C. R. Acad. Sci. Paris, t. 327, SCrie I, p. 189-192, 1998 GCom6trie alg6briquelAlgebraic Geometry

Effective bounds for Nori’s connectivity theorem

Jan NAGEL

Universitiit GH Essen, FB 6 Mathematik, 45117 Essen, Allemagne E-mail: jan.nagel@uni-essen.de

(ReGu le 5 juin 1998, accept& le 15 juin 1998)

Abstract. We announce an effective version of Nori’s connectivity theorem. Using this theorem, we obtain effective versions of results of Green, Voisin and Mtiiler-Stach on the image of the Abel-Jacobi and regulator maps for very general complete intersections. 0 Academic des SciencesElsevier. Paris

Bornes eRectives pour le thkorkme de connectivitk de Nori

RCsumC. On annonce une version effective du the’okne de connectivitt de Nori. A l’aide de ce thkor2me on obtient des versions effectives des rtkdtats de Green, de Voisin et de Miller-Stach sur l’image de l’application d’Abel-Jacobi et de l’application rkgulateur pour les intersections complktes trks g&ne’rales. 0 Academic des SciencesJElsevier, Paris

Version francake abrkgke

Soit (Y, Qr( 1)) une variete polarisee complexe et lisse de dimension n + T + 1 et soit E = @FE’=, &(di) (/I& > 0, i = 0,. . . , r). On pose S = HI=‘=, PHO(Y, C?y(&)). Soit Xs c Y x S la famille universelle des intersections completes de multidegre (da, . . . , d,) dans Y, et soit g : T + S un morphisme lisse. On definit Xr = XS xs T, YT = Y x T.

TH~OR~ME 1 (Nori [12], Thm. 4). - Pour tout nombre entierpositif c 5 n, il existe un nombre entier positifN = N(Y, 13,(l)! c) tel que Hn+k(Yr, XT; Q) = 0 pour k 5 c si min(do, . . . : d,) > N.

Dans [13], K. Paranjape a demontre une version effective du theoreme de Nori en utilisant la regularit de Castelnuovo-Mumford (voir aussi [2] et [14]). Soit rni la regularit de 0:. On pose my = nrax{mi - i - 1 / 0 5 i 5 dimY}.

THBOR~ME 2 (Paranjape). - Avec les notations du thkorkme 1, on a :

NW OY(~)! c) I my + n + c + 1.

Note prCsentCe par Jean-Pierre DEMAILLY.

07644442/98/03270189 0 AcadCmie des Sciences/Elsevier. Paris 189

J. Nagel

Dans cette Note on impose une condition un peu plus forte sur le changement de base, qui suffit pour dkduire les conskquences gComCtriques du thkorkme de Nori ; on obtient des bornes sur les degrks qui sont plus prkises que celles du theorkme 2. Les details paraitront dans [ 1 I]. Soit U c $HO(Y. E) l’ouvert paramktrant les intersections compktes lisses. Pour un nombre entier positif c 5 71, on d&nit :

n+c b,= ~ [ 1 2

, b,.+ = b, - i. l<i<c-1.

THBORCME 3. - Soit T - Ii un changement de base lisse, soit c < n un nombre entier positif et soient 61: . . . , b, d&G comme ci-dessus. On suppose que do > . . > d,,. Si les conditions suivantes sont ve’rifie’es :

(C) d,. 2 7n1' + n + 2(c - b,.) + 2, (C,) CIzi d, + (b,. - c + 1:)dr > rrbl’ + dirnY + c - % pour 0 < i < min(c - 1: T),

alors Hn+k (I’T, XT; Q) = 0 pour tout nombre entier k 5 c.

Les conditions (C;) peuvent &tre remplackes par la condition : (D) (b, - c + r + l)d, > 7rl1, + n + c + r + 1.

La condition (D) Cquivaut 2 celle de Paranjape si 7’ = 0 et n - 1 5 c 5 71 ; dans les autres cas, elle est plus prkise. Si (Y$ Oy(1)) satisfait la condition :

H”(Y,Ri,(k)) = 0. pour tous les entiers i > 0, j > 0, k > 0, (2)

on peut omettre la condition (C). Soit J,“,,,(Y) c JJ’(Y) 1 a sous-vari& abklienne associCe 2 la sous-structure de Hodge maximale contenue dans F”-lH2p-l(Y) fl H2PW1(Y, Q), et soit

n : @V,Q(d) - Hg(X, Q(p))/i*J&,(Y) la projection.

TH~ORI~ME 4. - Soit (Y, 0>-(l)) une vari&e’ polarisPe et soient X C Y une intersection complkte de multidegre’ (do. . . , d,.) (do 2 . . . > d,,) dans Y, et % : X --+ Y l’inclusion. Soit c < 71 un nombre entier posit$ Si X est tr?s &e’rale et si (Y> 1317 (1)) ve’r$e les conditions (2) et (C,), 0 5 % < min(c - l,r), alors on a :

(i) l’image de l’application 7r o c1n.s : CH”(X), -+ H~(X;Q(p))/,~*.I~,,(Y) coincide avec l’image de l’application r 0 i* o cl~.l- : CH”(Y)Q - H, “‘(X, Q(p))/~*J&,,(Y) si 21, 5 7L + c - 1 ;

(ii) l’image de l’application rkgulateur cP,m : CHp(X, m)$ - Hg-“‘(X, Q(p)) est contenue dans l’image de ,i* : Hz-““(Y, Q(p)) --+ Hg-“’ (X,Q(p)) si 2p - m < 71 + c - 1.

Let (Y, O,-(l)) be a smooth polarized complex variety of dimension n. + T + I and set E = $i’l, O,(di) (di > 0, % = 0,. . . , r). Set 5’ = nl_O PH’(Y, &(d;)), and let XS c Ys = Y x S be the universal family of complete intersections of multidegree (do, . . , d,,) in Y. Let T - S be a smooth morphism. Define XT = Xs xs T, YT = Y x T. In [ 121, M. Nori proved the following result, which is a relative version of the classical Lefschetz hyperplane theorem:

THEOREM 1 (Nori) [12], Theorem. 4. - For every natural number c 2 n, there exists a natural numberN=N(Y,0~~(1),c)suchthatH”fk(Y~~X~~Q)=O~~~rallk<r~henrnin(do,...~tl,.)>N.

Remark. - Theorem 1 remains valid if one replaces 5’ by PH’(Y! E) or U = PH’(Y, E) \ a, the complement of the discriminant locus (see [6], Lecture 8).

The proof of Theorem 1 uses the mixed Hodge structure on the relative cohomology groups H7’+k(YT,X~; C) constructed by Deligne [4]. As the weight filtration satisfies Gr~HVL+IC(YT,XT) =O

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Effective bounds for Nori’s connectivity theorem

for all i < 7~ + F - 1, it follows that H”+k (YT, X,) = 0 if F”” Hnflc (Yr, X,) = 0 for some integer 0 5 bl, 5 [?I. I n is paper, Nori chooses bk = k for all k 5 c; our choice is h’

n+c bc= 2, [ I k-i = b, - i, 1 5 i 5 c - 1. (1)

K. Paranjape proved in [ 131 an effective version of Theorem 1 using Castelnuovo-Mumford regularity (see also [2] and [14]). Let rn, be the regularity of the vector bundle ai-, i.e., rni = min{k 1 Hj(Y,R&(Ic-j)) = 0 f or all .i > 0}, and set ml- = rnax{ 7~ - i - 1 1 0 5 i 5 dim Y }. For example, rnjlr = 0 if Y is a projective space and ml, = 1 if Y is a quadric.

THEOREM 2 (Paranjape). - With notation us in Theorem 1, we have:

N(Y, 0,.(l), c) < ml, + n + c + 1.

In this Note, we impose a somewhat stronger condition on the base change, which suffices to deduce the geometric consequences of Nori’s theorem; the degree bounds that we obtain are more precise than those of Theorem 2. The proof of this result is based on unpublished manuscripts of Green and Mtiller-Stach [8] and on a remark of Nori ([ 121, Remark 3.10). We use Koszul cohomology computations and Jacobi modules. Details will appear in [ 1 11.

THEOREM 3. - Let do > > d,,, U = PH’(Y, E) \ A be the complement of the discriminant locus and T --+ U be a smooth morphism. Let c < n be a natural number and bl, . . . ! b, be dejned as in (1). Zfi

(C) d,. 2 ml, + n + 2(c - b,) + 2, and (Ci) Cc=, 4 + (bc - c + i)d, 2 ml. + dimY + c - 1: for all 0 5 1. 5 min(c - 1, r.),

then H”+IC(YT, Xr) = 0 for all k 5 c.

Remarks

I. Theorem 3 has been proved independently by M. Asakura and S. Saito [1] for the case Y = lP71+p+1; their approach is similar to ours, but they use a different version of the Jacobi module.

2. Conditions (Co), . . . ,(C,.-1) can be replaced by the less precise condition:

(D) (b, - c + T + l)d,. 2 ml, + n + c + T + 1.

Condition (D) is equivalent to Paranjape’s bound if 7’ = 0 and 7~ - 1 < c < n; in the other cases it is more precise.

If (Y, c31~( 1)) is a polarized variety that satisfies

Hi(Y,O&(k)) = 0 for all i > 0, j > 0, k > 0, (2)

we can omit condition (C). As a corollary of Theorem 3, we obtain an effective version of a result of Green and Miiller-Stach that describes the image of the Deligne cycle class map for complete intersections (see [7]) and its subsequent generalization to higher Chow groups (see [9], Theorem. 6.2). Let J&,(Y) c Jp(Y) be the abelian subvariety associated to the maximal sub-Hodge structure contained in F”-1H2p-1(Y) n H ““-‘(Y> Q), and let 7r : Hz(X, Q(I))) - Hg(X, Q(p))/i*$,,(E’) be the projection map.

THEOREM 4. - Let (Y, C?J~( 1)) b e u smooth polarized variety of dimension n + r + 1, and let c 5 n,

be a natural number. Let X = V(do, . . : d,) n Y (do > . . . > d,) be a smooth complete intersection

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of dimension n in Y, with i : X ---+ Y the inclusion map. If X is general and if (Y, 0, (1)) satis$es conditions (2) and (Ci), 0 5 i 5 min(c - 1, T), then:

(i) the image of the map TT o cl~.~ : CHp(X)o -+ Hg(X, Q(p))/i*Jz,(Y) coincides with the image ofthe map T o i* o cln,. : CHp(Y)o - Hg(X, Q(p))/i*Jga,(Y) if2p 5 rr. + c - 1;

(ii) the image of the regulator map cP,,,, : CH’(X, m)o - Hz-“‘(X, Q(y)) is contained in the image of i* : Hz-““(Y! Q(p)) - Hz-“(X: Q(p)) if2p - m 5 71 + c - 1.

Examples

1. (Y,Oy(l)) = (P n+r+l c?p( 1)). In this case, condition (2) is satisfied by the Bott vanishing theorem. Taking c = ; in Theorem 4, we obtain the Noether-Lefschetz theorem (see [3]). For c = 2, we obtain a generalization of a theorem of Green-Voisin [5] on the image of the Abel-Jacobi map to the case of complete intersections (see also [IO]).

2. Let (Y?&(l)) b e a smooth polarized toric variety. Condition (2) is then satisfied by the Bott-Danilov-Steenbrink vanishing theorem.

3. Let (Y, 0~ (1)) be a polarized abelian variety. Condition (2) is then satisfied by the Kodaira vanishing theorem. In this case, condition (Ci) can be replaced by the weaker condition:

(C;) CL=, d, + (b, - c + i)d,. > c - i - 1.

This condition is empty if c 5 2.

If Y is a homogeneous space (e.g. a Grassmann variety or a quadric) one can sometimes compute reasonably precise degree bounds using the Bott vanishing theorem. For instance, let Y c P7 be a smooth quadric; explicit computations show that if X = V(do, dI) f? Y is very general and dl = min(da, dl) > 5, then the conclusion of Theorem 3 holds for c = 3. Using a theorem of Nori ([12], Theorem l), we find that the Griffiths group Griff”(X) 8 Q is nonzero if dI 2 5.

References

[I] Asakura M., Saito S., Filtration on Chow groups and generalized normal functions, Preprint, 1996. [2] Braun R., Mttller-Stach S., Effective bounds for Nori’s connectivity theorem, in: Higher dimensional complex varieties,

Proc. Int. Conf., Trento, Italy, Walter de Gruyter, 1996, pp. 83-88. [3] Deligne P., Le theoreme de Noether, Expose XIX dans: P. Deligne, N. Katz, Groupes de monodromie en GComCtrie

algebrique, SGA 711, Lect. Notes in Math. 340, Springer-Verlag, 1973. [4] Deligne P., Theorie de Hodge II, III, Publ. Math. Inst. Hautes Etudes Sci. 40 (1971) 5-57 et 44 (1974) 5-77. [5] Green M., Griffiths’ infinitesimal invariant and the Abel-Jacobi map, J. Differ. Geom. 29 (1989) 545-555. [6] Green M., Infinitesimal methods in Hodge theory, in: Algebraic cycles and Hodge theory, Lect. Notes in Math. 1594,

Springer-Verlag, 1994. [7] Green M., Milk-Stach S., Algebraic cycles on a general complete intersection of high multi-degree, Compos. Math. 100

(I 996) 305-309. [8] Green M., Mtiller-Stach S., Algebraic cycles on a general hypersurface section of high degree of a smooth projective

variety, non publie. [9] Mtiller-Stach S., Constructing indecomposable motivic cohomology classes on algebraic surfaces, J. Alg. Geom. 6 (1997)

513-543. [IO] Nagel J., The Abel-Jacobi map for complete intersections. Indag. Math. 8 (1997) 95-l 13. [I I] Nagel J., Effective bounds for Hodge-theoretic connectivity, Preprint. [ 121 Nori M.V., Algebraic cycles and Hodge-theoretic connectivity, Invent. Math. 111 (1993) 349-373. [13] Paranjape K., Cohomological and cycle-theoretic connectivity, Ann. Math. 140 (1994) 64-660. [14] Ravi M.S., An effective version of Nori’s theorem, Math. Z. 214 (1993) l-7.

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