D-modules and finite monodromyHélène Esnault, Freie Universität Berlin

Sackler Lectures, Nov. 2016

Hélène Esnault, Freie Universität Berlin

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Acknowledgements
Thank you tothe mathematicians of Tel Aviv university for the kind invitation to deliver
the Sackler Distinguished Lectures;Hélène Esnault, Freie Universität Berlin

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Acknowledgements
Thank you tothe mathematicians of Tel Aviv university for the kind invitation to deliver
the Sackler Distinguished Lectures;Mark Kisin for constructive comments and remarks on the slides.

Hélène Esnault, Freie Universität Berlin

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Table of Contents

1

D-modules in characteristic 0

2

D-modules in characteristic p and Tannakian approach3

Two theorems under the stronger D-module assumption.

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Differential operators in complex geometry
Complex geometry• In complex geometry, we know well what is the sheaf D of differential
operators on a manifold X : it is a sheaf of O-algebras, generated, locallywhere one has coordinates (x1 , . . . , xn ), by differential operators D≤N oforder ≤ N, which are written as finite sums with coefficients
Pn in theholomorphic functions O of operators ∂xm11 ◦ . . . ◦ ∂xmnn ,i=1 mi ≤ N. In
particular, D≤0 = O, D≤1 = O ⊕ T , where T is the sheaf of tangent
vectors. The splitting D → O is in fact independent of the choice of thelocal coordinates, that is it is global, and defined by P 7→ P(1), where 1 is
the global constant holomorphic function equal to 1 everywhere.

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Differential operators in complex geometry
Complex geometry• In complex geometry, we know well what is the sheaf D of differential
operators on a manifold X : it is a sheaf of O-algebras, generated, locallywhere one has coordinates (x1 , . . . , xn ), by differential operators D≤N oforder ≤ N, which are written as finite sums with coefficients
Pn in theholomorphic functions O of operators ∂xm11 ◦ . . . ◦ ∂xmnn ,i=1 mi ≤ N. In
particular, D≤0 = O, D≤1 = O ⊕ T , where T is the sheaf of tangent
vectors. The splitting D → O is in fact independent of the choice of thelocal coordinates, that is it is global, and defined by P 7→ P(1), where 1 is
the global constant holomorphic function equal to 1 everywhere.
• As we see on the local description, D as an O-algebra is spanned by D≤1 .

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D-modules in complex geometry
Complex geometry• D acts on O. Any sheaf of O-modules E which has the property that
the O-action factors through a D-action is called a sheaf of D-modules.
So O is a D-module, called the trivial D-module.

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D-modules in complex geometry
Complex geometry• D acts on O. Any sheaf of O-modules E which has the property that
the O-action factors through a D-action is called a sheaf of D-modules.
So O is a D-module, called the trivial D-module.
P• Writing ∇(e) = ∂xi (e)dxi defines a connection∇ : E → Ω1 ⊗O E ,and the relation ∂xi ∂xj (e) = ∂xj ∂xi (e) for all i, j translates into theintegrability of ∇.

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D-modules in complex geometry
Complex geometry• D acts on O. Any sheaf of O-modules E which has the property that
the O-action factors through a D-action is called a sheaf of D-modules.
So O is a D-module, called the trivial D-module.
P• Writing ∇(e) = ∂xi (e)dxi defines a connection∇ : E → Ω1 ⊗O E ,and the relation ∂xi ∂xj (e) = ∂xj ∂xi (e) for all i, j translates into theintegrability of ∇.
• If one assumes E is O-coherent, this implies E is locally free, that is avector bundle, so ∇ becomes a linear differential equation:∂xi ej =rXaij` e` , r = rank(E ).

`=1
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D-modules in complex geometry
Complex geometry• One sets M := (E , ∇).

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D-modules in complex geometry
Complex geometry• One sets M := (E , ∇).
• More is true: there is a tensor product M ⊗ N, an (internal) dual M ∨ ,Hom(M, N) is a complex vector space, End(O) = C: the category of such
M is tannakian, and so is the subcategory hMi of subquotients of tensors
M n ⊗ (M ∨ )m of M and its dual.

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D-modules in complex geometry
Complex geometry• One sets M := (E , ∇).
• More is true: there is a tensor product M ⊗ N, an (internal) dual M ∨ ,Hom(M, N) is a complex vector space, End(O) = C: the category of such
M is tannakian, and so is the subcategory hMi of subquotients of tensors
M n ⊗ (M ∨ )m of M and its dual.
• If we fix a complex point x, we have the monodromy representationρ : π1top (X , x) → GL(r , C) of M.

Hélène Esnault, Freie Universität Berlin

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D-modules in complex geometry
Complex geometry• One sets M := (E , ∇).
• More is true: there is a tensor product M ⊗ N, an (internal) dual M ∨ ,Hom(M, N) is a complex vector space, End(O) = C: the category of such
M is tannakian, and so is the subcategory hMi of subquotients of tensors
M n ⊗ (M ∨ )m of M and its dual.
• If we fix a complex point x, we have the monodromy representationρ : π1top (X , x) → GL(r , C) of M.
• The category hMi is equivalent to the category of algebraic
representations in finite dimensional complex vector spaces of a complex
group scheme G (M), called its Tannaka group, which is the Zariski closureρ(π1top (X , x)) ⊂ GL(r , C) of the monodromy group. (It is a consequence of the
Riemann-Hilbert correspondence.)Hélène Esnault, Freie Universität Berlin

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Tannakian category over a field k of characteristic 0
Tannaka group• The same definition in the Zariski topology yields O-coherent
D-modules, which are vector bundles with an integrable connection.

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Tannakian category over a field k of characteristic 0
Tannaka group• The same definition in the Zariski topology yields O-coherent
D-modules, which are vector bundles with an integrable connection.
• The category hMi is Tannakian.

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Tannakian category over a field k of characteristic 0
Tannaka group• The same definition in the Zariski topology yields O-coherent
D-modules, which are vector bundles with an integrable connection.
• The category hMi is Tannakian.
• Clearly, one can no longer compute its Tannaka group G (M) as before,
as one no longer has a topological fundamental group at disposal.

Hélène Esnault, Freie Universität Berlin

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Tannakian category over a field k of characteristic 0
Tannaka group• The same definition in the Zariski topology yields O-coherent
D-modules, which are vector bundles with an integrable connection.
• The category hMi is Tannakian.
• Clearly, one can no longer compute its Tannaka group G (M) as before,
as one no longer has a topological fundamental group at disposal.
• One computes it by fixing a rational point x ∈ X (if there are no rational points, one
makes a field extension and computes there). Its k-points consist of all automorphisms of
M|x which respect all Homs of subquotients of M n ⊗ (M ∨ )m |x .

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Tannakian category over a field k of characteristic 0
Deligne’s Riemann-Hilbert correspondence
says that if k = C, if ∇ extends to a connection with logarithmic poles¯ : M̄ → Ω1 (log(∞)) ⊗ M̄ along infinity, i.e. if ∇ has regular singular∇poles, then the two definitions of G (M) coincide.

ExampleIn particular, it applies when X is projective.

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Algebraic solutions
Algebraic solutions
To say M has algebraic solutions as in the first lecture is equivalent tosaying that G (M) is a 0-dimensional group-scheme, or, equivalent tosaying that the monodromy group ρ(π1top (X (C), x)) is finite. (Here k ⊂ C).

Proof
The monodromy group ρ(π1top (X (C), x)) is finite if and only if M
trivializes over a finite étale cover h : Y → X (C). Using that the algebraic
closure of ρ(π1top (X (C), x)) is itself, and the Tannaka formalism over k,
one sees that h is in fact defined over k. Then solutions trivialize on Y .
So solutions of the differential system of equations, locally in aneighborhood on X , consist of some of the regular functions on thepull-back neighborhood on Y .

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Recall Grothendieck’s p-curvature conjecture
Grothendieck’s p-curvature conjecture
Let X be a smooth projective curve defined over a number field K . Then asystem of linear differential equations M has algebraic solutions if and only
if it has a full set of solutions modulo p for almost all p.
Here (XS , MS ) is a model of (X , M) over a non-trivial open of thespectrum of the number ring OK .

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Tannakian wish
Tannakian thought
In view of the Tannakian characterization of M having algebraic solutions,
it is natural to wish to be able to argue as follows.

Hélène Esnault, Freie Universität Berlin

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Tannakian wish
Tannakian thought
In view of the Tannakian characterization of M having algebraic solutions,
it is natural to wish to be able to argue as follows.
• One has the model (XS , MS ).

Hélène Esnault, Freie Universität Berlin

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Tannakian wish
Tannakian thought
In view of the Tannakian characterization of M having algebraic solutions,
it is natural to wish to be able to argue as follows.
• One has the model (XS , MS ).
• On XK , one has the group-scheme G (MK ) over K , of which one wants
to show that it is 0-dimensional, i.e. the set (G (MK )(K̄ ) of K̄ -points isfinite, where K̄ is an algebraic closure of K .

Hélène Esnault, Freie Universität Berlin

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Tannakian wish
Tannakian thought
In view of the Tannakian characterization of M having algebraic solutions,
it is natural to wish to be able to argue as follows.
• One has the model (XS , MS ).
• On XK , one has the group-scheme G (MK ) over K , of which one wants
to show that it is 0-dimensional, i.e. the set (G (MK )(K̄ ) of K̄ -points isfinite, where K̄ is an algebraic closure of K .
• One would like to construct a model GS (MK ) of G (MK ) over S, suchthat the fiber at a closed point s ∈ S is equal or at least related to G (Ms )over k(s).

Hélène Esnault, Freie Universität Berlin

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Tannakian wish
Tannakian thought
In view of the Tannakian characterization of M having algebraic solutions,
it is natural to wish to be able to argue as follows.
• One has the model (XS , MS ).
• On XK , one has the group-scheme G (MK ) over K , of which one wants
to show that it is 0-dimensional, i.e. the set (G (MK )(K̄ ) of K̄ -points isfinite, where K̄ is an algebraic closure of K .
• One would like to construct a model GS (MK ) of G (MK ) over S, suchthat the fiber at a closed point s ∈ S is equal or at least related to G (Ms )over k(s).
• Then one would say: GS (MK ) has relative dimension 0 over S if andonly if G (Ms ) has relative dimension 0 over s for all closed points s ∈ S.

Hélène Esnault, Freie Universität Berlin

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Tannakian wish
Tannakian thought
In view of the Tannakian characterization of M having algebraic solutions,
it is natural to wish to be able to argue as follows.
• One has the model (XS , MS ).
• On XK , one has the group-scheme G (MK ) over K , of which one wants
to show that it is 0-dimensional, i.e. the set (G (MK )(K̄ ) of K̄ -points isfinite, where K̄ is an algebraic closure of K .
• One would like to construct a model GS (MK ) of G (MK ) over S, suchthat the fiber at a closed point s ∈ S is equal or at least related to G (Ms )over k(s).
• Then one would say: GS (MK ) has relative dimension 0 over S if andonly if G (Ms ) has relative dimension 0 over s for all closed points s ∈ S.
• This would then reduce the problem to a study in pure characteristicp > 0, over a finite field.
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From the wish to reality
The bundle with connection Ms does not span a Tannakian category• Write Ms = (Es , ∇s ). If ∇s (ϕ) = 0, ϕ ∈ Es , then ∇s (λp ϕ) = 0 for all
λ ∈ OXs , thus the abelian subsheaf Es∇s ⊂ Es of solutions is an(1)(1)
OX (1) -module Es , where Xs is the Frobenius twist.
s

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From the wish to reality
The bundle with connection Ms does not span a Tannakian category• Write Ms = (Es , ∇s ). If ∇s (ϕ) = 0, ϕ ∈ Es , then ∇s (λp ϕ) = 0 for all
λ ∈ OXs , thus the abelian subsheaf Es∇s ⊂ Es of solutions is an(1)(1)
OX (1) -module Es , where Xs is the Frobenius twist.
s• So having a full set of solutions (or equivalently having p-curvature 0) is just saying(1)(Es , ∇s ) = (F −1 Es(1)⊗F −1 O (1) OXs , ∇s = 1 ⊗ d) = (F ∗ Es , ∇can ),Xsthat is Ms has one time Frobenius descent.

Hélène Esnault, Freie Universität Berlin

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From the wish to reality
The bundle with connection Ms does not span a Tannakian category• Write Ms = (Es , ∇s ). If ∇s (ϕ) = 0, ϕ ∈ Es , then ∇s (λp ϕ) = 0 for all
λ ∈ OXs , thus the abelian subsheaf Es∇s ⊂ Es of solutions is an(1)(1)
OX (1) -module Es , where Xs is the Frobenius twist.
s• So having a full set of solutions (or equivalently having p-curvature 0) is just saying(1)(Es , ∇s ) = (F −1 Es(1)⊗F −1 O (1) OXs , ∇s = 1 ⊗ d) = (F ∗ Es , ∇can ),Xsthat is Ms has one time Frobenius descent.
(1)• So any OX (1) -submodule V of Essyields a subconnection F ∗ V ⊂ Ms ,with quotient F ∗ (E (1) /V ) which is not necessarily locally free. So thesubquotients of tensors of Ms and Ms∨ can not span a Tannakian category.

Hélène Esnault, Freie Universität Berlin

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From the wish to reality
The bundle with connection Ms does not span a Tannakian category• Write Ms = (Es , ∇s ). If ∇s (ϕ) = 0, ϕ ∈ Es , then ∇s (λp ϕ) = 0 for all
λ ∈ OXs , thus the abelian subsheaf Es∇s ⊂ Es of solutions is an(1)(1)
OX (1) -module Es , where Xs is the Frobenius twist.
s• So having a full set of solutions (or equivalently having p-curvature 0) is just saying(1)(Es , ∇s ) = (F −1 Es(1)⊗F −1 O (1) OXs , ∇s = 1 ⊗ d) = (F ∗ Es , ∇can ),Xsthat is Ms has one time Frobenius descent.
(1)• So any OX (1) -submodule V of Essyields a subconnection F ∗ V ⊂ Ms ,with quotient F ∗ (E (1) /V ) which is not necessarily locally free. So thesubquotients of tensors of Ms and Ms∨ can not span a Tannakian category.
• End of dream. Awakening.
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D-modules in characteristic p > 0
Grothendieck’s definition of D-modules• The differential operators are defined as additive endomorphisms
O → O. It is a sheaf of rings, it is filtered, and [Dn , Dm ] has degree≤ n + m − 1 if Dn has degree n.

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D-modules in characteristic p > 0
Grothendieck’s definition of D-modules• The differential operators are defined as additive endomorphisms
O → O. It is a sheaf of rings, it is filtered, and [Dn , Dm ] has degree≤ n + m − 1 if Dn has degree n.
• In characteristic p > 0, not only ∂xm11 ◦ . . . ◦ ∂xmnn span the sheaf ofdifferential operators over O, but also e.g. ∂xp1 /p. Unlike in characteristic
0, D is not spanned by D≤1 over O, and in fact not even by D≤N for any
N.

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D-modules in characteristic p > 0
Grothendieck’s definition of D-modules• The differential operators are defined as additive endomorphisms
O → O. It is a sheaf of rings, it is filtered, and [Dn , Dm ] has degree≤ n + m − 1 if Dn has degree n.
• In characteristic p > 0, not only ∂xm11 ◦ . . . ◦ ∂xmnn span the sheaf ofdifferential operators over O, but also e.g. ∂xp1 /p. Unlike in characteristic
0, D is not spanned by D≤1 over O, and in fact not even by D≤N for any
N.
• As a consequence, a OXs -coherent DXS -module is more than just an
integrable connection.

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Katz’s theorems
Theorem (Riemann-Hilbert correspondence in characteristic p > 0)
An integrable connection Ms = (Es , ∇s ) is a DXs -module if(0) Es is a vector bundle;(i) Ms has a full set of solutions;(ii) One has a further infinite Frobenius descent: there are vector bundles
(i)
(i)
(i)
Es on the Frobenius twists Xs such that F ∗ Esi+1 ∼= Es .

RemarkOne should put the data of those isomorphisms in the definitions but as Xs
is projective, this is irrelevant (Katz).

Theorem (Tannaka theory)
OXs -coherent DXs -modules build a Tannakian category over k(s).
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Properties of O-coherent D-modules over a smoothprojective variety over a finite field
We study those properties, which are independent of whether the
O-coherent D-module comes from characteristic 0 or not. So let (Xs , Ms )be an O-coherent D-module defined over a smooth projective variety Xs
over a finite field k(s).

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Properties of O-coherent D-modules over a smoothprojective variety over a finite field
We study those properties, which are independent of whether the
O-coherent D-module comes from characteristic 0 or not. So let (Xs , Ms )be an O-coherent D-module defined over a smooth projective variety Xs
over a finite field k(s).

Theorem (E-Kisin)
Then G (Ms ) is 0-dimensional and étale. Said differently, there is a finiteétale cover hs : Ys → Xs (which is even a torsor under G (Ms )) such that hs∗ Ms is trivial,
i.e. hs∗ Ms ∼= ⊕r1 (O, d). In simple terms: (Xs , Ms ) is isotrivial.

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Proof
It is pure algebraic geometry. Sketch:(i)(i+1) ∼ i(i) The system of Frobenius divided bundles {Es , F ∗ Es
= Es }i≥0(i)(i+1) ∼ i∗
has the property for some n0 ≥ 0, {Es , F Es
= Es }i≥n0 is anextension of such objects, with the property that the underlying(i)vector bundles Es are all stable of degree 0.

(ii) Enough to show isotriviality of one such object (extension of trivial
objects by trivial objects are isotrivial).
(iii) Use existence of quasi-projective moduli (Langer) to show there are(i)repetitions on the moduli points of the Es .
(iv) Use that k(s) is a finite field to conclude that there are repetitions
(i)
(i)
(i)
among the Es , i.e. F a Es ∼= Es for some a and i ≥ i0 for somelarge i0 (i.e. the Brauer obstruction vanishes).
(v) Then hs is a Lang torsor trivializing those bundles.

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On track for the Tannaka thought
Reinforcing Grothendieck’s requirement from full set of solutions to
D-module.
Given the previous theorem, one can try to develop the Tannakian dreampresented above: compare the group-scheme G (MK ) over the number field
K and the group-schemes G (Ms ) over the finite fields k(s), under thestronger assumption that not only Ms has a full set of solutions for almost
all s, that is one has one time Frobenius descent on Ms , but there is a
D-module structure on Ms , that is one has a further infinite Frobenius
descent.

Hélène Esnault, Freie Universität Berlin

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On track for the Tannaka thought
Reinforcing Grothendieck’s requirement from full set of solutions to
D-module.
Given the previous theorem, one can try to develop the Tannakian dreampresented above: compare the group-scheme G (MK ) over the number field
K and the group-schemes G (Ms ) over the finite fields k(s), under thestronger assumption that not only Ms has a full set of solutions for almost
all s, that is one has one time Frobenius descent on Ms , but there is a
D-module structure on Ms , that is one has a further infinite Frobenius
descent.

Dependency(1)

On the other hand, while the bundle of solutions Es = Es∇s , which is thefirst Frobenius descent, depends only on the restriction Ms of MS , theD-module structure on Ms , i.e. the further Frobenius descents, is a choice.
Yet one has the following theorem.
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Independency
Let again (Xs , Ms ) be an O-coherent D-module defined over a smoothprojective variety over a finite field k(s).

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Independency
Let again (Xs , Ms ) be an O-coherent D-module defined over a smoothprojective variety over a finite field k(s).

Theorem (E-Kisin)
Then the moment Ms carries a D-module structure, G (Ms ) is independentof the choice, thus is prescribed by G (MK ) in case Ms is the reduction ats of MK = (EK , ∇K ) defined over a number field.

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Independency
Let again (Xs , Ms ) be an O-coherent D-module defined over a smoothprojective variety over a finite field k(s).

Theorem (E-Kisin)
Then the moment Ms carries a D-module structure, G (Ms ) is independentof the choice, thus is prescribed by G (MK ) in case Ms is the reduction ats of MK = (EK , ∇K ) defined over a number field.

Comment
It is more precise than this: the underlying vector bundle Es is strongly(semi)-stable, hEs i defines a Tannaka category, and G (Ms ) ∼= G (Es ).

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General assumption in the sequel(X , M) shall be an O-coherent D-module defined over a smooth projective
variety over a number field K . We shall assume that for almost all s of amodel (XS , MS ), Ms descends to a D-module.

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First theorem
Theorem (E-Kisin)
If there is a dense set of closed points s ∈ S such that the order of Gs isprime to p, then M is isotrivial, that is has algebraic solutions.

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First theorem
Theorem (E-Kisin)
If there is a dense set of closed points s ∈ S such that the order of Gs isprime to p, then M is isotrivial, that is has algebraic solutions.

Comment
It would be better not to have to assume this non-divisibility, conjecturally
it should be automatic. Still, the assumption is much much weaker thanassuming that the order of Gs is bounded independently of s. Under thislatter assumption, Matzat-van der Put conjectured that M is isotrivial. So
we give a positive answer to their conjecture if X is smooth projective.
However, we also give a negative answer if X is not proper.

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On proof of the First Theorem• Theorem of Camille Jordan, together with the theorem that the Gs are
finite étale, together with the fact that X being proper, its geometric étale
fundamental group in characteristic 0 is topologically finitely generated,enables one to replace X by a finite étale cover of X over a finite extensionof K and to assume that for a dense set of closed points s, Gs is abelian.

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On proof of the First Theorem• Theorem of Camille Jordan, together with the theorem that the Gs are
finite étale, together with the fact that X being proper, its geometric étale
fundamental group in characteristic 0 is topologically finitely generated,enables one to replace X by a finite étale cover of X over a finite extensionof K and to assume that for a dense set of closed points s, Gs is abelian.
• Algebraic geometry enables one to then go up to characteristic 0 to
conclude that M is a successive extension of rank 1 integrable connections
M 0.

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On proof of the First Theorem• Theorem of Camille Jordan, together with the theorem that the Gs are
finite étale, together with the fact that X being proper, its geometric étale
fundamental group in characteristic 0 is topologically finitely generated,enables one to replace X by a finite étale cover of X over a finite extensionof K and to assume that for a dense set of closed points s, Gs is abelian.
• Algebraic geometry enables one to then go up to characteristic 0 to
conclude that M is a successive extension of rank 1 integrable connections
M 0.
• One then concludes using the theorems of
Chudnosky-Chudnovsky-André mentioned in the first lecture.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

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Second theorem
Theorem (E-Kisin)
If M is an abstract polarized Z-variation of Hodge structure, then it isisotrivial.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201622 / 22

Second theorem
Theorem (E-Kisin)
If M is an abstract polarized Z-variation of Hodge structure, then it isisotrivial.

On proof
One uses stability of the Es and the Simpson correspondence.

Hélène Esnault, Freie Universität Berlin

Finite monodromy

Sackler Lectures, Nov. 201622 / 22