Stratified bundles and étale fundamental group (new version)
aa r X i v : . [ m a t h . AG ] D ec STRATIFIED BUNDLES AND ´ETALE FUNDAMENTALGROUP
H´EL`ENE ESNAULT AND XIAOTAO SUN
Abstract. On X projective smooth over an algebraically closedfield of characteristic p >
0, we show that all irreducible stratifiedbundles have rank 1 if and only if the commutator [ π , π ] of the´etale fundamental group π is a pro- p -group, and we show thatthe category of stratified bundles is semi-simple with irreducibleobjects of rank 1 if and only if π is abelian without p -power quo-tient. This answers positively a conjecture by Gieseker [4, p. 8]. Introduction
Let X be a smooth projective variety defined over an algebraicallyclosed field k of characteristic p >
0. In [4], stratified bundles are de-fined and studied. It is shown that they are a characteristic p > π is trivial.(ii) If all the irreducible stratified bundles have rank 1, then [ π , π ]is a pro- p -group.(iii) If every stratified bundle is a direct sum of stratified line bun-dles, then π is abelian without non-trivial p -power quotient.Here π is the ´etale fundamental group based at some geometric point.He conjectures that in the three statements, the “ if ” can be replacedby “ if and only if ”. The aim of this note is to give a positive answerto Gieseker’s conjecture (see Theorem 3.9).The converse to (i) is the main theorem of [3], and is analog toMalcev-Grothendieck theorem ([10], [5]) asserting that if the ´etale fun-damental group of a smooth complex projective variety is trivial, thenthere is no non-trivial bundle with a flat connection. Date : December 20, 2011.The first author is supported by the SFB/TR45 and the ERC AdvancedGrant 226257. The second author is supported by NBRPC 2011CB302400,NSFC60821002/F02 and NSFC 10731030.
The complex analog to the converse to (iii) relies on the following.If X is an abelian variety over a field k , Mumford [11, Section 16]showed that a non-trivial line bundle L which is algebraically equiv-alent to 0 fulfills H i ( X, L ) = 0 for all i ≥
0. If k = C , the field ofcomplex numbers, L carries a unitary flat connection with underlyinglocal system ℓ , and H ( X, L ) = 0 implies that H ( X, ℓ ) = 0. In fact,more generally, if X is any complex manifold with abelian topologicalfundamental group, and ℓ is a non-trivial rank 1 local system, then H ( X, ℓ ) = 0 (see Remark 3.11).Finally the complex analog to the converse to (ii) and (iii) togethersays that if X is a smooth complex variety, then all irreducible complexlocal systems on X have rank 1 if and only if π is abelian. However,the semi-simplicity statement has a different phrasing: all irreduciblecomplex local systems on X have rank 1 and the category is semi-simple if and only if π is a finite abelian group (see Claim 3.13 for aprecise statement).The proof of the converse to (ii) is done in Section 2. It relies on aconsequence of the proof of the main theorem in [3], which is formu-lated in Theorem 2.3. It is a replacement for the finite generation ofthe topological fundamental group in complex geometry (see the analogstatement in Remark 3.12). The proof of the converse to (iii) is done inSection 3. The key point to show the converse to (iii) is Theorem 3.4.Its proof relies on the construction (Proposition 3.5) of a moduli spaceof non-trivial extensions, with a rational map f induced by Frobeniuspullback, which allows us to use a theorem of Hrushovski. Acknowledegements:
We thank the department of Mathematics of Har-vard University for its hospitality during the preparation of this note.We thank Nguy˜ˆen Duy Tˆan for a very careful reading of a first ver-sion, which allowed us to improve the redaction. We thank the refereeof the first version who pointed out a mistake, corrected in Section 3.The second named author would like to thank his colleague Nanhua Xifor discussions related to Claim 3.3.2.
Stratified bundles
Let k be an algebraically closed field of characteristic p >
0, and X a smooth connected projective variety over k . A stratified bundle on X is by definition a coherent O X -module E with a homomorphism ∇ : D X → E nd k ( E )of O X -algebras, where D X is the sheaf of differential operators actingon the structure sheaf of X . By a theorem of Katz (cf. [4, Theorem TRATIFIED BUNDLES AND ´ETALE FUNDAMENTAL GROUP 3 F -divided sheaves in [2]). Definition 2.1.
A stratified bundle on X is a sequence of bundles E = { E , E , E , · · · , σ , σ , . . . } = { E i , σ i } i ∈ N where σ i : F ∗ X E i +1 → E i is a O X -linear isomorphism, and F X : X → X is the absolute Frobenius.A morphism α = { α i } : { E i , σ i } → { F i , τ i } between two stratifiedbundles is a sequence of morphisms α i : E i → F i of O X -modules suchthat F ∗ X E i +1 σ i (cid:15) (cid:15) F ∗ X α i +1 / / F ∗ X F i +1 τ i (cid:15) (cid:15) E i α i / / F i is commutative. The category str ( X ) of stratified bundles is abelian,rigid, monoidal. To see it is k -linear, it is better to define stratifiedbundles and morphisms in the relative version: objects consist of E ′ = { E = E ′ , E ′ , E ′ , · · · , σ ′ , σ ′ , . . . } = { E ′ i , σ ′ i } i ∈ N where E ′ i is a bundle on the i -th Frobenius twist X ( i ) of X , σ ′ i : F ∗ i,i +1 E ′ i +1 → E ′ i is a O X ( i ) -linear isomorphism, and F i,i +1 : X ( i ) → X ( i +1) is the rela-tive Frobenius, the morphisms are the obvious ones. A rational point a ∈ X ( k ) yields a fiber functor ω a : str ( X ) → vec k , E a ∗ E with values in the category of finite dimensional vector spaces. Thus( str ( X ) , ω a ) is a Tannaka category ([2, Section 2.2]), and one has anequivalence of categories str ( X ) ω a ∼ = −−−→ rep k ( π str )(2.1)where π str = Aut ⊗ ( ω a )is the Tannaka group scheme, and rep k ( π str ) is the category of finitedimensional k -representations of π str . Let π := π ´et1 ( X, a ) be the ´etalefundamental group of X . In [4, Theorem 1.10], D. Gieseker proved thefollowing theorem. Theorem 2.2. (Gieseker): Let X be a smooth projective variety overan algebraically closed field k , then (i) If every stratified bundle is trivial, then π is trivial. H´EL`ENE ESNAULT AND XIAOTAO SUN (ii)
If all the irreducible stratified bundles have rank , then [ π , π ] is a pro- p -group. (iii) If every stratified bundle is a direct sum of stratified line bun-dles, then π is abelian with no p -power order quotient. Then Gieseker conjectured that the converse of the above statementsmight be true. In [3], it is proven that the converse of statement (i) istrue. The aim of this section is to prove the converse of (ii). We provethe converse of (iii) in Section 3.The proof relies on the following theorem extracted from [3], andwhich plays a similar rˆole as the finite generation of the topologicalfundamental group π top1 in complex geometry (see Remark 3.12).Fixing an ample line bundle O X (1) on X , recall that E is said to be µ -stable if for all coherent subsheaves U ⊂ E one has µ ( U ) < µ ( E ),where µ ( E ) is the slope which is defined to be the degree of E withrespect to O X (1) divided by the rank of E . Theorem 2.3.
Let X be a smooth projective variety over an alge-braically closed field k . If there is a stratified bundle E = ( E n , σ n ) n ∈ N of rank r ≥ on X , where { E n } n ∈ N are µ -stable, then there existsan irreducible representation ρ : π → GL ( V ) with finite monodromy,where V is a r -dimensional vector space over ¯ F p .Proof. By the proof of [3, Theorem 3.15], there is a smooth projec-tive model X S → S of X → Spec k , with model a S ∈ X S ( S ) of a ∈ X ( k ), where S is smooth affine over F p , together with a quasi-projective model M S → S of the moduli of stable vector bundles con-sidered in [3, Section 3], there is a closed point u → M S of residue field F q , corresponding to a bundle E over X s , stable over X ¯ s = X s ⊗ ¯ F p ,where s is the closed point u viewed as a closed point of S , such that( F m ) ∗ E ∼ = E , for some m ∈ N \ { } , where F is the Frobenius of X s / F q . Thus, as explained in [3, Theorem 3.15], E trivializes over aLang torsor Y → X s [9, Satz 1.4], thus its base change E ¯ s to X ¯ s triv-ializes over Y × s ¯ s → X ¯ s . This trivialization defines an irreduciblerepresentation ρ ′ : π ´et1 ( X ¯ s , b ) → GL ( E b ), where b = a S ⊗ ¯ s . As thespecialization homomorphism sp : π → π ´et1 ( X ¯ s , b ) is surjective ([13,Expos´e X, Th´eor`eme 3.8]), the composite ρ = sp ◦ ρ ′ is a represen-tation of π with the same finite irreducible monodromy. This is asolution to the problem. (cid:3) We now use the following two elementary lemmas.
Lemma 2.4 (See Chapter 8, Proposition 26 of [12]) . Let G be a finite p -group and ρ : G → GL ( V ) be a representation on a vector space TRATIFIED BUNDLES AND ´ETALE FUNDAMENTAL GROUP 5 V = 0 over a field k of characteristic p . Then V G = { v ∈ V | ρ ( g ) v = v, ∀ g ∈ G } 6 = 0 . Lemma 2.5.
Let G be a finite group and ρ : G → GL ( V ) be anirreducible representation on a finite dimensional vector space V overan algebraically closed field k of characteristic p > . If the commutator [ G, G ] of G is a p -group, then dim( V ) = 1 .Proof. If [
G, G ] is a p -group, by Lemma 2.4, there exists a 0 = v ∈ V such that ρ ( g ) ρ ( g ) v = ρ ( g ) ρ ( g ) v for any g , g ∈ G . Since V is anirreducible G -module, the sub-vector space spanned by the orbit ρ ( G ) v is V itself. Thus for all w ∈ V , all g ∈ G , there are a g ( w ) ∈ k , suchthat w = X g ∈ G a g ( w ) ρ ( g ) v. On the other hand, for all g , g , g ∈ G , one has g g = g g c for some c ∈ [ G, G ], thus g g g = g g cg = g g gc ′ , for some c ′ ∈ [ G, G ]. Weconclude that ρ ( g ) ρ ( g ) w = ρ ( g ) ρ ( g ) w for any g , g ∈ G , that is ρ ( G ) is abelian. As ρ is irreducible, V must have dimension 1. (cid:3) Theorem 2.6.
Let X be a smooth projective variety over an alge-braically closed field k . Then all irreducible stratified bundles on X have rank if and only if the commutator [ π , π ] of π is a pro- p -group.Proof. One direction is the (ii) in Theorem 2.2. We prove the converse.Assume that [ π , π ] is a pro- p -group. Let E = ( E n , σ n ) n ∈ N be an irre-ducible stratified bundle of rank r ≥ X . Assume r ≥
2. Then by[3, Proposition 2.3], there is a n ∈ N such that the stratified bundle E ( n ) := ( E n , σ n ) n ≥ n is a successive extension of stratified bundles U = ( U n , τ n ) with underlying bundles U n being µ -stable. But E be-ing irreducible implies that E ( n ) is irreducible as well. Hence all the E n are µ -stable for n ≥ n . By Theorem 2.3, there is an irreduciblerepresentation ρ : π → GL ( V ) of dimension r ≥ p >
0. ByLemma 2.5, this is impossible. Thus r = 1. (cid:3) Extensions of stratified line bundles
In this section, we prove the following theorem.
Theorem 3.1.
Let X be a smooth projective connected variety overan algebraically closed field k of characteristic p > . If the ´etale H´EL`ENE ESNAULT AND XIAOTAO SUN fundamental group π of X is abelian and has no non-trivial p -powerorder quotient, then any extension → L → V → L ′ → in str (X) is split when L and L ′ are rank objects. We start the proof by the following proposition, which can be con-sidered as a generalization of [3, Proposition 2.4].
Proposition 3.2.
Let L be a line bundle on X such that ( F ∗ X ) a L isisomorphic to L , for some non zero natural number a , where F X : X → X is the absolute Frobenius map. Then ( F ∗ X ) a : H ( X, L ) → H ( X, L ) is nilpotent if π is abelian without non-trivial p -power quotient.Proof. We choose an isomorphism ( F ∗ X ) a L ∼ = L , or equivalently anisomorphism L p a − ∼ = O X . This define a Kummer cover φ : Y → X with Galois group H = Z / ( p a −
1) such that φ ∗ L = O Y . One has an exact sequence (recall that π := π ´et1 ( X, ¯ a ))1 → π ´et( Y, ¯ b )1 → π → H → , where ¯ b is a geometric point of Y above the geometric point ¯ a of X . Claim 3.3. π ´et1 ( Y, ¯ b ) is abelian without p -power quotient. Proof.
Since π is commutative, the kernel of any quotient map π ´et1 ( Y, ¯ b ) ։ K is normal in π . Taking for K a finite p -group defines the push-outexact sequence 1 → K → G → H → G is a quotient of π , thus is commutative and has no p -powerquotient. Since K splits in G , K = { } . (cid:3) Then, by [3, Proposition 2.4], there is an integer
N > F ∗ Y ) N : H ( Y, O Y ) → H ( Y, O Y )is a zero map, which implies that φ ∗ · ( F ∗ X ) Na = ( F ∗ Y ) Na · φ ∗ : H ( X, L ) → H ( Y, O Y )is a zero map. But φ ∗ : H ( X, L ) ֒ → H ( Y, φ ∗ L ) = H ( Y, O Y ) isinjective, thus ( F ∗ X ) Na : H ( X, L ) → H ( X, L ) is a zero map. (cid:3)
TRATIFIED BUNDLES AND ´ETALE FUNDAMENTAL GROUP 7
Proof of Theorem 3.1.
By twisting with ( L ′ ) − , we may assume that L ′ = I , the trivial object in str ( X ). We prove Theorem 3.1 by contra-diction. If there exists a nontrivial extension0 → L → V → I → str ( X ), then, by definition, there is a set E X = { ( L i ֒ → V i ։ O X ) } i ∈ N of isomorphism classes of non-trivial extensions on X such that( L i ֒ → V i ։ O X ) ∼ = ( F ∗ X L i +1 ֒ → F ∗ X V i +1 ։ F ∗ X O X )for any i ∈ N modulo scale . This means that there is are λ i ∈ k × =Γ( X, O × X ) such that the push down of ( L i ֒ → V i ։ O X ) by L i λ i −→ L i is isomorphic as an extension to ( F ∗ X L i +1 ֒ → F ∗ X V i +1 ։ O X ). Inparticular, L i is isomorphic as a line bundle to F ∗ X ( L i +1 ).If E X is a finite set, then there are only finitely many non-isomorphic L i , thus there is a a ∈ N \ { } such that ( F ∗ X ) a L i ∼ = L i for all i . Weobtain a contradiction by Proposition 3.2.If E X is a infinite set, by Theorem 3.4 below, there is a nontrivialextension 0 → L → V → O X ¯ s → X ¯ s of X (thus over F p ) such that for an integer a > → ( F ∗ X ¯ s ) a L → ( F ∗ X ¯ s ) a V → ( F ∗ X ¯ s ) a O X ¯ s → X ¯ s , we obtain again a contradiction since π ´et1 ( X ¯ s , a ¯ s ),which is a quotient of π via the specialization map, is also abelianwithout non-trivial p -power quotient. Here a ¯ s is a specialization of thegeometric point ¯ a used to define π . (cid:3) Theorem 3.4.
Let X be a smooth projective variety over an alge-braically closed field k . If there exists an infinite set E X of equivalentclasses of nontrivial extensions ( L i ֒ → V i ։ O X ) , i ∈ N satisfying ( L i ֒ → V i ։ O X ) ∼ = ( F ∗ X L i +1 ֒ → F ∗ X V i +1 ։ F ∗ X O X ) for any i ∈ N , modulo scale, then there exists a nontrivial extension → L → V → O X ¯ s → on a good reduction X ¯ s of X (over F p ) such that for some a > (cid:0) ( F ∗ X ¯ s ) a L ֒ → ( F ∗ X ¯ s ) a V ։ ( F ∗ X ¯ s ) ∗ O X ¯ s (cid:1) ∼ = ( L ֒ → V ։ O X ¯ s ) modulo scale. H´EL`ENE ESNAULT AND XIAOTAO SUN
We now prove Theorem 3.4. Let X S → S be a smooth projectivemodel of X/k , endowed with a section, where S is a smooth affineirreducible variety over F q with H ( X, O X S ) = H ( S, O S ). So X S → S has smooth projective geometrically irreducible fibers. Then we have Proposition 3.5.
There exists a reduced S -scheme M → S of finitetype and a rational map f : M M over S such that (1) For any field extension K ⊃ F q , the set M ( K ) = { e } of K -valued points consists of isomorphism classes of non-trivial ex-tensions L ֒ → V ։ O X s modulo scale on X s , where X s is thefiber of X S → S at the image s ∈ S ( K ) of e ∈ M ( K ) . (2) The set E X of Theorem 3.4 minus finitely many elements liesin M ( k ) . (3) The rational map f : M M is well-defined at e = ( L ֒ → V ։ O X s ) ∈ M ( K ) if and only if the pull-back extension → F ∗ X s L → F ∗ X s V → F ∗ X s O X s → under the absolute Frobenius F X s does not split, in which case f ( e ) = ( F ∗ X s L ֒ → F ∗ X s V ։ F ∗ X s O X s ) ∈ M ( K ) . Proof.
The line bundles { L i } i ∈ N occurring in E X must satisfy L i ∼ = L pi +1 , ∀ i ∈ N , so they are infinitely p -divisible, thus lie in Pic τ ( X )( k ), the group ofnumerically trivial line bundles over k ([8, 9.6]).We first assume L = I . We shrink S so that H ( O X S ) is locally freeand commutes with base change. Let H ( O X S ) denote the first directimage of O X S under X S → S , then π S : M = P ( H ( O X S ) ∨ ) → S is the moduli space of isomorphism classes of non-trivial extensions O X ¯ s ֒ → V ։ O X ¯ s . In the diagram X S F XS % % F / / ! ! ❇❇❇❇❇❇❇❇❇ X ′ S / / (cid:15) (cid:15) X S (cid:15) (cid:15) S F S / / S,F S , F X S are the absolute Frobenius and F : X S → X ′ S is the relativeFrobenius. Then the ”Frobenius pullback” induces F ∗ : F ∗ S H ( O X S ) = H ( O X ′ S ) → H ( O X S )(3.3) TRATIFIED BUNDLES AND ´ETALE FUNDAMENTAL GROUP 9 which is O S -linear and commutes with base change. Let π ∗ S H ( O X S ) ∨ → O M (1) → F M : M → M , defines the O M -linear map π ∗ S F ∗ S H ( O X S ) ∨ → O M ( p ) → F ∗ M π ∗ S = π ∗ S F ∗ S and F ∗ M O M (1) := O M ( p )). Combining the dual π ∗ S ( F ∗∨ ) : π ∗ S H ( O X S ) ∨ → π ∗ S F ∗ S H ( O X S ) ∨ of (3.3) with (3.4) defines the O M -linear map π ∗ S H ( O X S ) ∨ → O M ( p ) . By the universal property, this is the same as a rational map f : M M over S . To see that f satisfies (3), let e → M be a point with image s → S (under M π S −→ S ) , which corresponds a nontrivial extension e = ( O X s ֒ → V ։ O X s )where X s is the fiber of X S → S at s → S . Then f is well-defined at e → M if and only if the O M -linear map π ∗ S H ( O X S ) ∨ → O M ( p )at e → M is surjective. By definition, this is equivalent to saying thatthe O S -linear map (3.3) at s → S , which is F ∗ : F ∗ k ( s ) H ( O X s ) = H ( O X ′ s ) → H ( O X s ) , is not trivial at F ∗ k ( s ) ([ e ]) ∈ H ( O X ′ s ) (i.e. F ∗ ( F ∗ k ( s ) ([ e ])) = 0), where[ e ] ∈ H ( O X s )denotes the corresponding class of e = ( O X s ֒ → V ։ O X s ). Since F ∗ X s : H ( O X s ) F ∗ k ( s ) −−−→ H ( O X ′ s ) F ∗ −→ H ( O X s ) , we have F ∗ X s ([ e ]) = F ∗ ( F ∗ k ( s ) ([ e ])). Thus f is well-defined at e = ( O X s ֒ → V ։ O X s ) ∈ M if and only if ( F ∗ X s O X s ֒ → F ∗ X s V ։ F ∗ X s O X s ) is not splitting, and f ( e ) = ( F ∗ X s O X s ֒ → F ∗ X s V ։ F ∗ X s O X s ) . The objects ( O X ֒ → V i ։ O X ) ∈ E X ( i ∈ N ) define points e i → M over s = Spec ( k ) → S , and f is well-defined at e i with f ( e i ) = e i − for i >
1. This finishes the proof if L = I . If L = I , after removing a finite number of elements in the set E X ,we can assume that all line bundles { L i } i ∈ N occurring in E X satisfy H ( X, L i ) = 0 , H ( L i ) = 0 i ∈ N . Let Pic τX S → S be the torsion component of the identity of the Picardscheme [8, 9.6]. Let L be the universal line bundle on X S × S Pic τX S .We define N = { t ∈ Pic τX S | H ( L t ) = 0 } with its reduced structure. By semi-continuity of cohomology, it is aclosed sub-scheme of Pic τX S . If N i is defined, let N i +1 = { t ∈ N i | H ( L p i +1 t ) = 0 } . By the noetherian property, N ⊇ N ⊇ · · · N i ⊇ N i +1 ⊇ · · · termi-nates. Then, there is a k ≥ N i = N k , ∀ i ≥ k . The line bundles { L i + p k } i ∈ N occurring in E X ( p k ), where E X ( p k ) := { ( L i + p k ֒ → V i + p k ։ O X ) ∈ E X } i ∈ N , are k -points of N k = N i for all i ≥ k .We define T ⊂ N k → S with its reduced structure to be the sub-scheme T = { t ∈ N k | H ( L t ) = 0 } . It is open in N k . Let L be the restriction of universal line bundle on X S × S T (thus, for any t ∈ T , H ( L p i t ) = 0 for i ≥ R p T ∗ ( L ) is neither commuting with base change nor locally free ingeneral, where p T : X S × S T → T is the projection. However, E = R n − p T ∗ ( L ∨ ⊗ p ∗ X S ω X S /S ) , n = dim( X )may not be locally free, but does commute with base change since H n ( L ∨ t ⊗ ω X S ×{ t } ) = H ( L t ) ∨ = 0 , ∀ t ∈ T. There exists a quotient scheme π : M = P ( E ) → T together with π ∗ E → O M (1) → , which represents the functor that sends a T -scheme p W : W → T tothe set of isomorphic classes of quotients p ∗ W E → Q →
0, where Q is aline bundle on W ([7, Thm 2.2.4]). By definition, M π −→ T → S satisfiesthe properties (1) and (2) in the proposition (replacing M by M red ifnecessary). The rest of the proof is devoted to the construction of arational map f : M M TRATIFIED BUNDLES AND ´ETALE FUNDAMENTAL GROUP 11 over S , which satisfies the property (3) in the proposition.Let F : X S × S T → ( X S × S T ) ′ denote the relative Frobenius mor-phism over T . We write X S × S T F XS × ST ) ) F / / p T ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ ( X S × S T ) ′ / / p ′ T (cid:15) (cid:15) X S × S T p T (cid:15) (cid:15) T F T / / T. (3.5)One has ( X S × S T ) ′ = X ′ S × S T thus we can project to X ′ S , further to X S . Let L ′ be the pullback of L under the projection( X S × S T ) ′ p −→ X S × S T and ω (resp. ω ′ ) be the relative canonical line bundle of X S × S T p T −→ T (resp. ( X S × S T ) ′ p ′ T −→ T ). The line bundle F ∗ L ′ on X S × S T defines a S -morphism ν : T → Pic τX S . Define T = { t ∈ T | H ( L pt ) = 0 } ⊂ T with its reduced scheme structure. It is open in T . By definition of T , ν restricted to T factors through T , thus induces ν : T → T such that (1 × ν ) ∗ L ⊗ p ∗ T η = F ∗ L ′ on X S × S T , with X S × S T p T −−→ T , and where η is a line bundleon T . We abuse notations and still write ω for the relative dualizingsheaf ω X S × S T /T of p T , ω ′ for the relative dualizing sheaf ω X ′ S × S T /T of p ′ T ‘ : X ′ S × S T → T . Since ω = p ∗ X S ω X S /S = (1 × ν ) ∗ ω X S × S T/T ,we have R n − p T ∗ ( ω ⊗ F ∗ L ′∨ ) = η − ⊗ ν ∗ E , R n − p ′ T ∗ ( ω ′ ⊗ L ′∨ ) = F ∗ T E , and those identities commute with base change. By Lemma 3.6, wehave φ : η − ⊗ ν ∗ E → F ∗ T E . Together with the universal quotient π ∗ E → O M (1) →
0, restrictedto M := M × S T , where π : M → T , T → S factors through thecomposition of the open embedding T ⊂ T with the map T → S , thisinducesΦ : π ∗ ν ∗ E φ −→ π ∗ η ⊗ π ∗ F ∗ T E = π ∗ η ⊗ F ∗ M π ∗ E → π ∗ η ⊗ O M ( p ) . Here F T , F M are the absolute Frobenius morphisms satisfying M F M −−−→ M y π y π T F T −−−→ T Let M ⊂ M be the reduced open set consisting of points q ∈ M such that Φ is surjective at q ∈ M (which implies M ⊂ M ). Thenthere exists a unique morphism f : M → M corresponding via the universal property to f ∗ ( π ∗ E → O M (1) →
0) =( π ∗ ν ∗ E → π ∗ η ⊗ O M ( p ) → M f −−−→ M y π y π T ν −−−→ T. Then Lemma 3.6 below implies that the rational map f : M M satisfies the requirement (3) in the proposition. (cid:3) For any point t → T , let s → S be its image under T → S , and X s be the fiber of X S → S at s → S . Then the diagram (3.5) specializesto X s × t F Xs × t ( ( F t / / & & ▲▲▲▲▲▲▲▲▲▲▲▲ ( X s × t ) ′ / / (cid:15) (cid:15) X s × t (cid:15) (cid:15) t / / t and F t induces the k ( t )-linear map F ∗ t : H ( L ′ t ) → H (( F ∗ L ′ ) t ), whichinduces a k ( t )-linear map( F ∗ t ) ∨ : H (( F ∗ L ′ ) t ) ∨ → H ( L ′ t ) ∨ . By Serre duality, we have the induced k ( t )-linear map φ t : H n − (( ω ⊗ F ∗ L ′∨ ) t ) → H n − (( ω ′ ⊗ L ′∨ ) t ) . (3.6)Serre duality allows to make (3.6) in families: TRATIFIED BUNDLES AND ´ETALE FUNDAMENTAL GROUP 13
Lemma 3.6.
There exists a homomorphism φ : R n − p T ∗ ( ω ⊗ F ∗ L ′∨ ) → R n − p ′ T ∗ ( ω ′ ⊗ L ′∨ ) such that R n − p T ∗ ( ω ⊗ F ∗ L ′∨ ) ⊗ k ( t ) τ −−−→ H n − (( ω ⊗ F ∗ L ′∨ ) t ) y φ ⊗ k ( t ) y φ t R n − p ′ T ∗ ( ω ′ ⊗ L ′∨ ) ⊗ k ( t ) τ −−−→ H n − (( ω ′ ⊗ L ′∨ ) t ) is commutative for any t → T , where τ is the canonical base changeisomorphism, and φ t is the homomorphism in (3.6) .Proof. Let ω (resp. ω ′ ) be the relative canonical line bundle of X T := X S × S T p T −→ T (resp. X ′ T := ( X S × S T ) ′ p ′ T −→ T ). Let F : X T → X ′ T be the relativeFrobenius morphism in (3.5), which is flat and Cohen-Macaulay as X S → S is smooth. The injection L ′ ֒ → F ∗ F ∗ L ′ of vector bundlesinduces the surjection of vector bundles H om ( F ∗ F ∗ L ′ , ω ′ ) ։ H om ( L ′ , ω ′ ) . (3.7)Duality theory for the Cohen-Macaulay map F : X T → X ′ T implies anisomorphism F ∗ H om ( F ∗ L ′ , ω ) ∼ = H om ( F ∗ F ∗ L ′ , ω ′ ) . (3.8)Equations (3.7) and (3.8) imply that one has a surjection F ∗ H om ( F ∗ L ′ , ω ) ։ H om ( L ′ , ω ′ )(3.9)of vector bundles on X ′ S . Since F is a finite morphism, taking R n − p ′ T ∗ ((3.9))induces φ : R n − p T ∗ H om ( F ∗ L ′ , ω ) → R n − p ′ T ∗ H om ( L ′ , ω ′ ) . (3.10)For any t → T , let X t be the fiber of X T → T , and F t : X t → X ′ t bethe relative Frobenius. Then (3.10) ⊗ k ( t ) induces (through τ ) φ ⊗ k ( t ) : H n − ( X t , H om ( F ∗ t L ′ t , ω t )) → H n − ( X ′ t , H om ( L ′ t , ω ′ t ))which is induced by ϕ := (3.9) ⊗ k ( t ). Then the surjection of vectorbundles on X ′ t ϕ : ( F t ) ∗ H om ( F ∗ t L ′ t , ω t ) ։ H om ( L ′ t , ω ′ t )(3.11)is dual, by taking H om ( · , ω ′ t ), to the canonical injection L ′ t ֒ → ( F t ) ∗ F ∗ t L ′ t (3.12) of vector bundles on X ′ t . As the dual H n − ( ϕ ) ∨ of H n − ( ϕ ) = φ ⊗ k ( t )is the k ( t )-linear map F ∗ t : H ( X ′ t , L ′ t ) → H ( X t , F ∗ t L ′ t )induced by (3.12), this finishes the proof. (cid:3) To prove Theorem 3.4, the key tool is a theorem of Hrushovski. Infact, we only need a special case of his theorem [6, Corollary 1.2].
Theorem 3.7 (Hrushovski, [6, Corollary 1.2]) . Let Y be an affinevariety over F q , and let Γ ⊂ ( Y × F q Y ) ⊗ F q ¯ F q be an irreducible subvarietyover F q . Assume the two projections Γ → X are dominant. Then,for any closed subvariety W Y , there exists x ∈ Y ( F q ) such that ( x, x q m ) ∈ Γ and x / ∈ W for large enough natural number m . Remark 3.8.
Writing Y ⊂ A r , then x = ( x , ..., x r ) ∈ ( F q ) r = A r ( F q )and x q m := ( x q m , ..., x q m r ) ∈ A r ( F q ). In our application, Γ is the Zariskiclosure of the graph of a dominant rational map f : Y Y and W isthe locus where f is not well-defined. Proof of Theorem 3.4. (Compare with [3, Thm 3.14].) Let f : M M be as in Proposition 3.5. So M → S is a S -reduced scheme of finitetype, where S is smooth affine over F q , the normalization of F p in H ( S, O S ). Let M k be the general fiber of M → S at Spec ( k ) → S .We define Z k to be the intersection of the Zariski closures E X ( m ) ofthe E X ( m ) ⊂ M k , where E X ( m ) = { ( L i + m ֒ → V i + m ։ O X ) ∈ E X } i ∈ N .By the noetherian property, there exists a m > Z k = E X ( m ) ⊂ M k ( k ) = M ( k )for any m ≥ m . As E X is assumed to be infinite, all components of Z k have dimension ≥
1. Indeed, if it had a component of dimension 0,then there would be a m > m such that this 0-dimensional componentwould not lie on E X ( m ), a contradiction. The fields of constants of theirreducible components of Z k lies between F q ( S ) and k . Let S ′ → S be affine with S ′ smooth affine irreducible such that all componentsof Z k are defined over S ′ , yielding models of them over S ′ which aregeometrically irreducible over F q , and a model Z → S ′ of Z k . Wereplace now X S → S and M → S by X S ′ and the corresponding M S ′ → S ′ . We abuse notations, set S = S ′ , and have Z ⊂ M → S .Let M ⊂ M be the largest open subset where f is well-defined.Notice that E X (1) ⊂ M k TRATIFIED BUNDLES AND ´ETALE FUNDAMENTAL GROUP 15 and f ( E X ( m )) = E X ( m −
1) for any m ≥
1. Thus f ( Z k ∩ M k ) containsa dense subset of Z k . On the other hand, f ( Z k ∩ M k ) ⊂ Z k , thus f : M M induces a dominant rational map f : Z k Z k , and thus a rational dominant map f : Z Z. We consider one irreducible component of Z k , and its model Z irr → S ,which is a geometrically irreducible component of Z over F q . There isan integer a > f a : Z irr Z irr is a dominant rationalmap over a finite field F q .To prove that Z irr contains a periodic point of f a , without loss ofgenerality, we can assume that Y := Z irr ⊂ A r is an affine variety and f a : Y Y is defined by rational functions f , ..., f r ∈ F q ( Y ). LetΓ ⊂ Y × F q Y be the Zariski closure of the graph of f a , which is irreducible over F q as Y is by assumption. The two projections Γ → Y are dominantsince f a : Y Y is dominant. By Theorem 3.7, there exists apoint x ∈ Y ( F q ) such that f is well-defined at x = ( x , ..., x r ) and f a ( x ) = ( x q m , ..., x q m r ) := x q m for some large m . Recall that f i ∈ F q ( Y )( i = 1 , ..., r ) have coefficients in F q , we have f a ( f a ( x )) = ( f ( x q m , ..., x q m r ) , ... , f ( x q m , ..., x q m r ))= ( f ( x , ..., x r ) q m , ... , f ( x , ..., x r ) q m )= ( x q m , ... , x q m r ) = x q m . Thus f a a ( x ) = x q a m = x when a is large enough. The point x ∈ Y ( F q )determines a nontrivial extension 0 → L → V → O X ¯ s → X ¯ s of X (over F q ). That f a ( x ) = x , a = a a , means byProposition 3.5 (3) (cid:0) ( F ∗ X ¯ s ) a L ֒ → ( F ∗ X ¯ s ) a V ։ ( F ∗ X ¯ s ) ∗ O X ¯ s (cid:1) ∼ = ( L ֒ → V ։ O X ¯ s )up to scale. This finishes the proof of Theorem 3.4 and thus of Theo-rem 3.1. (cid:3) Theorem 3.9.
Let X be a smooth projective variety over an alge-braically closed field k of characteristic p > , then: (i) Every stratified bundle on X is trivial if and only if π is trivial. (ii) All the irreducible stratified bundles have rank if and only if [ π , π ] is a pro- p -group. (iii) Every stratified bundle is a direct sum of stratified line bundles,that is str ( X ) is a semi-simple category with irreducible objectsof rank , if and only if π is abelian with no non-trivial p -powerquotient.Proof. (i) is the main theorem of [3]. (ii) is Theorem 2.6. To show(iii), assume that π is abelian with no p -power order quotient. Let E = ( E n , σ n ) n ∈ N be a stratified bundle on X . By (ii), any irreduciblestratified bundle has rank 1. Thus there is a filtration 0 = E ⊂ E ⊂ . . . ⊂ E r = E in str ( X ) such that L v = E v /E v − , 1 ≤ v ≤ r , are rankone stratified bundles. Then, by Theorem 3.1, the filtration splits and E is a direct sum of rank 1 objects. (cid:3) We now comment on analogs of (ii) and (iii) in complex geometry.For this reason, we include here the following lemma, which may havean independent interest.
Lemma 3.10.
Let G be a commutative group scheme over an alge-braically closed field k such that all its quotients in G m are smooth.Let ℓ be a non-trivial character of G . Then H ( G, ℓ ) = 0 .Proof.
Let χ : G → G m = Aut( ℓ ) be the non-trivial character, and let σ : G → ℓ be a cocycle representing a class in H ( G, ℓ ). By definitionof a cocycle, one has σ ( gh ) = χ ( g ) σ ( h ) + σ ( g ). The commutativity of G implies σ ( hg ) = χ ( h ) σ ( g ) + σ ( h ) = σ ( gh ) = χ ( g ) σ ( h ) + σ ( g ) . (3.13)As χ is non-trivial, and Im( χ ) ⊂ G m is assumed to be smooth, there isa h ∈ G ( k ) such that0 = ( χ ( h ) − ∈ End( ℓ ) = k. Thus ( χ ( h ) − ∈ k × = G m ( k ) = Aut( ℓ ) . Set v = ( χ ( h ) − − σ ( h ) ∈ ℓ . Then, by (3.2), one has σ ( g ) = χ ( g ) v − v ,which means that σ is a coboundary. Thus H ( G, ℓ ) = 0. (cid:3)
Remark 3.11.
The same proof shows that if X is a smooth complexvariety, with abelian topological fundamental group, and if ℓ is a non-trivial rank 1 local system, then H ( X, ℓ ) = 0. Indeed, G is now π top1 ( X, a ), the topological fundamental group based at a complex point a ∈ X ( C ), and non-triviality implies the existence of h ∈ G with( χ ( h ) − ∈ C × = Aut( ℓ ). One then concludes identically. TRATIFIED BUNDLES AND ´ETALE FUNDAMENTAL GROUP 17
This fact ought to be well known, but we could not find a referencein the literature.
Remark 3.12.
As already mentioned in the introduction, the Malcev-Grothendieck theorem ([10], [5]) asserts that the ´etale fundamentalgroup π of a smooth complex projective variety is trivial if and only itsstratified fundamental group π str is trivial, where π str is the pro-affinecomplex algebraic group of the Zariski closures of the monodromies ofcomplex linear representations of the topological fundamental group π top1 . By going to the associated Galois cover, it implies that π str isfinite if and only if π is finite (in which case π str = π ). One also hasthat π str is abelian if and only if π is abelian. Indeed, by definition π str is abelian if and only irreducible complex linear representationshave dimension 1. But if ρ : π top1 → GL ( r, C ) is a representation, since π top1 is spanned by finitely many elements, ρ has values in GL ( r, A )for A a ring of finite type over Z . Then if ρ is irreducible, there is aclosed point s ∈ Spec ( A ) such that ρ ⊗ κ ( s ) : π top1 → GL ( r, κ ( s )) isirreducible as well, where κ ( s ) is the residue field of s . As ρ ⊗ κ ( s ) isfinite, it factors through π , thus r = 1. Since a linear representationwith finite monodromy is semi-simple, one can summarize as follows: Claim 3.13. π is1) finite,2) resp. abelian,3) resp. abelian and finiteif and only if π str1
1) finite2) resp. abelian,3) resp. abelian and finiteif and only if1) local systems have finite monodromy,2) resp. irreducible local systems have rank 1,3) resp. local systems are direct sums of rank 1 local systems.While the main result of [3] is an analog in characteristic p >
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Mathematik, Universit¨at Duisburg-Essen, 45117 Essen, Germany
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