The Nori Fundamental Gerbe

Motivation Main results An application Beyond the proﬁnite fundamental gerbe

The Nori fundamental gerbe

Niels Borne1 Angelo Vistoli2

1Université Lille 1

2Scuola Normale Superiore di Pisa

Spring school on the fundamental group scheme, 5.5.2014

Niels Borne, Angelo Vistoli The Nori fundamental gerbe Motivation Main results An application Beyond the profinite fundamental gerbe Plan Motivation The étale fundamental group The Nori fundamental group Main results Reminder on gerbes Existence of the profinite fundamental gerbe An application Curves with no Galois sections Proof Beyond the profinite fundamental gerbe Pro-algebraic fundamental gerbe C -fundamental gerbe

et π1 (X, x) = Aut(Fx )

where Fx : Cov X / Sets

(Y → X) / Y (x) is the ﬁber functor associated to x. If G is a ﬁnite group, then

et π1 (X, x) → G ←→ {étale G-covers (Y , y) → (X, x)}

F : Cov X → Sets

is a transitive groupoid. More elaborate description (Deligne): if X is deﬁned over k, and Y → X is a Galois cover, then

P = IsomAut Y (pr∗ Y , pr∗ Y ) X Y X×k X 2 1 ⇒

is an étale transitive groupoid acting on X, and P = lim P is bX ←−Y Y Deligne fundamental (absolute) groupoid. Note that if x : spec Ω → X are geometric points, i = 1, 2, then i PbX = Isom(Fx1 , Fx2 ). (x1,x2)

Niels Borne, Angelo Vistoli The Nori fundamental gerbe Motivation Main results The étale fundamental group An application The Nori fundamental group Beyond the profinite fundamental gerbe Definition and existence statement Let X/k be a scheme, endowed with a rational point x ∈ X(k). Nori considers triples (G, Y → X, y), where G is a finite group-scheme, Y → X is a G-torsor, and y ∈ Y (k) is a lift of x. Definition X admits a fundamental group scheme if there is a triple (π(X, x), Xfx , x˜) which is pro-universal among such triples. Theorem (Nori) If X is reduced and connected, then X admits a fundamental group scheme. Sketch of a proof. Show that the category of triples is co-filtered and put π(X, x) = lim G. ←−(G,Y →X,y)

Niels Borne, Angelo Vistoli The Nori fundamental gerbe Motivation Main results The étale fundamental group An application The Nori fundamental group Beyond the profinite fundamental gerbe The Nori fundamental groupoid Let X/k be proper and reduced. Esnault-Hai: even if no x ∈ X(k) is given, the category EF Vect X of essentially finite vector bundles on X is a (non neutral) tannakian category. One can use the universal fiber functor: F : EF Vect X → Vect X and consider the transitive groupoid: ∗ ∗ PX = IsomX×k X (pr2 F, pr1 F) ⇒ X For any x ∈ X(k), one recovers Nori’s group in the cartesian diagram: π(X, x) / PX

spec k / X ×k X (x,x) Niels Borne, Angelo Vistoli The Nori fundamental gerbe Motivation Main results Reminder on gerbes An application Existence of the profinite fundamental gerbe Beyond the profinite fundamental gerbe Affine gerbes Any scheme X/k is characterized by its functor of points op hx :(Sch /k) / Sets

S / Homk (S, X) One can consider more generally pseudo-functors: op (Sch /k) / Groupoids The key example: the classifying stack of a group-scheme:

B G : S / {G torsors over S}

Definition A pseudo functor Γ:(Sch /k)op → Groupoids is an affine gerbe if Γ is a (fpqc) stack, Γ 6= ∅, two objects are locally isomorphic, and the automorphism group of any object is affine. Niels Borne, Angelo Vistoli The Nori fundamental gerbe Motivation Main results Reminder on gerbes An application Existence of the profinite fundamental gerbe Beyond the profinite fundamental gerbe More examples Fix a central extension: 0 → A → G → H → 1 of algebraic k-groups, and T /k a H-torsor. Then

Γ: S / {reductions of structure group of TS to G} is an afﬁne gerbe, banded by A.

gerbes endowed with a chart ←→ transitive groupoids (S → Γ) → (S ×Γ S ⇒ S) afﬁne gerbes with a given object over k ←→ afﬁne groups/k

(Γ, ξ ∈ Γ(k)) → Autk (ξ) B G ← G

The deﬁnition of a gerbe makes sense over any topology. A typical example from deformation theory: Fix i : X0 → X a thickening of order one deﬁned by an ideal I of square 0, and E0 a vector bundle on X0. Then:

0 n 0o Γ:(X → X0) / liftings of E0 0 to X 0 |X0

is a gerbe, banded by I ⊗ End(E0). It is neutral if and only if the deformation problem is unobstructed, and in this case the stack of obstructions is isomorphic, after choosing a lifting, to B (I ⊗ End(E0)).

Niels Borne, Angelo Vistoli The Nori fundamental gerbe Motivation Main results Reminder on gerbes An application Existence of the profinite fundamental gerbe Beyond the profinite fundamental gerbe Construction as an inverse limit Theorem (B.-Vistoli) Let X/k be geometrically (connected and reduced) scheme. There exists a morphism X → πX/k universal among morphisms to (pro)finite affine gerbes over k. Sketch of a proof. Say a morphism X → Γ is Nori-reduced if for any factorisation X → Γ0 → Γ where Γ0 → Γ is faithful, then Γ0 ' Γ. Then the category of Nori-reduced morphisms X → Γ is (equivalent to) a co-filtered category, so we can put:

π = Γ X/k ←−lim X→Γ Nori reduced that by construction has the wanted universal property. Niels Borne, Angelo Vistoli The Nori fundamental gerbe Motivation Main results Reminder on gerbes An application Existence of the profinite fundamental gerbe Beyond the profinite fundamental gerbe Tannakian interpretation The Tannaka correspondence, in Rivano’s form, reads: Affine gerbes over k ←→ (Non neutral) tannakian categories/k

Γ → Vectk Γ

(S → { fibre functors of CS}) ← C Theorem (B.-Vistoli) Let X/k be a proper scheme. 1. If X is geometrically (connected and reduced), then the pullback along X → πX/k identifies the non-neutral tannakian category Vect πX/k with EF Vect X. 2. EF Vect X is tannakian if and only if every morphism X → F, where F is a finite stack, factors as X → Γ → F, where Γ is a finite gerbe, and Γ → F is a closed immersion.

Niels Borne, Angelo Vistoli The Nori fundamental gerbe Motivation Main results Curves with no Galois sections An application Proof Beyond the proﬁnite fundamental gerbe Section conjecture One is interested in producing curves whose homotopy exact sequence

1 → π1(Xk , x) → π1(X, x) → Galk → 1 does not split. Indeed, these sections are conjecturally related to rational points. By functoriality, x ∈ X(k) induces a section sX (x), well deﬁned up to conjugacy by an element of π1(Xk , x). One thus get:

sX : X(k) → Sπ1(X/k)

Conjecture (Section conjecture, Grothendieck 1983) If X a smooth, proper curve of genus greater than 2 over a ﬁeld k of ﬁnite type over Q, the application sX is one to one.

Niels Borne, Angelo Vistoli The Nori fundamental gerbe Motivation Main results Curves with no Galois sections An application Proof Beyond the proﬁnite fundamental gerbe Curves with non trivial homotopy exact sequences n Let P/k be a Brauer-Severi variety (that is, Pk sep ' Pk sep ). Denote by r its period: ∗ r = min {l ∈ N /O(l) is deﬁned over k}

Theorem (B.-Vistoli) Assume that the characteristic of k is 0, that there is a given morphism f : X → P, where P is a Brauer-Severi variety of period r. Assume moreover that there exists a prime divisor p of r, an invertible sheaf Λ on X, and an isomorphism ⊗p ∗ Λ ' f OP (r). Then the fundamental exact sequence

et et 1 → π1 (Xk , x) → π1 (X, x) → Galk → 1 does not split. Niels Borne, Angelo Vistoli The Nori fundamental gerbe Motivation Main results Curves with no Galois sections An application Proof Beyond the proﬁnite fundamental gerbe Sketch of a proof The ﬁrst step of the proof relies on the correspondence

Isomorphism classes in πX/k (k) ←→ Sπ1(X/k)

So to show that Sπ1(X/k) = ∅, it is enough to produce a morphism X → Γ, where Γ is a non neutral ﬁnite gerbe. Indeed by the universal property, such a morphism factors as X → πX/k → Γ, and the hypothesis on Γ means that Γ(k) = ∅. To construct such a gerbe, consider the dual Brauer-Severi variety P∨, and put

n ∨ ⊗p o Γ(S) = (L, ψ), L invertible sheaf on P , ψ : L 'O ∨ (r) S PS

A morphism X → Γ exists by construction.

Niels Borne, Angelo Vistoli The Nori fundamental gerbe Motivation Main results Pro-algebraic fundamental gerbe An application C -fundamental gerbe Beyond the proﬁnite fundamental gerbe A pro-algebraic fundamental gerbe? Let X/k be a proper geometrically (connected and reduced) alg scheme. Does there exists a morphism X → πX/k universal among morphisms X → Γ, where Γ is an algebraic gerbe ? The answer is no, even for very standard k-schemes X, for instance X = P1. One reason is the following: if G is an algebraic group, and H is a subgroup, then the diagram

spec k / B G O O G/H / B H

is cartesian. Assume that H is a Borel, so that G/H is projective, and ﬁx a non constant morphism X → G/H. Such a morphism cannot factor trough a gerbe over k, since the scheme-theoretic image of a gerbe is a point.

Niels Borne, Angelo Vistoli The Nori fundamental gerbe Motivation Main results Pro-algebraic fundamental gerbe An application C -fundamental gerbe Beyond the profinite fundamental gerbe Pro-C gerbe If C is a class of algebraic groups over k, we say that a gerbe Γ is a C-gerbe if it is banded by an element of C. So we can speak of unipotent gerbes, abelian gerbes, and so on. C A C-fundamental gerbe X → πX/k is a morphism (pro-)universal among morphisms X → Γ to a C-gerbe. In a work in progress with A.Vistoli, we have worked out a technical condition on C that ensures the existence of a C-fundamental gerbe under mild assumptions on X/k 0 (geometrically reduced, concentrated, H (X, OX ) = k). The two largest classes for which we can show the condition is respected are the classes of virtually abelian (resp. virtually unipotent) groups. We say that an affine group scheme of finite type G over a perfect field k is virtually abelian (resp. virtually 0 unipotent) if Gred is abelian (resp. unipotent).

The (virtually-)unipotent fundamental gerbe has a natural and standard tannakian interpretation. uni Nori has proved in his Phd that Vect πX/k is equivalent to the category of vector bundles on X that are successive extensions of the structure sheaf OX . virt−uni Similarly, it should not be too hard to show that Vect πX/k is equivalent to the category of vector bundles on X that are successive extensions of essentially ﬁnite bundles. The (virtually)-abelian fundamental gerbe is much more mysterious. The main issue is that the pullback functor ab Vect πX/k → Vect X is not fully faithful. We have no idea of a tannakian interpretation at the moment.

Niels Borne, Angelo Vistoli The Nori fundamental gerbe