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The Gauß–Manin connection on the Hodge–Tate structures
C. R. Acad. Sci. Paris, t. 333, Série I, p. 333–337, 2001 Geométrie algébrique/Algebraic Geometry
The Gauß–Manin connection on the Hodge–Tate structures Marat ROVINSKY Independent University of Moscow, 121002 Moscow, B. Vlasievsky Per. 11, Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia E-mail: [email protected] (Reçu le 6 février 2001, accepté après révision le 25 juin 2001)
Abstract.
For each simplicial complex scheme X• the singular cohomology H ∗ (X• ) carries a canonical mixed Hodge structure. There exists a canonical homomorphism ∇ : H ∗ (X• ) → Ω1C/Q ⊗ H ∗ (X• ), the Gauß–Manin connection. We show that there is a unique functorial connection on each mixed Hodge–Tate structure having certain properties of the Gauß– Manin connection. This connection is non-integrable in general, and therefore, its integrability is a non-trivial condition on the Hodge structure to be geometric. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
La connexion de Gauß–Manin sur les structures de Hodge–Tate Résumé.
Pour tout schéma simplicial complexe X• la cohomologie singulière à coefficients rationnels H ∗ (X• ) est munie d’une structure de Hodge mixte canonique. Il existe une application canonique ∇ : H ∗ (X• ) → Ω1C/Q ⊗ H ∗ (X• ), la connexion de Gauß–Manin. Nous montrons qu’il existe une unique connexion fonctorielle sur toute structure de Hodge– Tate mixte ayant certaines propriétés de la connexion de Gauß–Manin. Cette connexion n’est pas intégrable en général, et donc son intégrabilité est une condition non triviale pour qu’une structure de Hodge–Tate soit géométrique. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Version française abrégée Soit X• une variété simpliciale algébrique lisse complexe. La connexion de Gauß–Manin sur la ∇ q q cohomologie de de Rham HdR/C (X• ) → Ω1C ⊗C HdR/C (X• ) est la première differentielle dans la suite q p+q spectrale de Leray E1p,q = ΩpC ⊗C HdR/C (X• ) qui converge vers HdR/Q (X• ). Il n’est pas difficile d’établir les propriétés habituelles de la connexion de Gauß–Manin (la transversalité de Griffiths, l’intégrabilité, la compatibilité avec la filtration de poids). Soient H une structure de Hodge mixte et W• la filtration de poids sur H ⊗ Q. Une structure H est dite de Hodge–Tate si la structure de Hodge pure grW j := Wj /Wj−1 est une somme directe de sous-structures de Hodge de rang un pour tout entier j.
Note présentée par Christophe S OULÉ. S0764-4442(01)02064-X/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés
333
M. Rovinsky
ˇ Par un calcul direct sur le complexe de Cech–de Rham, on trouve la connexion de Gauß–Manin sur toute structure de Hodge logarithmique, c’est-à-dire, sur la cohomologie relative H 1 (Gm , {1, a}; Q) de Gm modulo {1, a} pour a ∈ C× quelconque (voir formule (1) ci-dessous). Dans le cas d’une structure de Hodge–Tate quelconque on utilise un cas particulier du scindage fonctoriel des structures de Hodge dû à Deligne (voir (2) ci-dessous) pour identifier l’espace sous-jacent complexifié de toute structure de Hodge–Tate avec une somme directe de sous-espaces des espaces sous-jacents complexifiés des structures de Hodge–Tate logarithmiques. Soit H une catégorie abélienne de structures de Hodge–Tate sur Q contenant les structures logarithmiques, contenant les structures H(m) := H ⊗ Q(m) pour chaque H ∈ Ob(H) et tout entier m, et contenant toutes les sous-structures de Hodge de chacun de ses objets. la catégorie dont les objets sont les objets de H munis d’une connexion qui satisfait la Soit H transversalité de Griffiths et les morphismes sont les morphismes de structures de Hodge qui commutent avec les connexions. P ROPOSITION 1. – → H tel que pour – Il existe un unique foncteur H → (H, ∇H ) inverse à droite au foncteur d’oubli H toute structure logarithmique L la connexion ∇L coïncide avec la connexion de Gauß–Manin, et ∇H induit la même connexion sur H ⊗ C = H(1) ⊗ C que ∇H(1) . W × W W – Soit ξj la classe de W2j /W2j−4 dans Ext1 (grW 2j , gr2j−2 ) = C ⊗ Hom(gr2j (1), gr2j−2 ). Soit ψj l’application composée W W 1 W × W ⊗ C× ⊗ Hom grW Ext1 grW 2j+2 , gr2j ⊗ Ext gr2j , gr2j−2 −→ C 2j+2 (2), gr2j−2 d log ∧d log ⊗id W −−−−−−−−−−→ Ω2C ⊗ Hom grW 2j+2 (2), gr2j−2 . Alors, ∇H est intégrable si et seulement si ψj (ξj+1 ⊗ ξj ) = 0 pour tout entier j. On en déduit la proposition suivante. P ROPOSITION 2. – Soit Ω2C,log le sous-groupe de Ω2C engendré par a−1 da ∧ b−1 db pour tous a, b ∈ C. Alors, dans la catégorie des structures de Hodge–Tate plates on a Ext2 (Q(0), Q(2)) = Ω2C,log .
1. Introduction Let X• be a smooth simplicial algebraic variety over a field k of characteristic zero. The Gauß–Manin ∇ q q connection Ωpk ⊗k HdR/k (X• ) −→ Ωp+1 ⊗k HdR/k (X• ) is the first differential in the Leray spectral k q p+q sequence E1p,q = Ωpk ⊗k HdR/k (X• ) converging to HdR/Q (X• ). 2 The Gauß–Manin connection is integrable: ∇ = 0; it respects the weight filtration; it satisfies the Griffiths transversality: q q (X• ) ⊆ Ω1k ⊗k F p−1 HdR/k (X• ); ∇ F p HdR/k
the Leibniz rule for cup-products holds; the connection on the cohomology of a product coincides with the tensor product of the connections on the cohomology of the multiples. If one believes in the Hodge conjecture then for a given pure Hodge structure H there is at most one connection ∇ such that H is a Hodge substructure of a cohomology group of a smooth projective complex variety with ∇ induced by the Gauß–Manin connection. Independently of the Hodge conjecture, it follows from results of Katz [3] and Deligne [2] that there are at most countably many connections ∇ on a given
334
The Gauß–Manin connection on the Hodge–Tate structures
pure Hodge structure H such that H is a Hodge substructure of a cohomology group of a smooth projective complex variety with ∇ induced by the Gauß–Manin connection. Let H be a mixed Hodge structure, and W• the weight filtration on H ⊗ Q. Then, H is called a Hodge– Tate structure if the pure Hodge structure grW j := Wj /Wj−1 is a direct sum of Hodge substructures of rank one for any integer j. In Proposition 3.1 below we show that there is a unique functorial connection on each mixed Hodge– Tate structure satisfying the Griffiths transversality, compatible with Tate twists and coinciding with the Gauß–Manin connection on the logarithmic structures. This connection is non-integrable in general, so, if any Hodge substructure of a geometric Hodge–Tate structure is again geometric, its integrability gives a non-trivial necessary condition for a Hodge–Tate structure to be geometric. 2. The connection on the logarithmic structures Consider the first relative cohomology group of Gm modulo {1, a} for some a ∈ C× . To calculate this group we present Gm as the complement of P1 to the divisor (0) + (∞) and then 1 HdR/C Gm , {1, a} = H1 OP1 −(1) − (a) −→ Ω1P1 /C (0) + (∞) . ˇ For the covering P1 = U0 ∪ U1 with U0 = P1 \{a} and U1 = P1 \{1} the 1-cocycles in the Cech– de Rham complex are collections (f01 , ω0 , ω1 ) with df01 = ω0 − ω1 , where f01 ∈ O(U0 ∩ U1 ) and ωi ∈ Ω1P1 /C ((0) + (∞))(Ui ). There is an isomorphism 1 Gm , {1, a} = b, c · z −1 dz | b, c ∈ C , HdR/C ˇ where (1, 0) =: e0 denotes the 1-cocycle in the Cech–de Rham complex presented by the function 1 on 1 U0 ∩ U1 , and F 1 HdR/C (Gm , {1, a}) = Γ(P1 , Ω1P1 /C ((0) + (∞))) = z −1 dzC . ˇ Rham complex, one has As e0 lifts tautologically to a 1-cocycle in the first term of the absolute Cech–de −1 −1 ∇e0 = 0. To calculate ∇(z dz) we lift the (relative) form z dz to a section ηj of the sheaf of absolute 1-forms vanishing at 1 and a over each element Uj of the covering, say, η0 = z −1 dz, η1 = (a/z) d(z/a). Then the coboundary of the 1-cochain (ηj ) is (a−1 da, 0), so ∇(z −1 dz) = a−1 da ⊗ e0 . One can translate this calculation in Hodge-theoretic terms as follows. Let H be the Hodge–Tate structure H 1 (Gm , {1, a}; Q). Its weights are 0 and 2. Let ξ be the preimage of a rational element in C under the mod(W0 )C
isomorphism F 1 −−−−−−→ Q(−1) ⊗ C = C. Then ∇e0 = 0 and ∇ξ = e−z0 dez0 ⊗ e0
(1)
for any non-zero e0 ∈ W0 , where ξ − z0 e0 ∈ H(1). Clearly, this is independent of e0 . 3. The case of an arbitrary Hodge–Tate structure Let H be an Abelian category of Hodge–Tate structures over Q containing all logarithmic structures, invariant with respect to the Tate twists and containing each Hodge substructure of each of its objects. whose objects are objects of H equipped with a connection satisfying the Consider the category H Griffiths’ transversality and morphisms are morphisms of Hodge structures commuting with connections. P ROPOSITION 3.1. – → H such that for – There is a unique functor H → (H, ∇H ) right inverse to the forgetful functor H any logarithmic structure L the connection ∇L coincides with the Gauß–Manin connection calculated above, and ∇H induces the same connection on H ⊗ C = H(1) ⊗ C as ∇H(1) .
335
M. Rovinsky W × W W – Let ξj be the class of W2j /W2j−4 in Ext1 (grW 2j , gr2j−2 ) = C ⊗ Hom(gr2j (1), gr2j−2 ). Let ψj be the composition W W 1 W × × W ⊗ Hom grW Ext1 grW 2j+2 , gr2j ⊗ Ext gr2j , gr2j−2 −→ C ⊗ C 2j+2 (2), gr2j−2 d log ∧d log ⊗id W −−−−−−−−→ Ω2C ⊗ Hom grW 2j+2 (2), gr2j−2 .
Then ∇H is integrable if and only if ψj (ξj+1 ⊗ ξj ) = 0 for all integer j. Proof. – It is a particular case of Deligne’s functorial splitting of Hodge structures (see, e.g., [1], definition and Proposition 2.6) that for any Hodge–Tate structure H the inclusion maps induce the decomposition ∼ F j ∩ (W2j ⊗ C) −→ H ⊗ C. (2) j
Then the canonical projections ϕj : F j ∩ (W2j ⊗ C) → grW 2j ⊗ C are isomorphisms. It follows from the functoriality, applied to the morphism Wj → H, that ∇ respects the weight filtration. One knows from the calculation for the logarithmic structures, that the connection on the structure Q(0) is zero. This implies that ∇ : W2j (j) → Ω1C ⊗ W2j−2 . Combining these with the Griffiths’ transversality ∇ : F j → Ω1C ⊗C F j−1 , we get W ∇ 1 W gr2j −→ ΩC ⊗C F j−1 ∩ (W2j−2 )C = Ω1C ⊗ (2πi)j−1 ϕ−1 (3) (2πi)j ϕ−1 j j−1 gr2j−2 . By the functoriality and compatibility with the Tate twists, to construct that map for some j we may identify the space F j ∩ (W2j )C with the space F 1 H ∩ (W2 H )C , and F j−1 ∩ (W2j−2 )C with F 0 H ∩ (W0 H )C , where H = (W2j /W2j−4 )(j − 1). Then H can be identified with a Hodge substructure of a direct sum of logarithmic Hodge structures where we have fixed the connection. Let ξj be the image of ξj under the map d log ⊗id W W 1 W C× ⊗ Hom grW 2j (1), gr2j−2 −−−−−−→ ΩC ⊗ Hom gr2j (1), gr2j−2 . W One can rewrite formula (1) as ∇H (e) = (id ⊗ ϕj−1 )−1 (ξj (ϕj (e))) for any e ∈ (2πi)j ϕ−1 j (gr2j ). Here 1 j 1 W id ⊗ ϕj is the isomorphism ΩC ⊗C (F ∩ (W2j )C ) → ΩC ⊗ gr2j . One concludes from the existence of the decomposition (2) that the vectors e generate HC , when j varies. The statement on the integrability is clear from the above explicit formula for ∇H . ✷
Remark. – It is easy to see that the connection constructed in Proposition 3.1 on the tensor product of two Hodge–Tate structures coincides with the tensor product of the connections on these Hodge–Tate structures. P ROPOSITION 3.2. – Let Ω2C,log be the subgroup of Ω2C generated by a−1 da ∧ b−1 db for all a, b ∈ C. Then, in the category of the flat Hodge–Tate structures one has Ext2 (Q(0), Q(2)) = Ω2C,log . Proof. – By the standard argument, each class in Extm (Q(0), Q(m)) can be presented by an acyclic complex Q(m) → Bm → · · · → B1 → Q(0), where Bj has weights −2j and 2 − 2j. The existence of such presentations implies that Ext1 Q(0), Q(1) ⊗ Ext1 Q(1), Q(2) ⊗ · · · ⊗ Ext1 Q(m − 1), Q(m) → Extm Q(0), Q(m) is surjective. Note, that Ext1 (Q(j), Q(j + 1)) = C× ⊗ Q for any integer j. s ∪ In the case m = 2 this gives a surjection C× ⊗ C× −→ Ext2 (Q(0), Q(2)). Let j=1 aj ⊗ bj be an element in its kernel. Denote by Hb a Hodge–Tate structure with the class (b1 , . . . , bs ) ∈ (C× )sQ = Ext1 (Q(0), Q(1)s ). So Hb fits into an exact sequence 0 → Q(1)s → Hb → Q(0) → 0 which, after taking
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The Gauß–Manin connection on the Hodge–Tate structures
RHom from it to Q(2) in the category of flat Hodge–Tate structures, gives an exact sequence 0 → Ext1 Q(0), Q(2) → Ext1 Hb , Q(2) → Ext1 Q(1)s , Q(2) → Ext2 Q(0), Q(2) . It follows from the integrability criterion of Proposition 3.1 that in the category of flat Hodge–Tate structures one has
s dAj dbj Ext1 (Hb , Q(2)) × s , . . . , A ) ∈ (C ) ⊗ Q ∧ = 0 . = (A 1 s Aj bj Ext1 (Q(0), Q(2)) j=1 This implies that there is a natural embedding
s dAj dbj × s × s (C ) ∧ = 0 → Ext2 Q(0), Q(2) , (A1 , . . . , As ) ∈ (C ) Aj bj j=1 and thus s
daj j=1
aj
∧
dbj = 0, bj
so the kernel of ∪ coincides with the kernel of the map C× ⊗ C× → Ω2C given by a ⊗ b → implies that Ext2 (Q(0), Q(2)) = Ω2C,log . ✷
da a
∧
db b .
This
Acknowledgement. I would like to thank P. Deligne for pointing out non-integrability of the connection constructed in Proposition 3.1. I am grateful to Andrey Levin for inspiring discussions, to the I.H.E.S. for its hospitality and to the European Post-Doctoral Institute for its support.
References [1] Brylinski J.-L., Zucker S., An overview of recent advances in Hodge theory. Several complex variables, VI, in: Encyclopaedia Math. Sci., Vol. 69, Springer, Berlin, 1990, pp. 39–142. [2] Deligne P., Théorie de Hodge II, III, Inst. Hautes Études Sci. Publ. Math. 40 (1972) 5–58, 44 (1974) 3–77. [3] Katz N., Nilpotent connections and themonodromy theorem, Applications of a result of Turritin, Inst. Hautes Études Sci. Publ. Math. 39 (1971) 175–232.
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The Gauß–Manin connection on the Hodge–Tate structures Marat ROVINSKY Independent University of Moscow, 121002 Moscow, B. Vlasievsky Per. 11, Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia E-mail: [email protected] (Reçu le 6 février 2001, accepté après révision le 25 juin 2001)
Abstract.
For each simplicial complex scheme X• the singular cohomology H ∗ (X• ) carries a canonical mixed Hodge structure. There exists a canonical homomorphism ∇ : H ∗ (X• ) → Ω1C/Q ⊗ H ∗ (X• ), the Gauß–Manin connection. We show that there is a unique functorial connection on each mixed Hodge–Tate structure having certain properties of the Gauß– Manin connection. This connection is non-integrable in general, and therefore, its integrability is a non-trivial condition on the Hodge structure to be geometric. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
La connexion de Gauß–Manin sur les structures de Hodge–Tate Résumé.
Pour tout schéma simplicial complexe X• la cohomologie singulière à coefficients rationnels H ∗ (X• ) est munie d’une structure de Hodge mixte canonique. Il existe une application canonique ∇ : H ∗ (X• ) → Ω1C/Q ⊗ H ∗ (X• ), la connexion de Gauß–Manin. Nous montrons qu’il existe une unique connexion fonctorielle sur toute structure de Hodge– Tate mixte ayant certaines propriétés de la connexion de Gauß–Manin. Cette connexion n’est pas intégrable en général, et donc son intégrabilité est une condition non triviale pour qu’une structure de Hodge–Tate soit géométrique. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Version française abrégée Soit X• une variété simpliciale algébrique lisse complexe. La connexion de Gauß–Manin sur la ∇ q q cohomologie de de Rham HdR/C (X• ) → Ω1C ⊗C HdR/C (X• ) est la première differentielle dans la suite q p+q spectrale de Leray E1p,q = ΩpC ⊗C HdR/C (X• ) qui converge vers HdR/Q (X• ). Il n’est pas difficile d’établir les propriétés habituelles de la connexion de Gauß–Manin (la transversalité de Griffiths, l’intégrabilité, la compatibilité avec la filtration de poids). Soient H une structure de Hodge mixte et W• la filtration de poids sur H ⊗ Q. Une structure H est dite de Hodge–Tate si la structure de Hodge pure grW j := Wj /Wj−1 est une somme directe de sous-structures de Hodge de rang un pour tout entier j.
Note présentée par Christophe S OULÉ. S0764-4442(01)02064-X/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés
333
M. Rovinsky
ˇ Par un calcul direct sur le complexe de Cech–de Rham, on trouve la connexion de Gauß–Manin sur toute structure de Hodge logarithmique, c’est-à-dire, sur la cohomologie relative H 1 (Gm , {1, a}; Q) de Gm modulo {1, a} pour a ∈ C× quelconque (voir formule (1) ci-dessous). Dans le cas d’une structure de Hodge–Tate quelconque on utilise un cas particulier du scindage fonctoriel des structures de Hodge dû à Deligne (voir (2) ci-dessous) pour identifier l’espace sous-jacent complexifié de toute structure de Hodge–Tate avec une somme directe de sous-espaces des espaces sous-jacents complexifiés des structures de Hodge–Tate logarithmiques. Soit H une catégorie abélienne de structures de Hodge–Tate sur Q contenant les structures logarithmiques, contenant les structures H(m) := H ⊗ Q(m) pour chaque H ∈ Ob(H) et tout entier m, et contenant toutes les sous-structures de Hodge de chacun de ses objets. la catégorie dont les objets sont les objets de H munis d’une connexion qui satisfait la Soit H transversalité de Griffiths et les morphismes sont les morphismes de structures de Hodge qui commutent avec les connexions. P ROPOSITION 1. – → H tel que pour – Il existe un unique foncteur H → (H, ∇H ) inverse à droite au foncteur d’oubli H toute structure logarithmique L la connexion ∇L coïncide avec la connexion de Gauß–Manin, et ∇H induit la même connexion sur H ⊗ C = H(1) ⊗ C que ∇H(1) . W × W W – Soit ξj la classe de W2j /W2j−4 dans Ext1 (grW 2j , gr2j−2 ) = C ⊗ Hom(gr2j (1), gr2j−2 ). Soit ψj l’application composée W W 1 W × W ⊗ C× ⊗ Hom grW Ext1 grW 2j+2 , gr2j ⊗ Ext gr2j , gr2j−2 −→ C 2j+2 (2), gr2j−2 d log ∧d log ⊗id W −−−−−−−−−−→ Ω2C ⊗ Hom grW 2j+2 (2), gr2j−2 . Alors, ∇H est intégrable si et seulement si ψj (ξj+1 ⊗ ξj ) = 0 pour tout entier j. On en déduit la proposition suivante. P ROPOSITION 2. – Soit Ω2C,log le sous-groupe de Ω2C engendré par a−1 da ∧ b−1 db pour tous a, b ∈ C. Alors, dans la catégorie des structures de Hodge–Tate plates on a Ext2 (Q(0), Q(2)) = Ω2C,log .
1. Introduction Let X• be a smooth simplicial algebraic variety over a field k of characteristic zero. The Gauß–Manin ∇ q q connection Ωpk ⊗k HdR/k (X• ) −→ Ωp+1 ⊗k HdR/k (X• ) is the first differential in the Leray spectral k q p+q sequence E1p,q = Ωpk ⊗k HdR/k (X• ) converging to HdR/Q (X• ). 2 The Gauß–Manin connection is integrable: ∇ = 0; it respects the weight filtration; it satisfies the Griffiths transversality: q q (X• ) ⊆ Ω1k ⊗k F p−1 HdR/k (X• ); ∇ F p HdR/k
the Leibniz rule for cup-products holds; the connection on the cohomology of a product coincides with the tensor product of the connections on the cohomology of the multiples. If one believes in the Hodge conjecture then for a given pure Hodge structure H there is at most one connection ∇ such that H is a Hodge substructure of a cohomology group of a smooth projective complex variety with ∇ induced by the Gauß–Manin connection. Independently of the Hodge conjecture, it follows from results of Katz [3] and Deligne [2] that there are at most countably many connections ∇ on a given
334
The Gauß–Manin connection on the Hodge–Tate structures
pure Hodge structure H such that H is a Hodge substructure of a cohomology group of a smooth projective complex variety with ∇ induced by the Gauß–Manin connection. Let H be a mixed Hodge structure, and W• the weight filtration on H ⊗ Q. Then, H is called a Hodge– Tate structure if the pure Hodge structure grW j := Wj /Wj−1 is a direct sum of Hodge substructures of rank one for any integer j. In Proposition 3.1 below we show that there is a unique functorial connection on each mixed Hodge– Tate structure satisfying the Griffiths transversality, compatible with Tate twists and coinciding with the Gauß–Manin connection on the logarithmic structures. This connection is non-integrable in general, so, if any Hodge substructure of a geometric Hodge–Tate structure is again geometric, its integrability gives a non-trivial necessary condition for a Hodge–Tate structure to be geometric. 2. The connection on the logarithmic structures Consider the first relative cohomology group of Gm modulo {1, a} for some a ∈ C× . To calculate this group we present Gm as the complement of P1 to the divisor (0) + (∞) and then 1 HdR/C Gm , {1, a} = H1 OP1 −(1) − (a) −→ Ω1P1 /C (0) + (∞) . ˇ For the covering P1 = U0 ∪ U1 with U0 = P1 \{a} and U1 = P1 \{1} the 1-cocycles in the Cech– de Rham complex are collections (f01 , ω0 , ω1 ) with df01 = ω0 − ω1 , where f01 ∈ O(U0 ∩ U1 ) and ωi ∈ Ω1P1 /C ((0) + (∞))(Ui ). There is an isomorphism 1 Gm , {1, a} = b, c · z −1 dz | b, c ∈ C , HdR/C ˇ where (1, 0) =: e0 denotes the 1-cocycle in the Cech–de Rham complex presented by the function 1 on 1 U0 ∩ U1 , and F 1 HdR/C (Gm , {1, a}) = Γ(P1 , Ω1P1 /C ((0) + (∞))) = z −1 dzC . ˇ Rham complex, one has As e0 lifts tautologically to a 1-cocycle in the first term of the absolute Cech–de −1 −1 ∇e0 = 0. To calculate ∇(z dz) we lift the (relative) form z dz to a section ηj of the sheaf of absolute 1-forms vanishing at 1 and a over each element Uj of the covering, say, η0 = z −1 dz, η1 = (a/z) d(z/a). Then the coboundary of the 1-cochain (ηj ) is (a−1 da, 0), so ∇(z −1 dz) = a−1 da ⊗ e0 . One can translate this calculation in Hodge-theoretic terms as follows. Let H be the Hodge–Tate structure H 1 (Gm , {1, a}; Q). Its weights are 0 and 2. Let ξ be the preimage of a rational element in C under the mod(W0 )C
isomorphism F 1 −−−−−−→ Q(−1) ⊗ C = C. Then ∇e0 = 0 and ∇ξ = e−z0 dez0 ⊗ e0
(1)
for any non-zero e0 ∈ W0 , where ξ − z0 e0 ∈ H(1). Clearly, this is independent of e0 . 3. The case of an arbitrary Hodge–Tate structure Let H be an Abelian category of Hodge–Tate structures over Q containing all logarithmic structures, invariant with respect to the Tate twists and containing each Hodge substructure of each of its objects. whose objects are objects of H equipped with a connection satisfying the Consider the category H Griffiths’ transversality and morphisms are morphisms of Hodge structures commuting with connections. P ROPOSITION 3.1. – → H such that for – There is a unique functor H → (H, ∇H ) right inverse to the forgetful functor H any logarithmic structure L the connection ∇L coincides with the Gauß–Manin connection calculated above, and ∇H induces the same connection on H ⊗ C = H(1) ⊗ C as ∇H(1) .
335
M. Rovinsky W × W W – Let ξj be the class of W2j /W2j−4 in Ext1 (grW 2j , gr2j−2 ) = C ⊗ Hom(gr2j (1), gr2j−2 ). Let ψj be the composition W W 1 W × × W ⊗ Hom grW Ext1 grW 2j+2 , gr2j ⊗ Ext gr2j , gr2j−2 −→ C ⊗ C 2j+2 (2), gr2j−2 d log ∧d log ⊗id W −−−−−−−−→ Ω2C ⊗ Hom grW 2j+2 (2), gr2j−2 .
Then ∇H is integrable if and only if ψj (ξj+1 ⊗ ξj ) = 0 for all integer j. Proof. – It is a particular case of Deligne’s functorial splitting of Hodge structures (see, e.g., [1], definition and Proposition 2.6) that for any Hodge–Tate structure H the inclusion maps induce the decomposition ∼ F j ∩ (W2j ⊗ C) −→ H ⊗ C. (2) j
Then the canonical projections ϕj : F j ∩ (W2j ⊗ C) → grW 2j ⊗ C are isomorphisms. It follows from the functoriality, applied to the morphism Wj → H, that ∇ respects the weight filtration. One knows from the calculation for the logarithmic structures, that the connection on the structure Q(0) is zero. This implies that ∇ : W2j (j) → Ω1C ⊗ W2j−2 . Combining these with the Griffiths’ transversality ∇ : F j → Ω1C ⊗C F j−1 , we get W ∇ 1 W gr2j −→ ΩC ⊗C F j−1 ∩ (W2j−2 )C = Ω1C ⊗ (2πi)j−1 ϕ−1 (3) (2πi)j ϕ−1 j j−1 gr2j−2 . By the functoriality and compatibility with the Tate twists, to construct that map for some j we may identify the space F j ∩ (W2j )C with the space F 1 H ∩ (W2 H )C , and F j−1 ∩ (W2j−2 )C with F 0 H ∩ (W0 H )C , where H = (W2j /W2j−4 )(j − 1). Then H can be identified with a Hodge substructure of a direct sum of logarithmic Hodge structures where we have fixed the connection. Let ξj be the image of ξj under the map d log ⊗id W W 1 W C× ⊗ Hom grW 2j (1), gr2j−2 −−−−−−→ ΩC ⊗ Hom gr2j (1), gr2j−2 . W One can rewrite formula (1) as ∇H (e) = (id ⊗ ϕj−1 )−1 (ξj (ϕj (e))) for any e ∈ (2πi)j ϕ−1 j (gr2j ). Here 1 j 1 W id ⊗ ϕj is the isomorphism ΩC ⊗C (F ∩ (W2j )C ) → ΩC ⊗ gr2j . One concludes from the existence of the decomposition (2) that the vectors e generate HC , when j varies. The statement on the integrability is clear from the above explicit formula for ∇H . ✷
Remark. – It is easy to see that the connection constructed in Proposition 3.1 on the tensor product of two Hodge–Tate structures coincides with the tensor product of the connections on these Hodge–Tate structures. P ROPOSITION 3.2. – Let Ω2C,log be the subgroup of Ω2C generated by a−1 da ∧ b−1 db for all a, b ∈ C. Then, in the category of the flat Hodge–Tate structures one has Ext2 (Q(0), Q(2)) = Ω2C,log . Proof. – By the standard argument, each class in Extm (Q(0), Q(m)) can be presented by an acyclic complex Q(m) → Bm → · · · → B1 → Q(0), where Bj has weights −2j and 2 − 2j. The existence of such presentations implies that Ext1 Q(0), Q(1) ⊗ Ext1 Q(1), Q(2) ⊗ · · · ⊗ Ext1 Q(m − 1), Q(m) → Extm Q(0), Q(m) is surjective. Note, that Ext1 (Q(j), Q(j + 1)) = C× ⊗ Q for any integer j. s ∪ In the case m = 2 this gives a surjection C× ⊗ C× −→ Ext2 (Q(0), Q(2)). Let j=1 aj ⊗ bj be an element in its kernel. Denote by Hb a Hodge–Tate structure with the class (b1 , . . . , bs ) ∈ (C× )sQ = Ext1 (Q(0), Q(1)s ). So Hb fits into an exact sequence 0 → Q(1)s → Hb → Q(0) → 0 which, after taking
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The Gauß–Manin connection on the Hodge–Tate structures
RHom from it to Q(2) in the category of flat Hodge–Tate structures, gives an exact sequence 0 → Ext1 Q(0), Q(2) → Ext1 Hb , Q(2) → Ext1 Q(1)s , Q(2) → Ext2 Q(0), Q(2) . It follows from the integrability criterion of Proposition 3.1 that in the category of flat Hodge–Tate structures one has
s dAj dbj Ext1 (Hb , Q(2)) × s , . . . , A ) ∈ (C ) ⊗ Q ∧ = 0 . = (A 1 s Aj bj Ext1 (Q(0), Q(2)) j=1 This implies that there is a natural embedding
s dAj dbj × s × s (C ) ∧ = 0 → Ext2 Q(0), Q(2) , (A1 , . . . , As ) ∈ (C ) Aj bj j=1 and thus s
daj j=1
aj
∧
dbj = 0, bj
so the kernel of ∪ coincides with the kernel of the map C× ⊗ C× → Ω2C given by a ⊗ b → implies that Ext2 (Q(0), Q(2)) = Ω2C,log . ✷
da a
∧
db b .
This
Acknowledgement. I would like to thank P. Deligne for pointing out non-integrability of the connection constructed in Proposition 3.1. I am grateful to Andrey Levin for inspiring discussions, to the I.H.E.S. for its hospitality and to the European Post-Doctoral Institute for its support.
References [1] Brylinski J.-L., Zucker S., An overview of recent advances in Hodge theory. Several complex variables, VI, in: Encyclopaedia Math. Sci., Vol. 69, Springer, Berlin, 1990, pp. 39–142. [2] Deligne P., Théorie de Hodge II, III, Inst. Hautes Études Sci. Publ. Math. 40 (1972) 5–58, 44 (1974) 3–77. [3] Katz N., Nilpotent connections and themonodromy theorem, Applications of a result of Turritin, Inst. Hautes Études Sci. Publ. Math. 39 (1971) 175–232.
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