Semistable Modules over Lie Algebroids

in Positive Characteristic

Adrian Langer^{1}

Received: November 13, 2013 Revised: February 18, 2014

Communicated by Takeshi Saito

Abstract. We study Lie algebroids in positive characteristic and moduli spaces of their modules. In particular, we show a Langton’s type theorem for the corresponding moduli spaces. We relate Langton’s construction to Simpson’s construction of gr-semistable Griffiths transverse filtration. We use it to prove a recent conjecture of Lan-Sheng-Zuo that semistable systems of Hodge sheaves on liftable varieties in positive characteristic are strongly semistable.

2010 Mathematics Subject Classification: 14D20, 14G17, 17B99 Keywords and Phrases: Lie algebroids, Langton’s theorem, sheaves with connection, Higgs sheaves, positive characteristic

Introduction

In this paper we give a general approach to relative moduli spaces of modules over Lie algebroids. As a special case one recovers Simpson’s “non-abelian Hodge filtration” moduli space (see [Si4] and [Si5]). This allows to consider Higgs sheaves and sheaves with integrable connections at the same time as objects corresponding to different fibers of the relative moduli space of modules over a deformation of a Lie algebroid over an affine line. A large part of the paper is devoted to generalizing various facts concerning vector bundles with connections to modules over Lie algebroids. In particular, we introduce restricted Lie algebroids, which generalize Ekedahl’s 1-foliations [Ek]. In positive characteristic we define a p-curvature for modules over restricted Lie algebroids. This leads to a deformation of the morphism given by p-curvature on the moduli space of modules to the Hitchin morphism corresponding to the trivial Lie algebroid structure. In the special case of bundles with connections on curves this deformation was already studied by Y. Laszlo and Ch. Pauly [LP]. We prove Langton’s type theorem for the moduli spaces of modules over Lie algebroids. We compare it via Rees’ construction with Simpson’s inductive construction of gr-semistable Griffiths transverse filtration (see [Si5]), concluding that the latter must finish. This leads to the main application of our results. Namely, we obtain a canonical gr-semistable Griffiths transverse filtration on a module over a Lie algebroid. This implies a recent conjecture of Lan-Sheng-Zuo that semistable systems of Hodge sheaves on liftable varieties in positive characteristic are strongly semistable. The rank 2 case of this conjecture was proven in [LSZ], the rank 3 case in [Li]. Recently, independently of the author Lan, Sheng, Yang and Zuo [LSYZ] also proved the Lan-Sheng-Zuo conjecture using a similar approach. However, they give a different proof that Simpson’s inductive construction must finish. They also obtain a slightly weaker result proving their conjecture only for an algebraic closure of a finite field.

1Author’s work was partially supported by Polish National Science Centre (NCN) contract number 2012/07/B/ST1/03343.

Documenta Mathematica 19 (2014) 509–540

The results of this paper are used in [La3] to prove Bogomolov’s type inequality for Higgs sheaves on varieties liftable modulo p^{2}.

0.1 Notation

If X is a scheme and E is a quasi-coherent O_{X}-module then we set E^{∗} = H om_{O_{X}}(E,O_{X}) and 𝕍(E) = Spec(S^{•}E). Let S be a scheme of characteristic p (i.e., O_{S} is an 𝔽_{p}-algebra). By F_{S}^{r} : S → S we denote the r-th absolute Frobenius morphism of S which corresponds to the p^{r}-th power mapping on O_{S}. If X is an S-scheme, we denote by X^{(1/S)} the fiber product of X and S over the (1-st) absolute Frobenius morphism of S. The absolute Frobenius morphism of X induces the relative Frobenius morphism F_{X/S}: X → X^{(1/S)}.

Let X be a projective scheme over some algebraically closed field k. Let O_{X}(1) be an ample line bundle on X. For any coherent sheaf E on X we define its Hilbert polynomial by P(E)(n) = χ(X,E(n)) for n ∈ ℤ. If d is the dimension of the support of E then we can write

P(E)(n) = \frac{r(E)n^{d}}{d!}+ lower order terms in n.

The (rational) number r = r(E) is called the generalized rank of E (note that if X is not integral then the generalized rank of a sheaf depends on the polarization). The quotient p(E) = \frac{P(E)}{r(E)}is called the normalized Hilbert polynomial of E. In case X is a variety then for a torsion free sheaf E the generalized rank r(E) is a product of the degree of X with respect to O_{X}(1) and of the usual rank. If X is normal and E is a rank r torsion free sheaf on X then we define the slope μ(E) of E as the quotient of the degree of detE = (⋀^{r}E)^{∗∗} with respect to O_{X}(1) by the rank r. In some cases we consider generalized slopes defined with respect to a fixed 1-cycle class, coming from a collection of nef divisors on X.

Let us recall that E is slope semistable if for every subsheaf E′⊂ E we have μ(E′) ≤ μ(E).

1 Moduli spaces of modules over sheaves of rings of differential operators

In this section we recall some definitions and the theorem on existence of moduli spaces of modules over sheaves of rings of differential operators. This combines the results of Simpson [Si2] with the results of [La1] and [La2]. Let S be a locally noetherian scheme and let f : X → S be a scheme of finite type over S. A sheaf of (associative and unital) O_{S}-algebras A on X is a sheaf A on X of (possibly non-commutative) rings of O_{X}-bimodules such that the image of f^{−1} O_{S} → A is contained in the center of A. Let us recall after [Si2] that a sheaf of rings of differential operators on X over S is a sheaf Λ of O_{S}-algebras on X, with a filtration Λ_{0} ⊂ Λ_{1} ⊂ ... by subsheaves of abelian subgroups satisfying the following properties:

1. Λ = ⋃ _{i=0}^{∞} Λ_{i} and Λ_{i}· Λ_{j} ⊂ Λ_{i+j} ,

2. the image of O_{X} → Λ is equal to Λ_{0},

3. the left and right O_{X}-module structures on Gr_{i}(Λ) := Λ_{i}/Λ_{i−1} coincide and the O_{X}-modules Gr_{i}(Λ) are coherent,

4. the sheaf of graded O_{X}-algebras Gr(Λ) := ⨁ _{i=0}^{∞} Gr_{i}(Λ) is generated in degree 1, i.e., the canonical graded morphism from the tensor O_{X}-algebra T^{•}Gr_{1}(Λ) of Gr_{1}(Λ) to Gr(Λ) is surjective.

Note that in positive characteristic, the sheaf of rings of crystalline differential operators (see [BMR] or Subsection 2.2) is a sheaf of rings of differential operators, but the sheaf of rings of usual differential operators is not as it almost never is generated in degree 1.

Assume that S is a scheme of finite type over a universally Japanese ring R. Let f : X → S be a projective morphism of R-schemes of finite type with geometrically connected fibers and let O_{X}(1) be an f-very ample line bundle. Let Λ be a sheaf of rings of differential operators on X over S. A Λ-module is a sheaf of (left) Λ-modules on X which is quasi-coherent with respect to the induced O_{X}-module structure. Let T → S be a morphism of R-schemes with T locally noetherian over S. Let us set X_{T} = X ×_{S}T and let p be the projection of X_{T} onto X. Then Λ_{T} = O_{X_{T}}⊗_{p^{−1} O_{X}} p^{−1} Λ has a natural structure of a sheaf of rings of differential operators on X_{T} over T. Moreover, if E is a Λ-module on X then the pull back E_{T} = p^{∗}E has a natural structure of a Λ_{T}-module. Note that if E is a Λ-module and E′⊂ E is a quasi-coherent O_{X}-submodule such that Λ_{1}·E′⊂ E′then E′has a unique structure of Λ-module compatible with the Λ-module structure on E (i.e., such that E′is a Λ-submodule of E).

Let Y be a projective scheme over an algebraically closed field k (with fixed polarization) and let Λ_{Y} be a sheaf of rings of differential operators on Y. Let E be a Λ_{Y}-module which is coherent as an O_{Y}-module. E is called Gieseker (semi)stable if it is of pure dimension as an O_{Y}-module (i.e., all its associated points have the same dimension) and for any Λ_{Y}-submodule F ⊂ E we have inequality p(F) < p(E) (p(F) ≤ p(E), respectively) of normalized Hilbert polynomials. Every Gieseker semistable Λ_{Y}-module E has a filtration 0 = E_{0} ⊂ E_{1} ⊂ ... ⊂ E_{m}= E by Λ_{Y}-submodules such that the associated graded ⊕_{i=0}^{m} E_{i}/E_{i−1} is a Gieseker polystable Λ_{Y}-module (i.e., it is a direct sum of Gieseker stable Λ_{Y}-modules with the same normalized Hilbert polynomial). Such a filtration is called a Jordan–Hölder filtration of this Λ_{Y}-module.

Now let us go back to the relative situation, i.e., Λ on X over S (over R). A family of Gieseker semistable Λ-modules on the fibres of p_{T} : X_{T} = X ×_{S} T → T is a Λ_{T}-module E on X_{T} which is T -flat (as an O_{X_{T}}-module) and such that for every geometric point t of T the restriction of E to the fibre X_{t} is pure and Gieseker semistable as a Λ_{t}-module. We introduce an equivalence relation ∼ on such families by saying that E ∼ E′if and only if there exists an invertible O_{T}-module L such that E′≃ E ⊗ p_{T}^{∗}L. Let us define the moduli functor

\underline{M}^{Λ}(X/S,P) : (Sch/S)^{o} → Sets

from the category of locally noetherian schemes over S to the category of sets by

\underline{M}^{Λ}(X/S,P)(T) = \{ \begin{matrix}∼ equivalence classes of families of Gieseker \\ semistable Λ-modules on the fibres of X_{T} → T, \\ which have Hilbert polynomial P. \end{matrix}\} .

Then we have the following theorem summing up the results of Simpson and the author (see [Si2, Theorem 4.7], [La1, Theorem 0.2] and [La2, Theorem 4.1]).

Theorem 1.1. Let us fix a polynomial P. Then there exists a quasi-projective Sscheme M^{Λ}(X/S,P) of finite type over S and a natural transformation of functors

φ : \underline{M}^{Λ}(X/S,P) → Hom_{S}(·,M^{Λ}(X/S,P)),

which uniformly corepresents the functor \underline{M}^{Λ}(X/S,P). For every geometric point s ∈ S the induced map φ(s) is a bijection. Moreover, there is an open scheme M^{Λ,s}(X/S,P) ⊂ M^{Λ}(X/S,P) that universally corepresents the subfunctor of families of geometrically Gieseker stable Λ-modules.

In general, for every locally noetherian S-scheme T we have a well defined morphism M^{Λ}(X/S,P) ×_{S}T → M^{Λ_{T}} (X_{T}/T,P) which is a bijection of sets if T is a geometric point of S. Let us recall that a scheme M^{Λ}(X/S,P) uniformly corepresents \underline{M}^{Λ}(X/S,P) if for every flat base change T → S the fiber product M^{Λ}(X/S,P) ×_{S} T corepresents the fiber product functor Hom_{S}(·,T) ×_{Hom_{S}(·,S)}\underline{M}^{Λ}(X/S,P).

2 Lie algebroids

2.1 Lie algebroids and de Rham complexes

Let f : X → S be a morphism of schemes. A sheaf of O_{S}-Lie algebras on X is a pair (L,[·,·]_{L}) consisting of a (left) O_{X}-module L (which is an f^{−1} O_{S}-bimodule) with a morphism of f^{−1} O_{S}-modules [·,·]_{L}: L ⊗_{f^{−1} O_{S}} L → L, which is alternating and which satisfies the Jacobi identity. A homomorphism of sheaves of O_{S}-Lie algebras on X is an O_{X}-linear morphism L → L′which preserves the Lie bracket. As usual for x ∈ L(U) we define adx: L(U) → L(U) by (adx)(y) = [x,y]_{L}.

Let T_{O_{S}}(L) = ⨁ _{n≥0} \overbrace{L ⊗_{f^{−1} O_{S}} ... ⊗_{f^{−1} O_{S}} L}^{n} be the tensor algebra of L over f^{−1} O_{S} (it is a non-commutative f^{−1} O_{S}-algebra). Let us recall that the universal enveloping algebra U_{O_{S}}(L) of a Lie algebra sheaf (L,[·,·]_{L}) is defined as the quotient of T_{O_{S}}(L) by the two-sided ideal generated by x ⊗ y − y ⊗ x − [x,y]_{L} for all local sections x,y ∈ L. The most important example of a sheaf of O_{S}-Lie algebras on X is the relative tangent sheaf T_{X/S}= Der_{O_{S}}(O_{X},O_{X}) with a natural bracket given by [D_{1},D_{2}] = D_{1}D_{2}−D_{2}D_{1} for local O_{S}-derivations D_{1}, D_{2} of O_{X}.

Definition 2.1. An O_{S}-Lie algebroid on X is a triple (L,[·,·]_{L},α) consisting of a sheaf of O_{S}-Lie algebras (L,[·,·]_{L}) on X and a homomorphism α : L → T_{X/S}, x → α_{x}, of sheaves of O_{S}-Lie algebras on X, which satisfies the following Leibniz rule

[x,fy]_{L} = α_{x}(f)y + f [x,y]_{L}

for all local sections f ∈ O_{X} and x,y ∈ L (in the formula we treat α_{x}as an O_{S}-derivation of O_{X}). We say that L is smooth if it is coherent and locally free as an O_{X}-module. L is quasi-smooth if it is coherent and torsion free as an O_{X}-module.

The map α in the above definition is usually called the anchor. A Lie algebroid is a sheaf of Lie-Rinehart algebras (see [Ri]). It is also a special case of the more general notion of a Lie algebra in a topos defined by Illusie (see [Il, Chapitre VIII, Definition 1.1.5]). A homomorphism of O_{S}-Lie algebroids L and L′on X is a homomorphism L → L′of sheaves of O_{S}-Lie algebras on X which commutes with the anchors. Note that an O_{S}-Lie algebroid on X with the zero anchor map corresponds to a sheaf of O_{X}-Lie algebras.

Definition 2.2. A de Rham complex on X over S is a pair (⋀^{•}M,d_{M}^{•}) consisting of the exterior algebra ⋀^{•}M := ⋀ _{O_{X}}^{•}M of an O_{X}-module M and an O_{S}-anti-derivation d_{M}^{•} : ⋀^{•}M → ⋀^{•}M of degree 1 (i.e., d_{M}^{•}(x ∧ y) = (d_{M}^{•}x) ∧ y + (−1)^{j}x ∧ d_{M}^{•}y for all local sections x ∈ ⋀^{j} M and y ∈ ⋀^{•}M) such that (d_{M}^{•})^{2} = 0. We say that (⋀^{•}M,d_{M}^{•}) is smooth if M is coherent and locally free.

A de Rham complex is a special case of a sheaf of graded-commutative differential graded algebras. A special case of a de Rham complex is the de Rham complex (Ω_{X/S}^{•},d_{X/S}^{•}), which is the unique de Rham complex extending the canonical O_{S}-derivation d_{X/S}: O_{X} → Ω_{X/S} (uniqueness follows because Ω_{X/S} is generated by d_{X/S}O_{X} as a left O_{X}-module). By the universal property of d_{X/S} we have Der_{O_{S}}(O_{X},M) ≃ Hom_{O_{X}}(Ω_{X/S},M) and hence for every de Rham complex (⋀^{•}M,d_{M}^{•}) we have a unique morphism of de Rham complexes (Ω_{X/S}^{•},d_{X/S}^{•}) → (⋀^{•}M,d_{M}^{•}). This morphism induces a well defined map on the hypercohomology groups:

H_{DR}^{i}(X/S) := ℍ^{•}(Ω_{X/S}^{•}) → ℍ^{•}(⋀ ^{•}M).

To every O_{S}-Lie algebroid (L,[·,·]_{L},α) on X we can associate a de Rham complex (⋀^{•}M,d_{M}^{•}) on X over S for M = L^{∗}. This is done by the following well known formula generalizing the usual exterior differential:

\begin{aligned}(d_{M}m)(l_{1},...,l_{k+1}) & = & ∑ _{i=1}^{k+1}(−1)^{i+1} α_{l_{i}}(m(l_{1},...,l̂_{i},...,l_{k+1})) \\ & + & ∑_{1≤i<j≤k+1} (−1)^{i+j} m([l_{i},l_{j}]_{L},l_{1},...,l̂_{i},...,l̂_{j},...,l_{k+1}) \end{aligned}

for m ∈ ⋀^{k}M and l_{1},...,l_{k+1} ∈ L. This gives a functor from the category of Lie algebroids to the category of de Rham complexes. On the other hand, to every de Rham complex (⋀^{•}M,d_{M}^{•}) on X over S we can associate a Lie algebroid structure on L = M^{∗}. The anchor L → T_{X/S}= (Ω_{X/S})^{∗} is obtained as the transpose of the O_{X}-homomorphism Ω_{X/S}→ M corresponding to the O_{S}-derivation d_{M} : O_{X} → M. The bracket on L can be read off the above formula defining d_{M} : M → ⋀^{2}M. This provides a functor in the opposite direction: from the category of de Rham complexes to the category of Lie algebroids. These functors are quasi-inverse on subcategories of smooth objects.

If L is a smooth O_{S}-Lie algebroid on X then the corresponding de Rham complex is denoted by (Ω_{L}^{•},d_{L}^{•}). In this case we set

H_{DR}^{i}(L) := ℍ^{i}(Ω_{L}^{•},d_{L}^{•}).

We have the following standard spectral sequence associated to the de Rham complex of L:

E_{1}^{ij} = H^{j}(X/S,Ω_{L}^{i}) ⇒ H_{DR}^{i+j}(L).

2.2 Universal enveloping algebra of differential operators

Definition 2.3. A sheaf of O_{S}-Poisson algebras on X is a pair (A ,{·,·}) consisting of a sheaf A of commutative, associative and unital O_{X}-algebras with a Poisson bracket {·,·} such that (A ,{·,·}) is a sheaf of O_{S}-Lie algebras on X satisfying the Leibniz rule

{x,y · z} = {x,y} · z + y · {x,z}

for all x,y,z ∈ A.

Let Λ be a sheaf of rings of differential operators on X over S such that Λ_{0} = O_{X}. Let us assume that Λ is almost commutative, i.e., the associated graded Gr(Λ) is a sheaf of commutative O_{X}-algebras. Then Gr(Λ) has a natural structure of a sheaf of O_{S}-Poisson algebras on X with the Poisson bracket given by

{[x],[y]}:= (xy − yx modΛ_{i+j−2} ) ∈ Gr_{i+j−1} (Λ),

where [x] ∈ Gr_{i}(Λ) is the class of x ∈ Λ_{i} and [y] ∈ Gr_{j}(Λ) is the class of y ∈ Λ_{j}. The Poisson bracket induces an O_{S}-Lie algebroid structure on Gr_{1}(Λ). The Lie bracket on Gr_{1}(Λ) is equal to the Poisson bracket and the anchor map α : Gr_{1}(Λ) → T_{X/S}is given by sending [x] to the O_{S}-derivation y → {[x],y}, y ∈ O_{X} = Gr_{0}(Λ). On the other hand, if L is an O_{S}-Lie algebroid on X then we can associate to L a sheaf of rings of differential operators on X over S in the following way. We define an O_{S}-Lie algebra structure on L̃= O_{X}⊕ L by setting

[f + x,g + y]L̃ = α_{x}(g) − α_{y}(f) + [x,y]_{L}

for all local sections f,g ∈ O_{X} and x,y ∈ L. Let U_{O_{S}}(L̃) be the universal enveloping algebra of L̃ and let Ũ_{O_{S}}(L̃) be the sheaf of subalgebras (without unit!) generated by the image of the canonical map iL̃: L̃→ U_{O_{S}}(L̃) (note that in general this map need not be injective). We define Λ_{L} as the quotient of Ũ_{O_{S}}(L̃) by the two-sided ideal generated by all elements of the form iL̃(f)iL̃(x) − iL̃(fx) for all f ∈ O_{X} and x ∈ L̃. Let Λ_{L,j} be the left O_{X}-submodule of Λ_{L} generated by products of at most j elements of the image of L in Λ_{L}. This defines a filtration of Λ_{L} equipping it with structure of sheaf of rings of differential operators (since the canonical graded morphism S^{•}Gr_{1}(Λ_{L}) → Gr(Λ_{L}) is surjective, the constructed Λ_{L} is almost commutative). We call Λ_{L} the universal enveloping algebra of differential operators associated to L. By the Poincare-Birkhoff-Witt theorem, if the Lie algebroid L is smooth then L → Gr_{1}(Λ_{L}) is an isomorphism and the canonical epimorphism S^{•}L → Gr(Λ_{L}) is an isomorphism of sheaves of graded O_{X}-algebras (see [Ri, Theorem 3.1]). This implies that if L is quasi-smooth then the canonical map L → Λ_{L} is injective. If L = T_{X/S}and the anchor map is identity, then Λ_{L} is denoted by D_{X/S}and it is called the sheaf of crystalline differential operators (see [BMR]). In [BO] the authors call it the sheaf of PD differential operators. In the characteristic zero case the sheaf Λ_{L} and the correspondence between Lie algebroids and sheaves of rings of differential operators was studied by Simpson in [Si2, Theorem 2.11] with subsequent corrections by Tortella in [To, Theorem 4.4].

We can also consider twisted versions of sheaves of rings of differential operators associated to a Lie algebroid (see [BB] and [To]). Let Λ be an almost commutative sheaf of rings of differential operators on X over S such that Λ_{0} = O_{X}. Then Λ_{1} has an O_{S}-Lie algebra structure on X given by the usual Lie bracket [·,·] coming from Λ and the anchor map given by sending x ∈ Λ_{1} to f → [x,f]. Then Λ_{1} → Gr_{1}(Λ) is a homomorphism of O_{S}-Lie algebras with kernel being the sheaf O_{X} (with a trivial O_{S}-Lie algebroid structure). The following definition is motivated by [BB, Definition 2.1.3]:

Definition 2.4. A generalized O_{S}-Picard Lie algebroid on X is an O_{S}-Lie algebroid L̃equipped with a section 1L̃ of L̃inducing an exact sequence of O_{S}-Lie algebroids

0 → O_{X} → L̃→ L → 0,

where O_{X} is taken with the trivial O_{S}-Lie algebroid structure.

To any generalized O_{S}-Picard Lie algebroid L̃ we can associate an almost commutative sheaf of rings of differential operators Λ̃L̃ on X over S such that Λ̃_{L̃,0} = O_{X} and Λ̃_{L̃,1} = L̃. Λ̃L̃ is constructed as a quotient of the universal enveloping algebra of differential operators ΛL̃ by the two-sided ideal generated by 1L̃− 1. As in [BB, Lemma 2.1.4], this defines a fully faithful functor from the category of generalized Picard Lie algebroids to the category of almost commutative sheaves of rings of differential operators. The analogous construction can be also found in [To], where the author constructs Λ̃L̃ by gluing local pieces.

3 Modules over Lie algebroids

3.1 Modules with generalized connections

Let X be an S-scheme. Let M be a coherent O_{X}-module with an O_{S}-derivation d_{M} : O_{X} → M. A d_{M}-connection on a coherent O_{X}-module E is an O_{S}-linear morphism ∇: E → E ⊗_{O_{X}}M satisfying the following Leibniz rule

∇(fe) = f∇(e) + e ⊗ d_{M}(f)

for all local sections f ∈ O_{X} and e ∈ E. Note that notion of d_{M}-connection depends on the choice of derivation d_{M} and not only the sheaf M. For example if M = Ω_{X/S}then the standard derivation d_{X/S}leads to a sheaf with a usual connection whereas the zero derivation leads to a Higgs sheaf (but without any integrability condition).

3.2 Generalized Higgs sheaves

Assume that (⋀^{•}M,d_{M}^{•}) is a de Rham complex and let E be a coherent O_{X}-module. Then a d_{M}-connection ∇ : E → E ⊗ M can be extended to a morphism ∇_{i} : E ⊗_{O_{X}} ⋀^{i}M → E ⊗_{O_{X}}⋀^{i+1} M by setting

∇_{i}(e ⊗ ω) = e ⊗ d_{M}ω + (−1)^{i}∇(e) ∧ ω,

where e ∈ E and ω ∈ ⋀^{i}M are local sections. As usually one can check that the curvature K = ∇_{1} ∘ ∇ is O_{X}-linear and ∇_{i+1} ∘ ∇_{i}(e ⊗ ω) = K(e) ∧ ω. We say that (E,∇) is integrable if the curvature K = 0. If (E,∇) is integrable then the sequence

0 → E →^{∇} E ⊗ M →^{∇_{1}} E ⊗ ⋀ ^{2}M → ...

becomes a complex. The hypercohomology groups of this complex are denoted by H_{DR}^{i}(X,E) := ℍ^{i}(E ⊗ ⋀^{•}M,∇).

Let ⋀^{•}M be the de Rham complex corresponding to the exterior O_{S}-algebra of M with zero anti-derivation d_{M}. Then a coherent O_{X}-module with an integrable d_{M}- connection θ : E → E ⊗_{O_{X}}M is called an M-Higgs sheaf. The corresponding homomorphism θ is O_{X}-linear and it is called an M-Higgs field (or just a Higgs field). A system of M-Hodge sheaves is an M-Higgs sheaf (E,θ) with decomposition E = ⨁ E^{j} such that θ : E^{j} → E^{j−1} ⊗ M. For M = Ω_{X/S}we recover the usual notions of a Higgs sheaf and a system of Hodge sheaves.

To be consistent with notation in the characteristic zero case, the hypercohomology groups ℍ^{i}(E ⊗ ⋀^{•}M,θ) of the complex associated to an M-Higgs sheaf are denoted by H_{Dol}^{i}(X,E). The following lemma can be proven in the same way as [Si1, Lemma 2.5]:

Lemma 3.1. Let X be a smooth d-dimensional projective variety over an algebraically closed field k and let (E,θ) be an M-Higgs sheaf. Then we have χ_{Dol}(X,E) = rkE · χ_{Dol}(X,O_{X}). Moreover, if E is locally free then we have a perfect pairing

H_{Dol}^{i}(X,E) ⊗ H_{Dol}^{2d−i} (X,E^{∗}) → k

induced by Serre’s duality.

3.3 Modules over Lie algebroids and coHiggs sheaves

Let L be an O_{S}-Lie algebroid on X and let E be an O_{X}-module. Let us recall that a (left) Λ_{L}-module structure on E is the same as an L-module structure, i.e., a homomorphism ∇ : L → E nd_{O_{S}}E of sheaves of O_{S}-Lie algebras on X (in particular, ∇ is O_{X}-linear) satisfying Leibniz’s rule

∇(x)(fe) = α_{x}(f)e + ∇(fx)(e)

for all local sections f ∈ O_{X}, x ∈ L and e ∈ E. One can also look at L-modules E as modules E over the sheaf of O_{S}-Lie algebras L̃= O_{X}⊕ L on X defined in Subsection 2.2, which satisfy equality (fy)e = f(ye) for all local sections f ∈ O_{X}, y ∈ L′ and e ∈ E. Proof of the following easy lemma is left to the reader:

Lemma 3.2. Let L be a smooth O_{S}-Lie algebroid L and let (⋀^{•}Ω_{L},d_{Ω_{L}}^{•} ) be the associated de Rham complex. Then we have an equivalence of categories between the category of L-modules and coherent O_{X}-modules with integrable d_{Ω_{L}} -connection.

Let L be a coherent O_{X}-module. Let us provide it with the trivial O_{S}-Lie algebroid structure, i.e., we take zero bracket and zero anchor map. In this case we say that L is a trivial Lie algebroid. For a trivial Lie algebroid the corresponding sheaf of rings of differential operators Λ_{L} is equal to the (commutative) symmetric O_{X}-algebra S^{•}(L). In this case an L-coHiggs sheaf is a (left) Λ_{L}-module, coherent as an O_{X}-module. If L is smooth then giving an L-coHiggs sheaf is equivalent to giving an Ω_{L}-Higgs sheaf.

If L is smooth then 𝕍(L) → X is a vector bundle and we can take its projective completion π : Y = ℙ(L ⊕ O_{X}) → X. The divisor at infinity D = Y − 𝕍(L) is canonically isomorphic to ℙ(L). On Y we have the tautological relatively ample line bundle O_{ℙ(L⊕O_{X})}(1). If O_{X}(1) is an S-ample polarization on X then for sufficiently large n the line bundle A = O_{ℙ(L⊕O_{X})}(1) ⊗ π^{∗}(O_{X}(n)) is also S-ample. By definition any L-coHiggs sheaf gives rise to a coherent O_{𝕍(L)}-module. The following lemma describes image of the corresponding functor (cf. [Si3, Lemma 6.8 and Corollary 6.9]):

Lemma 3.3. We have an equivalence of categories between L-coHiggs sheaves and coherent sheaves on Y, whose support does not intersect D. Under this equivalence pure sheaves correspond to pure sheaves of the same dimension and the notions of (semi)-stability are the same when considered with respect to polarizations O_{X}(1) on X and A on Y.

This lemma suggests another construction of the moduli space M_{Dol}^{L}(X/S,P) = M^{Λ_{L}}(X/S,P) of Gieseker semistable L-coHiggs sheaves (with fixed Hilbert polynomial P) on X/S using construction of the moduli space M(Y /S,P) of Gieseker semistable sheaves of pure dimension n = dim(X/S) on Y /S (with Hilbert polynomial P). Namely, M(Y /S,P) is constructed as a GIT quotient R//G, where R is some parameter space and G is a reductive group acting on R. Then M_{Dol}^{L}(X/S,P) can be constructed as the quotient R′//G, where R′is the G-invariant subscheme of R corresponding to subsheaves whose support does not intersect D.

3.4 Modules on varieties over fields

In this subsection we take as S the spectrum of an algebraically closed field k. We also assume that X is normal and projective with fixed polarization O_{X}(1). We say that a sheaf with an M-connection (E,∇) is slope semistable if E is torsion free as an O_{X}-module and if for any O_{X}-submodule E′⊂ E such that ∇(E′) ⊂ E′⊗_{O_{X}}M we have

μ(E′) ≤ μ(E).

We say that (E,∇) is slope stable if we have stronger inequality μ(E′) < μ(E) for every proper O_{X}-submodule E′⊂ E preserved by ∇ and such that rkE′< rkE. In much the same way we can introduce notions of slope (semi)stability for M-Higgs sheaves and systems of M-Hodge sheaves. In each case to define (semi)stability we use only subobjects in the corresponding category.

Let us fix a smooth k-Lie algebroid L on X. We have a natural action of 𝔾_{m} on Ω_{L}- Higgs sheaves given by sending (E,θ) to (E,tθ) for t ∈ 𝔾_{m}. The following lemma is a simple generalization of the well known fact in case of usual Higgs bundles (see, e.g., [Si1, Lemma 4.1]) but we include proof for completeness. The assertion in the positive characteristic case is slightly different to that of [Si1, Lemma 4.1]. The difference comes from the fact that for k = 𝔽̄_{p} every t ∈ k^{∗} is a root of unity.

Lemma 3.4. A rank r torsion free Ω_{L}-Higgs sheaf (E,θ) is a fixed point of the 𝔾_{m}- action if and only if it has a structure of system of Ω_{L}-Hodge sheaves.

Proof. Taking reflexivization we can assume that E is reflexive. By assumption for every t ∈ 𝔾_{m} there exists an isomorphism of O_{X}-modules f : E → E (depending on t) such that fθ = tθf. On the subset U where E is locally free, the coefficients of the characteristic polynomial of f define sections of O_{X}. Since X is normal and projective we have O_{X}(U) = O_{X}(X) = k, so they are constant. Hence we can decompose E into eigensubsheaves E = ⨁ E_{λ}, where E_{λ} = ker(f − λ)^{r} for λ ∈ k^{∗} (eigenvalue 0 does not occur as f is an isomorphism). Since (f − tλ)^{r}θ = t^{r}θ(f − λ)^{r}, the Higgs field θ maps E_{λ} to E_{tλ}. If we take t such that t^{j} ≠ 1 for j = 0,...,r then for every eigenvalue λ the elements λ,tλ,...,t^{r}λ are pairwise distinct. So there exists j_{0} such that t^{j_{0}}λ is an eigenvalue but t^{j_{0}−1} λ is not an eigenvalue. Then E^{i} = ⨁ _{j_{0}≤j≤i} E_{t^{j}λ} defines a system of Ω_{L}-Hodge sheaves which is a direct summand of (E,θ). So we can complete the proof by induction on the rank r of E.

Corollary 3.5. A system of Ω_{L}-Hodge sheaves (E,θ) is slope (or Gieseker) semistable if and only if it is slope (respectively, Gieseker) semistable as an Ω_{L}-Higgs sheaf.

Proof. It is sufficient to prove that the maximal destabilizing Ω_{L}-Higgs subsheaf of a system of Ω_{L}-Hodge sheaves (E,θ) is a system of Ω_{L}-Hodge sheaves. This follows from the above lemma and the fact that the maximal destabilizing Ω_{L}-Higgs subsheaf is unique so it is preserved by the natural 𝔾_{m}-action.

3.5 Hitchin’s morphism for moduli spaces of L-coHiggs sheaves

Let G be a quasi-coherent O_{S}-module. Consider the functor which to an S-scheme T associates Hom_{O_{T}}(G_{T},O_{T}). It is representable by the S-scheme 𝕍(G). In particular, for π : T = 𝕍(G) → S we get the tautological homomorphism

λ_{G} ∈ Hom_{O_{𝕍(G)}}(π^{∗}G,O_{𝕍(G)}) = Hom_{O_{S}}(G,π_{∗}O_{𝕍(G)}) = Hom_{O_{S}−alg}(S^{•}G,S^{•}G)

corresponding to the identity on S^{•}G. If G is a locally free sheaf of finite rank then 𝕍(G) → S is a vector bundle with sheaf of sections isomorphic to G^{∗}.

The following lemma was explained to the author by C. Simpson:

Lemma 3.6. Let f : X → S be a flat projective morphism of noetherian schemes and let G be a locally free sheaf on X. Then the functor \underline{H}^{0}(X/S,G) which to an S-scheme h: T → S associates H^{0}(X_{T}/T,G_{T}) is representable by an S-scheme.

Proof. Since certain twist of G^{∗} by a relatively very ample line bundle is relatively globally generated, we can embed G as a subbundle into a direct sum K_{1} of relatively very ample line bundles. Then we can again embed the quotient K_{1}/G into K_{2} with K_{2} a direct sum of relatively very ample bundles. Then for any S-scheme T we have an exact sequence

0 → \underline{H}^{0}(X/S,G)(T) → \underline{H}^{0}(X/S,K_{1})(T) → \underline{H}^{0}(X/S,K_{2})(T).

But we can assume that all the higher direct images of K_{1} vanish and then by the Grauert’s theorem \underline{H}^{0}(X/S,K_{1}) is representable by the bundle 𝕍(f_{∗}K_{1}) → S. Similarly, \underline{H}^{0}(X/S,K_{1}) is representable by the bundle 𝕍(f_{∗}K_{2}) → S. Therefore \underline{H}^{0}(X/S,G) is represented by the kernel of the map between bundles. This is a vector subscheme of 𝕍(f_{∗}K_{1}) → S.

We will also need the following well-known lemma:

Lemma 3.7. Let f : X → S be a flat family of irreducible d-dimensional schemes satisfying Serre’s condition (S_{2}). Let E be an S-flat coherent O_{X}-module such that E ⊗k(s) is pure of dimension d for every point s ∈ S. Then there exists a relatively big open subset j: U ⊂ X such that E^{∗∗} → j_{∗}(E|_{U}) is an isomorphism.

Consider a flat projective morphism f : X → S of noetherian schemes. Let L be a smooth O_{S}-Lie algebroid on X and let us recall that Ω_{L} = L^{∗}. Consider the functor which to an S-scheme h: T → S associates

⨁_{i=1}^{r} H^{0}(X_{T}/T,S^{i}Ω_{L,T}).

By Lemma 3.6 this functor is representable by an S-scheme 𝕍^{L}(X/S,r) → S. Let us also assume that X/S is a family of d-dimensional varieties satisfying Serre’s condition (S_{2}). If T is an S-scheme then X_{T}/T is also a flat family of d-dimensional varieties satisfying Serre’s condition (S_{2}). Assume that L is a trivial O_{S}-Lie algebroid and consider a family (E,θ : E → E ⊗ Ω_{L,T}) of L-coHiggs sheaves of pure dimension d = dim(X/S) on the fibres of X_{T} → T. Then there exists an open subset U ⊂ X_{T} such that E is locally free on U and the intersection of U with any fiber of X_{T} → T has a complement of codimension at least 2. Let us consider ⋀^{i}θ|_{U} : ⋀^{i}(E|_{U}) → ⋀^{i}(E|_{U} ⊗_{O_{U}} Ω_{L,T}|_{U}). We have a well defined surjection ⋀^{i}(E|_{U}⊗_{O_{U}} Ω_{L,T}|_{U}) → ⋀^{i}E|_{U}⊗_{O_{U}} S^{i}Ω_{L,T}|_{U}, given by

(e_{1}⊗ λ_{1}) ∧ ... ∧ (e_{i}⊗ λ_{i}) → (e_{1}∧ ... ∧ e_{i}) ⊗ (λ_{1}...λ_{i}),

where e_{1},...,e_{i} ∈ E and λ_{1},...,λ_{i} ∈ Ω_{L,T}. So we get a morphism of sheaves

O_{U} → E nd_{O_{X}}(⋀ ^{i}E)|_{U}⊗_{O_{U}} S^{i}Ω_{L,T}|_{U} −→^{(−1)^{i}Tr⊗id} S^{i}Ω_{L,T}|_{U}

The corresponding section σ_{i}(θ|_{U}) ∈ H^{0}(U,S^{i}Ω_{L,T}|_{U}) is just an evaluation of the ith elementary symmetric polynomial on θ|_{U}. By Lemma 3.7 this section extends uniquely to section σ_{i}(θ) ∈ H^{0}(X_{T}/T,S^{i}Ω_{L,T}). In this way we can define a T -point σ(E,θ) = (σ_{1}(θ),...,σ_{r}(θ)) of 𝕍^{L}(X/S,r). Let P be a polynomial of degree d = dim(X/S) corresponding to (some) rank r torsion free sheaves on the fibres of X → S. Consider the moduli space M_{Dol}^{L}(X/S,P) of Gieseker semistable L-coHiggs sheaves with Hilbert polynomial P. Then the above construction defines a morphism of functors inducing the corresponding morphism of coarse moduli spaces H_{L} : M_{Dol}^{L}(X/S,P) → 𝕍^{L}(X/S,r). This morphism is called Hitchin’s morphism. There is also a stack theoretic version of Hitchin’s morphism. The moduli stack of L-coHiggs sheaves is defined as a lax functor between 2-categories by

M_{Dol}^{L}(X/S,P) : (Sch/S) → (groupoids)
T  →  M (T),

where M (T) is the category whose objects are T -flat families of pure d-dimensional L-coHiggs sheaves with Hilbert polynomial P on the fibres of X_{T} → T, and whose morphisms are isomorphisms of coherent sheaves. Then M_{Dol}^{L}(X/S,P) is an algebraic stack for the fppf topology on (Sch/S). As above we can construct Hitchin’s morphism M_{Dol}^{L}(X/S,P) → 𝕍^{L}(X/S,r). By abuse of notation, we also denote this morphism by H_{L}.

As in the usual Higgs bundle and characteristic zero case, one can construct the total spectral scheme 𝕎^{L}(X/S,r) ⊂ 𝕍(L) ×_{S} 𝕍^{L}(X/S,r), which is finite and flat over X ×_{S}𝕍^{L}(X/S,r). This subscheme has the property that for any family (E,θ : E → E ⊗ Ω_{L,T}) of L-coHiggs sheaves of pure dimension d on the fibres of X_{T} → T, the corresponding coherent sheaf on 𝕍(L_{T}) is set-theoretically supported on 𝕎^{L}(X/S,r)×_{𝕍^{L}(X/S,r)}T. This can be seen as follows. Let x be a geometric point of X at which E is locally free. Then S^{•}L⊗k(x) acts on V = E ⊗k(x) via θ(x). Let us recall that over an algebraically closed field any finitely dimensional vector space which is irreducible with respect to a set of commuting linear maps has dimension 1. Therefore V has a filtration 0 = V_{0} ⊂ V_{1} ⊂ ... ⊂ V_{r} = V with quotients V^{i} = V_{i}/V_{i−1} of dimension 1 over k(x) and such that θ(x) acts on V^{i} as multiplication by λ_{i} ∈ (L ⊗ k(x))^{∗}. It is clear from our definition that τ ∈ L ⊗ k(x) acts on V via θ_{τ} := θ(x)^{T}(τ) in such a way that in the characteristic polynomial

det(t · I − θ_{τ}) = t^{r} + σ_{1}(θ_{τ})t^{r−1} + ... + σ_{r}(θ_{τ})

we have σ_{i}(θ_{τ}) = (−1)^{i}∑_{1≤j_{1}<...<j_{i}≤r} λ_{j_{1}}...λ_{j_{i}}. This and the Cayley–Hamilton theorem show that the coherent sheaf on 𝕍(L_{T}) corresponding to (E,θ) has a schemetheoretic support contained in 𝕎^{L}(X/S,r) ×_{𝕍^{L}(X/S,r)}T and it coincides with it settheoretically.

Note that in the curve case there exists a different interpretation of Higgs bundles using cameral covers. Such an approach allows to deal with general reductive groups (see [DG] for the characteristic zero case). In positive characteristic the analogous construction requires some restrictions on the characteristic of the base field.

The following theorem can be proven in a similar way as the usual characteristic zero version [Si3, Theorem 6.11]. It also follows from Langton’s type Theorem 5.3.

Theorem 3.8. Hitchin’s morphism H_{L}: M_{Dol}^{L}(X/S,P) → 𝕍^{L}(X/S,r) is proper.

3.6 Deformation of a Lie algebroid over an affine line.

Let R be a commutative ring with unity. Let f : X → S be a morphism of R-schemes. Let 𝔸_{R}^{1}= SpecR[t] and let p_{1} : X ×_{R}𝔸_{R}^{1}→ X be the projection onto the first factor. Let us consider an O_{S}-Lie algebroid L on X and the morphism f × id : X ×_{R}𝔸_{R}^{1} → S ×_{R}𝔸_{R}^{1} of R-schemes. We can define an O_{S×_{R}𝔸_{R}^{1}}-Lie algebroid L^{R} on X ×_{R}𝔸_{R}^{1} by taking L^{R}:= p_{1}^{∗}L with Lie bracket given by [·,·]_{L^{R}}:= p_{1}^{∗}[·,·]_{L}⊗ t and the anchor map given by α^{R}:= p_{1}^{∗}α ⊗ t. The universal enveloping algebra of differential operators Λ_{L}^{R}:= Λ_{L^{R}} associated to L^{R} can be constructed as a subsheaf of p_{1}^{∗}Λ_{L} generated by sections of the form ∑ t^{i}λ_{i}, where λ_{i} are local sections of Λ_{L,i}. If R = k is a field and s ∈ 𝔸^{1}(k) − {0} then the restricted sheaf Λ_{L}^{R}|_{X×{s}} is naturally isomorphic to Λ_{L}. The sheaf Λ_{L}^{R}|_{X×{0}} is naturally isomorphic to the associated graded sheaf of algebras GrΛ_{L}. This gives a deformation of Λ_{L} to its associated graded sheaf of algebras (or a quantization of the commutative algebra GrΛ_{L}).

Let T be an S-scheme and let us fix λ ∈ H^{0}(T /R,O_{T}). Let E be a coherent O_{X_{T}}- module and let p_{X} and p_{T} be the projections of X ×_{S}T onto X and T, respectively. Let (M,d_{M}) be a coherent O_{X}-module with an O_{S}-derivation. Then we set M̃ = p_{X}^{∗}M and dM̃ = p_{X}^{∗}d_{M} · p_{T}^{∗}λ. A dM̃-connection on E is called a λ-d_{M}-connection. This generalizes the usual notion of λ-connection. For the constant section λ = 0 ∈ H^{0}(T /R,O_{T}) an integrable λ-d_{M}-connection is just an M-Higgs field. Similarly, for λ = 1 ∈ H^{0}(T /R,O_{T}) we recover the notion of a d_{M}-connection.

Assume that L is a smooth O_{S}-Lie algebroid on X. Let us fix a morphism of Rschemes T → S ×_{R}𝔸_{R}^{1} and let λ ∈ H^{0}(T /R,O_{T}) be the section corresponding to the composition of T → S ×_{R}𝔸_{R}^{1} with the canonical projection S ×_{R}𝔸_{R}^{1} → 𝔸_{R}^{1}. Since T ×_{S×_{R}𝔸_{R}^{1}} X ×_{R}𝔸_{R}^{1} = X_{T}, an L^{R}-module structure on a coherent O_{X_{T}}-module E is equivalent to giving an integrable λ-d_{Ω_{L}} -connection.

4 Lie algebroids in positive characteristic

4.1 Sheaves of restricted Lie algebras

Let R be a commutative ring (with unity) of characteristic p and let L be a Lie Ralgebra. We define the universal Lie polynomials s_{j} by the formula

s_{j}(x_{1},x_{2}) = −\frac{1}{j} ∑_{σ} adx_{σ(1)}...adx_{σ(p−1)} (x_{2})

in which we sum over all σ : {1,...,p − 1} → {1,2} taking j times value 1. Let A be an associative R-algebra. For x ∈ A we define ad(x) : A → A by the formula (ad(x))(y) = xy − yx for y ∈ A. Then we have the following well known Jacobson’s formulas:

ad(x^{p}) = ad(x)^{p}

(x + y)^{p} = x^{p} + y^{p} + ∑_{0<j<p} s_{j}(x,y).

Let X be a scheme over a scheme S of characteristic p > 0. A sheaf of restricted O_{S}- Lie algebras on X is a sheaf of O_{S}-Lie algebras (L,[·,·]) on X equipped with a p-th power operation L → L, x → x^{[p]}, which satisfies the following conditions:

1. (fx)^{[p]}= f^{p}x^{[p]}for all local sections f ∈ O_{S} and x ∈ L,

2. ad(x^{[p]}) = (ad(x))^{p} for x ∈ L,

3. (x + y)^{[p]}= x^{[p]}+ y^{[p]}+ ∑_{0<j<p} s_{j}(x,y) for all x,y ∈ L.

A homomorphism of sheaves of restricted O_{S}-Lie algebras φ : L → L′on X is such a homomorphism of sheaves of O_{S}-Lie algebras on X that φ(x^{[p]}) = φ(x)^{[p]}for all x ∈ L.

Let A be a sheaf of associative O_{S}-algebras on X. It has a natural structure of a sheaf of restricted O_{S}-Lie algebras on X with bracket [x,y] = xy − yx and p-th power operation x^{[p]}= x^{p} for local sections x,y ∈ A. Now let L be a sheaf of restricted O_{S}-Lie algebras on X. For any homomorphism φ : L → A of sheaves of O_{S}-Lie algebras on X we can define ψ : L → A by ψ(x) = (φ(x))^{p} − φ(x^{[p]}) for x ∈ L. The map ψ measures how far is φ from being a homomorphism of sheaves of restricted O_{S}-Lie algebras on X.

Lemma 4.1. The map ψ : L → A is additive and its image commutes with the image of φ. In particular, [ψ(L),ψ(L)] = 0.

Proof. Let us take sections x,y ∈ L(U) for some open subset U ⊂ X. From Jacobson’s formula in A we have

(φ(x + y))^{p} = φ(x)^{p} + φ(y)^{p} + ∑_{0<j<p} s_{j}(φ(x),φ(y))

On the other hand, from definition of a sheaf of restricted Lie algebras we have

φ((x + y)^{[p]}) = φ(x^{[p]}) + φ(y^{[p]}) + ∑_{0<j<p} s_{j}(φ(x),φ(y)),

so subtracting these equalities we get additivity of ψ. Now we need to prove that [ψ(x),φ(y)] = 0. But we have

[φ(x)^{p},φ(y)] = ad(φ(x)^{p})(φ(y)) = (adφ(x))^{p}(φ(y))

and

[φ(x^{[p]}),φ(y)] = φ([x^{[p]},y]) = φ(ad(x^{[p]})(y)) = φ(ad(x)^{p}(y)) = (adφ(x))^{p}(φ(y)),

so subtracting yields the required equality.

The restricted universal enveloping algebra U_{O_{S}}^{[p]}(L) of a sheaf of restricted O_{S}-Lie algebras L on X is the quotient of the universal enveloping algebra U_{O_{S}}(L) by the two-sided ideal generated by all elements of the form x^{p}−x^{[p]}for local sections x ∈ L. If S = X and L is locally free as an O_{X}-module then L is contained in U_{O_{X}}^{[p]}(L). Moreover, if x_{1},...,x_{r} are local generators of L as an O_{X}-module then x_{1}^{i_{1}}...x_{r}^{i_{r}} with 0 ≤ i_{j} < p for all j, form a local basis of U_{O_{X}}^{[p]}(L) as an O_{X}-module. In particular, U_{O_{X}}^{[p]}(L) is locally free of rank p^{rkL}. In this case for any sheaf A of associative algebras on X and any homomorphism φ : L → A of sheaves of Lie algebras on X, the map ψ : L → A is F_{X}^{∗}-linear, i.e., ψ(fx) = f^{p}ψ(x) for all f ∈ O_{X} and x ∈ L (this follows from the first condition in the definition of a sheaf of restricted Lie algebras). So by adjunction ψ induces an O_{X}-linear map F_{X}^{∗}L → A that by abuse of notation is also denoted by ψ. Then the restricted universal enveloping algebra U_{O_{X}}^{[p]}(L) has the following universal property. For any sheaf A of associative O_{X}-algebras and any homomorphism φ : L → A of sheaves of O_{X}-Lie algebras with ψ : L → A equal to zero, there exists a unique homomorphism φ̃: U_{O_{X}}^{[p]}(L) → A of sheaves of associative O_{X}-algebras such

that φ : L → A is the composition of the natural map L → U_{O_{X}}^{[p]}(L) with φ̃.

4.2 Restricted Lie algebroids

Note that the relative tangent sheaf T_{X/S} has a natural structure of a sheaf of restricted O_{S}-Lie algebras on X in which the p-th power operation on O_{S}-derivation D: O_{X} → O_{X} is defined as the derivation acting on functions as the p-th power differential operator D^{p}. In fact, T_{X/S}with the usual Lie bracket and this p-th power operation is a sheaf of restricted O_{S}-Lie subalgebras of the associative algebra E nd_{O_{S}}O_{X} taken with the natural structure of a sheaf of restricted O_{S}-Lie algebras on X. This motivates the following definition:

Definition 4.2. A restricted O_{S}-Lie algebroid on X is a quadruple (L,[·,·],·^{[p]},α) consisting of a sheaf of restricted O_{S}-Lie algebras (L,[·,·],·^{[p]}) on X and a homomorphism of sheaves of restricted O_{S}-Lie algebras α : L → T_{X/S}on X satisfying the Leibniz rule and the following formula:

(fx)^{[p]}= f^{p}x^{[p]}+ α_{fx}^{p−1} (f)x

for all f ∈ O_{X} and x ∈ L.

As in the non-restricted case we can define a trivial restricted Lie algebroid as a trivial Lie algebroid with the zero p-th power operation. T_{X/S}with the usual Lie bracket and p-th power operation will be called the standard restricted O_{S}-Lie algebroid on X. The last condition in the definition requires certain compatibility of the p-th power operation on L with the anchor map and O_{X}-module structure of L. It can be explained by the fact that, as expected, a restricted O_{S}-Lie algebroid on X with the zero anchor map is a sheaf of restricted O_{X}-Lie algebras. In fact, the formula in the definition comes from the following Hochschild’s identity:

Lemma 4.3. (see [Ho, Lemma 1]) Let A be an associative 𝔽_{p}-algebra and R ⊂ A a commutative subalgebra. If for an element x ∈ A we have (adx)(R) ⊂ R then for any element r ∈ R we have

(rx)^{p} = r^{p}x^{p} + (ad(rx))^{p−1} (r)x.

A similar formula can be found as [Ka1, Proposition 5.3] (although with a sign error as pointed out by A. Ogus in [Og]).

The following criterion allows us to check when a submodule of a restricted Lie algebroid is a restricted Lie subalgebroid. It generalizes well known Ekedahl’s criterion allowing to check when a submodule of the tangent bundle defines a 1-foliation (see [Ek, Lemma 4.2]).

Lemma 4.4. 1. Let L′be an O_{X}-submodule of an O_{S}-Lie algebroid L on X. Then the Lie bracket on L induces an O_{X}-linear map

⋀ ^{2}L′→ L/L′

sending x∧y to the class of [x,y]. If this map is the zero map then L′is an O_{S}-Lie subalgebroid of L.

2. If L′is an O_{S}-Lie subalgebroid of a restricted O_{S}-Lie algebroid L then the p-th power map induces an O_{X}-linear morphism F_{X}^{∗}L′→ L/L′. If this map is the zero map then L′is a restricted O_{S}-Lie subalgebroid of L.

Proof. Let us take f ∈ O_{X} and x,y ∈ L′. The first part follows from the equality

[x,fy] = f[x,y] + α_{x}(f)y ≡ f[x,y] modL′.

To prove the second part note that

(x + y)^{[p]}= x^{[p]}+ y^{[p]}+ ∑_{0<j<p} s_{j}(x,y) ≡ x^{[p]}+ y^{[p]} modL′,

since s_{j}(x,y) ∈ L′, as the s_{j} are Lie polynomials. Therefore F_{X}^{∗}L′→ L/L′is additive. Hence to prove that it is O_{X}-linear it is sufficient to note that

(fx)^{[p]}= f^{p}x^{[p]}+ α_{fx}^{p−1} (f)x ≡ f^{p}x^{[p]} modL′.

Let us consider the following commutative diagram

\xymatrix{𝕍(L)^{(1/X)} \ar[dr] \ar[r]^{=} & 𝕍(F_{X/S}^{∗}L′) \ar[d]^{π} \ar[r]^{F̃_{X/S}} & 𝕍(L′) \ar[r] \ar[d]^{π′} & 𝕍(L) \ar[d] \\  & X \ar[rr]_{F_{X}} \ar[r]^{F_{X/S}} & X′ \ar[r] & X}

in which L′is the pull back of L via X′→ X. The following lemma is an analogue of [BMR, Lemma 1.3.2]:

Lemma 4.5. Let L be a restricted O_{S}-Lie algebroid on X. Then the map ı: L → Λ_{L} sending x ∈ L to ı(x) := x^{p} − x^{[p]}∈ Λ_{L} is F_{X}^{∗}-linear and its image is contained in the center Z(Λ_{L}) of Λ_{L}. In particular, if L smooth then ı extends to an O_{X′}-linear inclusion S^{•}L′↪ F_{X/S,∗}Z(Λ_{L}).

Proof. Lemma 4.3 proves that the p-th power operation satisfies α_{x^{[p]}}= (α_{x})^{p} and (fx)^{[p]}− f^{p}x^{[p]}= (fx)^{p} − f^{p}x^{p} in Λ_{L} for all f ∈ O_{X} and x ∈ L. Hence ı is F_{X}^{∗}-linear. Lemma 4.1 implies that its image is contained in Z(Λ_{L}). For any f ∈ O_{X} and x ∈ L we have xf^{p} − f^{p}x = α_{x}(f^{p}) = 0 in Λ_{L}, as α_{x} is an O_{S}- derivation. Therefore O_{X′}⊂ F_{X/S,∗}Z(Λ_{L}) which together with the first part proves the required assertion.

Note that the above lemma shows that Λ_{L} contains a commutative subalgebra S^{•}(F_{X/S}^{∗}L′), so Λ_{L} defines a quasi-coherent sheaf Λ̃_{L} on 𝕍(F_{X}^{∗}L).

Let Λ_{L}^{[p]}be the quotient of Λ_{L} by the two-sided ideal generated by ı(x) for x ∈ L. We call it the restricted universal enveloping algebra of differential operators of L.

Lemma 4.6. Let L be smooth of rank m. Then Λ̃_{L} is a locally free O_{𝕍(L)}-module of rank p^{m}.

Proof. The canonical embedding j : L → Λ_{L} induces an embedding j̃: L → Λ_{L}^{[p]}. Let us take an open subset U ⊂ X such that L(U) is a free O_{X}(U)-module with generators x_{1},...,x_{m}. The kernel of Λ_{L}(U) → Λ_{L}^{[p]}(U) is generated by elements ı(x_{1}),...,ı(x_{m}) which are in the center of Λ_{L}(U). But ı(x_{i}) ≡ x_{i}^{p} modΛ_{L,p−1} , so by the Poincare-Birkhoff-Witt theorem Λ_{L}^{[p]}has local generators j̃(x_{1})^{i_{1}}...j̃(x_{m})^{i_{m}} for 0 ≤ i_{l} < p. Hence j(x_{1})^{i_{1}}...j(x_{m})^{i_{m}} for 0 ≤ i_{l} < p locally generate Λ_{L} as an S^{•}(F_{X}^{∗}L)-module and Λ̃_{L} is locally free of rank p^{m}.

Lemma 4.5 shows that if L is smooth then ı induces an O_{X′}-linear map L′→ F_{X/S,∗}Λ_{L} and a homomorphism of sheaves of O_{X′}-algebras

S^{•}(L′) → F_{X/S,∗}(Z(Λ_{L})) ⊂ Λ′_{L}:= F_{X/S,∗}Λ_{L}.

In particular, it makes Λ′_{L} into a quasi-coherent sheaf of S^{•}(L′)-modules. This sheaf defines on 𝕍(L′) a quasi-coherent sheaf of O_{𝕍(L′)}-algebras Λ̃′_{L}. Note that by construction

π′_{∗}Λ̃′_{L} = F_{X/S,∗}Λ_{L} = F_{X/S,∗}π_{∗}Λ̃_{L} = π′_{∗}F̃_{X/S,∗}Λ̃_{L},

so we have

Λ̃′_{L} = F̃_{X/S,∗}Λ̃_{L}.

By an explicit computation as in Lemma 4.6 one can prove the following theorem:

Theorem 4.7. Assume that X/S is smooth of relative dimension d and L is smooth of rank m. Then Λ̃′_{L} is a locally free O_{𝕍(L′)}-module of rank p^{m+d}.

By [BMR] in the special case when L = T_{X/S}is the standard O_{S}-Lie algebroid on X, the sheaf Λ̃′_{L} is a sheaf of Azumaya O_{𝕍(L′)}-algebras. In this case we have a canonical splitting

F̃_{X/S}^{∗}Λ̃′_{L} ≃ E nd_{O_{𝕍(F{∗ \atop X/S}L′)}}Λ̃_{L}.

4.3 Relation with groupoid schemes

This subsection contains a quick tour on relation between Lie algebroids and groupoid schemes of height ≤ 1. This is analogous to the well-known relation between restricted Lie algebras and group schemes of height ≤ 1.

Let us recall that a groupoid is a small category in which every morphism is an isomorphism. Let X and R be S-schemes. An S-groupoid scheme G is a quintuple of S-maps s,t : R → X (“source and target objects”), c: R ×_{(s,t)}R → R (“composition”), e: X → R (“identity map”) and i: R → R (“inverse map”) such that for every S-scheme T the quintuple s(T), t(T), c(T), e(T) and i(T) defines in a functorial way a groupoid with morphisms R(T) and objects X(T). For an S-groupoid scheme G we denote by J the kernel of s_{∗}O_{R} → O_{X}. We say that G is infinitesimal if s is an affine homeomorphism and J is a nilpotent ideal. An infinitesimal S-groupoid scheme is of height ≤ 1 if (s,t) : R → X ×_{S}X factors through the first Frobenius neighbourhood of the diagonal (i.e., through X ×_{X^{(1/S)}}X). An S-groupoid scheme is called finite (flat) if s is finite (respectively, flat). If X is smooth over a perfect field k then restricted k-Lie subalgebras L of the standard k-Lie algebroid T_{X/k} such that T_{X/k}/L is locally free are in bijection with finite flat height 1 morphisms X → Y (see [Ek, Proposition 2.4]). Note that a sheaf of restricted k-Lie subalgebras of T_{X/k}is automatically a restricted k-Lie subalgebroid of T_{X/k}. So the following proposition generalizes the above fact (and it corrects [Ek, Proposition 2.3]):

Proposition 4.8. Let X/S be a smooth morphism. Assume that for every point x ∈ X the set t(s^{−1} (x)) is contained in an affine open subset of X. Then there exists an equivalence of categories between the category of finite flat S-groupoid schemes of height ≤ 1 with X/S as a scheme of objects and with locally free “conormal sheaf” J /J^{2} and the category of smooth restricted O_{S}-Lie algebroids on X/S.

Proof. We sketch the proof leaving details to the reader. If G is a finite, flat, infinitesimal S-groupoid scheme then we define L as the Lie algebra of this groupoid, i.e., the dual of J /J^{2}. It has a natural structure of a sheaf of restricted O_{S}-Lie algebras. Since G has height ≤ 1, L is equipped with the anchor map. In the other direction, to a smooth restricted O_{S}-Lie algebroid L on X/S we associate Λ_{L}^{[p]}which comes with a canonical homomorphism of O_{S}-algebras Λ_{L}^{[p]}→ Λ_{T_{X/S}}^{[p]}. But

Λ_{T_{X/S}}^{[p]}is an O_{S}-subalgebra of the sheaf of rings of “true” differential operators and the “morphisms” R of the groupoid scheme can be defined as the spectrum of the dual of Λ_{L}^{[p]}.

4.4 Modules over restricted Lie algebroids

If E is a module over a restricted O_{S}-Lie algebroid L then ∇: L → E nd_{O_{S}}E leads to a morphism

ψ : L → E nd_{O_{S}}E

defined by sending x to (∇(x))^{p} − ∇(x^{[p]}) for x ∈ L. Let us set α_{x}^{0}(f) = f and (∇(x))^{0}(e) = e. Using Leibniz’ rule one can easily see that

(∇(x))^{m}(fe) = ∑_{i=0}^{m} \binom{m}{i} α_{x}^{i}(f)(∇(x))^{m−i}(e)

for any sections f ∈ O_{X}(U), x ∈ L(U) and e ∈ E(U) and any open subset U ⊂ X. In particular, we have

(∇(x))^{p}(fe) = α_{x}^{p}(f)e + f(∇(x))^{p}(e).

Since

∇(x^{[p]})(fe) = α_{x^{[p]}}(f)e + f∇(x^{[p]})(e)

and α_{x^{[p]}}= α_{x}^{p} we see that for any x ∈ L the image ψ(x) is O_{X}-linear. So we can consider ψ as the mapping ψ : L → E nd_{O_{X}}E. This mapping is called the p-curvature morphism of the L-module E. The following lemma generalizes [Ka1, Proposition 5.2]:

Lemma 4.9. The p-curvature morphism ψ : L → E nd_{O_{X}}E is F_{X}^{∗}-linear and its image commutes with the image of ∇ in E nd_{O_{S}}E.

Proof. By Lemma 4.1 we know that ψ is additive and its image commutes with the image of ∇. So it is sufficient to check that

ψ(fx) = f^{p}ψ(x)

for all local sections f ∈ O_{X} and x ∈ L. Applying Hochschild’s identity to elements f and ∇(x) in E nd_{O_{S}}E we obtain

(∇(fx))^{p} = f^{p}∇(x)^{p} + (ad(f∇(x)))^{p−1} (f)∇(x) = f^{p}∇(x)^{p} + α_{fx}^{p−1} (f)∇(x).

From the definition of a restricted O_{S}-Lie algebroid and O_{X}-linearity of ∇ : L → E nd_{O_{S}}E we have

∇((fx)^{[p]}) = f^{p}∇(x^{[p]}) + α_{fx}^{p−1} (f)∇(x).

Subtracting these equalities we get the required identity.

By the above lemma ψ defines an O_{X}-linear map L → F_{X,∗}E nd_{O_{X}}E and hence the adjoint O_{X}-linear map

ψ_{L}: F_{X}^{∗}L → E nd_{O_{X}}E,

which will also be called the p-curvature morphism. Note that ψ_{L} makes E into an F_{X}^{∗}L-coHiggs sheaf (integrability of the F_{X}^{∗}L-coHiggs field follows from the lemma). Another way of seeing it is that if E is a Λ_{L}-module then by Lemma 4.5 it has a structure of S^{•}(F_{X}^{∗}L)-module given by the p-curvature ψ_{L}.

Example 4.10. Let L be a smooth trivial restricted O_{S}-Lie algebroid on X. Then giving an L-module is equivalent to giving S^{•}L-module structure on E. In this case the p-curvature morphism ψ_{L}: F_{X}^{∗}L → E nd_{O_{X}}E is obtained by composing the canonical inclusion F_{X}^{∗}L → S^{p}L with the action map S^{p}L → E nd_{O_{X}}E.

Example 4.11. Let X be a smooth S-scheme and let us fix λ ∈ H^{0}(O_{S}). Let us denote by T_{X/S}^{λ}the restricted O_{S}-Lie algebroid structure on T_{X/S}with Lie bracket [·,·]_{T_{X/S}^{λ}}=

λ · [·,·]_{T_{X/S}}, anchor map α given by multiplication by λ and the p-th power operation given by

x_{T_{X/S}^{λ}}^{[p]}= λ^{p−1} · x_{T_{X/S}}^{[p]}

for x ∈ T_{X/S}. The apparently strange formula for the p-th power operation comes from the requirement

α(x_{T_{X/S}^{λ}}^{[p]}) = λ · x_{T_{X/S}^{λ}}^{[p]}= (α(x))^{[p]}= λ^{p} · x^{[p]}.

Giving a T_{X/S}^{λ}-module is equivalent to giving a coherent O_{X}-module E with an integrable λ-connection ∇: E → E ⊗_{O_{X}}Ω_{X/S}. In this case the above defined p-curvature of the T_{X/S}^{λ}-module gives a more conceptual approach to the p-curvature of an O_{X}- module with λ-connection (E,∇) defined in [LP, Definition 3.1].

Remark 4.12. If ∇_{1} and ∇_{2} are two L-module structures on E then φ = ∇_{1}−∇_{2} : L → E nd_{O_{S}}E is O_{X}-linear and its image lies in E nd_{O_{X}}E. In particular, if the p-curvatures ψ_{L}(∇_{1}) and ψ_{L}(∇_{2}) are equal then φ is zero on the kernel of Λ_{L} → Λ_{L}^{[p]}and hence it induces the homomorphism Λ_{L}^{[p]}→ E nd_{O_{X}}E of O_{X}-algebras.

Definition 4.13. We say that the p-curvature of (E,∇) is nilpotent of level less than l if (E,∇) satisfies one of the following equivalent conditions:

1. There exists a filtration M^{m} = 0 ⊂ M^{m−1} ⊂ ... ⊂ M^{0} = (E,∇) of length m ≤ l such that the associated graded L-module has p-curvature 0.

2. For any open subset U ⊂ X and any collection {x_{1},...,x_{l}} of sections of L(U) we have ψ_{L}(x_{1})...ψ_{L}(x_{l}) = 0.

We say that he p-curvature of (E,∇) is nilpotent of level l if it is nilpotent of level less than (l + 1) but not nilpotent of level less than l (for l = 0 we require simply that the p-curvature is nilpotent of level less than 1).

4.5 Deformation of Hitchin’s morphism for restricted Lie algebroids

This subsection contains a partial generalization of the results of Laszlo and Pauly [LP] to higher dimensions. Note that in general, the direct analogue of their [LP, Proposition 3.2] is not expected to be true.

Let S be a noetherian scheme of characteristic p and let X → S be a flat, projective family of d-dimensional varieties satisfying Serre’s condition (S_{2}). Let L be a smooth restricted O_{S}-Lie algebroid on X. Let us fix a polynomial P and a relatively ample line bundle on X/S. We define the moduli stack as a lax functor between 2-categories by

M^{L}(X/S,P) : (Sch/S) → (groupoids)
T  →  M (T),

where M (T) is the category whose objects are T -flat families of pure d-dimensional L-modules with Hilbert polynomial P on the fibres of X_{T} → T, and whose morphisms are isomorphisms of coherent sheaves. One can prove that M^{L}(X/S,P) is an algebraic stack for the fppf topology on (Sch/S). If M is a coherent O_{X}-module considered as an O_{S}-Lie algebroid on X with the trivial structure, then the corresponding moduli stack is denoted by M_{Dol}^{M}(X/S,P). The p-curvature defines a morphism of stacks

\begin{aligned}Ψ_{L}: & M^{L}(X/S,P) & → & M_{Dol}^{F_{X}^{∗}L}(X/S,P) \\ & (E,∇) & → & (E,ψ(∇)). \end{aligned}

Let us consider the deformation L^{R}of L over an affine line 𝔸^{1} over 𝔽_{p} (see Subsection 3.6). For simplicity of notation, in the following we skip writing 𝔽_{p}. L^{R} has a natural structure of a smooth restricted O_{S×𝔸^{1}} -Lie algebroid on X × 𝔸^{1} with the p-th power

operation given by ·_{L^{R}}^{[p]}= p_{1}^{∗}( ·_{L}^{[p]})⊗ t^{p−1} . We can treat L^{R} as a family of restricted

O_{S}-Lie algebroids on X parameterized by 𝔸^{1}. For example, if X/S is smooth and we fix λ ∈ H^{0}(O_{S}) = Hom(S,𝔸^{1}) then for L = T_{X/S}with the standard restricted O_{S}-Lie algebroid structure, the pull-back of L^{R} along (id_{S},λ) : S → S × 𝔸^{1} gives T_{X/S}^{λ} from Example 4.11. We have a commutative diagram

\xymatrix{M^{L^{R}}(X × 𝔸^{1}/S × 𝔸^{1},P) \ar[r]^{Ψ_{L^{R}}} & M_{Dol}^{F_{X×𝔸^{1}}^{∗} L^{R}}(X × 𝔸^{1}/S × 𝔸^{1},P) \ar[r]^{H_{F{∗ \atop X×𝔸^{1}} L^{R}}} & 𝕍^{F_{X×𝔸^{1}}^{∗} L^{R}}(X × 𝔸^{1}/S × 𝔸^{1},r) \\ M_{Dol}^{L}(X/S,P) \ar[r]^{H_{L}} \ar[u] & 𝕍^{L}(X/S,r) \ar[r] & 𝕍^{F_{X}^{∗}L}(X/S,r), \ar[u]}

where the vertical arrows are induced by the base change via the zero section 0 : S → S × 𝔸^{1} and 𝕍^{L}(X/S,r) → 𝕍^{F_{X}^{∗}L}(X/S,r) is the canonical morphism induced by the absolute Frobenius on X. Roughly speaking, this diagram says that the p-curvature morphism Ψ_{L} deforms to the p-th power of the Hitchin morphism.

Let N ilp^{L}(X/S,P) be the substack of M^{L}(X/S,P) of L-modules with nilpotent pcurvature. By definition Ψ̃ maps N ilp^{L}(X/S,P) into {0} × 𝔸^{1} = 𝔸^{1} and the corresponding map will be still denoted by Ψ̃. The stacks M^{L}(X/S,P) and N ilp^{L}(X/S,P) contain open substacks M^{L,ss}(X/S,P) and N ilp^{L,ss}(X/S,P) parametrizing slope semi-stable objects (openness of semistability is a standard exercise left to the reader).

By boundedness theorem (see [La1]) these substacks are of finite type. Theorem 5.1 implies that the morphisms Ψ̃^{ss} : M^{L^{R},ss}(X/S,P) → 𝕍^{F_{X}^{∗}L}(X/S,r) × 𝔸^{1} and N ilp^{L,ss}(X/S,P) → 𝔸^{1} are universally closed.

Let N ilp _{l}^{L,ss}(X/S,P) be the substack of N ilp^{L,ss}(X/S,P) parametrizing objects with nilpotent p-curvature of level < l. Note that it is a closed substack, since nilpotence of level < l is a closed condition. Therefore N ilp _{l}^{L,ss}(X/S,P) → 𝔸^{1} is universally closed (see [LP, Proposition 5.1] for a special case of this assertion). Let us note that the fiber of N ilp _{1}^{L,ss}(X/S,P) → 𝔸^{1} over 0 is equal to the moduli stack of semistable L-coHiggs sheaves (E,θ) with vanishing p-curvature (see Example 4.10). In particular, [LP, Remark 5.1] is false. On smooth projective curves of genus g ≥ 2 the proof of [LP, Lemma 5.1] shows that a vector bundle with a λ-connection of level less than l can be extended to a Higgs bundle with the Higgs field θ satisfying θ^{l} = 0. In particular, for l = 1 we get the zero Higgs field.

So one could hope that in this case, e.g., if { \atop N ilp}_{1}^{L,ss}(X/S,P) → 𝔸^{1} is the open substack of N ilp _{1}^{L,ss}(X/S,P) → 𝔸^{1}, which over 0 is the moduli substack of semistable

sheaves then { \atop N ilp}_{1}^{L,ss}(X/S,P) → 𝔸^{1} is also universally closed as suggested by [LP, Remark 5.1]. However, this expectation is false. In case of a smooth projective curve X of genus g ≥ 2 there exists a semistable bundle E whose Frobenius pull back F_{X}^{∗}E is not semistable. But F_{X}^{∗}E carries a canonical connection ∇_{can} and (F_{X}^{∗}E,∇_{can}) is semistable. After pulling back via X_{K} → X, where K = k((t)), and twisting by t, this provides a semistable vector bundle with a t-connection on X_{K} which cannot be extended to a semistable family on X_{k[[t]]}so that the Higgs field at the special fibre vanishes. Otherwise, we would get a contradiction with openness of the usual semistability of vector bundles.

5 Deformations of semistable sheaves and the Lan-Sheng-Zuo conjecture

5.1 Langton’s theorems

Let R be a discrete valuation ring with maximal ideal m generated by π ∈ R. Let K be the quotient field of R and let us assume that the residue field k = R/m is algebraically closed. Let X → S = SpecR be a smooth projective morphism and let L be a smooth O_{S}-Lie algebroid on X. Let us fix a collection (D_{0},D_{1},...,D_{n−1} ) of n relatively nef divisors on X/S such that D_{0} = D_{1}. In the following stability of sheaves on the fibers of X → S is considered with respect to this fixed collection. The following theorem generalizes well known Langton’s theorem [Lt, Theorem 2)]. We recall the proof as it is not available in the generality that we need. The notation introduced in this proof will be also used in proof of Theorem 5.5.

Theorem 5.1. Let F be an R-flat O_{X}-coherent L-module of relative pure dimension n such that the L_{K}-module F_{K} = F ⊗_{R}K is slope semistable. Then there exists an L-submodule E ⊂ F such that E_{K} = F_{K} and E_{k} is a slope semistable L_{k}-module on X_{k}.

Proof. First let us note that we can assume that F_{k} is torsion free as an O_{X_{k}}-module (this follows, e.g., from [HL, Proposition 4.4.2] or can be proven using a similar method as below). We use without warning the fact that for an R-flat F the degrees of F_{K} and F_{k} with respect to (D_{1},...,D_{n−1} ) coincide. This follows from the fact that F has a finite locally free resolution on X and intersection products are compatible with specialization (see [SGA6, Expose X, Appendice]). Let us set F^{0} := F. If F_{k}^{0} is not slope semistable then we take the maximal destabilizing L-submodule B^{0} in F_{k}^{0} and denote by F^{1} the kernel of the composition F^{0} → F_{k}^{0} → G^{0} := F_{k}^{0}/B^{0}. If F_{k}^{1} is semistable then we get the required submodule of F. Otherwise, we repeat the same procedure for F^{1}. In this way we construct a sequence of L-modules F = F^{0} ⊃ F^{1} ⊃ F^{2} ⊃ ... and the main point of the proof is to show that this process cannot continue indefinitely. Let us assume otherwise. First, let us note that we have short exact sequences

0 → G^{n} → F_{k}^{n+1} → B^{n} → 0,

where G^{n} = F_{k}^{n}/B^{n}. Let C^{n} be the kernel of the composition B^{n+1} → F_{k}^{n+1} → B^{n}. If C_{n} = 0 then B_{n+1} ⊂ B_{n} and hence μ(B^{n+1} ) ≤ μ(B^{n}). If C^{n} ≠ 0 then

μ(C^{n}) ≤ μ_{max}(G^{n}) < μ(B^{n}),

where the first inequality comes from the fact that C^{n}⊂ G^{n} and the second one follows from the fact that B^{n} ⊂ F_{k}^{n} is the maximal destabilizing subsheaf and G^{n} = F_{k}^{n}/B^{n}. We claim that μ(B^{n+1} ) < μ(B^{n}). If μ(C^{n}) ≥ μ(B^{n+1} ) then this inequality follows from the above inequality. If μ(C^{n}) < μ(B^{n+1} ) then μ(B^{n+1} ) < μ(B^{n+1} /C^{n}). But B^{n+1} /C^{n} is isomorphic to a subsheaf of B^{n} and B^{n} is semistable, so in this case we also have μ(B^{n+1} ) < μ(B^{n}). Therefore the sequence {μ(B^{n})} is non-increasing. But μ(B^{n+1} ) < μ(B^{n}) is possible for only finitely many n since r!μ(B^{n}) ∈ ℤ are bounded below by r!μ(F_{k}). Therefore for all large n we have C^{n} = 0, i.e., we have inclusions B^{n} ⊃ B^{n+1} ⊃ ... and G^{n} ⊂ G^{n+1} ⊂ .... For sufficiently large n these sequences consist of torsion free sheaves with the same slope, so they must stabilize to B and G, respectively. Then F_{k}^{n} = B⊕G for n ≫ 0. Set R̂:= lim_{←}R/π^{n}R and let K̂ be the quotient field of R̂. Note that F /F^{n} is R/π^{n}-flat and as O_{X_{k}}-module has a filtration with quotients isomorphic to G. Then Q̂:= lim_{←}F /F^{n} is a destabilizing quotient of FK̂. But the Harder–Narasimhan filtration is stable under base field extension and therefore F_{K} is also unstable, contradicting our assumption.

Our exposition of proof of Langton’s theorem is based on [HL] with some small changes (one of the inequalities in proof of [HL, Theorem 2.B.1] is false and we need to give a slightly different argument).

Note that in the above theorem we allow the case when all D_{i}’s are zero. In this case we claim that there exists an L-submodule E ⊂ F such that E_{K} = F_{K} and E_{k} is torsion free as O_{X_{k}}-module (by definition slope semistable sheaves are torsion free!).

Let us recall that every slope semistable L-module E has a Jordan–Hölder filtration E_{0} = 0 ⊂ E_{1} ⊂ ... ⊂ E_{m} = E by L-submodules such that the associated graded sheaf Gr(E) = ⊕E_{i}/E_{i−1} is slope polystable, i.e., a direct sum of slope stable (torsion free) L-modules of the same slope. The following theorem is motivated by theory of moduli spaces and it generalizes [Lt, Theorem 1)].

Theorem 5.2. Assume that the collection (D_{0},D_{1},...,D_{n−1} ) consists of relatively ample divisors. Let F be an R-flat O_{X}-coherent L-module of relative pure dimension n such that the L_{K}-module F_{K} = F ⊗_{R}K is slope semistable. Let E_{1} and E_{2} be Lsubmodules of F such that (E_{1})_{K}= (E_{2})_{K}= F_{K}, (E_{1})_{k} and (E_{2})_{k} are slope semistable. Then the reflexivizations of the associated graded slope polystable sheaves Gr((E_{1})_{k}) and Gr((E_{2})_{k}) are isomorphic. Moreover, if at least one of (E_{1})_{k} and (E_{2})_{k} is slope stable then there exists an integer n such that E_{1} = π^{n}E_{2}.

Proof. We prove only the second part, leaving proof of the first one to the reader. Assume that (E_{1})_{k} is slope stable. Consider the discrete valuation ring O_{X,η}, where η is the generic point of X_{k}. Multiplying E_{1} by some power of π, we can assume that E_{1}⊗_{O_{X}}O_{X,η} ⊂ E_{2}⊗_{O_{X}}O_{X,η} and the induced map E_{1}⊗k(η) → E_{2}⊗k(η) is non-zero. But E_{1} and E_{2} are torsion free so E_{1} ⊂ E_{2}. Since (E_{1})_{k} is slope stable the non-zero map (E_{1})_{k} → (E_{2})_{k} between slope semistable sheaves of the same slope must be an inclusion. Since the Hilbert polynomials of (E_{1})_{k} and (E_{2})_{k} coincide (from flatness of E_{1} and E_{2}), it must be an isomorphism.

Let Y be a projective scheme over a field k and let L_{Y} be a k-Lie algebroid on Y. Let us fix an ample line bundle O_{Y}(1) on Y. Let Coh_{d}^{L}(Y) be the full subcategory of the category of L-modules which are coherent as O_{Y}-modules and whose objects are sheaves supported in dimension ≤ d. Then we can consider the quotient category Coh_{d,d′}^{L}(Y) := Coh_{d}^{L}(Y)/Coh_{d′−1}^{L} (Y). For any object of Coh_{d,d′}^{L}(Y) one can define its Hilbert polynomial which can be used to define notion of (semi)stability in this category.

We can generalize Langton’s theorem to singular schemes at the cost of dealing with only one ample polarization. In this case compatibility of intersection product with specialization follows from computation of the Hilbert polynomial. One can also generalize Theorem 5.1 so that it works for other kinds of stability as defined above. Let X → SpecR be a projective morphism with relatively ample line bundle O_{X}(1) and let L be a smooth O_{S}-Lie algebroid on X. The following Langton’s type theorem generalizes [Si4, Theorem 10.1] and [HL, Theorem 2.B.1]:

Theorem 5.3. Let F be an R-flat O_{X}-coherent L-module of relative dimension d. Assume that the L_{K}-module F_{K} = F ⊗_{R}K is pure of dimension d and semistable in Coh_{d,d′}^{L}(X_{K}) for some d′< d. Then there exists an L-submodule E ⊂ F such that E_{K} = F_{K} and E_{k} is semistable in Coh_{d,d′}^{L}(X_{k}).

Proof. The proof is almost the same as the proof of [HL, Theorem 2.B.1]. However, there are a few small problems that we meet in the proof. The first one is that we need to define reflexive hulls of sheaves on the special fiber X_{k}. This can be done by embedding X into a fixed smooth R-scheme (e.g., use some multiple of the polarization O_{X}(1) to embedd X into some projective space over R). The second problem is the same as before: one of the inequalities in proof of [HL, Theorem 2.B.1] is false and we need to use a slightly different argument similar to the one used in proof of Theorem 5.1. We sketch the necessary changes using the notation of proof of [HL, Theorem 2.B.1]. If C^{n} ≠ 0 then we only have

p(C^{n}) ≤ p_{max}(G^{n}) < p(B^{n})modℚ[T]_{δ−1} .

Hence if p_{d,δ}(C^{n}) ≥ p_{d,δ}(B^{n+1} ) then p_{d,δ}(B^{n+1} ) < p_{d,δ}(B^{n}). If p_{d,δ}(C^{n}) < p_{d,δ}(B^{n+1} ) then we have p_{d,δ}(B^{n+1} ) < p_{d,δ}(B^{n+1} /C^{n}) ≤ p_{d,δ}(B^{n}). This proves that if C^{n} ≠ 0 then we always have p_{d,δ}(B^{n+1} ) < p_{d,δ}(B^{n}) as needed in the argument. The last problem is the use of Quot-schemes in [HL], which do not exist as projective schemes in our situation. This can be solved as in proof of Theorem 5.1.

Theorem 5.4. Let F be an R-flat O_{X}-coherent L-module of relative pure dimension d such that the L_{K}-module F_{K} = F ⊗_{R}K is semistable in Coh_{d,d′}^{L}(X_{K}) for some d′< d. Let E_{1} and E_{2} be L-submodules of F such that (E_{1})_{K} = (E_{2})_{K} = F_{K}, (E_{1})_{k} and (E_{2})_{k} are semistable in Coh_{d,d′}^{L}(X_{k}) and at least one of them is stable. Then there exists an integer n such that E_{1} = π^{n}E_{2} in Coh_{d,d′}^{L}(X_{K}).

5.2 Semistable filtrations on sheaves with connection

Let L be a smooth Lie algebroid on a normal projective variety X defined over an algebraically closed field k. Let us consider a torsion free coherent O_{X}-module E with an integrable d_{Ω_{L}} -connection ∇ (i.e., an L-module whose underlying sheaf is coherent and torsion free as an O_{X}-module). We say that a filtration E = N^{0} ⊃ N^{1} ⊃ ... ⊃ N^{m}= 0 satisfies Griffiths transversality if ∇(N^{i}) ⊂ N^{i−1} ⊗_{O_{X}}Ω_{L} and the quotients N^{i}/N^{i+1} are torsion free. For every such filtration the associated graded object Gr_{N}(E) := ⨁ _{i}N^{i}/N^{i+1} carries a canonical Ω_{L}-Higgs field θ defined by ∇. Note that (Gr_{N}(E),θ) is a system of L-Hodge sheaves. A convenient way of looking at this is by means of the Rees construction. More precisely, if N^{•} is a Griffiths transverse filtration on (E,∇) then we can consider the subsheaf

ξ(E,N^{•}) := ∑ t^{−i} N^{i}⊗ O_{X×𝔸^{1}} ⊂ p_{X}^{∗}E

on X × 𝔸^{1}. By Griffiths transversality of the filtration N^{•} the connection t∇ on ξ(E,N^{•})|_{X×𝔾_{m}} extends to a t-d_{Ω_{L}} -connection on X ×𝔸^{1} (i.e., we get an L^{R}-module on X × 𝔸^{1}). In the limit as t → 0 we get exactly the above described system of L-Hodge sheaves (Gr_{N}(E),θ).

In the remainder of this section to define semistability we use a fixed collection (D_{0},D_{1},...,D_{n−1} ) of nef divisors such that D_{0} = D_{1}. After Simpson [Si4] we say that a Griffiths transverse filtration N^{•} on (E,∇) is slope gr-semistable if the associated Ω_{L}-Higgs sheaf (Gr_{N}(E),θ) is slope semistable. A partial L-oper is a triple (E,∇,N^{•}) consisting of a torsion free coherent O_{X}-module E with an integrable d_{Ω_{L}} -connection ∇ and a Griffiths transverse filtration N^{•} which is slope gr-semistable.

Theorem 5.5. If (E,∇) is slope semistable then there exists a canonically defined slope gr-semistable Griffiths transverse filtration N^{•} on (E,∇) providing it with a partial L-oper structure. This filtration is preserved by the automorphisms of (E,∇).

Proof. Let R be a localization of 𝔸^{1} at 0 and let L̃ be the smooth Lie algebroid on X_{R} = X ×_{k}SpecR obtained by restricting of L^{R} from X ×_{k}𝔸^{1}. Consider the trivial filtration E = N^{0} ⊃ N^{1} = 0. It satisfies Griffiths tranversality so we can associate to it via the Rees construction and restricting to X_{R}, an R-flat O_{X_{R}}-coherent L̃-module F^{0} = F (in fact F = (p_{X}^{∗}E,π∇)). Now suppose that we have defined an L̃-submodule F^{n} ⊂ F coming by restriction from the Rees construction associated to a Griffiths transverse filtration N_{n}^{•} of E. If the associated Ω_{L}-Higgs sheaf F_{k}^{n} = (Gr_{N_{n}}(E),θ_{n}) is semistable then we get the required filtration. Otherwise, we consider its maximal destabilizing Ω_{L}-Higgs subsheaf B^{n}. But (Gr_{N_{n}}(E),θ_{n}) is a system of Ω_{L}-Hodge sheaves, so by Corollary 3.5 B^{n} is also a system of Ω_{L}-Hodge sheaves. Let us write B^{n} = ⨁ B_{m}^{n}, where B_{m}^{n} ⊂ Gr_{N_{n}}^{m}(E) = N_{n}^{m}/N_{n}^{m+1}. Then we can define a new Griffiths transverse filtration N_{n+1}^{•} on E by setting

N_{n+1}^{m} := ker( E → \frac{E/N_{n}^{m}}{B_{m−1}^{n}}) .

Let F^{n+1} denote the restriction to X_{R} of the L^{R}-module associated by the Rees construction to N_{n+1}^{•}. We need to prove that this procedure cannot continue indefinitely. To show it, it is sufficient to check that we follow the same procedure as the one described in the proof of Theorem 5.3. By construction πF^{n} ⊂ F^{n+1} ⊂ F^{n} and in particular F_{K}^{n+1} = F_{K}^{n}. On the other hand, on the special fiber of X_{R}→ SpecR we have a short exact sequence

0 → F_{k}^{n}/B^{n} → F_{k}^{n+1} = (Gr_{N_{n+1}}(E),θ_{n+1} ) → B^{n} → 0

coming from the definition of the filtration N_{n+1}^{•}. This shows that πF^{n} is the kernel of the composition F^{n+1} → F_{k}^{n+1} → B^{n}. But then F^{n+1} is the kernel of the composition F^{n} → F_{k}^{n} → F_{k}^{n}/B^{n}. Now the proof of Theorem 5.3 shows that this procedure must finish. Since the Harder–Narasimhan filtration is canonically defined, the above described procedure is also canonical and the obtained filtration is preserved by the automorphisms of (E,∇).

In the following the canonical filtration N^{•} from Theorem 5.5will be called Simpson’s filtration of (E,∇) and denoted by N_{S}^{•}. The reason is that apart from many spectacular results due to Simpson in non-abelian Hodge theory, the construction of the filtration described in the proof of the above theorem was done by Simpson in [Si4, Section 3] for the usual Higgs bundles on complex projective curves. However, our proof of the fact that the procedure stops is different.

Theorem 5.5 generalizes [Si4, Theorem 2.5] to higher dimensions as asked for at the end of [Si4, Section 3]. Indeed, in the characteristic zero case every vector bundle with an integrable connection has vanishing Chern classes. In particular, any saturated subsheaf of such a vector bundle which is preserved by the connection (is locally free and) has vanishing Chern classes. So any vector bundle with an integrable connection is slope semistable (with respect to an arbitrary polarization). This argument fails in the logarithmic case which shows that the above theorem is a correct analogue in this case.

Note that there can be many slope gr-semistable filtrations providing (E,∇) with a partial L-oper structure. This depends on the choice of the Griffiths transverse filtration at the beginning of our procedure (in the proof of Theorem 5.5 we used the canonical choice). In general, all the obtained filtrations are related as described by the following corollary which follows from Theorem 5.2:

Corollary 5.6. If N^{•} and M^{•} are two slope gr-semistable Griffiths transverse filtrations on (E,∇) then the reflexivizations of the associated-graded slope polystable Ω_{L}- Higgs sheaves obtained from their Jordan–Hölder filtrations are isomorphic. In particular, if the associated Ω_{L}-Higgs sheaf is slope stable then (E,∇) carries a unique gr-semistable Griffiths transverse filtration.

The above corollary generalizes [Si4, Corollary 4.2]. Note that Simpson’s proof does not work so easily in our situation as in higher dimensions we do not have appropriate moduli spaces at our disposal. Let us also note that any slope gr-semistable filtration can be refined so that the associated graded Ω_{L}-Higgs sheaf is slope polystable (in which case its reflexivization is uniquely determined by (E,∇) up to an isomorphism).

As an immediate application of Theorem 5.5 we also get the following interesting corollary:

Corollary 5.7. Let L be a smooth trivial Lie algebroid. Let (E,θ) be a torsion free, slope semistable Ω_{L}-Higgs sheaf on X. Then we can deform it to a slope semistable system of Ω_{L}-Hodge sheaves.

5.3 Higgs-de Rham sequences

Let k be an algebraically closed field of characteristic p > 0. Let X be a smooth projective k-variety of dimension n that can be lifted to a smooth scheme X over W_{2}(k). Let MIC_{p−1} (X/k) be the category of O_{X}-modules with an integrable connection whose p-curvature is nilpotent of level less or equal to p − 1. Similarly, let HIG_{p−1} (X/k) denote the category of Higgs O_{X′}-modules with a nilpotent Higgs sheaf of level less or equal to p − 1. In this case one of the main results of Ogus and Vologodsky (see [OV, Theorem 2.8]) says that:

Theorem 5.8. The Cartier operator

C_{X /S} : MIC_{p−1} (X/k) → HIG_{p−1} (X′/k) defines an equivalence of categories with quasi-inverse

C_{X /S}^{−1} : HIG_{p−1} (X/k) → MIC_{p−1} (X′/k).

A small variant of the following lemma can be found in proof of [OV, Theorem 4.17]:

Lemma 5.9. Let (E,θ) ∈ HIG_{p−1} (X′/S). Then

[C_{X /S}^{−1} (E)] = F_{X/S}^{∗}[E],

where [·] denotes the class of a coherent O_{X}-module in Grothendieck’s K-group K_{0}(X).

As a corollary to Theorem 5.8 and Lemma 5.9 we get the following:

Corollary 5.10. Let (E,θ) be a torsion free Higgs sheaf with nilpotent Higgs field of level less than p. Then it is slope semistable if and only if the corresponding sheaf with integrable connection (V,∇) := C_{X /S}^{−1} (E,θ) is slope semistable.

Now let (E,θ) be a rank r torsion free Higgs sheaf with nilpotent Higgs field. Let us assume that r ≤ p so that level of nilpotence of (E,θ) is less than p. Let us recall the following definition taken from [LSZ].

Definition 5.11. A Higgs–de Rham sequence of (E,θ) is an infinite sequence

\xymatrix{ & (V_{0},∇_{0}) \ar[dr]^{Gr_{N_{0}}} &  & (V_{1},∇_{1}) \ar[dr]^{Gr_{N_{1}}} \\ (E_{0},θ_{0}) = (E,θ) \ar@{-->}[ur]^{C^{−1}} &  & (E_{1},θ_{1}) \ar@{-->}[ur]^{C^{−1}} &  & ...}

in which C^{−1} = C_{X /S}^{−1} is the inverse Cartier transform, N_{i}^{•} is a Griffiths transverse filtration of (V_{i},∇_{i}) and (E_{i+1} := Gr_{N_{i}}(V_{i}),θ_{i+1} ) is the associated Higgs sheaf.

The following theorem proves the conjecture of Lan-Sheng-Zuo [LSZ, Conjecture 2.8]:

Theorem 5.12. If (E,θ) is slope semistable then there exists a canonically defined Higgs–de Rham sequence

\xymatrix{ & (V_{0},∇_{0}) \ar[dr]^{Gr_{N_{S}}} &  & (V_{1},∇_{1}) \ar[dr]^{Gr_{N_{S}}} \\ (E_{0},θ_{0}) = (E,θ) \ar@{-->}[ur]^{C^{−1}} &  & (E_{1},θ_{1}) \ar@{-->}[ur]^{C^{−1}} &  & ...}

in which each (V_{i},∇_{i}) is slope semistable and (E_{i+1} ,θ_{i+1} ) is the slope semistable Higgs sheaf associated to (V_{i},∇_{i}) via Simpson’s filtration.

Proof. The proof is by induction on index i. Once we defined slope semistable (E_{i},θ_{i}), we can construct (V_{i},∇_{i}), which is slope semistable by Corollary 5.10. So by Theorem 5.5 there exists Simpson’s filtration on (V_{i},∇_{i}) and hence we can construct a slope semistable Higgs sheaf (E_{i+1} ,θ_{i+1} ). Since (E_{i+1} ,θ_{i+1} ) is a system of Hodge sheaves and r ≤ p, it satisfies the nilpotence condition required to define C^{−1} .

In the above theorem slope semistability is defined with respect to an arbitrary fixed collection (D_{1},...,D_{n−1} ) of nef divisors on X.

Acknowledgements. The author would like to thank Carlos Simpson for explaining to him Lemma 3.6. The author would also like to thank Yanhong Yang and the referee for pointing out a small problem with the first formulation of Corollary 5.6. Finally, the author would like to thank Kang Zuo for inviting him to the Johannes Gutenberg University Mainz to lecture on the material contained in the present paper and SFB/TRR 45 “Periods, Moduli Spaces and Arithmetic of Algebraic Varieties” for supporting his visit.

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Adrian Langer

Institute of Mathematics University of Warsaw ul. Banacha 2, 02-097 Warszawa Poland alan@mimuw.edu.pl