Connections and Symmetric Differential Forms
Hélène Esnault, work in progress with Michael Groechenig

London, December 08, 2020

Hélène Esnault, work in progress with Michael Groechenig

Connections

over CX ∈ x sm proj var over C, π1top (X , x) top fund gr, r ∈ N>0

Hélène Esnault, work in progress with Michael Groechenig

Connections

over CX ∈ x sm proj var over C, π1top (X , x) top fund gr, r ∈ N>0
Simpson: MB (X , r ) !top MdR (X , r ) !top MDol (X , r )MB (X , r ) affine
Hitchin

MDol (X , r ) −−−−→ AN , N = ⊕ri=1 h0 (X , Symi Ω1 ) proper
Van: h0 (X , Symi Ω1 ) = 0 ∀i ∈ N>0=⇒Arapura : Van ⇒ Fin with Fin: [MB (X , r ) 0-dim’l]Fin ⇒ all complex local systems are rigid.

Hélène Esnault, work in progress with Michael Groechenig

Connections

Theorem (BKT ’13) in answer to a question I posed
TheoremVan ⇒ monodromy of all complex local systems is finite.

Hélène Esnault, work in progress with Michael Groechenig

Connections

Theorem (BKT ’13) in answer to a question I posed
TheoremVan ⇒ monodromy of all complex local systems is finite.
Proof relies on > 0ty in C-geom
Simpson: loc. syst. is a C-VHS
Katzarkov-Zuo: Van ⇒ is in fact a Z̄-factor of a Z-VHS
Zuo: Ω1 of image of period map big.

Hélène Esnault, work in progress with Michael Groechenig

Connections

Theorem (BKT ’13) in answer to a question I posed
TheoremVan ⇒ monodromy of all complex local systems is finite.
Proof relies on > 0ty in C-geom
Simpson: loc. syst. is a C-VHSKatzarkov-Zuo: Van ⇒ is in fact a Z̄-factor of a Z-VHSZuo: Ω1 of image of period map big.
Fin 6⇒ Van, Fin 6⇒ ThmMargulis superrigidity: Shimura var of rk ≥ 2: has Fin but by far
not Van and has inftly many loc syst with infinite monodromy.

Hélène Esnault, work in progress with Michael Groechenig

Connections

Integrality and F -isocrystals
Theorem (EG’18) in (partial) answer to Simpson’s integrality conjecture
Fin (in a given rank r ) ⇒(rank r ) complex loc syst are integral.
(E , ∇)W ⊗W K are isoc with a Frobenius structureHélène Esnault, work in progress with Michael Groechenig

Connections

Integrality and F -isocrystals
Theorem (EG’18) in (partial) answer to Simpson’s integrality conjecture
Fin (in a given rank r ) ⇒(rank r ) complex loc syst are integral.
(E , ∇)W ⊗W K are isoc with a Frobenius structure
So Fin (in a given rank r ) ⇒ (rank r ) Z̄-factor of a Z- VHS.

Hélène Esnault, work in progress with Michael Groechenig

Connections

Integrality and F -isocrystals
Theorem (EG’18) in (partial) answer to Simpson’s integrality conjecture
Fin (in a given rank r ) ⇒(rank r ) complex loc syst are integral.
(E , ∇)W ⊗W K are isoc with a Frobenius structure
So Fin (in a given rank r ) ⇒ (rank r ) Z̄-factor of a Z- VHS.
Proof purely algebraic, relies on the existence of companions.

Hélène Esnault, work in progress with Michael Groechenig

Connections

Integrality and F -isocrystals
Theorem (EG’18) in (partial) answer to Simpson’s integrality conjecture
Fin (in a given rank r ) ⇒(rank r ) complex loc syst are integral.
(E , ∇)W ⊗W K are isoc with a Frobenius structure
So Fin (in a given rank r ) ⇒ (rank r ) Z̄-factor of a Z- VHS.
Proof purely algebraic, relies on the existence of companions.
So: BKT ⇔ unitary mon ⇔ Higgs field = 0, seen in char. p > 0.

Hélène Esnault, work in progress with Michael Groechenig

Connections

Problems addressed
Van is a purely algebraic condition
having finite monodromy is a purely algebraic property
algebraically? : isocrystals and mod p connections?

Hélène Esnault, work in progress with Michael Groechenig

Connections

Problems addressed
Van is a purely algebraic condition
having finite monodromy is a purely algebraic property
algebraically? : isocrystals and mod p connections?
toy r = 1: analyticallyFin ⇒ MdR (X , 1) = Pic∇ (X ) = (Hodge Theory)
Picτ (X ) = NS(X )[torsion].

Hélène Esnault, work in progress with Michael Groechenig

Connections

Problems addressed
Van is a purely algebraic condition
having finite monodromy is a purely algebraic property
algebraically? : isocrystals and mod p connections?
toy r = 1: analyticallyFin ⇒ MdR (X , 1) = Pic∇ (X ) = (Hodge Theory)
Picτ (X ) = NS(X )[torsion].
toy r = 1: algebraically
L connection of rank 1; Fin ⇒ {Ln }n∈Z finite ⇒ Lm ∼= Ln forsome m 6= n ∈ N (preperiodicity) ⇒ Ln−m = 1 (m − n) 6= 0 so Ltorsion.

Hélène Esnault, work in progress with Michael Groechenig

Connections

Isocrystals
Proposition (EG’20)
X = XW ⊗W k, k = k̄ sm proj, Van/K ⇒1) all `-adic loc. syst have fin mon2) if k = F̄p ∃h : Y → X fin ét trivializing conv isocHélène Esnault, work in progress with Michael GroechenigConnections

Isocrystals
Proposition (EG’20)
X = XW ⊗W k, k = k̄ sm proj, Van/K ⇒1) all `-adic loc. syst have fin mon2) if k = F̄p ∃h : Y → X fin ét trivializing conv isoc
Proof1): π1ét (XC )  π1ét (Xk ) + BKT(EK , ∇K ) = (EW , ∇W ) ⊗W K , (EW , ∇W ) ⊗W k, nilp p-curv
F acts on isoc, Fin ⇒ preperiodicity F -orbit of any isoc
⇒ given (EK , ∇K ) (not nec conv), ∃N, (F N )∗ (EK , ∇K ) F -str
⇒ (Abe-E +K) ∃`-adic companion so 1) ⇒ ∃ h : Y → X sth∗ (F N )∗ (EK , ∇K ) trivialconv ⇒ 2): h∗ (EK , ∇K ) trivial as well.
Hélène Esnault, work in progress with Michael Groechenig

Connections

Problem on isocrystals
Problems
X /k, k = k̄ sm proj, Van/k ⇒? all Q̄` loc syst have fin monk = F̄p Van/k ⇒? all conv isoc are étale trivializableHélène Esnault, work in progress with Michael Groechenig

Connections

Problem on isocrystals
Problems
X /k, k = k̄ sm proj, Van/k ⇒? all Q̄` loc syst have fin monk = F̄p Van/k ⇒? all conv isoc are étale trivializable
RemarkX /k lifts to XW : Proposition ⇒ both problems have a > 0 answer

Hélène Esnault, work in progress with Michael Groechenig

Connections

/FqTheorem (EG’20)
X = XW2 (Fq ) ⊗ Fq sm proj, Van /Fq ⇒ rk 2 loc free ss deg 0(E , ∇) are étally trivializable.

Hélène Esnault, work in progress with Michael Groechenig

Connections

/FqTheorem (EG’20)
X = XW2 (Fq ) ⊗ Fq sm proj, Van /Fq ⇒ rk 2 loc free ss deg 0(E , ∇) are étally trivializable.
ProofVan ⇒ Hitchin base= {0} ⇒ p-curv nilpotent
preperiodic Higgs-dR flow (OV corr, Lan-Sheng-Zuo)
assume periodic period 1 (for talk): theneither (E , ∇) = (F ∗ E , ∇can ) ⇒ (Lang torsor) ét triv, or
0 → (F ∗ L<0 , can) → (E , ∇) → (F ∗ L>0 , can) → 0 (p-curv nil)
0 → L>0 → E → L<0 → 0 (F -Filt)
KS : L>0 ⊗ (L<0 )−1 ,→ Ω1 , (L<0 )p−1 ,→ O ,→ (L>0 )p−1
OX ,→ Symp−1 Ω1 ⊥ to Van.
Hélène Esnault, work in progress with Michael GroechenigConnections