Elmau, May 4th, 2018
p-CURVATURE of CONNECTIONS, GEOMETRICITY and INTEGRALITY ofLOCAL SYSTEMSby Hélène Esnault, FU Berlin
On a X be a smooth complex irreducible variety, one considers complex linear
systems V , that is conjugacy classes of irreducible linear representations
π1top (X, x) → GL(r, C)of the topological fundamental group. Equivalently, via the Riemann-Hilbert correspondence [Del70], one considers algebraic regular singular integrable connections∇an(E, ∇) associated to V , so in particular Ean= V . Among those are the geometricones.
From now on, all V ’s and (E, ∇)’s are assumed to be irreducible.
Definition 1. V, equivalently (E, ∇), is geometric if and only if there is a dense
Zariski open subvariety U ,→ X, there is a smooth projective morphism g : Y → Usuch that V |U is a subquotient as a local system of Rn g∗ C for some natural numbern.
We call the local system Rn g∗ C a Gauß-Manin local system.
Remark 2. Gauß-Manin local systems are defined over Z thus geometric localsystems are defined over the algebraic integers Z̄.
A geometric local system is in particular a variation of Hodge structures. As thecategory of variations of Hodge structures is semi-simple [Del71], V is geometric ifand only if it is a direct factor of a Rn g∗ C as in the definition.
Definition 3. V, equivalently (E, ∇), is finite if and only it is geometric and g in
Definition 1 is finite.
If V 6= 0 this implies that n = 0 and in addition, as π1top (U, x) → π1top (X, x) issurjective, that one can take g finite as above on U = X. A general problem is thefollowing:Problem 4. Characterize the geometric, resp. finite V (equivalently (E, ∇)).
The most general answer is given by the p-curvature conjecture. Let XS be amodel of X over a scheme S of finite type over Z, which we shall assume to besmooth over Z, over which (E, ∇) has a model (E, ∇)S . So in particular Es is avector bundle. For any closed point s ∈ S the p-curvature of (E, ∇)s is well definedψ((E, ∇)s ) : Es → Fs∗ Ω1Xs0 ⊗OXs Es .
Here Fs : Xs → Xs0 is the relative Frobenius. Then by Cartier descent, one has thefollowing ([Kat70]).
1) ψ((E, ∇)s ) is nilpotent if and only if there is a filtration Fil on (E, ∇)s suchthat grFil (E, ∇)s is generated by flat sections.
2) In particular, ψ((E, ∇)s ) is zero if and only if (E, ∇)s is generated by flat
sections, or equivalently if and only if there is a coherent vector bundle Es0on Xs0 such that (E, ∇)s = (Fs∗ Es0 , ∇can ) where ∇can = 1 ⊗ d on Fs∗ Es0 =Fs−1 Es0 ⊗F −1 OX 0 OXs .
s1

2Conjecture 5 (Grothendieck’s p-curvature conjecture).
1) (E, ∇) is geometric if and only if its p-curvatures are nilpotent on all closed points s of adense open S 0 ,→ S.
2) (E, ∇) is finite if and only if its p-curvatures are zero on all closed points sof a dense open S 0 ,→ S.
Historically, Grothendieck’s p-curvature conjecture was formulated only via 2)
and has been discussed and made public by Katz, see [Kat72], [Kat82]. The only
complete known result is in rank 1, for which the conjectures 1) and 2) are equivalent and are proven in [Chu85]. Even if we do not touch extensions in this note,
let us mention [And04] in which 2) is proven for successive extensions of rank 1
connections, [Kat72] in which 2) is proven for Gauß-Manin connections.
Remark 6. If 1) resp. 2) are true, then if (E, ∇) has nilpotent p-curvatures, then(E, ∇)K , the restriction of (E, ∇) to XK , with Spec(K) → XS a p-adic point, isinvariant under Frobenius. Here the Frobenius pull-back of (E, ∇)K is well defined,even if the Frobenius does not lift to XK , as the p-curvature is nilpotent.
Remark 7. If 2) is true, then if (E, ∇) has vanishing p-curvatures, then (Es , ∇s )is not only a connection but is also a DXs -module.
Theorem 8. ([EK18]) If X is projective, (E, ∇) has vanishing p-curvatures, and(Es , ∇)s is a DXs -module with Tannaka group of order prime to p for infinitely
many p’s, then (E, ∇) is finite.
Remark 9. The p-curvature conjecture implies in particular that if (E, ∇) has
nilpotent, resp. vanishing p-curvatures as in Conjecture 5, the monodromy representation ρσ defined as the underlying monodromy representation ρ : π1top (X, x) →GL(r, C) post-composed by σ ∗ : GL(r, C) → GL(r, C) coming from an automorphism σ of C, has associated connection (E, ∇)σ with nilpotent, resp. vanishing
p-curvatures.
We define the following notion.
Definition 10. (E, ∇) has absolute nilpotent, resp. vanishing p-curvature if andonly if for all σ, there is a dense open S σ ,→ S such that for all closed point s ∈ S σ ,ψ((E, ∇)s ) is nilpotent, resp. vanishing.
So we can subdivide Conjecture 5 as follows.
Conjecture 11.
A) (E, ∇) has nilpotent, resp. vanishing p-curvatures if andonly if it has absolute nilpotent, resp. absolute vanishing p-curvatures.
B) Absolute p-curvature conjecture: 1) (E, ∇) is geometric if and only if it has
absolute nilpotent p-curvatures.
2) (E, ∇) is finite if and only if it has absolute vanishing p-curvatures.
Theorem 12 ([EG18]). Let X be smooth projective. Then cohomologically rigid
finite determinant (E, ∇)’s with absolute vanishing p-curvature are finite.
Here a rigid connection, equivalently a rigid local system, is an isolated pointof the de Rham moduli, resp. Betti moduli constructed by Simpson in [Sim94].
It is cohomologically rigid if it is a smooth isolated point (i.e. H 1 (X, End0 (ρ)) =H 1 (X, End0 (E, ∇)) = 0.)
But Simpson conjectures more generally that rigid systems are all geometric.

3Conjecture 13 (Simpson’s geometricity conjecture). If V has finite determinantand quasi-unipotent monodromies at infinity, and is rigid, then V is geometric.
Here (cohomologically) rigid means that V is an isolated (a smooth isolated)point of the stack of finite type parametrizing such rigid local systems with fixed
determinant and given conjugacy classes of monodromy at infinity (see [EG18]).
Remarks 14.
1) Simpson’s conjecture implies that if V as in the conjectureis rigid, then the p-curvatures of (E, ∇) are absolutely nilpotent.
2) Simpson’s conjecture implies that if V as in the conjecture is rigid, then(E, ∇) restricted to XK , with Spec(K) → XS a p-adic point is invariant
under some power of Frobenius.
3) Simpson’s conjecture implies that if V as in the conjecture is rigid, then it
is integral.
Theorem 15 ([EG18], [EG18b]).
1) If V has quasi-unipotent monodromies
at infinity, has finite determinant, and is cohomologically rigid, then V isintegral.
2) If X is projective, and V has finite determinant and is rigid, then it has
absolute nilpotent p-curvatures, and (E, ∇)K is invariant under some powerof Frobenius.
Another way to formulate 2) (without the absolute condition) is to say that(E, ∇)K is an isocrystal with Frobenius structure. Using L. Lafforgue’s theorem
on curves ([Laf02]) together with its translation for overconvergent isocrystals withFrobenius structure ([Abe13]), this implies that if C ,→ X is a curve such that
π1top (C, x) → π1top (X, x) is surjective, then (E, ∇)|CK is geometric as an isocrystal.
Finally let us mention that on a smooth variety over a finite field, one has `-to`0companions as conjectured by Deligne [Del80] and proved by L. Lafforgue [Laf02]
and Drinfeld [Dri12], p-to-` companions ([AE16], see also [Ked18]), but we do notknow how to construct p-to-p or `-to-p companions, unless X has dimension 1([Abe13]). So we mention
Theorem 16 ([EG18b]). Cohomologically rigid connections on X projective, while
restricted to Xs , have a full set of companions.
References[Abe13][AE16][And04][Bos01][Chu85][Del70][Del71]

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