aa r X i v : . [ m a t h . AG ] F e b Hodge cohomology of invertible sheaves
H´el`ene Esnault and Arthur OgusOctober 29, 2018
Let k be an algebraically closed field and let X/k be a smooth projectiveconnected k -scheme. Let L be an invertible sheaf on X , and for each integer m , let H mHdg ( X/k, L ) := M a + b = m H b ( X, L ⊗ Ω aX/k ) . We wish to study how the dimensions of the k -vector spaces H mHdg ( X/k, L )and H b ( X, L ⊗ Ω aX/k ) vary with L . For example, if k has characteristic zero,Green and Lazarsefeld [4] proved that for given i, j, m , the subloci { L ∈ Pic ( X ) : dim H i ( X, Ω jX ⊗ L ) ≥ m } of Pic ( X ) are translates of abelian subvarieties, and Simpson [12] showedthat they in fact are translates by torsions points. Both these papers useanalytic methods, but Pink and Roessler [10] obtained the same results purelyalgebraically, using the technique of mod p reduction and the decompositiontheorem of Deligne-Illusie. A key point of their proof is the fact that if thatif L n ∼ = O X for some positive integer n , then for all natural numbers a with( a, n ) = 1 one hasdim H mHdg ( X/k, L ) = dim H mHdg ( X/k, L a ) (1)([10, Proposition 3.5]). They conjecture that equation 1 remains true incharacteristic p > X/k lifts to W ( k ) and has dimension ≤ p . Thepurpose of this note is to discuss a few aspects of this conjecture and somevariants. 1ur main result (see Theorem 7) says that the conjecture is true if n = p and X is ordinary in the sense of Bloch-Kato [2, Definition 7.2]. Wealso explain in section 2 some motivic variants of (1) and, in particular inProposition 1, a proof (due to Pink and Roessler) of the characteristic zerocase of (1), using the language of Grothendieck Chow motives. See [7, 9.3]for a discussion of a related problem using similar techniques. We shouldremark that there are also some log versions of these questions, which wewill not make explicit. Acknowledgements:
We thank D. Roessler for explaining to us his and R.Pink’s analytic proof of equation 1. We thank the referee for very useful,accurate and friendly remarks which helped us improving the exposition ofthis note.
Question 1
Let X be a smooth projective connected variety defined over analgebraically closed field k . Let L be an invertible sheaf on X and n a positiveinteger such that L n ∼ = O X . Is dim H mHdg ( X/k, L i ) = dim H mHdg ( X/k, L ) for every i relatively prime to n ? Let us explain how this question can be given a motivic interpretation.We refer to [11] for the definition of Grothendieck’s Chow motives over a field k . In particular, objects are triples ( Y, p, n ) where Y is a smooth projectivevariety over k , p is an element CH dim( Y ) ( Y × k Y ) ⊗ Q , the rational Chowgroup of dim( Y )-cycles, which, as a correspondence, is an idempotent, and n is a natural number.Let π : Y → X be a principal bundle under a k -group scheme µ , where X and Y are smooth and projective over k . Recall that this means that thereis a k -group scheme action µ × k Y → Y with the property that one has anisomorphism ( ξ, y ) ( y, ξy ) : µ × k Y ∼ = Y × X Y ⊆ Y × k Y. Thus a point ξ ∈ µ ( k ) defines a closed subset Γ ξ of Y × k Y , the graph of theendomorphism of Y defined by ξ . The map ξ Γ ξ extends uniquely to a2ap of Q -vector spacesΓ : Q [ µ ( k )] → CH dim( Y ) ( Y × k Y ) ⊗ Q . Here Q [ µ ( k )] is the Q -group algebra, so the product structure is induced bythe product of k -roots of unity. We can think of CH dim( Y ) ( Y × k Y ) ⊗ Q as a Q -algebra of correspondences acting on CH ∗ ( Y ) ⊗ Q , where for β ∈ CH s ( Y ) ⊗ Q , γ ∈ CH dim( Y ) ( Y × k Y ) ⊗ Q , one defines as usual γ · β := ( p ) ∗ ( γ ∪ p ∗ β ) . Then the map Γ is easily seen to be compatible with composition, as on closedpoints y ∈ Y one has Γ ξ ( y ) = ξ · y . In particular if ξ ∈ Q [ µ ] is idempotent inthe group ring Q [ µ ( k )], then Γ ξ ∼ = Y × ξ is idempotent as a correspondence.In this case we let Y ξ be the Grothendieck Chow motive ( Y, ξ, L be an n -torsion invertible sheaf on smooth irreducible projectivescheme X/k . Recall that the choice of an O X -isomorphism L n α −→ O X definesan O X -algebra structure on A := n − M i =0 L i (2)via the tensor product L i × L j → L i ⊗ O X L j = L i + j for i + j < n and itscomposition with the isomorphism L i × L j → L i ⊗ O X L j = L i + j α − −−→ L i + j − n for 0 ≤ i + j − n . Then the corresponding X -scheme π : Y := Spec X A → X is a torsor under the group scheme µ n of n th roots of unity. Indeed, locallyZariski on X , A ∼ = O X [ t ] / ( t n − u ) for a local unit u , the µ n -action is definedby A → A ⊗ Q [ ζ ] / ( ζ n − , t tζ , and the torsor structure is given by A ⊗ Q [ ζ ] / ( ζ n − ∼ = A ⊗ O X A , ( t, ζ ) ( t, tζ ). This construction definesan equivalence between the category of pairs ( L, α ) and the category of µ n -torsors over X . Assuming now that n is invertible in k , µ n is ´etale, hence π is ´etale and Y is smooth and projective over k . Note the character group X n := Hom( µ n , G m ) is cyclic of order n with a canonical generator (namely,the inclusion µ n → G m ). By construction, the direct sum decomposition (2)of A corresponds exactly to its eigenspace decomposition according to thecharacters of µ n .We can now apply the general construction of motives to this situation.Since µ n is ´etale over the algebraically closed field k , it is completely deter-mined by the finite group Γ := µ n ( k ), which is cyclic of order n . The group3lgebra Q [Γ] is a finite separable algebra over Q , hence is a product of fields: Q [Γ] = Y E e . Here E e = Q [ T ] / (Φ e ( T )) = Q ( ξ e ), where e is a divisor of n , Φ e ( T ) is thecyclotomic polynomial, and ξ e is a primitive e th root of unity. There is an(indecomposable) idempotent e corresponding to each of these fields, and foreach e we find a Chow motive Y e .The indecomposable idempotents of Q [Γ] can also be thought of as pointsof the spectrum T of Q [Γ]. If K is a sufficiently large extension of Q , then T ( K ) = Hom Alg ( Q [Γ] , K ) = Hom Gr (Γ , K ∗ ) , (3)and K ⊗ Q [Γ] ∼ = K [Γ] ∼ = K T ( K ) . (4)Thus T ( K ) can be identified with the character group X n of Γ, andis canonically isomorphic to Z /n Z , with canonical generator the inclusionΓ ⊆ k . Suppose that K/ Q is Galois. Then Gal( K/ Q ) acts on T ( K ), and thepoints of T correspond to the Gal( K/ Q )-orbits. By the theory of cyclotomicextensions of Q , this action factors through a surjective mapGal( K/ Q ) → ( Z /n Z ) ∗ and the usual action of ( Z /n Z ) ∗ on Z /n Z by multiplication. Thus the orbitscorrespond precisely to the divisors d of n ; we shall associate to each orbit S the index d of the subgroup of Z /n Z generated by any element of S . (Notethat in fact the image of d in Z /n Z belongs to S .) We shall thus identify theindecomposable idempotents of Q [Γ] and the divisors of n .Let us suppose that k = C . Then we can consider the Betti cohomologiesof X and Y , and in particular the group algebra Q [Γ] operates on H m ( Y, Q ).We can thus view H m ( Y, Q ) as a Q [Γ]-module, which corresponds to a co-herent sheaf ˜ H m ( Y, Q ) on T . If e is an idempotent of Q [Γ], then H m ( Y e , Q )is the image of the action of e on H m ( Y, Q ), or equivalently, it is the stalkof the sheaf ˜ H m ( Y, Q ) at the point of T corresponding to e , or equivalently,it is H m ( Y, Q ) ⊗ E e where the tensor product is taken over Q [Γ]. If K is asufficiently large field as above, then equation (4) induces an isomorphism of K -vectors spaces: H m ( Y e , Q ) ⊗ Q K ∼ = M { H m ( Y, K ) t : t ∈ T e ( K ) } , T e ( K ) means the set of points of T ( K ) in the Galois orbit corre-sponding to e , and H m ( Y, K ) t means the t -eigenspace of the action of Γ on H m ( Y, Q ) ⊗ Q K . The de Rham and Hodge cohomologies of Y e are defined inthe same way: they are the images of the actions of the idempotent e actingon the k -vector spaces H DR ( Y /k ) and H Hdg ( Y /k ).The following result is due to Pink and Roessler. Their article [10] con-tains a proof using reduction modulo p techniques and the results of [3]; thefollowing analytic argument is based on oral communications with them. Proposition 1
The answer to question 1 is affirmative if k is a field ofcharacteristic zero.Proof: As both sides of the equality in Question 1 satisfy base change withrespect to field extensions, we may assume that k = C . Let i → t i denote theisomorphism Z /n Z ∼ = T ( C ). For each divisor e of n there is a correspondingidempotent e of Q [Γ] ⊆ K [Γ], the sum over all i such that t i ∈ T e ( C ).Consider the Hodge cohomology of the motive Y e : H mHdg ( Y e / C ) := H mHdg ( Y / C ) ⊗ Q [Γ] E e ∼ = H mHdg ( Y / C ) ⊗ C [Γ] ( C ⊗ E e ) . ∼ = M { H mHdg ( Y / C ) i : i ∈ T e ( k ) } . Since π : Y → X is finite and ´etale, H b ( Y, Ω aY/ C ) ∼ = H b ( X, π ∗ π ∗ Ω aX/ C ) ∼ = H b ( X, Ω aX/ C ⊗ π ∗ O Y ) ∼ = M { H b ( X, Ω aX/ C ⊗ L i ) : i ∈ Z /n Z } . Thus H mHdg ( Y / C ) ∼ = M { H mHdg ( X, L i ) : i ∈ Z /n Z } , and hence from the explicit description of the action of µ n on A above itfollows that H mHdg ( Y e / C ) = M { H mHdg ( X, L i ) : i ∈ T e ( C ) } . The Hodge decomposition theorem for Y provides us with an isomorphism: H mHdg ( Y / C ) ∼ = C ⊗ H m ( Y, Q ) , compatible with the action of Q [Γ]. This gives us, for each idempotent e , anisomorphism of C ⊗ E e -modules. H mHdg ( Y e / C ) ∼ = C ⊗ H m ( Y e , Q ) . C ⊗ H m ( Y e , Q ) on the right just comes from the action of E e on H m ( Y e , Q ) by extension of scalars. Since E e is a field, H m ( Y e , Q ) is freeas an E e -module, and hence the C ⊗ E e -module H mHdg ( Y e / C ) is also free.It follows that its rank is the same at all the points t ∈ T e ( C ), affirmingQuestion 1.Let us now formulate an analog of Question 1 for the ℓ -adic and crystallinerealizations of the motive Y e in characteristic p . Question 2
Suppose that k is an algebraically closed field of characteristic p and ( n, p ) = 1 . Let ℓ be a prime different from p , let e be a divisor of n , andlet E e be the corresponding factor of Q [Γ] . Is it true that each H m ( Y e , Q ℓ ) is a free Q ℓ ⊗ E e -module? And is it true that H mcris ( Y e /W ) ⊗ Q is a free W ⊗ E e -module, where W := W ( k ) ? If K is an extension of Q ℓ (resp. of W ( k )) which contains a primitive n throot of unity, then as above we have a eigenspace decompositions: K ⊗ H m ( Y ´ et , Q ℓ ) ∼ = M { H m ( Y ´ et , K ) t : t ∈ T ( K ) } (5) K ⊗ H m ( Y cris /W ( k )) ∼ = M { H m ( Y cris , K ) t : t ∈ T ( K ) } , (6)and this question asks whether the K -dimension of the t -eigenspace is con-stant over the orbits T e ( K ) ⊆ T ( K ).We show in the sequel that the question has a positive answer.Suppose first that X/k lifts to characteristic zero, i.e. , that there exists acomplete discrete valuation ring V with residue field k and fraction field ofcharacteristic zero and a smooth proper ˜ X/V whose special fiber is
X/k . Let X m be the closed subscheme of ˜ X defined by π m +1 , where π is a uniformizingparameter of V . Choose a trivialization α of L n . It follows from Theorem18.1.2 of [6] that the ´etale µ n -torsor Y on X corresponding to ( L, α ) liftsto X m , uniquely up to a unique isomorphism, and hence that the same istrue for ( L, α ). This fact can also be seen by chasing the exact sequences ofcohomology corresponding to the commutative diagram of exact sequences6n the ´etale topology 0 (cid:15) (cid:15) (cid:15) (cid:15) O Xa π m a (cid:15) (cid:15) n ∼ = / / O Xa π m a (cid:15) (cid:15) / / µ n = (cid:15) (cid:15) / / O × X m (cid:15) (cid:15) n / / O × X m (cid:15) (cid:15) / / / / µ n / / O × X m − (cid:15) (cid:15) n / / O × X m − (cid:15) (cid:15) / /
11 1 (7)By Grothendieck’s fundamental theorem for proper morphisms, it followsthat (
L, α ) and Y lift to ( ˜ L, ˜ α ) and ˜ Y on ˜ X . Then by the ´etale to Bettiand Betti to crystalline comparison theorems, we see that under the liftingassumption, the answer to Question 2 is affirmative.In fact, the lifting hypothesis is superfluous, but this takes a bit morework. Claim 2
The answer to Question 2 is affirmative.Proof:
It is trivially true that H m ( Y e , Q ℓ ) is free over Q ℓ ⊗ E e if Q ℓ ⊗ E e is a field. If ( ℓ, n ) = 1, this is the case if and only if ( Z /e Z ) ∗ is cyclicand generated by ℓ . More generally, assuming ℓ is relatively prime to n ,there is a decomposition of Q ℓ [Γ] into a product of fields Q ℓ [Γ] ∼ = Q E ℓ,e ,where now e ranges over the orbits of Z /n Z under the action of the cyclicsubgroup of ( Z /n Z ) ∗ generated by ℓ . This is indeed the unramified lift ofthe decomposition of A = F ℓ [Γ] into a product of finite extensions of F ℓ ,corresponding to the orbits of Frobenius on the geometric points of A . Thisshows at least that the dimension of H m ( Y, K ) t in (5) is, as a function of t ,constant over the ℓ -orbits.For the general statement, let K be an algebraically closed field con-taining Q ℓ for all primes ℓ = p , and containing W ( k ). For ℓ = p let V ℓ := H m ( Y ´ et , Q ℓ ) ⊗ Q ℓ K , and let V p := H m ( Y cris , W ( k )) ⊗ W ( k ) K . Theneach V ℓ is a finite-dimensional representation of Γ, and the isomorphisms (5)7nd (6) are just its decomposition as a direct sum of irreducible representa-tions: V ℓ ∼ = M { n ℓ,i V i : i ∈ Z /n Z } , where V i = K , with γ ∈ Γ acting by multiplication by γ i . By [8, Theo-rem 2.2)] (and [1], [5] and [9] for the existence of cycle classes in crystallinecohomology) the trace of any γ ∈ Γ acting on V ℓ is an integer independent of ℓ , including ℓ = p . Since Γ is a finite group, it follows from the independenceof characters that for each i , n i := n ℓ,i is independent of ℓ . We saw abovethat n ℓ,ℓi = n ℓ,i if ( ℓ, n ) = 1 and ℓ = p , so that in fact n ℓi = n i for all ℓ = p with ( ℓ, n ) = 1. Since the group ( Z /n Z ) ∗ is generated by all such ℓ , it followsthat n i is indeed constant over the ℓ -orbits.What does this tell us about Question 1? If ( p, n ) = 1 and k is alge-braically closed, W [Γ] is still semisimple, and can be written canonicallyas a product of copies of W , indexed by i ∈ T ( W ) ∼ = Z /n Z . For every t ∈ T ( W ) ∼ = T ( k ), we have an injective base change map from crystalline tode Rham cohomology: k ⊗ H m ( Y /W ) t → H m ( Y /k ) t . Question 3
In the above situation, is H q ( Y /W ) torsion free when ( p, n ) =1 ? If the answer is yes, then the maps k ⊗ H m ( Y /W ) t → H m ( Y /k ) t are isomor-phisms, and this means that we can compute the dimensions of the de Rhameigenspaces from the ℓ -adic ones. Assuming also that the Hodge to de Rhamspectral sequence of Y /k degenerates at E , this should give an affirmativeanswer to Question 1. Note that if X/k lifts mod p , then Y /k lifts mod p as well, and if the dimension is less than or equal to p , the E -degenerationis true by [3].Of course, there is no reason for Question 3 to have an affirmative answerin general. Is there a reasonable hypothesis on X which guarantees it? Forexample, is it true if the crystalline cohomology of X/W is torsion free? p -torsion case in characteristic p Let us assume from now on that k is a perfect field of characteristic p > X ′ be the pull back of X via the Frobenius of k , let π : X ′ → X bethe projection, and let F : X → X ′ and F X : X → X be the relative andabsolute Frobenius morphisms. Then F ∗ X L = L p = F ∗ L ′ , where L ′ := π ∗ L .Then L p = F − L ′ ⊗ O X ′ O X is endowed with the Frobenius descent connection1 ⊗ d , i.e. the unique connection spanned by its flat sections L ′ . In general,for a given integrable connection ( E, ∇ ), we set H iDR ( X, ( E, ∇ )) = H i ( X/k, (Ω · X/k ⊗ E, ∇ )) , and we use again the notation H iHdg ( X/k, L ) = M a + b = i H b ( X, Ω aX/k ⊗ L )and write h iDR and h iHdg for the respective dimensions of these spaces. Proposition 3
Let L be an invertible sheaf on a smooth proper scheme X over k and let ∇ be the Frobenius descent connection on L p . Suppose that X/k lifts to W ( k ) and has dimension at most p . Then for every naturalnumber m , h mDR ( X/k, ( L p , ∇ )) = h mHdg ( X/k, L ) . Corollary 4
Under the assumtpions of Proposition 3, if L p ∼ = O X and ω := ∇ (1) , then for any integer a , h mHdg ( X/k, L a ) = h mDR ( X/k, ( O X , d + aω )) . Remark 5 If p divides a , this just means the degeneration of the Hodge tode Rham spectral sequence for ( O X , d ). Proof:
Let
Hdg · X ′ /k denote the Hodge complex of X ′ /k , i.e. , the direct sum ⊕ i Ω iX ′ /k [ − i ]. Recall from [3] that the lifting yields an isomorphism in thebounded derived category of O X ′ -modules: Hdg · X ′ /k ∼ = F ∗ (Ω · X/k , d ) . Tensoring this isomorphism with L ′ := π ∗ L and using the projection formulafor F , we find an isomorphism Hdg · X ′ /k ⊗ L ′ ∼ = F ∗ (Ω · X/k ⊗ L p , ∇ ) . H mHdg ( X/k, L ) F ∗ k ∼ = −−−→ H mHdg ( X ′ /k, L ′ ) F ∗ ∼ = ←−−− H mDR ( X, ( L p , ∇ )) . This proves the proposition. If L p = O X , the corresponding Frobenius de-scent connection ∇ on O X is determined by ω L := ∇ (1). It follows from thetensor product rule for connections that ω L a = aω L for any integer a .The corollary suggests the following question. Question 4
Let ω be a closed one-form on X and let c be a unit of k . Isthe dimension of H mDR ( X, ( O X , d + cω )) independent of c ? Remark 6
Some properness is necessary, since the p -curvature of d ω := d + ω can change from zero to non-zero as one multiplies by an invertible con-stant. If the p -curvature is non-zero, then the sheaf H (Ω · X/k , d ω ) vanishes,and hence so does H ( X, (Ω · X/k , d ω )). If the p -curvature vanishes, then H (Ω · X/k , d ω ) is an invertible sheaf L , which can have nontrivial sectionsif X is allowed to shrink. However, since by definition, L ⊂ O X , it can havea global section on a proper X only if L = O X .We can answer Question 4 under a strong hypothesis. Theorem 7
Suppose that
X/k is smooth, proper, and ordinary in the senseof Bloch and Kato [2, Definition 7.2]: H i ( X, B jX/k ) = 0 for all i, j , where B jX/k := Im (cid:16) d : Ω j − X/k → Ω jX/k (cid:17) . Then the answer to question 4 is affirmative. Hence if
X/k lifts to W ( k ) ,has dimension at most p , and if n = p , the answer to Question 1 is alsoaffirmative. We begin with the following lemmas.
Lemma 8
Let ω be a closed one-form on X , and let d ω := d + ω ∧ : Ω · X/k → Ω · +1 X/k . Then the standard exterior derivative induces a morphism of complexes: (Ω · X/k , d ω ) δ ✲ (Ω · X/k , d ω )[1] . roof: If α is a section of Ω qX/k , dd ω ( α ) = d ( dα + ω ∧ α )= ddα + dω ∧ α − ω ∧ dα = − ω ∧ dα. Since the sign of the differential of the complex (Ω · X/k , d ω )[1] is the negativeof the sign of the differential of (Ω · X/k , d ω ), d ω d ( α ) = − ( d + ω ∧ )( dα )= − ω ∧ dα Lemma 9
Let Z · := ker( d ) ⊆ (Ω · X/k , d ω ) and B · := Im( d )[ − ⊆ (Ω · X/k , d ω ) .Then for any a ∈ k ∗ , multiplication by a i in degree i induces isomorphisms ( Z · , d ω ) λ a ✲ ( Z · , d aω )( B · , d ω ) λ a ✲ ( B · , d aω ) . Proof:
It is clear that the boundary map d ω on Z · and on B · is just wedgeproduct with ω . Proof of Theorem 7
The morphism δ of Lemma 8 induces an exact sequence:0 → ( Z · , d ω ) → (Ω · X/k , d ω ) δ −→ ( B · , d ω )[1] → . (8)As X/k is ordinary, the E term of the first spectral sequence for ( B · , d ω ) is E i,j = H j ( X, B i ) = 0, and it follows that the hypercohomology of ( B · , d ω )vanishes, for every ω . Hence the natural map H q ( Z · , d ω ) → H q (Ω · X/k , d ω ) isan isomorphism. Since the dimension of H q ( Z · , d ω ) is unchanged when ω ismultiplied by a unit of k , the same is true of H q (Ω · X/k , d ω ). This completesthe proof of Theorem 7. Remark 10
A simple Riemann-Roch computation shows that on curves,question 1 has a positive answer with no additional assumptions. Indeed, if L is a nontrivial torsion sheaf, then its degree is zero and it has no globalsections. It follows that h ( L ) = g −
1. Since the same is true for L − , h ( L ⊗ Ω X ) = h ( L − ) = g −
1, and h ( L ⊗ Ω X ) = h ( L − ) = 0.11 emark 11 In the absence of the ordinarity hypothesis, one can ask if therank of the boundary map ∂ ω : H q +1 ( B · , ω ∧ ) → H q +1 ( Z · , ω ∧ )of (8) changes if ω is multiplied by a unit of k . To analyze this question, let c ω : ( B · , ω ∧ ) → ( Z · , ω ∧ )be the morphism in the derived category D ( X ′ , O X ′ ) defined by the exactsequence (8), so that ∂ ω can be identified with H q − ( c ω ). Similarly, the exactsequence 0 → ( Z · , ω ∧ ) → (Ω · , ω ∧ ) → ( B · , ω ∧ )[1] → a ω : ( B · , ω ∧ ) → ( Z · , ω ∧ )in D ( X ′ , O X ′ ) as well. There is also an inclusion morphism: b ω : ( B · , ω ∧ ) → ( Z · , ω ∧ ) . Then it is not difficult to check that c ω = a ω + b ω . If a ∈ k ∗ , we haveisomorphisms of complexes λ a : ( Z · , ω ∧ ) → ( Z · , aω ∧ ) λ a : ( B · , ω ∧ ) → ( B · , aω ∧ )Using these as identifications, one can check that c aω = a − a ω + b ω . Thiswould suggest a negative answer to Question 4, but we do not have an ex-ample. References [1] Berthelot, P.:
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