf with an integrable connection) E * * i is slope H-stable and ∆(E * * i )H n−2 = 0. So by Claim 2.7.2 all sheaves E * * i are locally free. Hence we can apply Lemma 1.12 to conclude that E is locally free. This finishes the proof that (1) implies (2).
To finish the proof that (1) implies (3) note that by Theorem 1.3 there exists a decreasing filtration E = N 0 ⊃ N 1 ⊃ ... ⊃ N m = 0 such that θ (N i ) ⊂ N i−1 ⊗ Ω X (log D) (in case of logarithmic connections we use Simpson's filtration) and the associated graded system (E 0 , θ 0 ) of logarithmic Hodge sheaves is slope Hsemistable. Let us recall that by Claim 2.7.1 we already know that ∆ i (E) = 0 for 2 ≤ i < n. Hence ∆ i (E 0 ) = 0 for 2 ≤ i < n and if we take large d and a general divisor Y ∈ |dH| then by Corollary 1.7 the restriction (E 0 ) Y satisfies (3) on Y . So by the induction assumption (E 0 ) Y is locally free, which by Lemma 1.14 implies that E 0 is locally free outside a finite number of points. Since we already know that (1) implies (2), we see that E * * 0 is locally free. Then Lemma 1.12 implies that E 0 is locally free. Now let us consider the canonical Higgs-de Rham sequence starting with  (1) implies (2) it is also locally free. Note also that ∆ i (E m ) = p im ∆(E) = 0 for i < n, so the same argument as in the case of E 0 shows that each sheaf E m is locally free outside a finite number of points. Now we prove by induction on m that E m and V m are locally free. For m = 0 we already know that E 0 = E is locally free and hence V 0 = C −1 (E 0 ) is also locally free. So let us assume that V m−1 is locally free. Then Lemma 1.12 implies that E m is locally free and hence also V m = C −1 (E m ) is locally free, which finishes the induction. Now let us write p m = rs m + q m for some non-negative integers s m and 0 ≤ q m < r. Let us Note also that c 1 (G m ) = q m c 1 (E) can take only finitely many values, so Theorem 1.2 implies that the family of locally free slope H-semistable logarithmic Higgs sheaves {(G m , θ G m )} m≥0 is bounded. In particular, the set {∆ n (G m )} m≥0 = {p nm ∆ n (E)} m≥0 is finite. Hence ∆ n (E) = 0, which finishes the proof of vanishing of ∆ i (E) for all 2 ≤ i ≤ n. Now Lemma 1.11 implies that for all m ≥ 1 . This finishes the proof that (1) implies (3). Clearly (2) implies (1), so it is sufficient to prove that (3) implies (1). Let us consider a rank r ≤ p slope H-semistable logarithmic Higgs sheaf (E, θ ) such that c m (E) = ( r m ) r m c 1 (E) m for all m ≥ 2. By Lemma 2.5 we have ∆(E * * )H n−2 = 0.
Since (1) implies (3) we know that (E * * ,θ ) satisfies c m (E * * ) = ( r m ) r m c 1 (E * * ) m for all m ≥ 1. As in the proof of Lemma 2.5 we see that c 1 (E * * ) = c 1 (E). So our assumptions imply that c m (E * * ) = c m (E) for all m ≥ 1. Since E and E * * have the same rank, the Riemann-Roch theorem implies that the Hilbert polynomials of E and E * * are equal. Let T = E * * /E. Then the short exact sequence shows that the Hilbert polynomial of T is trivial. So T = 0 and E is reflexive. In case of a sheaf with an integrable logarithmic connection the proof of implication (3) ⇒ (1) is exactly the same.
Lemma 2.7 and Theorem 2.2 immediately imply the following corollary: COROLLARY 2.8. Let (E, θ ) be a rank r ≤ p slope H-semistable logarithmic Higgs sheaf. Let us assume that E is reflexive and ∆(E)H n−2 = 0. If (G, θ G ) is a rank s factor in a slope H-Jordan-Hölder filtration of (E, θ ) then it is locally free and for all m ≥ 1 we have The following corollary is a direct generalization [La3,Theorem 11] to the logarithmic case. COROLLARY 2.9. Let (E, θ ) be a rank r ≤ p slope H-semistable logarithmic Higgs sheaf with ch 1 (E)H n−1 = 0 and ch 2 (E)H n−2 = 0. Assume that either E is reflexive or the normalized Hilbert polynomial of E is the same as that of O X . Then (E, θ ) has a filtration whose quotients are locally free slope H-stable logarithmic Higgs sheaves with vanishing Chern classes.
Proof. By Theorem 1.9 we have ∆(E)H n−2 ≥ 0. So by the Hodge index theorem we get Hence we have ∆(E)H n−2 = 0 and c 1 (E) 2 H n−2 = 0. Since c 1 (E)H n−1 = 0 this implies that c 1 (E) = 0 (see proof of Lemma 2.4). If E is reflexive then the corollary follows directly from Corollary 2.8. In the second case we argue as in the proof that (3) implies (1)  Note also that at the time of writing [Ar], Theorem 2.2 was not claimed in the logarithmic case that was used there. In the logarithmic case, even if k =F p and one has vanishing of all Chern classes, the method of proof of local freeness from [LSZ2, Proposition 3.12] does not apply.

Local freeness in characteristic zero
By a standard spreading-out argument Theorem 2.1, Theorem 2.2 and Corollary 2.9 imply the following generalization of [Si,Theorem 2] to the logarithmic case. THEOREM 2.13. Let X be a smooth projective variety defined over a field of characteristic zero and let D be a normal crossing divisor on X . Let H be an ample divisor on X and let (E, θ ) be a slope H-semistable logarithmic Higgs sheaf with ch 1 (E)H n−1 = 0 and ch 2 (E)H n−2 = 0. Then the following conditions are equivalent: 2. E is locally free, 3. the normalized Hilbert polynomial of E is the same as that of O X , 4. E has vanishing rational Chern classes, i.e., c m (E) = 0 in H 2m (X , Q) for all m ≥ 1,

(E, θ ) has a filtration whose quotients are locally free slope H-stable logarithmic Higgs sheaves with vanishing rational Chern classes.
Remark 2.14. The same theorems show that in the above theorem we can replace a logarithmic Higgs sheaf by a sheaf with an integrable logarithmic connection.
Remark 2.15. Let (V, ∇) be a polarized variation of Hodge structures on X − D with unipotent monodromy along the irreducible components of D. Let (Ṽ ,∇) be Deligne's canonical extension of (V, ∇) with nilpotent residues along the irreducible components of D. Then Schmid's nilpotent orbit theorem implies that the Hodge filtration on V extends to a filtration ofṼ with locally free subquotients. Note that it is easy to see thatṼ has vanishing Chern classes in the de Rham cohomology of X (as all residues are nilpotent) and hence it also has vanishing Chern classes in H 2 * (X , Q). Similarly, all the subobjects of (Ṽ ,∇) have vanishing rational Chern classes. In particular, (Ṽ ,∇) is slope semistable. Therefore Theorem 1.4 gives Simpson's filtration such that the associated graded (E, θ ) is slope semistable. Since E has vanishing rational Chern classes, Theorem 2.13 implies that E is locally free. Note that [La2,Corollary 5.6] implies that the associated graded of Simpson's filtration of (Ṽ ,∇) coincides with the associated graded of the filtration obtained by Schmid's theorem. Moreover, if the associated graded (E, θ ) is slope stable then the corresponding filtrations coincide.
Again, using spreading out, Theorem 2.2 implies the following theorem. However, we also give a different proof that deduces it from Theorem 2.13 that was already known in the non-logarithmic case (D = 0). Note also that Corollary 2.9 can be proven in a somewhat simpler way than Theorem 2.2. The difference is that if we follow the proofs of implications (1) ⇒ (2) and (1) ⇒ (3) in Theorem 2.2 under assumptions of Corollary 2.9 then we do not need to consider the family {G m } m≥0 and we can work directly with the family {E m } m≥0 . However, it should be stressed that similar arguments as below (showing that Theorem 2.16 follows from Theorem 2.13) do not allow to deduce Theorem 2.2 from Corollary 2.9 in positive characteristic. This is caused by the use of coverings that usually do not preserve liftability to W 2 (k). Another problem is that such covers are sometimes necessarily inseparable, in which case the pullback does not preserve semistability.
THEOREM 2.16. Let X be a smooth projective variety of dimension n ≥ 2 defined over a field of characteristic zero and let D be a normal crossing divisor on X . Let H be an ample divisor on X and let (E, θ ) be a slope H-semistable logarithmic Higgs sheaf with ∆(E)H n−2 = 0. If E is reflexive then it is locally free and HenceẼ is a slope f * H-semistable logarithmic Higgs sheaf with ch 1 (Ẽ)( f * H) n−1 = 0 and ch 2 (Ẽ)( f * H) n−2 = 0. By Theorem 2.13Ẽ is locally free and it has vanishing Chern classes. Therefore by the flat descent E is also locally free and we have for all m ≥ 2. Using the fact that f induces an injection H 2m (X , Q) → H 2m (X, Q), we get vanishing of ∆ m (E) for all m ≥ 2. Hence by Lemma 1.11 we get equalities r m c m (E) = r m c 1 (E) m . Now let (G, θ G ) be a rank s factor of a slope H-Jordan-Hölder filtration of (E, θ ). Then f * (G, θ G ) ⊗L is an extension of some factors of a slope f * H-Jordan-Hölder filtration of (Ẽ,θ ). In particular, it has a filtration whose quotients are locally free slope f * H-stable logarithmic Higgs sheaves with vanishing Chern classes. It follows that G is locally free, c 1 ( f * G) = −sc 1 (L) = s r c 1 ( f * E) and

Restriction theorem
The following theorem generalizes [La3,Theorem 12] to the logarithmic case and to arbitrary (Y, B).

be a proper morphism of smooth log pairs that has a good lifting to W 2 (k) (see Definition 5.1). Then the induced logarithmic Higgs sheaf
is slope A-semistable for any ample divisor A on Y .
Proof. By Theorem 1.3 we can deform (E, θ ) to a slope H-semistable system of Hodge sheaves (E 0 , θ 0 ). Moreover, by Theorem 2.2 E 0 is locally free. If f * (E 0 , θ 0 ) is semistable then by openness of semistability f * (E, θ ) is also semistable. So without loss of generality one can assume that (E, θ ) is a system of Hodge sheaves. The rest of the proof is the same as that of [La3, Theorem 12] using Theorem 1.5 instead of [La3,Theorem 5]. Here we also need to apply functoriality of the inverse Cartier transform in the logarithmic case (see Theorem 5.4).
Applying the above theorem to iterates of the Frobenius morphism we get the following corollary: COROLLARY 2.18. In the notation of the above theorem assume that (Y, B) = (X , D) and f is the Frobenius morphism. Then E is strongly A-semistable for any ample divisor A on X .
In formulation of [La3,Theorem 12] the author forgot to explicitly state the assumption on existence of a compatible lifting of C and X (even though it was used in the proof). The next example shows that this assumption is really necessary.
Example 2.19. Here we show an example of a smooth projective surface that is liftable to the Witt ring W (k) and a slope semistable Higgs sheaf, which is not semistable after restricting to the normalization of some projective curve on this surface.
Let us consider a smooth complex projective surface X which is a quotient of the product of upper half planes H × H by an irreducible, torsion free, cocompact lattice G in PGL (2, R) × PGL (2, R). Then Ω X = L ⊕ M, where L 2 = M 2 = 0, L and M are strictly nef (see [La2,Lemma 4.5]).
By openness of stability, the reduction of (E ′ , θ ′ ) modulo almost all primes is stable. Again this can be easily seen directly, because ampleness is an open condition and LH > 0 implies an analogous inequality for the reductions. Note also that for almost all reductions, X s lifts to W (k(s)).
Now for a large number of primes (of positive density) the reduction of L is not nef (see [La2,Example 5.6]). For such s there exists an irreducible curve C s such that L s .C s < 0. Let ν s :C s → C s be the normalization. Then ν * s (E ′ s , θ ′ s ) is not semistable because it has degree zero and it contains a Higgs subbundle (ν * s L −1 s , 0) of positive degree. This shows that ν s :C s → X s cannot be compatibly lifted to W 2 (k(s)), even though bothC s and X s can be lifted to W (k(s)).
By the usual spreading-out technique, Theorem 2.17 implies the following corollary.
COROLLARY 2.20. Let X be a smooth projective variety of dimension n defined over an algebraically closed field k of characteristic 0. Let D be a normal crossing divisor on X and let H be an ample divisor on X . Let E be a locally free O X -module with ∆(E)H n−2 = 0. If a logarithmic Higgs sheaf (E, θ ) is slope Hsemistable then for every smooth projective curve C not contained in D and a morphism f : C → X the Higgs bundle f * (E, θ ) is semistable.
Remark 2.21. In the case of complex projective manifolds and D = 0 the above corollary follows from Simpson's correspondence. A rough sketch of proof is as follows. A slope semistable Higgs bundle with vanishing rational Chern classes corresponds to a local system on X . So for any morphism f : C → X we get an induced local system on C. This again corresponds to a slope semistable Higgs bundle on C. By functoriality of Simpson's correspondence this is the pullback of the original Higgs bundle. The general case with ∆(E)H n−2 = 0 can be reduced to the above one by taking End E and using [Si,Theorem 2] (or Theorem 2.13). More precisely, if ∆(E)H n−2 = 0 then c 1 (End E) = 0 and ∆(End E)H n−2 = 0, so (End E, θ End E ) is semistable with vanishing rational Chern classes. Then f * (End E, θ End E ) is semistable, which implies semistability of f * (E, θ ).

Nearby-cycles
The main aim of this section is to understand the restriction of an integrable logarithmic connection (or a logarithmic Higgs sheaf) to the boundary divisor. In case of Hodge structures on complex varieties the analogous problem is realised by the construction of a nearby-cycle functor for the category of real graded-polarized families of mixed Hodge structures (see [Br1,Section 4]). Here we use a different approach that allows us to keep more information about the restrictions. As in [Br1] this construction is related to the standard constructions of a nearby-cycles functor coming back to Grothendieck, Deligne and Saito. In this section we will use some basic facts and definitions related to Lie algebroids for which we refer to [La2].

Nearby-cycles functor
Let X be a smooth projective variety of dimension n defined over an algebraically closed field k. Let D be a simple normal crossing divisor on X and let Y be an irreducible component of D.
Let ı : Y ֒→ X be the canonical embedding. We define a Lie algebroid L Y on Y as the triple (L, [·, ·], α), where L = ı * T X (log D) is a locally free O Y -module with the Lie algebra structure induced from the standard Lie algebra structure on T X and the anchor map α : [La2,Lemma 3.2]).
The usefulness of the above construction comes from the fact that the restriction to Y defines an obvious functor . By an abuse of notation we will often write Let L 0 Y be the trivial Lie algebroid underlying L Y (i.e., we consider the same L = ı * T X (log D) but with zero Lie bracket and zero anchor map). Similarly as above, we get the functor Φ Y : HIG (X , D) → L 0 Y -Mod from the category HIG (X , D) of coherent O X -modules with a logarithmic Higgs field on (X , D) to the category L 0 Y -Mod of coherent O Y -modules with an L 0 Ymodule structure. Note that L 0 Y -Mod is the same as the category of coherent O Y -modules with a Sym • (ı * T X (log D))-module structure. Similarly, as above we will often write (E Y , θ | Y ) to denote Φ(E, θ ).

General monodromy filtrations
Let Y be a smooth projective variety of dimension n defined over an algebraically closed field k. Let L be a smooth Lie algebroid on Y /k and let E be an L-module.
Let N : E → E be a nilpotent endomorphism of L-modules. By [De,Proposition 1.6.1] N induces on E a unique finite increasing filtration M • by L-submodules such that: We call M • the monodromy filtration for the L-module E.
Let us define the j-th primitive part P j (E) of E as the kernel of N j+1 : Gr M j E → Gr M − j−2 E for j ≥ 0 and P j (E) = 0 for j < 0. Then by [De,(1.6

.4)] we have the decomposition into primitive parts
LEMMA 3.

If E is torsion free (as an O Y -module) then all quotients Gr M j E are also torsion free.
Proof. The proof is by induction on the rank of E. If E has rank 1 then N is nilpotent if and only if N = 0, so the filtration is trivial. Now let us assume that the assertion holds for all sheaves of rank less than the rank of E. If N = 0 the assertion is trival, so we can assume that N = 0. Since for j ≥ 0 the map N j induces an isomorphism [De,Corollaire (1.6.6)] the associated graded of the filtration induced by M • on ker N satisfies But ker N ⊂ E is torsion free and since N is nilpotent, the rank of ker N is less than the rank of E. So by the induction assumption all quotients Gr M − j (ker N) are torsion free. Hence all P j (E) are torsion free and by the decomposition (3) all Gr M j E are also torsion free. Now let us fix an ample divisor H on Y . If an L-module E is slope Hsemistable then we always assume that it is torsion free as an O Y -module. The following lemma proves that the monodromy filtration (or the filtration by primitive cohomology) of a slope H-semistable L-module can be always refined to a Jordan-Hölder filtration.

LEMMA 3.2. Let E be a slope H-semistable L-module. Then every quotient Gr M j E of the monodromy filtration M
Proof. The proof is by induction on the rank of E. For rank 1 the assertion is clear so assume that it holds for all sheaves of rank less than the rank of E.
Let d be the largest integer such that

Residue maps
Let X be a smooth projective variety of dimension n defined over an algebraically closed field k. Let D be a simple normal crossing divisor on X and let Y be an irreducible component of D. We can write D = D ′ + Y for some divisor D ′ which does not contain Y . In the following we denote D ′ by D −Y and set Note that L Y (see Subsection 3.1) is equipped with the canonical map Res : given by the Poincaré residue. Using it for any L Ymodule E we can define the residue endomorphism Res E as a composition It is easy to check that Res E is an endomorphism of L Y -modules. In the same way we can define the residue endomorphism of an L 0 Y -module. Let L Y -Mod 0 (L 0 Y -Mod 0 ) be the full subcategory of L Y -Mod (L 0 Y -Mod ) containing as objects all L Y -modules E (L 0 Y -modules, respectively) with Res E = 0.
Similarly, let L Y -Mod nil (L 0 Y -Mod nil ) be the full subcategory of L Y -Mod (L 0 Y -Mod ) containing as objects all L Y -modules E (L 0 Y -modules, respectively) that have nilpotent residue Res E . Proof. The short exact sequence shows that an L Y -module E with Res E = 0 gives rise to a canonically defined is an element of MIC (Y, D Y ) then ∇ defines an integrable d Ω L Y -connection, so we get an L Y -module V with Res V = 0. If E is an L 0 Y -module with Res E = 0 then the same argument shows that E is a logarithmic Higgs sheaf on (Y, D Y ). This shows the first part of the lemma. Now let us assume that E is an L Y -module with nilpotent N = Res E . Let W • be the corresponding monodromy filtration (in the category of L Y -modules).

One can easily check that this map is O Y -linear and it factors through the restriction map
that we call the residue map of E along D i . In the same way we can define the residue maps along D i for any L 0 Y -module E.
Remark 3.4. Let (V, ∇) be an object of MIC(X , D) and let E = Ψ Y (V, ∇) be the corresponding L Y -module. Then the residue map Res E : E → E coincides with the residue map Res Y ∇ : coincides with the restriction of the residue map Res D i ∇ :

Compatibility of the Cartier transform with monodromy filtrations
Let X be a smooth projective variety of dimension n defined over an algebraically closed field k and let D be a simple normal crossing divisor on X . Let Y be an irreducible component of D.
) be the total space of the normal bundle of ı : Y ֒→ X and let π : Z → Y be the canonical projection. Let s : Y → Z be the zero section and let Y 0 be its image.
LEMMA 3.5. Let us set D Z = Y 0 + π −1 (D Y ). The short exact sequence Proof. Let us recall that the extension class of . Hence by [Wa,Proposition 3.3] the pull back of the above sequence to Z induces Let {D i } be the divisors corresponding to the irreducible components of D − Y . Now the required assertion follows from the following standard exact sequences: An alternative proof of the lemma can be obtained, e.g., by directly making local calculation and checking equality of the corresponding gluing conditions (cf. proof of [Wa,Proposition 3.3]).
Let (V, ∇) be a coherent O X -module with an integrable logarithmic connection ∇ : V → V ⊗ Ω X (log D). After restricting to Y we see that V Y acquires an integrable Ω L Y -connection. After further pull back to Z we get an induced integrable logarithmic connection The same construction allow us to associate to a logarithmic Higgs sheaf (E, θ ) on (X , D), a logarithmic Higgs sheaf (π * E Y , θ ′ ) on (Z, D Z ). Note that if θ is nilpotent then θ ′ is also nilpotent.
Remark 3.6. One could naively hope that one can work with logarithmic connections on projective varieties by pulling back the defines a different extension class. This can be seen by computing the extension class of both sequences after restricting to Y ∞ . This forces us to deal with nonprojective varieties, where the difficulty is that one cannot directly apply Theorem 2.2.
Let us assume that the base field k has characteristic p and (X , D) is liftable to W 2 (k). Let us fix a lifting (X,D). This lifting induces a lifting (Ỹ ,D Y ) of (Y, D Y ) to W 2 (k) and also a compatible lifting (Z,D Z ) of (Z, D Z ) to W 2 (k).
The following lemma is functoriality of Cartier transforms in a situation that is not covered by Theorem 5.4 (we do not even have a map (Z, D Z ) → (X , D)).
LEMMA 3.7. Let (E, θ ) be a reflexive logarithmic Higgs sheaf on (X , D) with a nilpotent Higgs field of level less or equal to p − 1.
is commutative.
Proof. Since E is reflexive and Y is smooth, E Y is torsion free. Since π is flat, π * E Y is also torsion free, so we can apply C −1 (Z,D Z ) to (π * E Y , θ ′ ). We will use the notation introduced in proof of Theorem 5.4.
Checking equality of gluing conditions is similar and left to the reader. Finally, note that since the isomorphism is functorial with respect to open embeddings V α ⊂ Z, it is sufficient to check the commutativity of the diagram only locally. In the local situation this follows easily from local equalities Res Y ∇| U α = F * (Res Y θ | U α ).
Let (E, θ ) be a logarithmic Higgs sheaf on (X , D) with a nilpotent Higgs field. Let M • be the monodromy filtration for Res Y θ . Then each quotient Gr M i E Y is endowed with a nilpotent logarithmic Higgs field θ M i on (Y, D Y ). Let (V, ∇) be an object of MIC (X , D). Assume that the residue Res Y (∇) is nilpotent and let W • be the monodromy filtration for Res Y ∇. Then each quotient PROPOSITION 3.8. Let (E, θ ) be a reflexive logarithmic Higgs sheaf on (X , D) with a nilpotent Higgs field of level less or equal to p−1 and let (V, ∇) = C −1 (X,D) (E, θ ). Let M • be the monodromy filtration for Res Y θ and let W • be the monodromy filtration for Res Y ∇. Then (Gr M i E Y , θ M i ) is a torsion free logarithmic Higgs sheaf on (Y, D Y ) with a nilpotent Higgs field of level less or equal to p − 1 and we have Proof. Note that π * M • is a filtration of (π * E Y , θ ′ ) by logarithmic Higgs submodules on (Z, D Z ). Moreover, quotients of this filtration are logarithmic Higgs modules on (Z, π −1 (D Y )). Similarly, π * W • is a filtration of (π * V Y , ∇ ′ ) by integrable logarithmic connections on (Z, D Z ) and the quotients are objects of MIC (Z, π −1 (D Y )). Lemma 3.7 and uniqueness of the monodromy filtrations imply that where ∇ ′ i and θ ′ i denote the restriction of ∇ ′ and θ ′ to the corresponding subsheaves. But this implies that Pulling back this equality by the zero section s : (Y, D Y ) → (Z, π −1 (Y )) and using functoriality of the Cartier transform, we get the required assertion.

Nearby-cycles in positive characteristic
Let X be a smooth projective variety of dimension n defined over an algebraically closed field k of characteristic p and let D be a simple normal crossing divisor on X . In this subsection we assume also that (X , D) is liftable to W 2 (k) and we fix a lifting (X,D).
Let H be an ample divisor on X and let us fix a class µ ∈ H 2 et (X , Q l ) for some l = p. We define the category MinHIG µ (X , D) of minimally semistable Higgs sheaves of slope µ as the full subcategory of the category HIG (X , D) of logarithmic Higgs sheaves on (X , D), whose whose objects are pairs (E, θ ), where • E is a locally free O X -module of rank r ≤ p, • c 1 (E) = rµ (i.e., the slope of E is equal to µ), • ∆(E)H n−2 = 0 (i.e., E has a minimal possible discriminant).
By Theorem 2.2 for any object (E, θ ) of MinHIG µ (X , D) we have c m (E) = r m µ m for all m ≥ 1. Taking in Theorem 2.17 as f identity, we see that the above category does not depend on the choice of polarization H.
Unfortunately, MinHIG µ (X , D) is not abelian as it does not contain direct sums of objects. However, by Theorem 2.2 it satisfies all other axioms of the abelian category. In particular, it contains kernels, images and cokernels (cf. [La4,Corollary 5]) and any morphism in this category admits a canonical decomposition.
Let Y be an irreducible component D and let us fix a class η ∈ H 2 et (Y, Q l ) for some l = p. Let us define the category Min-L 0 Y -Mod η as the full subcategory of the category L 0 Y -Mod (defined in Subsection 3.1), whose objects E satisfy the following conditions: • as an O Y -module E is locally free of rank r ≤ p, Replacing in the above definition L 0 Y by L Y one can also define the category Min-L Y -Mod η .
Let (E, θ ) be an object of MinHIG µ (X , D) and let us first assume that θ is nilpotent. Let (V, ∇) = C −1 (X,D) (E, θ ). Let us denote by S • (decreasing) Simpson's filtration on (V, ∇) and let (E 1 = Gr S (V ), θ 1 ) be the associated system of Hodge sheaves. Let . Then by Lemma 3.7 By Theorem 2.2 E 1 is locally free, so all S j are locally free. Thus we get an This filtration induces on V ′ a filtration that we denote by abuse of notation also by S • Y . In this way we get an L 0 Now let us consider the canonical Higgs-de Rham sequence starting with (E 0 , θ 0 ) = (E, θ ) (see Theorem 1.5) Since (E, θ ) is an object of MinHIG µ (X , D), Theorem 2.2 implies that (E m , θ m ) is an object of MinHIG p m µ (X , D) for all m ≥ 0. So we can apply the above described procedure at all levels of the Higgs-de Rham sequence. This allows us to construct Now we write p m = rs m + q m for some non-negative integers s m and 0 ≤ q m < r. Let us set (G m , θ G m ) := (E m , θ m ) ⊗ det E −s m . As in proof of Lemma 2.7.3 we see that the family of locally free slope H-semistable logarithmic Higgs sheaves {(G m , θ G m )} m≥0 is bounded. This implies that the family of sheaves {E m ⊗ det E −s m } m≥0 is bounded and hence the family of their restrictions to Y is bounded. Therefore the numbers Now let us consider the general case. Let (E, θ ) be an object of MinHIG µ (X , D). By Theorem 1.3 there exists a decreasing Griffiths transverse filtration N • of E such that the associated graded (E 0 , θ 0 ) := Gr N (E, θ ) is a slope H-semistable system of logarithmic Hodge sheaves (in particular, θ 0 is nilpotent). Moreover, by Theorem 2.2 E 0 is locally free. By the first part of the proof we know that the L 0 Let Min-L 0 Y -Mod η nil the the full subcategory of Min-L 0 Y -Mod η , whose objects . We also know that Gr M j E is endowed with a natural logarithmic Higgs field θ M j on (Y, D Y ), coming from the L 0 Y -action and triviality of the residue of Gr M j E. Since any logarithmic Higgs subsheaf of (Gr M j E, θ j ) has a canonical structure of an L 0 Ysubmodule, the pair (Gr M j E, θ j ) is slope H Y -semistable. Therefore by Theorem 2.1 all quotients Gr M j E are locally free with c m (Gr M j COROLLARY 3.11. Any element in the essential image of the functor has a filtration whose quotients are elements of MinHIG µ Y (Y, D Y ). D). In the last part of the proof of Theorem 3.9 we showed that there exists a decreasing Griffiths transverse filtration N • of (E, θ ) such that Φ 0 Y (Gr N (E, θ )) is an object of Min-L 0 Y -Mod η nil . In particular, by Theorem 3.10 every quotient in the monodromy filtration of Φ 0 Y (Gr N (E, θ )) is an element of MinHIG µ Y (Y, D Y ). The proof finishes by remarking that N • induces an analogous filtration on M.

Semistability and semipositivity
In this section we prove Theorem 0.2 and show some of its applications mentioned in the introduction.

General results on semistability
Unless otherwise stated, in this subsection (X , D) stands for a smooth log pair defined over an algebraically closed field kof positive characteristic. We assume that the pair (X , D) is liftable to W 2 (k) and we fix its lifting (X,D).
Let C be a smooth projective curve and let ν : C → X be a separable morphism. Let D ′ be the sum of irreducible components of D that do not contain ν(C) and let Definition 4.1. We say that ν : C → (X , D) is strongly liftable to W 2 (k), if there exists a good liftingν : In the above definition we write ν : C → (X , D) to keep in mind that being strongly liftable to W 2 (k) depends not only on ν : C → X but also on the choice of the normal crossing divisor D (in fact, it also depends on the choice of lifting (X.D) of (X , D)).
Proof. The proof is by induction on the dimension of X . In dimension n = 1 the required assertion follows from Theorem 2.17, so let us assume that n ≥ 2. As in the proof of Theorem 3.9 there exists a decreasing Griffiths transverse filtration N • of E such that the associated graded (E 0 , θ 0 ) := Gr N (E, θ ) is an object of MinHIG µ (X , D) with nilpotent θ 0 . Since Gr ν * N (ν * E, ν * θ ) = (ν * E 0 , ν * θ 0 ), by openness of semistability, if the Sym • ν * T X (log D)-module ν * E 0 is semistable then the Sym • ν * T X (log D)-module ν * E is semistable. So in the following we can assume that θ is nilpotent.
If ν(C) is not contained in D then ν * (E, θ ) is semistable by Theorem 2.17 (for this we do not need nilpotence of θ ). Since any Sym • ν * T X (log D)-submodule of ν * E defines a Higgs subsheaf of ν * (E, θ ), this implies that ν * E is semistable as a Sym • ν * T X (log D)-module.
If ν(C) is contained in D then we choose an irreducible component Y of D containing ν(C) and as before we set D Y = (D − Y )| Y . By definition of strong liftability, the morphism C → (Y, D Y ) is also strongly liftable to W 2 (k). By Theorem 3.9 nil . By Theorem 3.10 we know that E ′ has a filtration M • whose associated graded E ′′ = Gr M (E ′ ) is an element of MinHIG µ Y (Y, D Y ). Hence by the induction assumption the induced Y -submodules and the associated graded is equal to ν * E ′′ (here we use the fact that E ′′ is locally free). So by openness of semistability ν * E ′ is semistable as a ν * L 0 Y -module. This is equivalent to saying that ν * E is semistable as a Sym • ν * T X (log D)-module, which finishes the induction step.
The last part of the theorem follows from the fact that ker ν * θ with trivial action is a Sym • ν * T X (log D)-submodule of ν * E. COROLLARY 4.3. Let (E, θ ) be an object of MinHIG 0 (X , D). If E ′ is a locally split subsheaf of E contained in the kernel of θ then its dual (E ′ ) * is W 2 -nef.
Proof. If E ′ is a locally split subsheaf of E then for any smooth projective curve C and any morphism ν : C → X , ν * E ′ is a subsheaf of ν * E. Moreover, the image of ν * (ker θ ) in ν * E is contained in ker ν * θ , so ν * E ′ ⊂ ker ν * θ . So if ν is separable and liftable to W 2 (k), then by the above theorem any subsheaf of ν * E ′ has a nonpositive degree. Passing to the dual of ν * E ′ , we get the required assertion.
A standard spreading out arguments show that Theorem 4.2 implies the following result: THEOREM 4.4. Let (E, θ ) be a locally free logarithmic Higgs sheaf on a smooth log pair (X , D) defined over an algebraically closed field of characteristic zero. Assume that it has vanishing Chern classes in H 2 * (X , Q) and it is slope semistable with respect to some ample polarization. Let ν : C → X be any morphism from some smooth projective curve. Then the induced Sym • ν * T X (log D)-module ν * E is semistable. In particular, if G is a subsheaf of the kernel of ν * θ : Remark 4.5.
1. For the first part of Theorem 4.4 one can replace the assumption that E has vanishing Chern classes with assumption that r i c m (E) = r m (c 1 (E)) m for all m ≥ 2 in H 2 * (X , Q).
2. In Theorem 4.4 the assertion holds if we replace curve C by any smooth polarized variety. This immediately follows from the fact that semistability on a general complete intersection curve implies semistability on the original variety.
3. A posteriori one can see that it is possible to obtain proof of the above theorem without passing to positive characteristic. In case (E, θ ) comes from a real graded-polarized family of mixed Hodge structures it is possible to use Mochizuki's version of Simpson's correspondence to adapt Brunebarbe's proof [Br1,Theorem 4.5] to obtain the above theorem. This strategy can be also generalized to deal with arbitrary systems of logarithmic Hodge bundles. COROLLARY 4.6. Let (E, θ ) be a locally free logarithmic Higgs sheaf on a smooth log pair (X , D) defined over an algebraically closed field of characteristic zero. Assume that it has vanishing Chern classes in H 2 * (X , Q) and it is slope semistable with respect to some ample polarization. If E ′ is a locally split subsheaf of E contained in the kernel of θ then its dual (E ′ ) * is nef.

Geometric applications
In this subsection we give several geometric applications of Corollary 4.3 in more or less increasing degree of generality showing how to adjust some arguments. We fix the following notation. Let X and Y be smooth projective varieties defined over an algebraically closed field k of characteristic p and let f : X → Y be a surjective k-morphism of relative dimension d. Moreover, i and j are arbitrary non-negative integers.
COROLLARY 4.7. Assume that f is smooth d < p and there exists a liftingf : X →Ỹ of f to W 2 (k). Then (R i f dR * O X , ∇ GM ), where ∇ GM is the Gauss-Manin connection, is a locally free semistable sheaf with an integrable connection and vanishing Chern classes. In particular, R j f * ω X/Y is a W 2 -nef locally free sheaf on Y .
Proof. By [OV,Theorem 4.17] we have a canonical isomorphism Proof. Again the proof is the same as that of Corollary 4.7, except that now one needs to use [Il,Theorem 4.7] and check that the corresponding result describes an isomorphism Assumptions of this theorem are satisfied due to [Il,Corollary 2.4] and our assumption p > d + dimY . We leave checking cumbersome details to the interested reader.
In characteristic zero, the above result is almost the same as [Kw,Theorem 5].
Remark 4.11. One can also combine Corollaries 4.9 and 4.10 using [Il,4.22]). It is also possible to further generalize these results and deal with push-forwards of Fontaine modules as in [OV,Theorem 4.17] and the corresponding log versions.

Appendix: functoriality of the inverse Cartier transform
In this appendix we prove the functoriality of the inverse Cartier transform. In the non-logarithmic case functoriality follows from [OV,Theorem 3.22]. Unfortunately, although it seems very likely that an analogue of this result holds in the logarithmic case, this part of their paper was never generalized. In the following instead of dealing with a general theory that would demand a lot of space and additional notation, we deal only with the simple cases used in the paper. Instead of using the general framework of [Sc] that follows [OV], we use an explicit description of the Ogus-Vologodsky correspondence provided in [LSZ] and [LSYZ, Appendix].
Let k be an algebraically closed field of positive characteristic and let f : (Y, B) → (X , D) be a k-morphism of smooth log pairs over k.
Definition 5.1. We say that f has a good lifting to W 2 (k) if f lifts to a morphism of smooth log pairsf : (Ỹ ,B) → (X,D) over W 2 (k) such that locally in theétale topology onX,f admits compatible liftings of the Frobenius morphisms, i.e., we can coverX with images ofétale W 2 (k)-morphismsŨ →X andỸ with images of etale W 2 (k)-morphismsṼ →Ỹ so that 1. there existsF U :Ũ →Ũ lifting the Frobenius morphism F U , where U = U ⊗ W 2 (k) k, so thatF −1 U (D) = pD, 2. there existsF V :Ṽ →Ṽ lifting the Frobenius morphism F V , where V = V ⊗ W 2 (k) k, so thatF −1 U (B) = pB, 3. there existsf V :Ṽ →Ũ liftingf such that the diagram In this case we say thatf is a good lifting of f to W 2 (k).
Clearly, iff is an open embedding then it is a good lifting. Similarly, a composition of good liftings is a good lifting. It is also easy to see that the standard Frobenius morphism given by raising elements to their p-th power gives the following proposition: PROPOSITION 5.2. Assume f lifts to a morphism of smooth log pairsf : (Ỹ ,B) → (X,D) over S = SpecW 2 (k) such that locally in theétale topology onX,f is a composition of products of S-morphims of the following type: where a i j are some non-negative integers.
Thenf is a good lifting of f to W 2 (k).
Remark 5.3. It is easy to see that any log-smooth liftingf of f to W 2 (k) is a good lifting. One can also see that almost every reduction of a morphism of smooth log pairs from characteristic zero to positive characteristic gives rise to a good lifting. For example, in the caseB = 0 andD = 0 one can decompose any morphism of smooth schemes over an algebraically closed field into a composition of a closed embedding and a smooth morphism. A smooth closed subvariety of a smooth variety is locally in theétale topology a product of maps of type 2 and a smooth morphism is locally in theétale topology a product of maps of type 1.
Let HIG lf ≤p−1 (X , D) be the full subcategory of HIG (X , D) consisting of locally free logarithmic Higgs sheaves with nilpotent Higgs field of level less or equal to p − 1. Let MIC ≤p−1 (X , D) be the full subcategory of MIC (X , D) consisting of O X -modules with an integrable logarithmic connection whose logarithmic pcurvature is nilpotent of level less or equal to p − 1 and the residues are nilpotent of order less than or equal to p. Proof.
Step 1. Let us first assume that there exist global compatible logarithmic liftings of the Frobenius morphism on X and Y , i.e., 1. there existsF X :X →X lifting the Frobenius morphism F X so that The first condition implies that there exists a uniquely defined ζ X such that the diagramF * X Ω 1X (logD) is commutative. The second condition gives ζ Y with a similar diagram for (Ỹ ,B).
The third condition shows that we have a commutative diagram There exist an O X -linear map h X 12 such that the following diagram is commutative