We improve the known Hodge type bound for the exotic cohomology of complete
intersections. In the revised version, we included a simplification of our
original argument due to Pierre Deligne. The note appears in the C. R. de
l'Aca. des Sc. de Paris.Comment: 7 page

Esnault, Hélène

Wan, Daqing

English

arXiv

Hélène Esnault

Daqing Wan

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 336 (2003) 153–157Algebraic GeometryHodge type of the exotic cohomology of complete intersections✩Type de Hodge de la cohomologie exotique des intersectionscomplètesHélène Esnaulta, Daqing Wanba Mathematik, Universität Essen, FB 6, Mathematik, 45117 Essen, Germanyb Department of Mathematics, University of California, Irvine, CA 92697-3875, USAReceived 10 December 2002; accepted 16 December 2002Presented by Pierre DeligneAbstractIf X ⊂ Pn is a smooth complete intersection, its cohomology modulo the one ofPn is supported in middle dimension. If thecomplete intersection is singular, it might also carry exotic cohomology beyond the middle dimension. We show that for thisexotic cohomology, one can improve the known bound for the Hodge type of its de Rham cohomology.T cite this article:H. Esnault, D. Wan, C. R. Acad. Sci. Paris, Ser. I 336 (2003). 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. All rights reserved.RésuméSi X ⊂ Pn est une intersection complète lisse, sa cohomologie modulo celle dePn est supportée en dimension moitié. Sil’intersection complète est singulière, elle peut aussi avoir de la cohomologie exotique en dimension supérieure. Nous montronsqu’ on peut améliorer le type de Hodge de cette cohomologie de de Rham exotique.Pour citer cet article : H. Esnault, D. Wan,C. R. Acad. Sci. Paris, Ser. I 336 (2003). 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.Version française abrégéeSiX ⊂ Pn est une intersection complète lisse sur un corpsk, sa cohomologie modulo celle dePn est supportéeen dimension moitié. Si l’intersection complète est singulière, elle peut aussi avoir de la cohomologie exotique endimension supérieure. Si le degré deX vérified1  d2  · · ·  dr et si 0< κ  (n− d2 − · · · − dr)/d1, alors nousdisposons du théorème de Ax–Katz [7] pourk = Fq qui affirme que les zéros et pôles réciproques de la fonctionzêta dePn \X sont divisibles parqκ en tant que nombres algébriques. Son pendant en théorie de Hodge dit quele type de Hodge est alors κ [1,3,5,6]. Ces valeurs sont optimales. SiX n’est plus lisse, après normalisation✩ The first author is supported by the DFG-Schwerpunkt “Komplexe Mannigfaltigkeiten” while the second author is partially supported bythe NSF.E-mail addresses: esnault@uni-essen.de (H. Esnault), dwan@math.uci.edu (D. Wan).1631-073X/03/$ – see front matter 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.doi:10.1016/S1631-073X(03)00013-X154 H. Esnault, D. Wan / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 153–157convenable de la fonction zêta, on peut améliorer la divisibilité [8]. Dans cette Note, nous utilisons la méthode de[5] pour montrer l’amélioration correspondante du côté Hodge.1. IntroductionLetX ⊂ Pn be a complete intersection of multi-degreed1  d2  · · ·  dr  1 (1)over a fieldk of characteristic zero, where 1 r  n. It is well known thatHi(X)/H i(Pn)Hi+1c (Pn \X), for i  2(n− r),H ic(Pn \X)Hi(Pn), for i  2(n− r)+ 2. (2)Furthermore, one has the following vanishingHi(X)/H i(Pn)Hi+1c (Pn \X)= 0, for i < (n− r). (3)If X is also smooth, one has the further vanishingHi(X)/H i(Pn)Hi+1c (Pn \X)= 0, for 2(n− r) i > (n− r), (4)and so the cohomologyHi(X)/H i(Pn) is concentrated in the middle dimension= dim(X)= n− r in the smoothcase [1]. If the complete intersection is singular, it may have exotic cohomology fori > dim(X). That is, someof the groupsHi+1c (Pn \X)=Hi(X)/H i(Pn) might not vanish for somei with dim(X)= n− r < i  2(n− r).The Hodge type ofHi+1c (Pn \X), that is the largest non-negative integerµ for which the Hodge filtration fulfillsFµH i+1c (Pn \X)=Hi+1c (Pn \X), is closely related to the first slope of the zeta function ofPn \X over a finitefield.Let κ = max{0, [(n− d2 − · · · − dr)/d1]}, where[x] denotes the integer part ofx. If k = Fq is the finite fieldwith q elements of characteristicp, the theorem of Ax and Katz [7] says∣∣(Pn \X)(Fq)∣∣≡ 0 modqκ . (5)The zeta functionZ(Pn \X,T )= exp( ∞∑m=1|(Pn \X)(Fqm)|mTm)(6)is a rational function overQ, by Dwork’s rationality theorem [4]. We factor the zeta function over the algebraicclosure of thep-adic rational number fieldQp:Z(Pn \X,T )(−1)n−r = ∏ai=1(1− αiT )∏bi=1(1− βiT ), (7)where theαi and theβj are the reciprocal zeros and poles, respectively, ofZ(Pn \ X,T )(−1)n−r . The Ax–Katztheorem is equivalent to the following lower bound for the first slope of the zeta function:ordq(αi) κ, ordq(βj ) κ. (8)This result has a Hodge theoretical analogue. Ifk has characteristic 0, thenX has Hodge typeHTHT(Hic(Pn \X)) κ for all i. (9)(See [1] for the smooth case, [3] for the hypersurface case, [5] for the complete intersection case and [6] for thegeneral case.)H. Esnault, D. Wan / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 153–157 155The above bounds are sharp [7,5] among all theX defined byr equations of degreesd1  d2  · · ·  dr .However, it is proven in [8], Theorem 1.3, that ifX is a complete intersection, then the above slope boundcan be improved for theβj ’s. More precisely, ifκ1 = max{0, [(n− 1 − d2 − · · · − dr)/d1]}, then for completeintersectionsX, one hasordq(βj ) κ1 + 1. (10)If n− d2 − · · ·− dr > 0 andd1 divides(n− d2 − · · ·− dr), then this boundκ1 + 1= κ does not improve the one byAx and Katz. However, if eithern− d2 − · · · − dr  0 or d1 does not divide(n− d2 − · · · − dr), then one obtainsthe strict improvement:ordq(βj ) κ1 + 1 = κ + 1. (11)The purpose of this note is to show the corresponding Hodge theoretical improvement, which then concernsonly the exotic cohomology. One hasTheorem 1.1. Let X ⊂ Pn be a complete intersection of multi-degree d1  · · ·  dr  1. Letκ1 = max{0,[n− 1− d2 − · · · − drd1]}.Then for i  1, one hasHT(Hn+1−r+ic(Pn \X)) κ1 + 1. (12)Note that if i < 0, the cohomologyHn+1−r+ic (Pn \ X) vanishes. Ifi = 0, Ax–Katz’ bound (9) cannot beimproved in general as noted above. IfX is not assumed to be a complete intersection, for example think ofXbeing defined byr times the same equation, then the theorem, obviously, cannot be true. Remark 2.2 shows thatour bound on the Hodge type cannot be improved toκ1 + 2.Finally let us observe that, in the singular case, we do not know whether (5) implies that the eigenvalues ofFrobenius acting on-adic cohomology are divisible byqκ , due to possible cancellations in the representation ofthe zeta function as a rational function. While we know that the Hodge type is at leastκ for all cohomology groups,that is for all i in (9). Theorem 1.3 of [8] concerns only the slope improvement for the reciprocal poles, whileTheorem 1.1 concerns all cohomology groups beyond the middle dimension, whether of odd or of even dimension.We shall use the method of [5], Proposition 1.2, to reduce the theorem to a vanishing theorem which is a moreprecise version than the one shown in loc. cit, Proposition 2.1.2. The proof of the theoremLet I denote the ideal sheaf( 1, . . . , fr ) in Pn. We know by [5], Proposition 1.2, that it is enough to showHn+1−r+i(Pn,Iκ1+1 → Iκ1 ⊗Ω1 → ·· · → I ⊗Ωκ1)= 0, i  1. (13)This is of course implied by the vanishingProposition 2.1. Let (X,di, κ1, n, i) be as in Theorem 1.1. Then one hasHn+1−r+i−s(Pn,Iκ1+1−s ⊗Ωs)= 0, 0 s  κ1. (14)Proof. We give here a simplification of our original proof due to Pierre Deligne.By the standard resolution ofΩsPn0→ΩsPn→⊕nsOPn (−s)→⊕ns−1OPn(− (s − 1))→ ·· · →⊕n0OPn → 0 (15)156 H. Esnault, D. Wan / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 153–157for some positive integersni , we are reduced to showingHn+1−r+i−s−a(Pn,Iκ1+1−s(− (s − a)))= 0, 0  a  s  κ1. (16)For s = 0, this reduces toHn+1−r+i(Pn,Iκ1+1)= 0, i  1. (17)The exact sequence0→ Iκ1+1 →OPn →OPn/Iκ1+1 → 0 (18)induces the exact sequenceHn−r+i(Pn,OPn/Iκ1+1)→Hn+1−r+i(Pn,Iκ1+1)→Hn+1−r+i(Pn,OPn). (19)Sincen − r + i > dim(X), the term on the left vanishes by cohomological reasons, while the term on the rightvanishes sincen+ 1− r + i  1. This shows Proposition 2.1 fors = 0.Assume now thats  1. In particularκ1  1 and thusκ1d1  n− 1− d2 − · · ·− dr . For 0 a  s  κ1, one has(κ1 + 1− s)d1 + (s − a)  (κ1 + 1− s)d1 + s = κ1d1 − (s − 1)(d1 − 1)+ 1 κ1d1 + 1 n− d2 − · · · − dr . (20)Thus Proposition 2.1 of [5] allows us to concludeHm(Pn,Iκ1+1−s(− (s − a)))= 0, 0  a  s  κ1, m 0. (21)This finishes the proof. ✷Remark 2.2. If X is the union of two planesP2 ⊂ P3, thenκ1 = 1< nd = 32, andX is a normal crossing divisorto which one can apply directly [2]. One hasHic (P3 \X)= Hi (P3,Ω•(logX)(−X)). HoweverΩ2(logX)(−X)=O(−3)⊕O(−3)⊕O(−4). Since the Hodge to de Rham spectral sequence degenerates, one obtainsH 5c (P3 \X)=H 3(O(−4)) = 0,κ1+1= 2, and the Hodge type ofH 5c (P3\X) is 2 but not 3, which can also be checked by writingthatP3 \X is isomorphic toGm × A2. In contrast to this, ifκ1 = 0< nd = 33, andX is the union of three planesP2 ⊂ P3 in general position, one hasΩ1(logX)(−X)=O(−3)⊕O(−3)⊕O(−4), thus by the degeneration of theHodge to de Rham spectral sequence again, one seesH 4c (P3 \X)= H4(P3,Ω•(logX)(−X))=H 3(O(−4)) = 0.Thus the Hodge type ofH 4c (P3 \X) is 1 but not 2, which can also be checked by writing thatP3 \X is isomorphicto (Gm)2 × A1.We conclude the note by a remark which shows the pattern of Theorem 1.1.Theorem 2.3. Let X ⊂ Pn be a complete intersection of multi-degree d1  · · ·  dr  1. Let  be an integer suchthat 0   n+ 1− r . Letκ = max{0,[n− − d2 − · · · − drd1]}.Then for i  , one hasHT(Hn+1−r+ic(Pn \X)) κ + . (22)Proof. For i  n− r + 1, we haven+ 1− r + i  2(n− r)+ 2, thusHn+1−r+ic(Pn \X)=Hn+1−r+i(Pn). (23)H. Esnault, D. Wan / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 153–157 157This is 0 ifn+ 1− r + i is odd. Otherwise, it has Hodge type(n+ 1− r + i)/2. The last number is at least sincei   andn+ 1− r  . Thus, we may assume thatκ > 0, in which caseκ +  n− d2 − · · · − dr andn+ 1− r + i2 n+ 1− r  n− d2 − · · · − dr  κ + . (24)It remains to consider the casei  n− r. The proof goes as before. By loc. cit. it suffices to showHn−r+1+i−s(Pn,Iκ+−s ⊗Ωs)= 0, s < (κ + ). (25)Considering the exact sequence0→ Iκ+−s ⊗Ωs →Ωs →Ωs/Iκ+−s ⊗Ωs → 0 (26)one obtainsHn−r+1+i−s(Pn,Iκ+−s ⊗Ωs)= 0, for i > s, n− r + 1+ i − s = s. (27)On the other hand, ifn− r + 1− i = 2s ands  i− 1, one hasn− r + 1− i  2(i− 1) that isn− r + 3 i whichcontradictsi  n− r.Let now s  i  . Sinces < κ + , we must haveκ > 0. By the standard resolution (15) ofΩs , we arereduced to showingHn+1−r+i−s−a(Pn,Iκ+−s − (s − a))= 0, 0 a  s  κ. (28)One checks that fors   and 0 a  s  κ,(κ + − s)d1 + d2 + · · · + dr + s − a = κd1 + d2 + · · · + dr − (s − )(d1 − 1)+ (− a) κd1 + d2 + · · · + dr +  n. (29)The last inequality holds becauseκ > 0. The desired vanishing follows from Proposition 2.1 of [5]. This finishesthe proof. ✷Remark 2.4. The case= 0 of Theorem 2.3 is the main result of [5]. The case= 1 of Theorem 2.3 reduces toTheorem 1.1 in the above.AcknowledgementsWe thank the Morningside Center of Mathematics in Beijing for its hospitality. We thank Pierre Deligne for hisprecise and kind comments. He proposed a simplification of our original proof, which made transparent the proofof the generalization (15) of Theorem 1.1. Since our proof would have been too combinatorial, we had not includedTheorem 2.3 in our original manuscript.References[1] P. Deligne, Cohomologie des intersections complètes, in: SGA 7 XI, in: Lect. Notes in Math., Vol. 340, Springer, Berlin, 1973, pp. 39–61.[2] P. Deligne, Théorie de Hodge II, Publ. Math. IHES 40 (1972) 5–57.[3] P. Deligne, A. Dimca, Filtrations de Hodge et par l’ordre du pôle pour les hypersurfaces singulières, Ann. Sci. École Norm. Sup. (4) 23(1990) 645–656.[4] B. Dwork, On the rationality of the zeta function of an algebraic variety, Amer. J. Math. 82 (1960) 631–648.[5] H. Esnault, Hodge type of subvarieties ofPn of small degrees, Math. Ann. 288 (3) (1990) 549–551.[6] H. Esnault, M. Nori, V. Srinivas, Hodge type of projective varieties of low degree, Math. Ann. 293 (1) (1992) 1–6.[7] N. Katz, On a theorem of Ax, Amer. J. Math. 93 (1971) 485–499.[8] D. Wan, Poles of zeta functions of complete intersections, Chinese Ann. Math. 21B (2) (2000) 187–200.

Comptes Rendus Mathematique

Hodge type of the exotic cohomology of complete intersections

Numérisation de Documents Anciens Mathématiques

If X ⊂ ℙn is a smooth complete intersection, its cohomology modulo the one of ℙn is supported in middle dimension. If the complete intersection is singular, it might also carry exotic cohomology beyond the middle dimension. We show that for this exotic cohomology, one can improve the known bound for the Hodge type of its de Rham cohomology. © 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. All rights reserved

Hodge type of the exotic cohomology of complete intersections

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