On the $p$-supports of a holonomic $\mathcal{D}$-module

For a smooth variety $Y$ over a perfect field of positive characteristic, the sheaf $D_Y$ of crystalline differential operators on $Y$ (also called the sheaf of $PD$-differential operators) is known to be an Azumaya algebra over $T^*_{Y'},$ the cotangent space of the Frobenius twist $Y'$ of $Y.$ Thus to a sheaf of modules $M$ over $D_Y$ one can assign a closed subvariety of $T^*_{Y'},$ called the $p$-support, namely the support of $M$ seen as a sheaf on $T^*_{Y'}.$ We study here the family of $p$-supports assigned to the reductions modulo primes $p$ of a holonomic $\mathcal{D}$-module. We prove that the Azumaya algebra of differential operators splits on the regular locus of the $p$-support and that the $p$-support is a Lagrangian subvariety of the cotangent space, for $p$ large enough. The latter was conjectured by Kontsevich. Our approach also provides a new proof of the involutivity of the singular support of a holonomic $\mathcal{D}$-module, by reduction modulo $p.$

La section 4 contient la réduction de la démonstration du théorème principal au cas de l'espace affine. On y utilise l'image directe des D X/S -modules.

Conventions
Schemes are assumed to be noetherian, positive characteristics to be non zero and morphisms of algebras to send 1 → 1. Local coordinates of a smooth scheme X/S mean local étale relative coordinates in the neighborhood of a point of X. As a rule we define notions and state results for left modules, we often omit to mention that they easily adapt to right modules.

D X/S
Let S be a scheme and let X be a smooth S-scheme of relative dimension n. Let the tangent sheaf T X/S be the O X -module dual to the locally free sheaf of relative differentials Ω 1 X/S , it is endowed with a Lie bracket. The ring of PDdifferential operators of X/S [6, §4], noted D X/S , is a sheaf of noncommutative rings on X. It is the enveloping algebra of the Lie algebroid T X/S [8, 1.2]. Thus D X/S is generated by the structure sheaf O X and the tangent sheaf T X/S , subject to relations f.
for f and ∂, ∂ ′ local sections of O X and T X/S respectively. Note that it is compatible with base change. In terms of local étale relative coordinates {x 1 , ..., x n }, for the sections {∂ 1 , ..., ∂ n } of T X/S , dual to {dx 1 , ..., dx n }, one has D X/S = I O X .∂ I , summing over non negative multi-indices. By definition of an enveloping algebra, endowing an O X -module with a compatible left D X/S -module structure or with an integrable connection are equivalent. Left multiplication by O X makes D X/S into a quasi-coherent O X -module. Moreover, Proposition 1.2.1. The sheaf of rings D X/S has a natural positive filtration D X/S = m≥0 D X/S,≤m , defined by D X/S,≤0 := O X and D X/S,≤m+1 := T X/S .D X/S,≤m + D X/S,≤m , whose associated graded sheaf of rings grD X/S is canonically isomorphic to O T * (X/S) , the structure sheaf of the cotangent bundle of X/S.
where the cotangent bundle of X/S is T * (X/S)/X := Spec X (Sym OX T X/S ), also denoted V (T X/S ) [22, 1.7.8]. Therefore D X/S is a sheaf of coherent noetherian rings [3, 2.2.5 and 3.1.2]. One has the familiar finiteness condition for modules, [21, 0.5.3] if and only if it is quasi-coherent as an O X -module and its module of sections over any open of an affine covering is a finitely generated left module over the ring of sections Furthermore on X affine the functor of global sections is an equivalence from the category of coherent left D X/S -modules to the category of finitely generated left modules over the global sections of D X/S [3, 3.1.3 (iii)]. Note also that coherence is preserved under base change.
Similar results hold for right D X/S -modules.
There is also a notion of good filtration on a coherent left D X/S -module [5, 5.2.3.]. Namely recall that by 1.2.1, D X/S is a positively filtered sheaf of rings, then Definition 1.2.3. A filtration on a coherent left D X/S -module, that is a filtration by coherent sub-O X -modules compatible with the filtration on D X/S is said to be good if it is bounded below and if the associated graded module over grD X/S ∼ = O T * (X/S) is coherent.
Note that coherent left D X/S -modules admit good filtrations [5, 5.2.3 (iv)] and that for a good filtration Γ on a module M , the support of the graded coherent grD X/S ∼ = O T * (X/S) -module gr Γ M in T * (X/S) is independent of Γ, [11, V 2.2, lemma]. Hence the following is well-defined, The singular support of a coherent left D X/S -module M is the support of the associated graded module to a good filtration, it is a welldefined closed subset SS(M ) of T * (X/S).
The singular support behaves well under exact sequences, indeed, here are some easy consequences of coherence, Assume that S is the spectrum of a field of characteristic zero, then D X/S is the usual algebra of differential operators D X , also noted D. Moreover the natural filtration is the usual filtration by the order of differential operators and the notions of coherent module, good filtration and singular support specialize to their D-module counterparts [11,VI §1]. Recall that in characteristic zero the dimension of the singular support satisfies the following fundamental inequality, [11,

D X/S in positive characteristic
If S is a scheme of positive characteristic p, then so is X and let X (p/S) , or simply X ′ , be the base change of X/S by the Frobenius endomorphism of S, raising the local sections to their p-th power. There is a S-morphism F X/S : X → X (p/S) associated to the Frobenius endomorphism of X and called the relative Frobenius of X/S [36, §1]. Moreover, as the p-th iterate of a derivation is again a derivation, one may associate to a local section ∂ of T X/S the local section ∂ [p] of T X/S corresponding to its p-th iterate. Comparing it with the p-th power of the element corresponding to ∂ in D X/S , one gets a p-linear map c : ∂ → ∂ p −∂ [p] from T X/S to D X/S , which actually lands in the center Z(D X/S ) of D X/S [8, 1.3.1]. By adjunction one deduces from c an O X ′ -linear morphism c ′ : T X ′ /S → F X/S * Z(D X/S ).
It turns D X/S := F X/S * D X/S into a central O T * (X ′ /S) -algebra. Note that in local étale coordinates as above one has F X/S * Z(D X/S ) = I O X ′ .∂ pI hence D X/S is a locally free O T * (X ′ /S) -module of rank p 2n . Moreover, D X/S is an Azumaya algebra of rank p n on T * (X ′ /S).
Recall that an Azumaya algebra is a relative central simple algebra. The notion has many equivalent characterizations [24, 5.1] one of which is that an Azumaya algebra of rank r on a scheme Y is a sheaf of O Y -algebras, coherent as an O Y -module and isomorphic to a rank r matrix algebra M r (O Y ) on a flat covering.
In the case of D X/S , let A X/S be the centralizer of O X in D X/S and let's denote (D X/S ) A X/S the rank p n locally free A X/S -module D X/S , A X/S acting by right multiplication. Then A X/S is a faithfully flat F X/S * Z(D X/S )-algebra and the morphism D X/S ⊗ F X/S * Z(D X/S ) A X/S → End A X/S ((D X/S ) A X/S ) given by left multiplication by D X/S and right multiplication by A X/S is an isomorphism [8, 2.2.2] thus realizing D X/S as an Azumaya algebra of rank p n on T * (X ′ /S).

Symplectic geometry of the cotangent bundle
Let S be a scheme and let Y be a smooth S-scheme of relative dimension n.
Recall that the cotangent bundle of Y /S is the Y -scheme T * (Y /S) and hence that the sheaf of germs of Ysections of T * (Y /S)/Y is canonically identified with Ω 1 Y /S [22, 1.7.9]. Moreover T * (Y /S) is a smooth Y -scheme of relative dimension n [23, 17.3.8], smooth of relative dimension 2n as an S-scheme. For f : X → Y a S-morphism of smooth S-schemes, the pullback of differentials Ω 1 [23, 16.4.3.6] gives rise to the X-morphism T * (X/S) It is part of the cotangent diagram of f , where f π is the canonical projection. Let U ⊂ Y be an open subset, one sees right-away on the definitions that if s α is the section of T * (Y /S)/U corresponding to α ∈ Γ(U, is an immersion (resp. a closed immersion) then f d is smooth and surjective and f π is an immersion (resp. a closed immersion). Moreover f d admits a section locally on X.
The cotangent bundle of Y /S carries a canonical global S-relative 1-form θ Y /S corresponding to the section T * (Y /S) .., y n } be local étale coordinates on Y , then in terms of the associated local étale coordinates {y 1 , ..., y n ; ξ 1 , ..., ξ n } on T * (Y /S), where {ξ 1 , ..., ξ n } are dual to {dy 1 , ..., dy n }, θ Y /S = i=n i=1 ξ i dy i . Note that the canonical form is compatible with base change and with the cotangent diagram, the latter in the sense that f π * θ Y /S = f d * θ X/S . Suppose that S is the spectrum of a field k and let's drop the reference to the base S from the notations. So Y is a smooth k-scheme of pure dimension n. The nondegenerate global exact 2-form ω Y := dθ Y on T * Y is called the symplectic form.
preserve open dense subsets, the lemma follows.

Differential calculus in positive characteristic
Let Y be a smooth equidimensional scheme over a field k of positive characteris- Let's call it the p-curvature exact sequence and W ⋆ − C Y the p-curvature operator.

The p-support
Let X be a smooth scheme over a field k of positive characteristic p. Recall from 1.3, often dropping the base from the notations, that there is an It is a closed subset p-supp(M ) of T * (X ′ ), endowed with its reduced subscheme structure.
Further recall that T * (X ′ ) carries a nondegenerate exact 2-form ω X ′ which is called the symplectic form.
The symplectic form ω X ′ or rather the corresponding Poisson bracket has a natural deformation theoretic interpretation related to the lifting of X modulo p 2 .

The statement
Let S be a scheme of finite type over Z. Recall that the closed points of S have finite residue fields and that there are lots of them, indeed every nonempty locally closed subset of S contains a closed point [23, 10.4

About the proof
A complete proof is given in 7.4 and rests on most of the results of the paper. Let us roughly outline the argument, which has two parts, in accordance with the definition of a lagrangian subvariety. A first part bears on the dimension and equidimension of the p-support, notions which are of cohomological nature (vanishing of some double Ext's), and occupies section 3.
A second part handles the vanishing of the symplectic form on the regular locus of the p-support.
Its starting point is twofold, namely, there is a natural map from 1-forms to the Brauer group which sends the canonical form to the class of the Azumaya algebra of differential operators (subsection 6.2), and the latter splits on the regular locus of the p-support (theorem 6.1.4).
Thus the restriction of the canonical form to the regular locus of the psupport is in the kernel of the above map, which may be described in terms of the p-curvature operator (proposition 6.2.3). Further considering the action of the p-curvature operator on the order of poles along the boundary of a compactification (proposition 7.3.1), one shows that the restriction of the symplectic form, that is the exterior derivative of the canonical form, has logarithmic poles. The result then follows from the vanishing of globally exact forms with logarithmic poles, in characteristic zero.
Let us mention that crucial to the argument is an estimate of some degrees and ranks of modules, in the case of the affine space (theorem 5.3.2).
Note finally that in the above sketch, we should have written "for p large enough" several times.

First reductions
Here we carry out some standard reductions ( [23, 6.12.6]. It is non empty since the generic stalk is a field and hence is regular. Thus we may assume that the fiber of M at the generic point of S is non zero. The proof is deferred till 3.3. It is based on the notion of pure coherent sheaf (3.2) and its characterization in terms of duality (3.2.3). This is relevant as purity implies equidimensionality (3.2.2) and holonomic D-modules satisfy strong duality properties (1.2.8). One concludes using the Azumayaness of the algebra of differential operators (3.3.4).

Dimension of the p-supports
In view of 2.4.1, we may and shall assume that S and X are regular, integral and affine.

Pure coherent sheaves
Recall that the (co)dimension of a coherent sheaf is the (co)dimension of its support and let us call a coherent sheaf equidimensional if its support is equidimensional. There is a strengthening of equidimensionality which has a very convenient interpretation in terms of duality theory. Indeed, let us fix an affine scheme Y, Here's the interpretation in terms of duality theory,

Equidimensionality of the p-supports
Recall the notations and hypotheses of 3.1.1 and use 2.4.1. In particular X/S is smooth of relative dimension n with S and X regular, affine and integral.
Here are two lemmas and two propositions, preliminary to the proof of theorem 3.1.1, which follows.
is an isomorphism, where the subscript s denotes restriction to the fiber.
Proof: Since coherent left D X/S -modules form an abelian category, the proof of [23, 9.4.3] goes through here. Indeed provided the above abelianity, the proof of [23, 9.4.2] carries reducing to associated graded to good filtrations and using [30, A.17] and lemma 3.3.1 to conclude. There are only finitely many l's to consider since by [10, A:IV 4.5], both target and domain of the above morphism are zero for l > dimT * (X/S) ≥ dimT * X s , T * (X/S) and T * X s being the respective spectra of the regular rings grD X/S and grD Xs .
where M := F Y /k * M and l is an integer.
Both of them are coherent sheaves on T * (Y ′ ) (1.3.2) hence their respective vanishings may be checked on a flat covering U U induces an equivalence between the categories of coherent O U -modules and coherent left M r (O U )-modules. Note that the coherent sheaf

Reduction to A n
It is convenient (see section 5) to further reduce the proof of theorem 2.2.1 to modules on A n S . In order to do so we shall use the direct image of D X/S -modules.

Direct image of D X/S -modules for a closed immersion
Let X/S be smooth of relative dimension n, then the invertible O X -module Ω X/S := ∧ n Ω 1 X/S is endowed with a right D X/S -module structure, defined via the Lie derivative [30, 1.4(a)] (see also [4, 1.2.1]). Moreover for M a left D X/Smodule and for N , N ′ right D X/S -modules, N ⊗ OX M (resp. Hom OX (N, N ′ )) is naturally a right (resp. left) D X/S -module, [11,VI 3.4]. In particular, denote by M r the right D X/S -module Ω X/S ⊗ OX M and by N l the left D X/S -module Hom OX (Ω X/S , N ). In local étale relative coordinates {x 1 , ..., x n }, trivializing Ω X/S , exchanging left and right, that is going from M to M r and from N to N l , is expressed by making a differential operator P = I P I ∂ I ∈ I O X .∂ I ≃ D X/S act through its adjoint P t := I (−1) |I| ∂ I P I , where |I| is the length I 1 + ... + I n of the multi-index I, [4, 1.2.7].
Let Y /S be a smooth morphism of relative dimension m and let X is naturally endowed with a left D X/S -module structure, via the morphism Further

Reduction
Recall 2.4.1, in particular X/S is smooth of relative dimension n and one may assume that X and S are affine. Hence for some m, there is a closed immersion

A bound on degrees and ranks
Thanks to 4.2, in the proof of theorem 2.2.1, one may restrict one's attention to modules M over D A n S /S . In addition to the natural filtration (1.2.1), D A n S /S has a filtration whose associated graded pieces are finite over S, the Bernstein filtration. We use it to refine part of the comparison (3.1.1) between fibers of M at closed points and at the generic point of S.
More precisely, we get an estimate (theorem 5.3.2), bounding the degrees of the p-supports (for a suitable projective embedding) as well as the generic ranks, of the fibers at "almost all" closed points of a module M as above. These bounds are crucial in the proofs of theorems 6.1.3 and 2.2.1.

Bernstein filtration
Let S = spec(R) (2.4.1) and A n S = spec(R[x 1 , ..., x n ]), then the ring of global , the n-th Weyl algebra with coefficients in R, A n (R). The filtration on A n (R) = α,β Rx α ∂ β , α, β multi-indices, by the total order in x and ∂, B l A n (R) := |α|+|β|≤l Rx α ∂ β is called the Bernstein filtration. Note that the associated graded ring gr B A n (R) is the R-algebra of polynomials in the classes x 1 , ..., x n , y 1 , ..., y n of x 1 , ..., x n , ∂ 1 , ..., ∂ n , respectively, graded by the order of polynomials. In particular the B l A n (R)/B l−1 A n (R) are finite free modules over R.
A good filtration Γ on a left A n (R)-module M is an increasing exhaustive filtration on M, compatible with B, which is bounded below and such that the associated graded module gr Γ M is finite over the algebra of polynomials gr B A n (R).
is the rank of the free module Γ l M/Γ l−1 M | U over U .

Induced filtration over the center
Let K be a field of positive characteristic p. Then the center ZA n (K) of It is graded by the degree of polynomials, where degree(x p i ) = degree(∂ p j ) = 1 and the associated increasing filtration is denoted C. The Rees ring R n (C) of the filtered ring (ZA n (K), C) is the naturally graded ring i=∞ i=0 C i ZA n (K). Note that the graded algebra morphism ., x p n , ∂ p 1 , ..., ∂ p n ] is an isomorphism. Keeping the same notation for t 0 and its image under this isomorphism, the natural map R n (C)/t 0 R n (C) → gr C ZA n (K) is an isomorphism of graded algebras. Note also that summing components induces an isomorphism R n (C) (t0) →ZA n (K) where R n (C) (t0) is the subring of degree 0 elements of the graded ring R n (C) t0 .
An increasing C-compatible filtration G on a ZA n (K)-module M is said to be good if the associated Rees module R(M, G) := i=∞ i=−∞ G i M over the Rees ring i=∞ i=0 C i ZA n (K) is finitely generated. This implies in particular that G is bounded below. Moreover one easily sees that a filtration G on M is good if and only if G is bounded below and the associated graded module gr G M is finitely generated over gr C ZA n (K), [10, A:III 1.29].
Let Γ be a filtration on the left A n (K)-module M , then pΓ, (pΓ) l M := Γ pl M endows M with the structure of a filtered module over the center (ZA n (K), C). Lemma 5.2.1. Let Γ be a good filtration on the left A n (K)-module M , then pΓ is a good filtration on M seen as a (ZA n (K), C)-module.
Proof: Since Γ is bounded below then so is pΓ. Let's show that the gr C ZA n (K)-module gr pΓ M is finitely generated. For this let F be the filtration on ZA n (K) induced by B, in particular degree F (x p i ) = degree F (∂ p j ) = p and let F (Γ) be the F -compatible filtration on M defined by F (Γ) l M := Γ pm M , where pm is the greatest integer multiple of p such that pm ≤ l. Note that class of x p i (resp. ∂ p j ) → class of x p i (resp. ∂ p j ) induces an isomorphism of K-algebras gr C ZA n (K) → gr F ZA n (K) with which the K-module isomorphism gr pΓ M → gr F (Γ) M defined by (pΓ) l M/(pΓ) l−1 M = Γ pl M/Γ p(l−1) M = F (Γ) pl M/F (Γ) pl−1 M is compatible. Hence gr pΓ M is finitely generated over gr C ZA n (K) if and only if gr F (Γ) M is finitely generated over gr F ZA n (K). Consider the finite exhaustive filtration of gr F (Γ) M by graded sub-gr F ZA n (K)- with pm, as above, the greatest integer multiple of p such that pm ≤ l. Denote by Note also that gr Γ M seen as a module over gr F ZA n (K) ֒→ gr B A n (K) decomposes as a direct sum of graded sub-gr F ZA n (K)-modules . Then for all i, the morphism of graded gr F ZA n (K)- p(m−1)+i M is an isomorphism. These induce an isomorphism of graded gr F ZA n (K)-modules Since gr Γ M is finitely generated over gr B A n (K) by hypothesis and gr B A n (K) ∼ = K[x 1 , ..., x n , y 1 , ..., y n ] is finite over gr F ZA n (K) ∼ = K[x p 1 , ..., x p n , y p 1 , ..., y p n ], F * gr Γ M is finitely generated over gr F ZA n (K). Then gr F (Γ) M has an exhaustive finite filtration whose subquotients are finitely generated over gr F ZA n (K), hence it is finitely generated, giving the lemma.
Let M be a left A n (K)-module and let Γ be a good filtration on M. Since pΓ is a good filtration on the (ZA n (K), C)-module M (5.2.1) the Rees module of (M, pΓ), which is a finitely generated graded module over the Rees ring R n (C) ≃

The bound
There is a geometric picture of the Rees construction in which the affine scheme spec(ZA n (K)) ∼ = spec(R n (C) (t0) ) is identified with the affine open D + (t 0 ) associated to t 0 in the homogeneous prime spectrum P roj(R n (C)) of R n (C) [22, 2.4.1] and in which its complement, the reduced closed subscheme V + (t 0 ) is identified with the closed subscheme P roj(gr C ZA n (K)) ∼ = P roj(R n (C)/t 0 R n (C)) of P roj(R n (C)), [22, 2.9.2 (i)]. Making these identifications, let G be a good filtration on a finitely generated (ZA n (K), C)-module M then the coherent sheaf R(M, G) on P roj(R n (C)) extends M and its restriction to the complement P roj(gr C ZA n (K)) of spec(ZA n (K)) is isomorphic to gr G M . Moreover it's easy to see that the support of R(M, G) is the closure of supp( M ) in P roj(R n (C)). Note that here The leading coefficient of the Hilbert polynomial of R(M, G) is related to the top-dimensional irreducible components of its support through the following, Proof: The proof reduces to the case S is integral and affine = spec(R). Moreover F A n /k(u) * M u being a left module over an Azumaya algebra of rank where k(z) is an algebraic closure of k(z). Hence there is a finite dimensional is divisible by p n , thus proving the theorem.

The Brauer group and differential forms
Here we prove, in a first part, that "the Azumaya algebra of differential operators splits on the regular locus of the p-support of a holonomic D-module, for p large enough" (theorem 6.1.4).
In a second part, we consider a map arising from the p-curvature exact sequence (1.5), which sends 1-forms to the Brauer group. It maps the canonical form to the class of the Azumaya algebra of differential operators (prop. 6.2.4) and we describe its kernel (prop. 6.2.3).

Splittings of Azumaya algebras on the support of modules
Let Y be a scheme and let A be an Azumaya algebra on Y . Since A is a coherent Proof: By hypothesis there is a morphism of O Y -algebras A → End OY (V), 1 → 1. It is injective by [21, 0.5.5.4] since the fiber of A at each point of Y is a simple algebra [24, 5.1 (i)]. Therefore one may view A as a subalgebra of End OY (V) and in particular consider  [24, §2] and the theorem follows As in 4.2, theorem 6.1.3 implies the apparently more general Proof: By [24, 2.1] applied to the Zariski site and [23, 21.11.1] the case i = 2 of lemma 6.1.5 below implies that on a regular (noetherian) scheme for an Azumaya algebra to be split is a Zariski local condition. Therefore by remark 2.4.1 one may further assume that S and X are regular integral and affine and in particular that there is a closed immersion X f ֒→ A m S over S. Specializing to a closed point u of positive characteristic p of S it follows from the description of p-supp(f 0 (M u )) in 4.1 and from [23, 17.5 Since by lemma 1.4.1 f ′ d Zariski locally admits a section there is a Zariski open covering of X ′ above which the pullback of Brauer classes f ′ d * is injective and therefore to split being Zariski local on a regular noetherian scheme, f ′ d * induces an injective morphism Br(p-supp(M u ) reg ) →  Proof: By definition of the sheaf Div Y of Cartier divisors there's an exact sequence of abelian sheaves If Y is locally factorial then it is the sum of its (finitely many) irreducible components, each of which is integral [21, 4.5.5] Yi is isomorphic to the constant sheaf associated to k(y i ) * for y i the generic point of Y i . In particular K * Y is flasque. Since Div Y is flasque by [23, 21.6.11], K * Y → Div Y is a flasque right resolution of O * Y . This gives the result as sheaf cohomology may be computed using flasque resolutions.

The Brauer group via the p-curvature sequence
Let Y be a smooth equidimensional scheme over a perfect field K of positive characteristic p. Composing the coboundary morphisms of the étale cohomology long exact sequences of the two short exact sequences of étale sheaves It depends functorially on Y, namely where f ′ * on the left (resp. on the right) is the pullback of classes in the Brauer group (resp. pullback of forms) by f ′ .
Further diagram chasing through the cohomology long exact sequences gives control over the kernel of φ Y , say Zariski locally, Proposition 6.2.3. Suppose further that Y is affine. Then there is an exact sequence (compatible with restriction to affine open subsets) Proof: It is a special case of [28, 1.7].
For Y = T * X where X is a smooth equidimensional K-scheme, φ T * X relates the canonical 1-form θ X ′ on T * (X ′ ) to the class of the Azumaya algebra F X/K * D X (1.3) in Br(T * (X ′ )). Indeed, note that the pullback of forms [23, 16.4.5] hence induces an isomorphism (T * X) ′ → T * (X ′ ), use it to identify (T * X) ′ and T * (X ′ ) and denote the resulting K-scheme T * X ′ , then by [35, prop. 4.4 and 4.2] we have the Proposition 6.2.4. φ T * X (θ X ′ ) = [F X/K * D X ] ∈ Br(T * X ′ ).

Lagrangianity
In this section, we complete the proof of the main theorem 2.2.1.
Namely, the first bound of theorem 5.3.2 allows one to construct a smooth compactification of an open dense subset of the p-support (prop. 7.2.1), "uniformly in p", by reducing the problem to characteristic zero. From the description of the canonical form restricted to the p-support in terms of the p-curvature operator, given in section 6, and the analysis of the latter's action on the order of poles of differential forms (prop. 7.3.1), we get that the symplectic form has logarithmic poles along the boundary of the above compactification. We then conclude using Hodge theory in characteristic zero (7.4).

Poles and logarithmic poles
Let S be a scheme, let Y be a smooth S-scheme and let D be a closed subscheme of Y . The closed subscheme D is said to be a divisor with normal crossings relative to S if there is an étale covering U By noetherian induction the statement holds for the reduced closed subscheme of S whose underlying space is S − W. Hence combining with the above on W, the statement holds for S. This proves the proposition.

Action of the p-curvature operator on the order of poles
Let Y be a smooth scheme over a field k of positive characteristic p, let D be a divisor with normal crossings relative to k (7.1) and let Y j ֒→ Y be the inclusion of the open subscheme Y −D. Base changing by the Frobenius endomorphism of k, one sees that the closed subscheme D ′ ⊂ Y ′ is a divisor with normal crossings relative to k and that Y is equidimensional and let Im(W ⋆ − C Y ) be the image of the morphism . Then there's the following inclusion of abelian subsheaves of j ′ * Ω 2 Proof: Let η be a local section of Ω 1 − → Y be an étale covering as in the definition of a divisor with normal crossings (7.1). Since the pullback π ′ * η is a local section of where j U is the open immersion U − π −1 D ֒→ U and since being a section of Ω 2 Y ′ /k (logD ′ ) is an étale local condition, one may assume that at each point y ∈ Y there are local étale coordinates {y 1 , ..., y n } : V y → A n k in which the closed subscheme D is described by the equation y 1 ...y r = 0 for some r ≤ n, n and r depending on y.
Let us prove that are the exact 1-forms. It implies the proposition. Let η be a local section of j ′ * Im(W ⋆ − C Y ), by lemma 7.3.2 below the canonical inclusion Imj ′ ]. Note also that the étale coordinates {y 1 , ..., y n } : V y → A n k above determine a splitting on Y ′ ∩ V ′ y of the canonical short exact sequence given in terms of local sections by Ω 1 It induces a splitting on V ′ y of the direct image j ′ * of the above short exact sequence and hence a local section ζ of j ′ * Z 1 (F /k * Ω • Y /k )| V ′ y may uniquely be written as a sum ζ = β Indeed the proofs of both assertions reduce to (Y , D) = (A n k = spec(k[y 1 , ..., y n ]), {y 1 ...y r = 0}) with étale coordinates {y 1 , ..., y n } since the pullback by {y ′ 1 , ..., y ′ n } : V ′ y → A n ′ k preserves the splitting and the order of poles. There the second assertion is a direct consequence of the factoriality of rings of polynomials with coefficients in a field while the first can be proved as follows.
Let ζ = dg + Σ i=n i=1 F /k * (a i )y i p−1 dy i be the above decomposition of a closed form on an affine open {f = 0}, for a rational function g and a polyno- Indeed, multiplying by a high enough power of f p , one may assume that ζ is a global section. Moreover by uniqueness of the decomposition and the corresponding splitting for forms without poles, one may also assume that dg = Σ i=n i=1 ∂ i (g)dy i and Σ i=n i=1 F /k * (a i )y i p−1 dy i are sections of F /k * Ω 1 Y /k (pD). Further using uniqueness of the decomposition and the splitting for forms without poles, and multiplying by (y 1 ...y r ) p , if Σ i=n i=1 F /k * (a i )y i p−1 dy i was not a section of F /k * Ω 1 Y /k ((p − 1)D) then there would be an i such that y 1 ...y r divides ∂ i (g) + F /k * (a i )y i p−1 but not F /k * (a i )y i p−1 , where g and a i are polynomials. In particular there should be 1 ≤ j ≤ r such that j = i and y j divides ∂ i (g) + F /k * (a i )y i p−1 but not F /k * (a i )y i p−1 . Expressing as polynomials in y j and considering the degree zero terms would provide an equality ∂ i (g 0 ) + F /k * ((a i ) 0 )y i p−1 = 0 with (a i ) 0 = 0. Since this cannot happen in characteristic p, the assertion holds.
Proof: The exact sequence of abelian sheaves on Y ′ (1.5) provides two short exact sequences The associated long exact sequence to the first one shows that the lemma follows from the vanishing of R 1 j ′ * cokerF /k * which in turn by the long exact sequence associated to the second short exact sequence would follow from the vanishings of R 1 j ′ * (F /k * O * Y ) and R 2 j ′ * O * Y ′ . Since the direct image F /k * preserves flasque sheaves and is exact as F /k is a homeomorphism, R q F /k * (G) = 0 for all abelian sheaves G and all q > 0 by [26,III 8.3] and

Conclusion
Recall the statement 2.2.1 Theorem. Let S be an integral scheme dominant and of finite type over Z, let X be a smooth S-scheme of relative dimension n and let M be a coherent left D X/S -module. Suppose that the fiber of M at the generic point of S is a holonomic left D-module. Then there is a dense open subset U of S such that the p-support of the fiber of M at each closed point u of U is a lagrangian subscheme of (T * (X ′ u ), ω X ′ u ). Proof: By the remark 2.4.3, we may assume that the fiber of M at the generic point of S is non zero. Hence by theorem 3.1.1, there is a dense open subset U a of S such that the p-support of the fiber of M at each closed point u of U a is equidimensional of dimension n. Since by 4.2 one may further suppose that X/S = A n S /S, theorem 6.1.3 implies that there is a dense open subset U b of U a such that for each closed point u ∈ U b and each z generic point of an irreducible component of p-supp(M u ) the Azumaya algebra F A n /k(u) * D A n k(u) on T * (A n ′ k(u) ) splits on ({z} red ) reg . Therefore, k(u) being perfect (2.2), by 6.2.4, 6.2.2 and 6. ֒→ Y z into a smooth projective scheme, which is the complement of a divisor D z with normal crossings relative to k(u). In addition there is a nonnegative integer m, independent of u, such that the restriction of the canonical form to Y z has poles of order at most m along D z . Hence inverting all primes ≤ m, one may suppose that m ≤ char(k(u)) − 1 and thus by proposition 7.3.1, ω A n ′ k(u) | Yz := dθ A n ′ k(u) | Yz has logarithmic poles along D z .
Actually by proposition 7.2.1, there is a finite set Ξ such that for each i ∈ Ξ there are an integral scheme S i whose generic point is of characteristic zero, a smooth S i -scheme Y i , an open immersion Y i j ֒→ Y i into a smooth projective S ischeme which is the complement of a divisor D i with normal crossings relative to S i and a relative 1-form θ i ∈ Ω 1 Yi/Si (mD i ) such that for each closed point u ∈ U c and each z generic point of an irreducible component of p-supp(M u ), there is i(z) ∈ Ξ such that Y z j ֒→ Y z and θ A n ′ k(u) | Yz are deduced from Y i(z) j ֒→ Y i(z) and θ i(z) , base changing by a k(u)-point of S i(z) . Moreover by construction of the S i 's (7.2.1), if dθ A n ′ k(u) | Yz ∈ Ω 2 Yz /k (logD z ) then dθ i(z) ∈ Ω 2 Yi/Si (logD i ), hence in particular dθ 0 ∈ Ω 2 Y 0 (logD 0 ), where here and below, the subscript 0 denotes restriction to the generic fiber. As the generic fiber is over a field of characteristic zero, the canonical inclusion Ω • Y 0 (logD 0 ) ⊂ (j 0 ) * Ω • Y0 is a quasi-isomorphism [17, 3.1.8], implying that the class of dθ 0 in the hypercohomology of the logarithmic de Rham complex is zero. Hence dθ 0 vanishes by the degeneracy at E 1 of the logarithmic Hodge to de Rham spectral sequence [17, 3.2.13 (ii) and 3.2.14] and so, by construction of the S i 's (7.2.1), the symplectic form ω A n ′ k(u) vanishes on the open dense subset Y z of ({z} red ) reg . Thus by the above there is a dense open subset U of U c ⊂ U a such that for each closed point u of U, the symplectic form vanishes on a dense open subset of p-supp(M u ). Since by definition of U a , p-supp(M u ) is equidimensional of dimension n for all closed points u ∈ U a , this concludes the proof of the theorem.