Some Fundamental Groups in Arithmetic GeometryHélène Esnault, Freie Universität Berlin

Utah, 27-29-30 July 2015

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

1 / 60Acknowledgements
Thank you tothe organisers for the kind invitation, and much more generally for thefriendly and efficient organisation of the whole conference. I can’t speak forthe first week, but can for the second one. It was wonderful. Thank you.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

2 / 60Acknowledgements
Thank you tothe organisers for the kind invitation, and much more generally for thefriendly and efficient organisation of the whole conference. I can’t speak forthe first week, but can for the second one. It was wonderful. Thank you.

Thank you to
Moritz Kerz, Lars Kindler, Takeshi Saito and Atsushi Shiho for theirconstructive comments and remarks on the slides.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

2 / 60Table of Contents

1

Deligne’s conjectures: `-adic theory

2

Deligne’s conjectures: crystalline theory3

Malčev-Grothendieck theorem; Gieseker conjecture; de Jong conjecture

4

Relative 0-cycles

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

3 / 60Finiteness theorem over C
Theorem (Deligne ’87)
X /C smooth connected variety, r ∈ N>0 given. Then there are finitely
many rank r Q-local systems which are direct factors of Q-variations ofpolarisable pure Hodge structures of a given weight, definable over Z.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

4 / 60Finiteness theorem over C
Theorem (Deligne ’87)
X /C smooth connected variety, r ∈ N>0 given. Then there are finitely
many rank r Q-local systems which are direct factors of Q-variations ofpolarisable pure Hodge structures of a given weight, definable over Z.

Example (Faltings’ finiteness of abelian schemes on X , ’83)
In general, this is a generalisation the version over C of Faltings’ theorem.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

4 / 60Finiteness theorem over Fq
Theorem (Deligne ’12)
X /Fq smooth quasi-projective variety, r ∈ N>0 given; D ⊂ X̄ an effective
Cartier divisor of a normal compactification with support X̄ \ X , andr ∈ N>0 given. Then there are finitely many irreducible Weil (resp. étale)rank r lisse Q̄` -sheaves with ramification bounded by D, up to twist with
Weil (resp. étale) characters of Fq . The number does not depend on the
choice of `.

Corollary (Deligne ’07, Deligne’s conjecture, Weil II, 1.2.10)
Given an étale lisse Q̄` -sheaf V with finite determinant, the subfield of Q̄`spanned by the EV of the Frobenii Fx at closed points x ∈ |X | acting on
Vx̄ is a number field.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

5 / 60Lefschetz theorem over C
Theorem over C is in fact a theorem on X of dimension 1: fixing a goodcompactification X̄ ⊃ X , with a s.n.c.d. at infinity, then a curve C̄ ,complete intersection of ample divisors in X̄ in good position, fulfils theLefschetz theorem
π1top (C := X ∩ C̄ )  π1top (X ).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

6 / 60Lefschetz theorem over C
Theorem over C is in fact a theorem on X of dimension 1: fixing a goodcompactification X̄ ⊃ X , with a s.n.c.d. at infinity, then a curve C̄ ,complete intersection of ample divisors in X̄ in good position, fulfils theLefschetz theorem
π1top (C := X ∩ C̄ )  π1top (X ).
For X of dimension ≥ 2 in char. p > 0, we do not have a Lefschetz
theorem at disposal. So Theorem over Fq does not reduce to X ofdimension 1.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

6 / 60Lefschetz theorem over C
Theorem over C is in fact a theorem on X of dimension 1: fixing a goodcompactification X̄ ⊃ X , with a s.n.c.d. at infinity, then a curve C̄ ,complete intersection of ample divisors in X̄ in good position, fulfils theLefschetz theorem
π1top (C := X ∩ C̄ )  π1top (X ).
For X of dimension ≥ 2 in char. p > 0, we do not have a Lefschetz
theorem at disposal. So Theorem over Fq does not reduce to X ofdimension 1.
Yet one has:Hélène Esnault, Freie Universität Berlin

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6 / 60Tame Lefschetz theorem in char. p > 0
Theorem (Drinfeld ’11)
Let X̄ ⊃ X be a projective normal compactification of X smooth over a
field k, Σ ⊂ X̄ be closed of codimension ≥ 2 such that (X̄ \ Σ) and(X̄ \ Σ) ∩ (X̄ \ X ) are smooth, C̄ ⊂ X̄ \ Σ be a smooth projective curve,
complete intersection of ample divisors, meeting X̄ \ X transversally. Then
π1t (C = C̄ ∩ X )  π1t (X ).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

7 / 60Tame Lefschetz theorem in char. p > 0
Theorem (Drinfeld ’11)
Let X̄ ⊃ X be a projective normal compactification of X smooth over a
field k, Σ ⊂ X̄ be closed of codimension ≥ 2 such that (X̄ \ Σ) and(X̄ \ Σ) ∩ (X̄ \ X ) are smooth, C̄ ⊂ X̄ \ Σ be a smooth projective curve,
complete intersection of ample divisors, meeting X̄ \ X transversally. Then
π1t (C = C̄ ∩ X )  π1t (X ).
No need of a good compactification.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

7 / 60Tame Lefschetz theorem in char. p > 0
Theorem (Drinfeld ’11)
Let X̄ ⊃ X be a projective normal compactification of X smooth over a
field k, Σ ⊂ X̄ be closed of codimension ≥ 2 such that (X̄ \ Σ) and(X̄ \ Σ) ∩ (X̄ \ X ) are smooth, C̄ ⊂ X̄ \ Σ be a smooth projective curve,
complete intersection of ample divisors, meeting X̄ \ X transversally. Then
π1t (C = C̄ ∩ X )  π1t (X ).
No need of a good compactification.

Proof.
Bertini to get that restriction to C of connected finite étale cover of X isconnected, tameness and transversality to keep smoothness, thus
irreducibility.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

7 / 60Tame Lefschetz theorem in char. p > 0 II
If X̄ \ X is a s.n.c.d. compactification, Kindler enhances the theorem: if
S̄ ⊂ X̄ is a smooth projective surface, complete intersection of divisors ingood position, then∼=
π1t (S = S̄ ∩ X ) −→ π1t (X ).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 20158 / 60Wild Lefschetz theorems in char. p > 0
Theorem (Wiesend ’06, Drinfeld ’11)
Over X quasi-projective smooth over Fq , with S ⊂ |X | finite:1) let V be an irreducible Q̄` -Weil or -étale lisse sheaf, then there is asmooth curve C → X with S ⊂ |C |, such that V |C is irreducible;
2) let H ⊂ π1 (X ) be an open normal subgroup, then there is a smooth
curve C → X with S ⊂ |C |, such that π1 (C )  π1 (X )/H.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

9 / 60Wild Lefschetz theorems in char. p > 0
Theorem (Wiesend ’06, Drinfeld ’11)
Over X quasi-projective smooth over Fq , with S ⊂ |X | finite:1) let V be an irreducible Q̄` -Weil or -étale lisse sheaf, then there is asmooth curve C → X with S ⊂ |C |, such that V |C is irreducible;
2) let H ⊂ π1 (X ) be an open normal subgroup, then there is a smooth
curve C → X with S ⊂ |C |, such that π1 (C )  π1 (X )/H.

Proof.
Uses Hilbert irreducibility à la Wiesend.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

9 / 60

Corollaries of the wild Lefschetz theorems: weights andcompanions
Corollary (Drinfeld ’11, Deligne’s conjecture in Weil II, 1.2.10)
1) if det(V ) is torsion, then V has weight 0;2) if V is an irreducible Weil lisse Q̄` -sheaf with determinant of finite order, andσ ∈ Aut(Q̄` /Q), there is an irreducible Weil lisse Q̄` -sheaf Vσ , called the
σ-companion of V , with determinant of finite order, such that the characteristic
polynomials fV , fVσ ∈ Q̄` [t] of the local Frobenii Fx satisfy fVσ = σ(fV ).

Hélène Esnault, Freie Universität Berlin

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Corollaries of the wild Lefschetz theorems: weights andcompanions
Corollary (Drinfeld ’11, Deligne’s conjecture in Weil II, 1.2.10)
1) if det(V ) is torsion, then V has weight 0;2) if V is an irreducible Weil lisse Q̄` -sheaf with determinant of finite order, andσ ∈ Aut(Q̄` /Q), there is an irreducible Weil lisse Q̄` -sheaf Vσ , called the
σ-companion of V , with determinant of finite order, such that the characteristic
polynomials fV , fVσ ∈ Q̄` [t] of the local Frobenii Fx satisfy fVσ = σ(fV ).

Proof.
Reduce the problem to curves. Then consequence of Lafforgue’s Langlands duality:
1) existence of weights on curves;2) existence of companions on curves.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

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Abelian Lefschetz theorem
Theorem (Kerz-S.Saito ’14)
Let X be a smooth quasi-projective variety over a perfect field k, let
X ⊂ X̄ be a projective s.n.c.d. compactification, D be an effective divisor
with support in X̄ \ X . Define π1ab (X , D) by the condition that a characterχ : π1 (X ) → Q/Z factors through π1ab (X , D) iff the Artin conductor of χ
pulled-back to any curve C → X is bounded by the pull-back of D via
C̄ → X̄ . Then Lefschetz holds: for i : Ȳ ⊂ X̄ very very ample and in good
position w.r.t. X̄ \ X , one has:i∗ : π1ab (Y , Ȳ ∩ D) → π1ab (X , D)
is an isomorphism if dim Y ≥ 2, surjective if dim Y = 1.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

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Corollary of Abelian Lefschetz theorem: abelian finitenessover Fq
Corollary (Raskind ’92, this formulation by Kerz-S.Saito ’14)k = Fq , then Ker π1ab (X , D) → π1ab (k) is finite. (So in particular, this
implies Deligne’s finiteness for sums of rank 1 lisse sheaves).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

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Corollary of Abelian Lefschetz theorem: abelian finitenessover Fq
Corollary (Raskind ’92, this formulation by Kerz-S.Saito ’14)k = Fq , then Ker π1ab (X , D) → π1ab (k) is finite. (So in particular, this
implies Deligne’s finiteness for sums of rank 1 lisse sheaves).

Proof.
Reduce to curves via de Jong’s alterations (in the more general case X̄ is anormal compactification) plus the Theorem and apply then CFT.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

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Right fundamental group with ramification bounded by aQ≥0 -divisor
Questions
One has the notion of a lisse étale Q̄` -sheaf π1 (X ) → Aut(V ) withramification bounded by D, a positive Q-divisor (Hu-Yang: does not needa good compactification; as for Drinfeld’s Lefschetz theorem for π1t (X )).
How does one define a quotient π1 (X )  π1 (X , D) generalising
π1ab (X , D)? Then one could ask for a Lefschetz theorem
π1 (C , DC )  π1 (X , D) for a suitable curve C which would reflect
Deligne’s finiteness theorem.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

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Table of Contents

1

Deligne’s conjectures: `-adic theory

2

Deligne’s conjectures: crystalline theory3

Malčev-Grothendieck theorem; Gieseker conjecture; de Jong conjecture

4

Relative 0-cycles

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

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The various categories of isocrystals under consideration I
X smooth geometrically connected over a perfect field k, W := W (k) thering of Witt vectors, K = Frac(W ) its field of fractions.

Hélène Esnault, Freie Universität Berlin

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The various categories of isocrystals under consideration I
X smooth geometrically connected over a perfect field k, W := W (k) thering of Witt vectors, K = Frac(W ) its field of fractions.
One defines the crystalline sites X /Wn as PD-thickenings(U ,→ T /Wn , δ), coverings coming from U ⊂ X Zariski;Hélène Esnault, Freie Universität Berlin

Fundamental Groups

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The various categories of isocrystals under consideration I
X smooth geometrically connected over a perfect field k, W := W (k) thering of Witt vectors, K = Frac(W ) its field of fractions.
One defines the crystalline sites X /Wn as PD-thickenings(U ,→ T /Wn , δ), coverings coming from U ⊂ X Zariski;
X /W as limit.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

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The various categories of isocrystals under consideration II• category of crystals (i.e. sheaves of OX /W -modules of finite
presentation, with transition maps which are isomorphisms) Crys(X /W ),which is W -linear;Hélène Esnault, Freie Universität Berlin

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The various categories of isocrystals under consideration II• category of crystals (i.e. sheaves of OX /W -modules of finite
presentation, with transition maps which are isomorphisms) Crys(X /W ),which is W -linear;• Q-linearisation Crys(X /W )Q =category of isocrystals, which is K -linear,tannakian;Hélène Esnault, Freie Universität Berlin

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The various categories of isocrystals under consideration II• category of crystals (i.e. sheaves of OX /W -modules of finite
presentation, with transition maps which are isomorphisms) Crys(X /W ),which is W -linear;• Q-linearisation Crys(X /W )Q =category of isocrystals, which is K -linear,tannakian;• absolute Frobenius F acts on Crys(X /W )Q ;Hélène Esnault, Freie Universität Berlin

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The various categories of isocrystals under consideration II• category of crystals (i.e. sheaves of OX /W -modules of finite
presentation, with transition maps which are isomorphisms) Crys(X /W ),which is W -linear;• Q-linearisation Crys(X /W )Q =category of isocrystals, which is K -linear,tannakian;• absolute Frobenius F acts on Crys(X /W )Q ;• largest full subcategory on which every object is F ∞ -divisible isConv(X /K ) ⊂ Crys(X /W )Q , the K -tannakian subcategory of convergent
isocrystals (Berthelot-Ogus); (Ogus defines the site of enlargements from
X /W , then convergent isocrystals are crystals of OX /K -modules of finite
presentation).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

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The various categories of isocrystals under consideration III∼=• F − Conv(X /K ) → Conv(X /K ), (E, Φ : F ∗ E −→ E) 7→ E;

Hélène Esnault, Freie Universität Berlin

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The various categories of isocrystals under consideration III∼=• F − Conv(X /K ) → Conv(X /K ), (E, Φ : F ∗ E −→ E) 7→ E;• F − Conv(X /K ) Qp -linear tannakian;Hélène Esnault, Freie Universität Berlin

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The various categories of isocrystals under consideration III∼=• F − Conv(X /K ) → Conv(X /K ), (E, Φ : F ∗ E −→ E) 7→ E;• F − Conv(X /K ) Qp -linear tannakian;fully faithful Kedlaya• F − Overconv(X /K ) −−−−−−−−−−−−−→ F − Conv(X /K );Hélène Esnault, Freie Universität Berlin

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The various categories of isocrystals under consideration III∼=• F − Conv(X /K ) → Conv(X /K ), (E, Φ : F ∗ E −→ E) 7→ E;• F − Conv(X /K ) Qp -linear tannakian;fully faithful Kedlaya• F − Overconv(X /K ) −−−−−−−−−−−−−→ F − Conv(X /K );• F − Overconv(X /K ) consists of those E which have ”unipotent local
monodromy” after alteration (Kedlaya);Hélène Esnault, Freie Universität Berlin

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The various categories of isocrystals under consideration IV• Brieskorn over C: Gauß-Manin connections have quasi-unipotent local
monodromies;Hélène Esnault, Freie Universität Berlin

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The various categories of isocrystals under consideration IV• Brieskorn over C: Gauß-Manin connections have quasi-unipotent local
monodromies;
so become unipotent after a surjective finite cover of X , possibly ramified(Kawamata’s trick);Hélène Esnault, Freie Universität Berlin

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The various categories of isocrystals under consideration IV• Brieskorn over C: Gauß-Manin connections have quasi-unipotent local
monodromies;
so become unipotent after a surjective finite cover of X , possibly ramified(Kawamata’s trick);• Grothendieck over Fq : lisse Q̄` -sheaves have quasi-unipotent local
monodromies (action of local inertia Z` (1)).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201518 / 60

The various categories of isocrystals under consideration V• Kedlaya over k (not necessarily perfect): E ∈ F − Overconv(X /K ) has’unipotent monodromy’ (in a suitable sense) at infinity after an alteration(uses André-Kedlaya-Mebkhout local result).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

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The various categories of isocrystals under consideration V• Kedlaya over k (not necessarily perfect): E ∈ F − Overconv(X /K ) has’unipotent monodromy’ (in a suitable sense) at infinity after an alteration(uses André-Kedlaya-Mebkhout local result).
There are blow-ups at infinity: analog to resolution of turning points overC (Kedlaya/T. Mochizuki).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201519 / 60

The various categories of isocrystals under consideration V• Kedlaya over k (not necessarily perfect): E ∈ F − Overconv(X /K ) has’unipotent monodromy’ (in a suitable sense) at infinity after an alteration(uses André-Kedlaya-Mebkhout local result).
There are blow-ups at infinity: analog to resolution of turning points overC (Kedlaya/T. Mochizuki).
• From the definition:

On X proper, F − Overconv(X /K ) = F − Conv(X /K ).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

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The various categories of isocrystals under consideration IV• Over Fq , q = p s , define FFq = F s − Overconv(X /K ), so
K = FracW (Fq )-linear; abuse of notations F − Overconv(X /K ).

Hélène Esnault, Freie Universität Berlin

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The various categories of isocrystals under consideration IV• Over Fq , q = p s , define FFq = F s − Overconv(X /K ), so
K = FracW (Fq )-linear; abuse of notations F − Overconv(X /K ).
• L-linearisation, for K ⊂ L ⊂ Q̄p , L → Q̄p , defines the category
F − Overconv(X /K )Q̄p .

Hélène Esnault, Freie Universität Berlin

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Analogy C ↔ Fq , `-adic, expected analogy /Fq , p-adic
Irreducible objects in F − Overconv(X /K )Q̄p with finite determinant;Hélène Esnault, Freie Universität Berlin

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Analogy C ↔ Fq , `-adic, expected analogy /Fq , p-adic
Irreducible objects in F − Overconv(X /K )Q̄p with finite determinant;are analog to irreducible lisse Q̄` -sheaves with finite determinant;Hélène Esnault, Freie Universität Berlin

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Analogy C ↔ Fq , `-adic, expected analogy /Fq , p-adic
Irreducible objects in F − Overconv(X /K )Q̄p with finite determinant;are analog to irreducible lisse Q̄` -sheaves with finite determinant;Upon bounding ramification at infinity (correct notion for
F − Overconv(X /K )Q̄p ?),Hélène Esnault, Freie Universität Berlin

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Analogy C ↔ Fq , `-adic, expected analogy /Fq , p-adic
Irreducible objects in F − Overconv(X /K )Q̄p with finite determinant;are analog to irreducible lisse Q̄` -sheaves with finite determinant;Upon bounding ramification at infinity (correct notion for
F − Overconv(X /K )Q̄p ?),are analog over C to irreducible Q-variations of polarisable pure Hodge
structures of pure weight definable over Z.

Hélène Esnault, Freie Universität Berlin

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Crystalline version (petits camarades cristallins) on curves
Theorem (Abe, Crystalline version of Lafforgue’s theorem ’13)
Let X be a smooth curve over Fq . Then
1) an irreducible overconvergent Q̄p -F -isocrystal with finite determinantis ι-pure of weight 0;2) an irreducible lisse Q̄` - étale sheaf with finite determinant has an
overconvergent Q̄p -F –isocrystal companion and vice-versa.

Hélène Esnault, Freie Universität Berlin

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Crystalline version (petits camarades cristallins) on higher
dimensional varieties
No wild Lefschetz theorem for F -overconvergent isocrystals
In particularHélène Esnault, Freie Universität Berlin

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Crystalline version (petits camarades cristallins) on higher
dimensional varieties
No wild Lefschetz theorem for F -overconvergent isocrystals
In particular1) one does not know whether irreducible F -isocrystals with finite determinantare ι-pure of weight 0 on X smooth quasi-projective variety over Fq ;Hélène Esnault, Freie Universität Berlin

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Crystalline version (petits camarades cristallins) on higher
dimensional varieties
No wild Lefschetz theorem for F -overconvergent isocrystals
In particular1) one does not know whether irreducible F -isocrystals with finite determinantare ι-pure of weight 0 on X smooth quasi-projective variety over Fq ;2) a fortiori, one does not have a number field capturing the EV of FFq(x) acting
on the stalks at closed points Spec Fq(x) ;Hélène Esnault, Freie Universität Berlin

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Crystalline version (petits camarades cristallins) on higher
dimensional varieties
No wild Lefschetz theorem for F -overconvergent isocrystals
In particular1) one does not know whether irreducible F -isocrystals with finite determinantare ι-pure of weight 0 on X smooth quasi-projective variety over Fq ;2) a fortiori, one does not have a number field capturing the EV of FFq(x) acting
on the stalks at closed points Spec Fq(x) ;3) nor does one have a crystalline version of Deligne’s finiteness theorem.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

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Crystalline version (petits camarades cristallins) on higher
dimensional varieties
No wild Lefschetz theorem for F -overconvergent isocrystals
In particular1) one does not know whether irreducible F -isocrystals with finite determinantare ι-pure of weight 0 on X smooth quasi-projective variety over Fq ;2) a fortiori, one does not have a number field capturing the EV of FFq(x) acting
on the stalks at closed points Spec Fq(x) ;3) nor does one have a crystalline version of Deligne’s finiteness theorem.

So: no higher dimensional generalisation of Drinfeld/Deligne.

Hélène Esnault, Freie Universität Berlin

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Abe’s Čebotarev theorem
Theorem (Abe ’13)
On X quasi-projective smooth over Fq , ι-pure (or mixed, ι : Q̄p ∼= C)semi-simple objects in F − Overconv(X /K ) are determined by their local
EV at closed points.

Hélène Esnault, Freie Universität Berlin

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Table of Contents

1

Deligne’s conjectures: `-adic theory

2

Deligne’s conjectures: crystalline theory3

Malčev-Grothendieck theorem; Gieseker conjecture; de Jong conjecture

4

Relative 0-cycles

Hélène Esnault, Freie Universität Berlin

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Weaker analogies
Rather than considering analogies between some irreducible complex local
systems (’motivic’ ones) with some lisse Q̄` - sheaves (irreducible withfinite determinant) over Fq , and with some overconvergent Q̄p - F
isocrystals (irreducible with finite determinant), one can raise theHélène Esnault, Freie Universität Berlin

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Weaker analogies
Rather than considering analogies between some irreducible complex local
systems (’motivic’ ones) with some lisse Q̄` - sheaves (irreducible withfinite determinant) over Fq , and with some overconvergent Q̄p - F
isocrystals (irreducible with finite determinant), one can raise theQuestionwhat is the analog of complex local systems on X over C for X over aperfect field of characteristic p > 0?

Hélène Esnault, Freie Universität Berlin

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26 / 60

Infinitesimal site and crystals in characteristic 0
Infinitesimal site
X smooth over a char. 0 field k;

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201527 / 60

Infinitesimal site and crystals in characteristic 0
Infinitesimal site
X smooth over a char. 0 field k;
X∞ : U ,→ T infinitesimal thickening of a Zariski open U; coverings fromthe Us.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201527 / 60

Infinitesimal site and crystals in characteristic 0
Infinitesimal site
X smooth over a char. 0 field k;
X∞ : U ,→ T infinitesimal thickening of a Zariski open U; coverings fromthe Us.
{finitely presented crystals on X∞ } = {(E , ∇)}, E coherent sheaf and ∇flat connection (thus E is locally free);k-linear category (assume here k = field of constants of X , i.e. X
geometrically connected over k);

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201527 / 60

Malčev-Grothendieck’s theorem
Theorem (Malčev ’40-Grothendieck ’70)
X smooth over C; then π1ét (X ) = {1} implies there are no non-trivial crystals in theinfinitesimal site (with regular singularities at infinity in case X is not projective).

Proof.
Use Riemann-Hilbert correspondence to translate to finite dimensional complex
local systems.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201528 / 60

Malčev-Grothendieck’s theorem
Theorem (Malčev ’40-Grothendieck ’70)
X smooth over C; then π1ét (X ) = {1} implies there are no non-trivial crystals in theinfinitesimal site (with regular singularities at infinity in case X is not projective).

Proof.
Use Riemann-Hilbert correspondence to translate to finite dimensional complex
local systems.
Then π1top (X (C)) is an abstract group of finite type, so ρ : π1top (X (C)) → GL(r , C)factors through ρA : π1top (X (C)) → GL(r , A), A/Z of finite type, and ρ = 1 iff
ρA = 1 iff ρa : π1top (X (C)) → GL(r , κ(a)) ∀ closed point a ∈ Spec(A).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201528 / 60

Conservativity
So { finite étale category } trivial implies { infinitesimal crystals } (regular
singular) trivial.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201529 / 60

Conservativity
So { finite étale category } trivial implies { infinitesimal crystals } (regular
singular) trivial.

More modest question: analogs in char. p > 0 of this conservativity
theorem? (Terminology ’conservativity’ borrowed from Ayoub’s work).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201529 / 60

Infinitesimal site and crystals in characteristic p > 0
Infinitesimal site
As in char. 0: X smooth over a char. p > 0 field k; X∞ : U ,→ T
infinitesimal thickening of a Zariski open U; coverings from the Us.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201530 / 60

Infinitesimal site and crystals in characteristic p > 0
Infinitesimal site
As in char. 0: X smooth over a char. p > 0 field k; X∞ : U ,→ T
infinitesimal thickening of a Zariski open U; coverings from the Us.
{finitely presented crystals on X∞ } = {OX − coherent DX − modules}

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201530 / 60

Infinitesimal site and crystals in characteristic p > 0
Infinitesimal site
As in char. 0: X smooth over a char. p > 0 field k; X∞ : U ,→ T
infinitesimal thickening of a Zariski open U; coverings from the Us.
{finitely presented crystals on X∞ } = {OX − coherent DX − modules}= {F − d.c.s.} (divided coherent sheaves) (Cartier isomorphism, Katz’
theorem).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201530 / 60

Infinitesimal site and crystals in characteristic p > 0
Infinitesimal site
As in char. 0: X smooth over a char. p > 0 field k; X∞ : U ,→ T
infinitesimal thickening of a Zariski open U; coverings from the Us.
{finitely presented crystals on X∞ } = {OX − coherent DX − modules}= {F − d.c.s.} (divided coherent sheaves) (Cartier isomorphism, Katz’
theorem).
k-linear category (assume here k = field of constants of X , i.e. X
geometrically connected over k).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201530 / 60

Gieseker’s conjecture
Gieseker’s conjecture ’75
On X projective smooth over k = k̄ of char. p > 0, π1ét (X ) = {1} implies
that there are no non-trivial crystals in the infinitesimal site.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201531 / 60

Gieseker’s conjecture
Gieseker’s conjecture ’75
On X projective smooth over k = k̄ of char. p > 0, π1ét (X ) = {1} implies
that there are no non-trivial crystals in the infinitesimal site.

Theorem (E-Mehta ’10)
Conjecture has a positive answer.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201531 / 60

Variants of Gieseker’s conjecture• X not proper: theory of regular singular crystals in the infinitesimal site
developed by Kindler (’13), so that for those with finite monodromy, it
coincides with the notion of tame quotient of π1ét (X ).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201532 / 60

Variants of Gieseker’s conjecture• X not proper: theory of regular singular crystals in the infinitesimal site
developed by Kindler (’13), so that for those with finite monodromy, it
coincides with the notion of tame quotient of π1ét (X ).
• Yet no ramification theory, so far.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201532 / 60

Variants of Gieseker’s conjecture• X not proper: theory of regular singular crystals in the infinitesimal site
developed by Kindler (’13), so that for those with finite monodromy, it
coincides with the notion of tame quotient of π1ét (X ).
• Yet no ramification theory, so far.
• So far no extension of the conservativity theorem, except for the tameabelian quotient (Kindler ’13) and for X = smooth locus of a normal
projective variety and k = F̄q (E-Srinivas ’14, using an improvement ofGrothendieck’s LEF theorem by Bost, ’14).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201532 / 60

On the proof of Gieseker’s conjecture• E ∈ F − d.c.s. implies E ∈ Coh(X ) is locally free. In fact, this is a
Tannakian category over k.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201533 / 60

On the proof of Gieseker’s conjecture• E ∈ F − d.c.s. implies E ∈ Coh(X ) is locally free. In fact, this is a
Tannakian category over k.
2i (X /W ), i ≥ 1.
• E ∈ F − d.c.s. implies 0 = ci,crys (E ) ∈ Hcrys

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201533 / 60

On the proof of Gieseker’s conjecture• E ∈ F − d.c.s. implies E ∈ Coh(X ) is locally free. In fact, this is a
Tannakian category over k.
2i (X /W ), i ≥ 1.
• E ∈ F − d.c.s. implies 0 = ci,crys (E ) ∈ Hcrys• Bounded problem: a high enough Frobenius division of E ∈ F − d.c.s. isa semi-stable vector bundle of slope 0 (whatever polarisation).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201533 / 60

On the proof of Gieseker’s conjecture• E ∈ F − d.c.s. implies E ∈ Coh(X ) is locally free. In fact, this is a
Tannakian category over k.
2i (X /W ), i ≥ 1.
• E ∈ F − d.c.s. implies 0 = ci,crys (E ) ∈ Hcrys• Bounded problem: a high enough Frobenius division of E ∈ F − d.c.s. isa semi-stable vector bundle of slope 0 (whatever polarisation).
• Thus they have moduli points in Langer’s moduli of semi-stable puresheaves with trivial numerical Chern classes.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201533 / 60

On the proof of Gieseker’s conjecture• E ∈ F − d.c.s. implies E ∈ Coh(X ) is locally free. In fact, this is a
Tannakian category over k.
2i (X /W ), i ≥ 1.
• E ∈ F − d.c.s. implies 0 = ci,crys (E ) ∈ Hcrys• Bounded problem: a high enough Frobenius division of E ∈ F − d.c.s. isa semi-stable vector bundle of slope 0 (whatever polarisation).
• Thus they have moduli points in Langer’s moduli of semi-stable puresheaves with trivial numerical Chern classes.
• Hrushovsky’s theorem then guarantees the existence of a Frobenius
invariant vector bundle on a specialization of X over F̄p , which yields a
non-trivial finite étale cover of this one.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201533 / 60

Various crystalline sites and (iso)crystal categories
The categories considered were already presented. Again we assume X
projective smooth geometrically connected over k perfect of char. p > 0.
We consider

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201534 / 60

Various crystalline sites and (iso)crystal categories
The categories considered were already presented. Again we assume X
projective smooth geometrically connected over k perfect of char. p > 0.
We consider• k = Fq , F -convergent isocrystals
F − Overconv(X /K ) = F − Conv(X /K );Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201534 / 60

Various crystalline sites and (iso)crystal categories
The categories considered were already presented. Again we assume X
projective smooth geometrically connected over k perfect of char. p > 0.
We consider• k = Fq , F -convergent isocrystals
F − Overconv(X /K ) = F − Conv(X /K );• k perfect, convergent isocrystals and isocrystals:
Conv(X /K ) ,→ Crys(X /W )Q .

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201534 / 60

de Jong’s conjecturede Jong’s conjecture ’10
On X projective smooth over k perfect of char. p > 0, π1ét (X ⊗k k̄) = {1}implies that there are no non-trivial isocrystals.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201535 / 60

de Jong’s conjecturede Jong’s conjecture ’10
On X projective smooth over k perfect of char. p > 0, π1ét (X ⊗k k̄) = {1}implies that there are no non-trivial isocrystals.

Theorem (E-Shiho, ’15)
Partial positive answer.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201535 / 60

de Jong’s conjecturede Jong’s conjecture ’10
On X projective smooth over k perfect of char. p > 0, π1ét (X ⊗k k̄) = {1}implies that there are no non-trivial isocrystals.

Theorem (E-Shiho, ’15)
Partial positive answer.
In the sequel, we report on it, raising a few questions on the way.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201535 / 60

Abelian case
π1ét (X ⊗k k̄) = {1} implies H 1 (X , OX×/W ) = 0, thus rank 1 locally freecrystals, and thus isocrystals, are trivial,
as well as H 1 (X /W ) = 0, thus successive extensions of trivial rank 1
crystals, thus isocrystals, are trivial.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201536 / 60

Abelian case
π1ét (X ⊗k k̄) = {1} implies H 1 (X , OX×/W ) = 0, thus rank 1 locally freecrystals, and thus isocrystals, are trivial,
as well as H 1 (X /W ) = 0, thus successive extensions of trivial rank 1
crystals, thus isocrystals, are trivial.
Thus the conjecture essentially predicts a relation between the”non-abelian” part of π1ét (X ) and irreducible isocrystals of higher rank.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201536 / 60

Abelian case
π1ét (X ⊗k k̄) = {1} implies H 1 (X , OX×/W ) = 0, thus rank 1 locally freecrystals, and thus isocrystals, are trivial,
as well as H 1 (X /W ) = 0, thus successive extensions of trivial rank 1
crystals, thus isocrystals, are trivial.
Thus the conjecture essentially predicts a relation between the”non-abelian” part of π1ét (X ) and irreducible isocrystals of higher rank.
At least when p ≥ 3 it is so; for p = 2 those statements are less direct and follow from the whole proof.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201536 / 60

Gauß-Manin F -convergent isocrystal
Theorem (E-Shiho ’15)
Let f : Y → X be a smooth projective morphism over X smooth projective over kperfect. If π1ét (X ⊗k k̄) = {1}, then the F -convergent isocrystal R i f∗ is trivial inConv(X /K ).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201537 / 60

Gauß-Manin F -convergent isocrystal
Theorem (E-Shiho ’15)
Let f : Y → X be a smooth projective morphism over X smooth projective over kperfect. If π1ét (X ⊗k k̄) = {1}, then the F -convergent isocrystal R i f∗ is trivial inConv(X /K ).

Model of Proof
Assume f was an abelian scheme and k = Fq . May assume X has a rational pointx0 . Then (argument of Faltings): π1ét (X ) acts on R i f∗ Q` via Gal(F̄p /Fq ), thus by
the Honda-Tate theorem, all geometric fibres of f are isogeneous. Thus for all
closed points x ∈ |X |, H 1 (Yx /FracW (k(x))) = H 1 (Yx0 /K ) ⊗K FracW (k(x)), thus
the isocrystal R i f∗ is trivial in Conv(X /K ).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201537 / 60

Gauß-Manin F -convergent isocrystal: Proof
Proof.
Over k = Fq , one replaces the motivic argument (Honda-Tate) by Abe’ sČebotarev’s density theorem.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201538 / 60

Gauß-Manin F -convergent isocrystal: Proof
Proof.
Over k = Fq , one replaces the motivic argument (Honda-Tate) by Abe’ sČebotarev’s density theorem.
Yields triviality of the semi-simplification of R i f∗ in Conv(X /K ) after
specialisation over Fq .

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201538 / 60

Gauß-Manin F -convergent isocrystal: Proof
Proof.
Over k = Fq , one replaces the motivic argument (Honda-Tate) by Abe’ sČebotarev’s density theorem.
Yields triviality of the semi-simplification of R i f∗ in Conv(X /K ) after
specialisation over Fq .
Apply base change to get it over k. Yields triviality of thesemi-simplification of R i f∗ in Conv(X /K ) over k.
Thus R i f∗ ∈ Conv(X /K ) ⊂ Crys(X /W )Q is trivial, as we already saw
that there are no non-trivial extensions.
From now on, we discuss the general case.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201538 / 60

Existence of lattices
Lemma
X smooth over k perfect, E ∈ Crys(X /W )Q , there is a p-torsion-free
E ∈ Crys(X /W ) with EQ = E, called a lattice.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201539 / 60

Existence of lattices
Lemma
X smooth over k perfect, E ∈ Crys(X /W )Q , there is a p-torsion-free
E ∈ Crys(X /W ) with EQ = E, called a lattice.

Proof.
Given any E ∈ Crys(X /W ), with EQ = E, the surjective maps
E /Ker(p n+1 )  E /Ker(p n ) stabilise, as one sees locally on finitely many
open affines U, as then Crys(U/W ) ∼= MIC (ÛW /W )qn , thequasi-nilpotent flat connections on a formal lift.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201539 / 60

Locally free lattices
E is said to be locally free if it has a locally free lattice E , so EQ = E, thatis equivalently if EX , the value of E on X ,→ X , viewed in Coh(X ), islocally free.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

40 / 60

Locally free lattices
E is said to be locally free if it has a locally free lattice E , so EQ = E, thatis equivalently if EX , the value of E on X ,→ X , viewed in Coh(X ), islocally free.

Question
Are all E ∈ Crys(X /W )Q locally free?
A positive answer would ease the understanding of de Jong’s conjecture.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

40 / 60

Crystalline Chern classes of the value on X of lattices
Theorem (E-Shiho ’15)
Let E ∈ Crys(X /W ) be a lattice.
1) If E is locally free, then 0 = ci,crys (EX ) ∈ H 2i (X /W ), i ≥ 1.
2) If EQ ∈ Conv(X /K ), then 0 = ci,crys (EX ) ∈ H 2i (X /K ), i ≥ 1.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

41 / 60

Proof of the vanishing of the crystalline Chern classes ofthe value on X of lattices
Proof.
On 1): there are (at least) two ways;

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201542 / 60

Proof of the vanishing of the crystalline Chern classes ofthe value on X of lattices
Proof.
On 1): there are (at least) two ways;
Modified splitting principle: on X ⊂ D → PW PD-hull, one considers the quotientΩ•D → Ω̄•D of DGAs defined by dx [n] = x [n−1] dx. This defines the quotientΩ•P(ED ) → Ω̄•P(ED ) of DGAs by moding out by the ’same’ kernel, where ED is thevalue of E on X ,→ D. Let π : P(ED ) → D be the principal bundle.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201542 / 60

Proof of the vanishing of the crystalline Chern classes ofthe value on X of lattices
Proof.
On 1): there are (at least) two ways;
Modified splitting principle: on X ⊂ D → PW PD-hull, one considers the quotientΩ•D → Ω̄•D of DGAs defined by dx [n] = x [n−1] dx. This defines the quotientΩ•P(ED ) → Ω̄•P(ED ) of DGAs by moding out by the ’same’ kernel, where ED is thevalue of E on X ,→ D. Let π : P(ED ) → D be the principal bundle. One shows ∇τon ED defines a splitting Ω̄•D → Rπ∗ Ω̄•P(ED ) −→ Ω̄•D , defining a partial connection on
π ∗ ED with value in π ∗ Ω̄1D , which is shown to respect OP(ED ) (1).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201542 / 60

Proof of the vanishing of the crystalline Chern classes ofthe value on X of lattices II
Proof.
Equating O and dR-cohomology: for X smooth, define D• to be the×(n+1)
simplicial scheme defined by Dn =PD-hull of the diagonal in PW.
i (D )(:= H i (D , Ω̄• )) = H i (D , O).
Then H i (X /W ) = HdR
•
•
•
Zar
D•

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

43 / 60

Proof of the vanishing of the crystalline Chern classes ofthe value on X of lattices II
Proof.
Equating O and dR-cohomology: for X smooth, define D• to be the×(n+1)
simplicial scheme defined by Dn =PD-hull of the diagonal in PW.
i (D )(:= H i (D , Ω̄• )) = H i (D , O). For E
Then H i (X /W ) = HdR
•
•
•
Zar
D•
given, EX defines e : X• → BGL(r ), BGL(r ) simplicial scheme, X• comingfrom a Čech simplicial scheme associated to local trivialisations of EX .

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

43 / 60

Proof of the vanishing of the crystalline Chern classes ofthe value on X of lattices II
Proof.
Equating O and dR-cohomology: for X smooth, define D• to be the×(n+1)
simplicial scheme defined by Dn =PD-hull of the diagonal in PW.
i (D )(:= H i (D , Ω̄• )) = H i (D , O). For E
Then H i (X /W ) = HdR
•
•
•
Zar
D•
given, EX defines e : X• → BGL(r ), BGL(r ) simplicial scheme, X• comingfrom a Čech simplicial scheme associated to local trivialisations of EX . Doa simplicial version D• of the D• above. Then E has a value ED• . Soe ∗ : H 2i (BGL(r )/W ) → H 2i (X /W ) factors throughH 2i (D• /W ) = H 2n (D• , O), thus e factors throughH 2i (BGL(r )/W ) → H 2i (BGL(r ), O), which is trivial for i ≥ 1.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 2015

43 / 60

Proof of the vanishing of the crystalline Chern classes ofthe value on X of lattices III
Proof.
2) Show, the class of EX in K0 (X ), where E is a lattice ofE ∈ Crys(X /W )Q , depends only on E . Thus since E ∈ Conv(X /K ) is
F ∞ -divisible, chi,crys (EX ) ∈ H 2i (X /K ) is p i∞ -divisible, thus = 0, thus
0 = ci,crys (EX ) ∈ H 2i (X /K ).

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201544 / 60

Lift of trivial crystals over k
Lemma
Assume E ∈ Crys(X /W ) is a lattice, such that ∃m ∈ N≥0 such that
(F m )∗ EX ∈ MIC (X /k)qn is trivial. Then if π1ét,ab (X ⊗k k̄) = {1},E ∈ Crys(X /W ) is trivial.

Begin of Proof.
F ∗ : Crys(X /W )Q → Crys(X /W )Q is fully faithful, so may assume EX
trivial.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201545 / 60

Proof of lift II
Proof.
For D PD-hull of X ⊂ PW , with Dn = D ⊗W Wn , has
1
Ker MIC (Dn+m ) → MIC (Dn )) ∼(Dm ) 1 ≤∀ m ≤ n.
= M(r × r , HdR

Hélène Esnault, Freie Universität Berlin

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Utah, 27-29-30 July 2015(?)

46 / 60

Proof of lift II
Proof.
For D PD-hull of X ⊂ PW , with Dn = D ⊗W Wn , has
1
Ker MIC (Dn+m ) → MIC (Dn )) ∼(Dm ) 1 ≤∀ m ≤ n.
= M(r × r , HdR(?)

1 (D ) = 0 ∃ a ∈ N and
π1ét,ab (X ⊗k k̄) = {1} implies (F a )∗ H 1 (X /k) = (F a )∗ HdR
10

1 (D ) →1∃
that such that HdRn − HdR (D1 ) ∀n ≥ N ∈ N>0 .

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201546 / 60

Proof of lift II
Proof.
For D PD-hull of X ⊂ PW , with Dn = D ⊗W Wn , has
1
Ker MIC (Dn+m ) → MIC (Dn )) ∼(Dm ) 1 ≤∀ m ≤ n.
= M(r × r , HdR(?)

1 (D ) = 0 ∃ a ∈ N and
π1ét,ab (X ⊗k k̄) = {1} implies (F a )∗ H 1 (X /k) = (F a )∗ HdR
10

1 (D ) →1∃
that such that HdRn − HdR (D1 ) ∀n ≥ N ∈ N>0 .

Applying (?) to (n, m) = (1, 1), (2, 1), . . . , (N − 1, 1), we conclude that there isb ∈ N depending only on X such that ((F b )∗ E )DN is trivial. Replace E by (F b )∗ E ,may assume EDN is trivial.
Applying (?) to (n, m) = (2N, N) we conclude EDN+1 = image ED2N via
M(r × r , H 1 (DN )) → M(r × r , H 1 (D1 )), is trivial.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201546 / 60

Proof of lift II
Proof.
For D PD-hull of X ⊂ PW , with Dn = D ⊗W Wn , has
1
Ker MIC (Dn+m ) → MIC (Dn )) ∼(Dm ) 1 ≤∀ m ≤ n.
= M(r × r , HdR(?)

1 (D ) = 0 ∃ a ∈ N and
π1ét,ab (X ⊗k k̄) = {1} implies (F a )∗ H 1 (X /k) = (F a )∗ HdR
10

1 (D ) →1∃
that such that HdRn − HdR (D1 ) ∀n ≥ N ∈ N>0 .

Applying (?) to (n, m) = (1, 1), (2, 1), . . . , (N − 1, 1), we conclude that there isb ∈ N depending only on X such that ((F b )∗ E )DN is trivial. Replace E by (F b )∗ E ,may assume EDN is trivial.
Applying (?) to (n, m) = (2N, N) we conclude EDN+1 = image ED2N via
M(r × r , H 1 (DN )) → M(r × r , H 1 (D1 )), is trivial.
One continues, etc.

Hélène Esnault, Freie Universität Berlin

Fundamental Groups

Utah, 27-29-30 July 201546 / 60

Trivializing a crystal
Theorem (E-Shiho ’15)
Let X be smooth projective over k = k̄ of char. p > 0. Let E be ∈ Conv(X /K )or be locally free in Crys(X /W )Q . If π1ét (X ) = {1}, µmax (Ω1X ) < N(r ) for acertain positive number N(r ) discussed below, and the irreducible constituents of
the Jordan-Hölder filtration of E have rank ≤ r , then E is trivial.
N(1) = ∞, so complete theorem for extension of rank 1 isocrystals;Hélène Esnault, Freie Universität Berlin

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Trivializing a crystal
Theorem (E-Shiho ’15)
Let X be smooth projective over k = k̄ of char. p > 0. Let E be ∈ Conv(X /K )or be locally free in Crys(X /W )Q . If π1ét (X ) = {1}, µmax (Ω1X ) < N(r ) for acertain positive number N(r ) discussed below, and the irreducible constituents of
the Jordan-Hölder filtration of E have rank ≤ r , then E is trivial.
N(1) = ∞, so complete theorem for extension of rank 1 isocrystals; N(2) = 2,
N(3) = 1, N(r ) = 1/M(r ), M(r ) = max lcm(a, b), a, b ≥ 1, a + b ≤ r .
Restriction on E-in Conv(X /K ) or locally free- comes from the fact that thevanishing of the crystalline Chern classes of EX is proven under thisassumption.

Hélène Esnault, Freie Universität Berlin

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Trivializing a crystal
Theorem (E-Shiho ’15)
Let X be smooth projective over k = k̄ of char. p > 0. Let E be ∈ Conv(X /K )or be locally free in Crys(X /W )Q . If π1ét (X ) = {1}, µmax (Ω1X ) < N(r ) for acertain positive number N(r ) discussed below, and the irreducible constituents of
the Jordan-Hölder filtration of E have rank ≤ r , then E is trivial.
N(1) = ∞, so complete theorem for extension of rank 1 isocrystals; N(2) = 2,
N(3) = 1, N(r ) = 1/M(r ), M(r ) = max lcm(a, b), a, b ≥ 1, a + b ≤ r .
Restriction on E-in Conv(X /K ) or locally free- comes from the fact that thevanishing of the crystalline Chern classes of EX is proven under thisassumption.The rôle of the stability assumptions will become clear in the proof.

Hélène Esnault, Freie Universität Berlin

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Proof of the Theorem on trivialisation of a crystal
Proof.
• As in the proof of the triviality of the crystals on the infinitesimal site, one has tobound the problem.

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Proof of the Theorem on trivialisation of a crystal
Proof.
• As in the proof of the triviality of the crystals on the infinitesimal site, one has tobound the problem.
• One proves a Langton type theorem: starting with E irreducible, there is a lattice
E such that EX ∈ MIC (X /k) is semi-stable, so yields a point of Langer’s moduli
M of semi-stable connections on pure sheaves with vanishing Chern classes.

Hélène Esnault, Freie Universität Berlin

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Proof of the Theorem on trivialisation of a crystal
Proof.
• As in the proof of the triviality of the crystals on the infinitesimal site, one has tobound the problem.
• One proves a Langton type theorem: starting with E irreducible, there is a lattice
E such that EX ∈ MIC (X /k) is semi-stable, so yields a point of Langer’s moduli
M of semi-stable connections on pure sheaves with vanishing Chern classes.
• Using vanishing of crystals in the infinitesimal site (Theorem E-Mehta), andquasi-projectivity of M, one concludes that if (F a )∗ EX was semi-stable for a large
enough but finite, then (F a )∗ EX would be trivial. So the lifting lemma would finish
the proof.

Hélène Esnault, Freie Universität Berlin

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Proof of the Theorem on trivialisation of a crystal
Proof.
• As in the proof of the triviality of the crystals on the infinitesimal site, one has tobound the problem.
• One proves a Langton type theorem: starting with E irreducible, there is a lattice
E such that EX ∈ MIC (X /k) is semi-stable, so yields a point of Langer’s moduli
M of semi-stable connections on pure sheaves with vanishing Chern classes.
• Using vanishing of crystals in the infinitesimal site (Theorem E-Mehta), andquasi-projectivity of M, one concludes that if (F a )∗ EX was semi-stable for a large
enough but finite, then (F a )∗ EX would be trivial. So the lifting lemma would finish
the proof.
• The slope assumption enables one to assume semi-stability of (F a )∗ EX for acertain a ≥ 0.

Hélène Esnault, Freie Universität Berlin

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Table of Contents

1

Deligne’s conjectures: `-adic theory

2

Deligne’s conjectures: crystalline theory3

Malčev-Grothendieck theorem; Gieseker conjecture; de Jong conjecture

4

Relative 0-cycles

Hélène Esnault, Freie Universität Berlin

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Base change for étale cohomology for torsion coefficientsof order prime to p
SGA 4,5, IV Thm.1.2.: Let A be an henselian discrete valuation ring (d.v.r.), withresidue field k of characteristic p > 0. Let X /A be a scheme, (n, p) = 1. Then if
X /A is proper, one has base change, that is the restriction homomorphism
resti (X , Z/n) −i (Y , Z/n) is an isomorphism, where Y = X ⊗ k.
Hét−→ HétA

Hélène Esnault, Freie Universität Berlin

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Base change for étale cohomology for torsion coefficientsof order prime to p
SGA 4,5, IV Thm.1.2.: Let A be an henselian discrete valuation ring (d.v.r.), withresidue field k of characteristic p > 0. Let X /A be a scheme, (n, p) = 1. Then if
X /A is proper, one has base change, that is the restriction homomorphism
resti (X , Z/n) −i (Y , Z/n) is an isomorphism, where Y = X ⊗ k.
Hét−→ Hét
A

Question
What is a motivic version of the base change theorem?

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Base change for étale cohomology for torsion coefficientsof order prime to p
SGA 4,5, IV Thm.1.2.: Let A be an henselian discrete valuation ring (d.v.r.), withresidue field k of characteristic p > 0. Let X /A be a scheme, (n, p) = 1. Then if
X /A is proper, one has base change, that is the restriction homomorphism
resti (X , Z/n) −i (Y , Z/n) is an isomorphism, where Y = X ⊗ k.
Hét−→ Hét
A

Question
What is a motivic version of the base change theorem?

Theorem (Kerz-E-Wittenberg ’15)
Answer for relative 0-cycles.

Hélène Esnault, Freie Universität Berlin

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Base change for étale cohomology for torsion coefficientsof order prime to p
SGA 4,5, IV Thm.1.2.: Let A be an henselian discrete valuation ring (d.v.r.), withresidue field k of characteristic p > 0. Let X /A be a scheme, (n, p) = 1. Then if
X /A is proper, one has base change, that is the restriction homomorphism
resti (X , Z/n) −i (Y , Z/n) is an isomorphism, where Y = X ⊗ k.
Hét−→ Hét
A

Question
What is a motivic version of the base change theorem?

Theorem (Kerz-E-Wittenberg ’15)
Answer for relative 0-cycles.
In the sequel, we report in it, raising a few questions on the way.

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Why relative 0-cycles?
Examples• Let X /A be a K3-surface, with k = k̄ and A large enough so NS(XK̄ ) isdefined over K = Frac(A). Then NS(XK ) → NS(Y ) is an injection of
torsion-free lattices of possibly different (Néron-Severi) ranks, e.g. assume
Y is supersingular! Thus composite
rest. surj.
sp
Pic(X )/n −−−−−−→ Pic(XK )/n −→ Pic(Y )/n, which is the restriction
homomorphism to the special fiber, can’t be surjective.

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Why relative 0-cycles?
Examples• Let X /A be a K3-surface, with k = k̄ and A large enough so NS(XK̄ ) isdefined over K = Frac(A). Then NS(XK ) → NS(Y ) is an injection of
torsion-free lattices of possibly different (Néron-Severi) ranks, e.g. assume
Y is supersingular! Thus composite
rest. surj.
sp
Pic(X )/n −−−−−−→ Pic(XK )/n −→ Pic(Y )/n, which is the restriction
homomorphism to the special fiber, can’t be surjective.
• Rosenschon-Srinivas ’07 produced examples of 3-dimensional XK , K ap-adic field with CH 2 (XK )/n infinite, thus CH 2 (X )/n infinite for a regularrest. surj.

model X /A as CH 2 (X )/n −−−−−−→ CH 2 (XK )/n; yet CH 2 (Y )/n is finite
thus restriction CH 2 (X )/n → CH 2 (Y )/n can’t be injective.

Hélène Esnault, Freie Universität Berlin

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Why relative 0-cycles?
Examples• Let X /A be a K3-surface, with k = k̄ and A large enough so NS(XK̄ ) isdefined over K = Frac(A). Then NS(XK ) → NS(Y ) is an injection of
torsion-free lattices of possibly different (Néron-Severi) ranks, e.g. assume
Y is supersingular! Thus composite
rest. surj.
sp
Pic(X )/n −−−−−−→ Pic(XK )/n −→ Pic(Y )/n, which is the restriction
homomorphism to the special fiber, can’t be surjective.
• Rosenschon-Srinivas ’07 produced examples of 3-dimensional XK , K ap-adic field with CH 2 (XK )/n infinite, thus CH 2 (X )/n infinite for a regularrest. surj.

model X /A as CH 2 (X )/n −−−−−−→ CH 2 (XK )/n; yet CH 2 (Y )/n is finite
thus restriction CH 2 (X )/n → CH 2 (Y )/n can’t be injective.
• So restriction neither surjective nor injective.

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Half way: from ’motivic’ cohomology on X to étale
cohomology on Y
So if we keep this direct (naı̈ve ?) formulation, we should exclude from thestudy relative cycles of positive dimension.

Hélène Esnault, Freie Universität Berlin

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Half way: from ’motivic’ cohomology on X to étale
cohomology on Y
So if we keep this direct (naı̈ve ?) formulation, we should exclude from thestudy relative cycles of positive dimension.
For relative 0-cycles one has

Theorem (Sato-S.Saito ’10)
Assume A excellent, henselian discrete valuation ring, with finite or
separably closed residue field k of characteristic p > 0. Assume X /A
projective, irreducible strict normal crossings (s.n.c.) scheme (so X in
particular is regular) of relative dimension d. Then the cycle map
2d (X , Z/n(d)) is an isomorphism.
cX : CH1 (X )/n →− HétHélène Esnault, Freie Universität Berlin

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Formulation of the problem
Sato-Saito’s theorem deals with the cycle map cX . We want to lift thisinformation to an information of the following kind, possibly enlarging therange of applicability for more general A and k:

Hélène Esnault, Freie Universität Berlin

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Formulation of the problem
Sato-Saito’s theorem deals with the cycle map cX . We want to lift thisinformation to an information of the following kind, possibly enlarging therange of applicability for more general A and k:

CH1 (X )/n
cX

ρ ∼
=



/ C(Y )/n


(?)

cY

∼
= / 2d2d (X , Z/n(d))
Hét
H (Y , Z/n(d)).
rest ∼= étHélène Esnault, Freie Universität Berlin

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Formulation of the problem II
There are a priori two ways to think:Hélène Esnault, Freie Universität Berlin

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Formulation of the problem II
There are a priori two ways to think:1) We keep the cycle group CH1 (X )/n and have to define a cycle group C(Y )/nand the restriction ρ and show it is an isomorphism;Hélène Esnault, Freie Universität Berlin

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Formulation of the problem II
There are a priori two ways to think:1) We keep the cycle group CH1 (X )/n and have to define a cycle group C(Y )/nand the restriction ρ and show it is an isomorphism;
2d (X , Z/n(d)) and define
2) We replace CH1 (X )/n by motivic cohomology Hmot
2d (Y , Z/n(d)); then one hopes
C(Y )/n to be motivic cohomology Hmot
functoriality defines ρ and one hopes that one can show it is an isomorphism.

Hélène Esnault, Freie Universität Berlin

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Formulation of the problem II
There are a priori two ways to think:1) We keep the cycle group CH1 (X )/n and have to define a cycle group C(Y )/nand the restriction ρ and show it is an isomorphism;
2d (X , Z/n(d)) and define
2) We replace CH1 (X )/n by motivic cohomology Hmot
2d (Y , Z/n(d)); then one hopes
C(Y )/n to be motivic cohomology Hmot
functoriality defines ρ and one hopes that one can show it is an isomorphism.
2d (X , Z(d)) for the (A, k) we
On the other hand, one conjectures CH1 (X ) = Hmot
shall consider.

Hélène Esnault, Freie Universität Berlin

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Formulation of the problem II
There are a priori two ways to think:1) We keep the cycle group CH1 (X )/n and have to define a cycle group C(Y )/nand the restriction ρ and show it is an isomorphism;
2d (X , Z/n(d)) and define
2) We replace CH1 (X )/n by motivic cohomology Hmot
2d (Y , Z/n(d)); then one hopes
C(Y )/n to be motivic cohomology Hmot
functoriality defines ρ and one hopes that one can show it is an isomorphism.
2d (X , Z(d)) for the (A, k) we
On the other hand, one conjectures CH1 (X ) = Hmot
shall consider.

In fact we mix the two viewpoints: we want to construct a restriction
2d (Y , Z(d))/n,
homomorphism ρ : CH1 (X )/n → Hmot

Hélène Esnault, Freie Universität Berlin

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Formulation of the problem II
There are a priori two ways to think:1) We keep the cycle group CH1 (X )/n and have to define a cycle group C(Y )/nand the restriction ρ and show it is an isomorphism;
2d (X , Z/n(d)) and define
2) We replace CH1 (X )/n by motivic cohomology Hmot
2d (Y , Z/n(d)); then one hopes
C(Y )/n to be motivic cohomology Hmot
functoriality defines ρ and one hopes that one can show it is an isomorphism.
2d (X , Z(d)) for the (A, k) we
On the other hand, one conjectures CH1 (X ) = Hmot
shall consider.

In fact we mix the two viewpoints: we want to construct a restriction
2d (Y , Z(d))/n, which is then an isomorphism
homomorphism ρ : CH1 (X )/n → Hmot

Hélène Esnault, Freie Universität Berlin

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Formulation of the problem II
There are a priori two ways to think:1) We keep the cycle group CH1 (X )/n and have to define a cycle group C(Y )/nand the restriction ρ and show it is an isomorphism;
2d (X , Z/n(d)) and define
2) We replace CH1 (X )/n by motivic cohomology Hmot
2d (Y , Z/n(d)); then one hopes
C(Y )/n to be motivic cohomology Hmot
functoriality defines ρ and one hopes that one can show it is an isomorphism.
2d (X , Z(d)) for the (A, k) we
On the other hand, one conjectures CH1 (X ) = Hmot
shall consider.

In fact we mix the two viewpoints: we want to construct a restriction
2d (Y , Z(d))/n, which is then an isomorphism
homomorphism ρ : CH1 (X )/n → Hmotand lifts the base change isomorphism.

Hélène Esnault, Freie Universität Berlin

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Recall

CH1 (X )/n
cX

ρ ∼
=



/ C(Y )/n


(?)

cY

∼
= / 2d2d (X , Z/n(d))
Hét
H (Y , Z/n(d)).
rest ∼= étHélène Esnault, Freie Universität Berlin

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Theorem
Theorem (Kerz-E-Wittenberg ’15)
1) Let Y be a strict normal crossings variety of dimension d defined over
2d (Y , Z[ 1 ](d)) asa perfect field k. Then there is a description of Hmotpa quotient of Z[Y sm ] by explicit relations, and2d (Y , Z[ 1 ](d)) = H d (Y , KM )[ 1 ].
Hmot
Nis
d
p
p
2) Assume A excellent henselian d.v.r., with perfect char. p > 0 residue
field, and X /A be a projective s.n.c. scheme. Then the following
holds.
di) If A has equal char. then CH1 (X )/n = HNis
(X , KdM /n) (Kerz’
theorem), ρ is then defined via restriction on KdM and one has (?);
ii) If k is finite or algebraically closed, one has (?);
iii) If ((d − 1)!, n) = 1, in particular if d = 2, one has (?).

Hélène Esnault, Freie Universität Berlin

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On Proofs
Proof.
Ad 1): uses localisation in motivic cohomology on Y , then duality to
2drelate Hc,mot(Y sm , Z/n(d)) with Suslin homology = Z[Y sm ]/(RI ), RI
spanned by certain (C , g ), C ⊂ Y integral 1-dimensional subscheme notcontained in Y sing , g rational function which is a unit generically andequal to 1 along Y sing .

Hélène Esnault, Freie Universität Berlin

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On Proofs
Proof.
Ad 1): uses localisation in motivic cohomology on Y , then duality to
2drelate Hc,mot(Y sm , Z/n(d)) with Suslin homology = Z[Y sm ]/(RI ), RI
spanned by certain (C , g ), C ⊂ Y integral 1-dimensional subscheme notcontained in Y sing , g rational function which is a unit generically andequal to 1 along Y sing .
2d (Y , Z/n(d)) = Z[Y sm ]/(RI , RII ), with RII spanned by
Result: Hmot
(C , g ), C simple n.c. curve and g unit along Y sing .

Hélène Esnault, Freie Universität Berlin

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On Proofs II
Proof.
Ad 2): ρ uniquely defined by writing CH1 (X ) as a quotient of
Z1g (X ) ⊂ Z1 (X ) spanned by A-flat 1-cycles which intersect Y in Y sm .
i) A equal char.: one uses Kerz’ theorem showingM ) to define ρ;
CH1 (X )/n = H d (XNis , KX,dHélène Esnault, Freie Universität Berlin

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On Proofs II
Proof.
Ad 2): ρ uniquely defined by writing CH1 (X ) as a quotient of
Z1g (X ) ⊂ Z1 (X ) spanned by A-flat 1-cycles which intersect Y in Y sm .
i) A equal char.: one uses Kerz’ theorem showingM ) to define ρ;
CH1 (X )/n = H d (XNis , KX,dii) k finite or algebraically closed: one uses étale cohomology and the Kato
conjecture proven by Kerz-S.Saito to define ρ;Hélène Esnault, Freie Universität Berlin

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On Proofs II
Proof.
Ad 2): ρ uniquely defined by writing CH1 (X ) as a quotient of
Z1g (X ) ⊂ Z1 (X ) spanned by A-flat 1-cycles which intersect Y in Y sm .
i) A equal char.: one uses Kerz’ theorem showingM ) to define ρ;
CH1 (X )/n = H d (XNis , KX,dii) k finite or algebraically closed: one uses étale cohomology and the Kato
conjecture proven by Kerz-S.Saito to define ρ;iii) ((d − 1)!, n) = 1: one uses a Grothendieck-Riemann-Roch typeargument to define ρ.

Hélène Esnault, Freie Universität Berlin

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On Proofs II
Proof.
Ad 2): ρ uniquely defined by writing CH1 (X ) as a quotient of
Z1g (X ) ⊂ Z1 (X ) spanned by A-flat 1-cycles which intersect Y in Y sm .
i) A equal char.: one uses Kerz’ theorem showingM ) to define ρ;
CH1 (X )/n = H d (XNis , KX,dii) k finite or algebraically closed: one uses étale cohomology and the Kato
conjecture proven by Kerz-S.Saito to define ρ;iii) ((d − 1)!, n) = 1: one uses a Grothendieck-Riemann-Roch typeargument to define ρ.
In all cases, once ρ is defined, one uses geometry to show it is an
isomorphism.

Hélène Esnault, Freie Universität Berlin

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Some consequences
Corollary
1) If k is finite, then CH0 (XK )/n is finite (already a consequence ofSato-S.Saito);
2) A = k[[t]], k p-adic field, then CH0 (XK )/n is finite.

Proof.
Ad 1): This is the link to the first lectures: Class Field Theory plus the
Kato conjecture enable one to show finiteness of C(Y )/n. One thenapplies the theorem.

Hélène Esnault, Freie Universität Berlin

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Some consequences
Corollary
1) If k is finite, then CH0 (XK )/n is finite (already a consequence ofSato-S.Saito);
2) A = k[[t]], k p-adic field, then CH0 (XK )/n is finite.

Proof.
Ad 1): This is the link to the first lectures: Class Field Theory plus the
Kato conjecture enable one to show finiteness of C(Y )/n. One thenapplies the theorem.
Ad 2): One uses again the Kato conjecture (and a result of Forré).

Hélène Esnault, Freie Universität Berlin

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Higher relative dimension?
Questions• What would be a (conjectural) formulation for cycles of higher relative
dimension?

Hélène Esnault, Freie Universität Berlin

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Higher relative dimension?
Questions• What would be a (conjectural) formulation for cycles of higher relative
dimension?
• For K a p-adic field, for which motivic cohomology groups of XK does
one have finiteness, and for those for which one does not have finiteness,does one have meaningful quotients which are finite?

Hélène Esnault, Freie Universität Berlin

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Higher relative dimension?
Questions• What would be a (conjectural) formulation for cycles of higher relative
dimension?
• For K a p-adic field, for which motivic cohomology groups of XK does
one have finiteness, and for those for which one does not have finiteness,does one have meaningful quotients which are finite?
• What about mod p, and what about replacing Y by its thickenings Ym ?
Assuming Gersten conjecture for Milnor K -theory on X one has ad (Y , KM /n) (possibly p
restriction homomorphism CH1 (X )/n → limm HNismd←−divides n) and one could ask, when A is the ring of integers of a number
field, whether the prosystem is constant. This is related to
Colliot-Thélène’s conjecture on the structure CH0 (XK ), which should be of
the shape Z (for the degree) + a finite group + a free lattice over Zp + a
divisible group.
Hélène Esnault, Freie Universität Berlin

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