A non-abelian version of Deligne’s Fix part
Theorem
Joint work with Moritz Kerz
Für Gerd Faltings, in Freundschaft
Hélène Esnault (FU Berlin, Harvard, Copenhagen)Bonn, July 25, 2024

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

Abstract
We prove a non-abelian version of Deligne’s Fix Part Theorem. It
is a statement which is purely anchored in complex geometry. The
reason for the consideration is a vaster program which aims atunderstanding some aspects of the monodromy-weight conjecturein unequal characteristic by ’tilting it’ to a complex situation forwhich we have the tools developed notably by Morihiko Saito andTakuro Mochizuki. This lecture focuses on a small part of it. Inprogress.

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

Deligne’s Fix Part Theorem
Assumption
f : X → S smooth projective, S smooth quasi-projective /C =⇒ f
is a fibration =⇒ natural action of π1 (S, s) on the cohomology
H i (Xs , Q).

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

Deligne’s Fix Part TheoremAssumption
f : X → S smooth projective, S smooth quasi-projective /C =⇒ f
is a fibration =⇒ natural action of π1 (S, s) on the cohomology
H i (Xs , Q).
Hodge classξ ∈ H 2i (Z , Q), Z smooth quasi-projective over C, is a Hodge classifξ ∈ im(H 2i (Z , Z)) ∩ F i H 2i (Z , C) ∩ Wi H 2i (Z , C).

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

Deligne’s Fix Part Theorem
Theorem (Deligne ’71)
f : X → S smooth projective, S smooth quasi-projective /C, S isan Artin K π1 =⇒ the following conditions are equivalent(1) the orbit of π1 (S, s) · ξ in H i (Xs , Q) is finite;(2) ξ essentially i.e after a finite étale cover of S extends to a Hodge class in
H 2i (X , Q);
2i (Xb /S)
b lies in(3) the Gauß-Manin flat deformation of ξ to HdR
i2ibb
F H (X /S).
dRHélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix PartDeligne’s Fix Part Theorem
Theorem (Deligne ’71)
f : X → S smooth projective, S smooth quasi-projective /C, S isan Artin K π1 =⇒ the following conditions are equivalent(1) the orbit of π1 (S, s) · ξ in H i (Xs , Q) is finite;(2) ξ essentially i.e after a finite étale cover of S extends to a Hodge class in
H 2i (X , Q);
2i (Xb /S)
b lies in(3) the Gauß-Manin flat deformation of ξ to HdR
i
2ibbF H (X /S).
dR

Proof Q-version (1) (2): Deligne; (3) Griffiths
Z-version: apply the Leray spectral sequence to H 2i (X , N1 Z/Z)
and the following observation to L = R i f∗ ( N1 Z/Z).

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

Deligne’s Fix Part Theorem
Theorem (Deligne ’71)
f : X → S smooth projective, S smooth quasi-projective /C, S isan Artin K π1 =⇒ the following conditions are equivalent(1) the orbit of π1 (S, s) · ξ in H i (Xs , Q) is finite;(2) ξ essentially i.e after a finite étale cover of S extends to a Hodge class in
H 2i (X , Q);
2i (Xb /S)
b lies in(3) the Gauß-Manin flat deformation of ξ to HdRi2ibb
F H (X /S).
dR

Proof Q-version (1) (2): Deligne; (3) Griffiths
Z-version: apply the Leray spectral sequence to H 2i (X , N1 Z/Z)
and the following observation to L = R i f∗ ( N1 Z/Z).
L be a local system on S with finite fibers, S is an Artin K π1, j > 0 is an integer =⇒ ∃ a finite étale coverS ′ → S such that the map H i (S, L) → H i (S ′ , L) vanishes.
Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

Non-Abelian Version (E-Kerz ’24)
Definitions
A good holomorphic map f : X → S is the complement of a
relative normal crossings divisor in f¯: X̄ → S smooth projectivewith S a complex holomorphic manifold.
For K a field ∈ {Q, R, C} a (Z)K -PVHS is a polarized K -variation(Ls , Fs , Qs ) of Hodge structure on a K - local system Ls on Xs
definable over Z.

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

Non-Abelian Version (E-Kerz ’24)
Theorem 1
f : X → S good holomorphic, S quasi-projective, s ∈ S,
(Ls , Fs , Qs ) (Z)K -PVHS; S Artin K π1 or Xs hyperbolic curve.
Then the following conditions are equivalent:(1) the orbit π1 (S, s) · [Ls ] is finite;(2) (Ls , Fs , Qs ) essentially extends to a (Z)K -PVHS on X ;(3) (Ls , Fs , Qs ) extends to a (Z)K -PVHS on X∆ , ∆ ball ∋ s.

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix PartRemarksEarlier work by Jost-Zuo, Katzarkov-Pantev, Landesman-Litt.
We see the parallel between 1) 2) in the abelian andnon-abelian versions; as for 3) the non-abelian version over ∆[F absolute] is ‘weaker’ than the abelian formal version [Frelative]. We do not know (yet) how to prove a formal
non-abelian version.

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

Algebraically isomonodromic deformation
Definition
An algebraically isomonodromic deformation of Ls on Xs for fgood holomorphic is an extension L on X such that
Mo0 (Ls , x) ∼= Mo0 (L, x)
where Mo(L, x) ⊂ GL(Lx ) is the Zariski closure of themonodromy group of L (i.e. the Tannaka group of L in GL(Lx )based at x ∈ Xs in the category of K -local systems) and 0 is the1-component.

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

Algebraically Isomonodromic Deformations
Theorem 2With (f , (Ls , Fs , Qs )) as in Theorem 1, then (1) is equivalent to(2’) Ls on Xs essentially extends to an algebraicallyisomonodromic L on X .
Then L in (2’) extends to a (Z)K -PVHS on X (Theorem 1 (2)).
Moreovera) L in (2’) essentially unique;b) if Ls is absolutely simple, then any extension L to X which
has finite determinant is algebraically isomonodromic.

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

Non-abelian Hodge Locus
Simpson’s non-Abelian Hodge Locus
For f : X → S a good proper holomorphic map, and r ∈ N≥1 ,Simpson defines the Hodge locus NL(f , r ) ( in fact on the de Rham side, here we
present it on the Betti side ) as the set of point M in the total étale spaceT = S̃ × H 1 (Xs , GL(r , C)) /π1 (S, s)
to R 1 f∗ GL(r , C) such that M underlies a (Z)C-PVHS.

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

Non-abelian Hodge Locus
Simpson’s non-Abelian Hodge Locus
For f : X → S a good proper holomorphic map, and r ∈ N≥1 ,Simpson defines the Hodge locus NL(f , r ) ( in fact on the de Rham side, here we
present it on the Betti side ) as the set of point M in the total étale spaceT = S̃ × H 1 (Xs , GL(r , C)) /π1 (S, s)
to R 1 f∗ GL(r , C) such that M underlies a (Z)C-PVHS.
Theorem (Simpson ’97)
NL(f , r ) ⊂ T is closed analytic and finite over S.
He conjectures that NL(f , r ) ⊂ T is algebraic when S isquasi-projective.

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix PartNon-abelian Hodge Locus
Finiteness
Simpson’s main tool is Deligne’s finiteness theorem ’87: there are
only finitely many local systems L of bounded rank on a given Z
quasi-projective complex smooth /C which lift to a (Z)C-PVHS.
Deligne: The present work is the result of an effort to try tounderstand Faltings’ finiteness theorem for abelian varieties ’83.
Theorem 3
Simpson’s theorem is true for f a good (not necessarily proper)
holomorphic map.
In addition to Simpson’s proof, one needs a form of the nilpotent
orbit theorem which holds in families near the special fibre, whichenables one to recognize on a mixed Hodge structure when it arises
as a limiting mixed Hodge structure of a (Z)C-PVHS.
Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix PartThe Mumford-Tate group in a algebraically isomonodromic
deformation
Definition
For (Lx , Fx , Qx ) a (Z)Q PHS on x = Spec C, its Mumford-Tategroup MT(Lx , Fx ) is the smallest linear Q-algebraic subgroup ofGL(Lx ) such that its complex points comprise the maps
Lx,C → Lx,C sending λ to λi λ̄j for all (i, j) ∈ (N)2 withi + j =weight Lx . It does not depend on Qx .

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

The Mumford-Tate group in a algebraically isomonodromic
deformation
Definition
For (Lx , Fx , Qx ) a (Z)Q PHS on x = Spec C, its Mumford-Tategroup MT(Lx , Fx ) is the smallest linear Q-algebraic subgroup ofGL(Lx ) such that its complex points comprise the maps
Lx,C → Lx,C sending λ to λi λ̄j for all (i, j) ∈ (N)2 withi + j =weight Lx . It does not depend on Qx .
Proposition 4
X smooth quasi-projective /C, (L, F , Q) (Z)Q-PVHS on X ,x ∈ X . Then MT(Lx , Fx ) normalises Mo(L, x) and Mo0 (L, x) in
GL(Lx ).

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix PartIt yields our Definition
The Mumford-Tate group of the (Z)Q-PVHS (L, F , Q):
MT((L, F ), x) = Mo0 (L, x) · MT(Lx , Fx ) ⊂ GL(Lx ).
It does not depend on Q.

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix PartIt yields our Definition
The Mumford-Tate group of the (Z)Q-PVHS (L, F , Q):
MT((L, F ), x) = Mo0 (L, x) · MT(Lx , Fx ) ⊂ GL(Lx ).
It does not depend on Q.
Corollary
1) MT((L, F ), x) has parallel transport as x ∈ X ;2) in an algebraically isomonodromic situation, for x ∈ Xs has∼=MT((Ls , Fs ), x) −→ MT((L, F ), x).

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix PartIt yields our Definition
The Mumford-Tate group of the (Z)Q-PVHS (L, F , Q):
MT((L, F ), x) = Mo0 (L, x) · MT(Lx , Fx ) ⊂ GL(Lx ).
It does not depend on Q.
Corollary
1) MT((L, F ), x) has parallel transport as x ∈ X ;2) in an algebraically isomonodromic situation, for x ∈ Xs has∼=MT((Ls , Fs ), x) −→ MT((L, F ), x).
RemarkDeligne: outside of a thin set of points x ∈ X , iso-class ofMT(Lx , Fx ) is ‘constant’; 1) defines MT((L, F ), x) as a local
system across this thin subset.
Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix PartSketch of Proof of Theorem 2
Go to profinite completions: group theory
Main point is to construct L out of the finiteness of the orbit ofLs . Assumption K π1 or family of hyperbolic curves =⇒\π\1 (Xs ) ,→ π1 (X ). Pass to ℓ-adic local systems. It becomes apurely group theoretic statement. Assume for simplicity Ls,ℓabsolutely irreducible. Then it lifts uniquely to X as a projectiveℓ-adic system which we may essentially lift as an ℓ-adic system Lℓwith finite determinant. Restrict to the topological fundamentalgroup =⇒ L. A priori Mo0 (L) is semi-simple. Prove it is simple.

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

Sketch of Proof of Theorem 2 II
Simpson-T. Mochizuki correspondence
Assume Ls absolutely irreducible for simplicity =⇒ essential unique
L with finite determinant. h “unique metric which is fiber-wiseharmonic” on L ⊗ OX , so agrees with hs on Ls ⊗ OXs . It defines
Higgs which is log at infinity. Eigenvalues of C× are constant (logcompactification) so recognizable on Xs where they are cyclic, sodefines a sum of Hodge bundles with a nilpotent Higgs operator.
So the same on X . Has to go back: to such a nilpotent Higgs thecorresponding L is a C-PVHS. This is due to T. Mochizuki anddoes not request the full package of the Simpson correspondence.

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

Sketch of Proof of Proposition 4
Exact sequence
Point is the exact sequence (d’Addezio-E, weaker version byAndré) on X
1 → Mo(L, x) → π(⟨(L, F )⟩, x) → π(C, x) → 1of Q-algebraic groups. π for Tannaka group. C ⊂ ⟨(L, F )⟩ full subof those objects with trivial local system.

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part

Sketch of Proof of Proposition 4
Exact sequence
Point is the exact sequence (d’Addezio-E, weaker version byAndré) on X
1 → Mo(L, x) → π(⟨(L, F )⟩, x) → π(C, x) → 1of Q-algebraic groups. π for Tannaka group. C ⊂ ⟨(L, F )⟩ full subof those objects with trivial local system.
ThenMT((L, F ), x) = π(⟨(L, F )⟩, x).

Hélène Esnault (FU Berlin, Harvard, Copenhagen)Non-abelian Fix Part