A geometric application of Nori s connectivity theorem

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1 A geometric application o Nori s connectivity theorem arxiv:math/ v1 [math.ag] 27 Jun 2003 Claire Voisin Institut de mathématiques de Jussieu, CNRS,UMR Introduction Our purpose in this paper is to study rational maps rom varieties with small dimensional moduli space to general hypersuraces in projective space. In the last section, we shall eventually extend this to the study o correspondences instead o rational maps. Deinition 1 Let Y S be a amily o r-dimensional smooth projective varieties. We say that a d-dimensional variety X is rationally swept out by varieties parametrized by S, i there exist a quasiprojective variety B o dimension d r, a amily K B which is the pull-back o the amily Y via a morphism ψ : B S, and a dominant rational map φ : K X, (which is necessarily generically inite on the generic iber K b since dimk = dimx). Our main result in this paper concerns the problem o sweeping out general hypersuraces o degree N d+2 in P d+1 : Theorem 1 Fix an integer 1 r d. Let γ = r 1, r odd, or γ = r, r 2 2 even, that is γ is the round-up o r 1. Let Y S, dims = C, be a amily 2 o r-dimensional smooth projective varieties. Then the general hypersurace o degree N in P d+1 is not rationally swept out by varieties parameterized by S i (N +1)r 2d+C +2, (γ +1)N 2d r+1+c. (0.1) (Note that except or r = 1, the second inequality implies the irst.) Remark 1 One could o course prove a similar statement or suiciently ample hypersuraces in any smooth variety. In the case o projective space, the estimates on N are sharp, and allow applications to the Calabi-Yau case (see section 3). 1

2 The proo is Hodge theoretic. Unlike [4], [15], [7], [3], the result has nothing to do with the canonical bundle o the varieties Y t, t S. Instead, the key point is the act that the dimension o the moduli space S is small : i every X was rationally swept out by varieties parameterized by S, or ixed Y, there would be a generically inite rational map Y ŨY XŨY where XŨY is the pull-back via a morphism ρ : Ũ Y U o the universal hypersurace parameterized by U H 0 (O P d+1(n)), and Imρ is o codimension dim S. This will be shown to contradict Nori s connectivity theorem (Theorem 4). We shall apply this particularly to the case o Calabi-Yau hypersuraces. In the paper [11], Lang ormulates a number o conjectures concerning smooth projective complex varieties X. One o them is that the analytic closure o the union o the images o holomorphic maps rom C to X is equal to the union o the images o non constant rational maps rom an abelian variety to X. Another one is that this locus is equal to X itsel i and only i X is not o general type. Next, by a standard countability argument or Chow varieties, we see that, according to these conjectures, i X is not o general type, there should exist a quasiprojective varietyb, aamilyk B oabelianvarieties, andadominant rational map φ : K X, which is non constant on the generic iber K b, b B. Let us now consider the case where X is a Calabi-Yau variety, that is K X is trivial. We claim that i a map φ as above exists, then we may assume that φ Kb is generically inite, or generic b B. Indeed, because H 0 (X,K X ) 0, or generic b B, the image φ(k b ) has eective canonical bundle, in the sense that any desingularization o it has eective canonical bundle, as ollows rom adjunction ormula and the act that the φ(k b ) cover X. Now it is immediate to prove that any dominant rational map K b K b, where K b is an abelian variety and K b has eective canonical bundle, actors through the quotient map K b K b, where K b is an abelian variety, which is a quotient o K b, and has the same dimension as K b. Replacing the amily o abelian varieties (K b ) b B, by the amily (K b ) b B gives the desired φ. In other words, Lang s conjecture asserts in particular that a Calabi-Yau variety should be rationally swept out by r-dimensional abelian varieties, or some r 1. Our theorem 1 implies : Theorem 2 Let X be a general Calabi-Yau hypersurace in projective space P d+1, that is N = d+2. Then X is not rationally swept out by r-dimensional abelian varieties, or any r 2. 2

3 Hence, i Lang s conjecture is true, such an X should be swept out by elliptic curves. On the other hand, we also prove the ollowing Lemma 1 I X is a general Calabi-Yau hypersurace o dimension 2, X is not rationally swept out by elliptic curves o ixed modulus. By rationally swept out by elliptic curves o ixed modulus, we mean that the elliptic curves in the amily K B o deinition 1 have constant modulus. Hence, combining theorem 2 with the above lemma, we get the ollowing corollary, which was pointed out to us by J. Harris : Corollary 1 I Lang s conjecture is true, any Calabi-Yau hypersurace X o dimension 2 has a divisor which is uniruled. In dimension 3, this shows that Lang s conjecture and Clemens conjecture on the initeness o rational curves o ixed degree in a general quintic threeold, contradict. In the case o hypersuraces o general type, inequality (0.1) can be applied to give a non trivial estimate on the minimal genus o covering amilies o curves, but the estimate is not sharp and could be obtained directly by geometry. What is interesting however is that looking more precisely at the proo o Theorem 1, we shall see that the result concerns in act only the Hodge structure on H d (X) prim and not the eective geometry o X. In act we get as well : Theorem 3 Let X be a general hypersurace o degree N 2d 2+3g, g 2 or N 2d + 2, g = 1, in P d+1. Then there exists no non-zero morphism o Hodge structure H d (X,Q) prim H d (Y,Q), where Y is rationally swept out by curves o genus g. Combining this statement with the generalization o Mumords theorem [12], this implies in particular that or N 2d + 2, X general, there exists no correspondence Γ CH d (Y X) inducing a surjective map Γ : CH 0 (Y) 0 CH 0 (X) 0, where Y admits an elliptic ibration. Similarly, i g 2 and N 2d + 2, there exists no such correspondence Γ CH d (Y X) where Y admits a ibration whose generic iber is a genus g curve. One may wonder whether these statement are true or any such hypersurace or only or the general one. The paper is organized as ollows : in section 1, we recall briely the proo o Nori s connectivity theorem or hypersuraces in projective space, in order to extend it to amilies o hypersuraces parameterized by subvarieties o the moduli space which are o small codimension. This will show us that or any amily o hypersuraces parameterized by a subvariety o the moduli space 3

4 which is o small codimension, the Hodge level o the cohomology groups o the total space o the amily is small. The next section is devoted to the proo (by contradiction) o theorem 1. In section 3, we prove the applications o this result described above. Acknowledgements. I would like to thank J. Harris who started me thinking to these problems and to Herb Clemens or very interesting exchanges. This work has been essentially done at the University La Sapienza, and I would like to thanks the organizers o the trimester Moduli spaces, Lie Theory, interactions with Physics, or the excellent working conditions and atmosphere I ound there. 1 Nori s connectivity theorem or hypersuraces In this section, we summarize the main points o the proo o Nori s connectivity theorem or hypersuraces, in order to prove theorem 4, which is the precision o it that we will need. in [1], [14], a sharper study o similar explicit bounds can be ound. We consider hypersuraces o degree N in P d+1, and we assume that N d + 2. Fix an integer r such that 1 r d, and let γ be the round-up o r 1. Denote by U 2 H0 (O P d+1(n)) the open set parametrizing smooth hypersuraces. Let ρ : M U be a morphism, where M is smooth quasiprojective. We assume that Corankρ is constant equal to C. We also assume or simplicity that Imρ is stable under the action o Gl(N). Let X U be the universal hypersurace parametrized by U and Let X M := X U U M. j : X M M P d+1 be the natural embedding. X M is a smooth quasi-projective variety, hence its cohomology groups carry mixed Hodge structures with associated Hodge iltration F i H k (X M,C). Theorem 4 i) Assume that Then, the restriction map is surjective. (N +1)r 2d+C +2. (1.2) j : F d H 2d r (M P d+1,c) F d H 2d r (X M,C) 4

5 ii) I then or any i 1, the restriction map is surjective. (γ +1)N 2d+1 r +C, (1.3) j : H 2d r i (M P d+1,c) H 2d r i (X M,C) Proo. i) One irst reduces i), (see [13]), to proving that under the assumption (1.2), the restriction map j : H l (Ω k M P d+1 ) H l (Ω k X M ) is bijective, or l d r, k + l 2d r. This step uses the mixed Hodge structure on relative cohomology and the Fröhlicher spectral sequence. Denote respectively by π X, π P the natural maps X M M, M P d+1 M. A Leray spectral sequence argument shows that it suices to prove that under the assumption (1.2) one has : The restriction map j : R l π P (Ω k M P d+1 ) R l π X (Ω k X M ) is bijective (1.4) or l d r, k +l 2d r. Let Hprim, d H p,q prim, p+q = d, be the Hodge bundles associated to the variation o Hodge structure on the primitive cohomologyotheamilyπ X : X M M. Theininitesimal variation o Hodge structure on the primitive cohomology o the ibers o π X is described by maps : H p,q prim Hp 1,q+1 prim Ω M, and they can be iterated to produce a complex :...H p+1,q 1 prim Ω s 1 M H p,q prim Ωs M H p 1,q+1 prim Ω s+1 M... (1.5) One can show, using the iltration o Ω k X M by the subbundles π X Ωs B Ωk s X M and the associated spectral sequence, that (1.4) is equivalent to the ollowing The sequence (1.5) is exact at the middle or q d r, p+s+q 2d r. Note that since p+q = d, the last inequality reduces to s d r. It is convenient to dualize (1.5) using Serre duality, which gives : H q+1,p 1 prim s+1 s t TM H q,p prim T s t s 1 M H q 1,p+1 prim T s+1 M. (1.6) 5

6 We inally use Griiths, Griiths-Carlson description o the IVHS o hypersuraces ([10], [2]) to describe the complex (1.6) at the point M as ollows. We have the map ρ : T M, T U, = S N, where S is the polynomial ring in d + 2 variables. Next the residue map provides isomorphisms R d 2+N(p+1) = H q,p prim (X ), where R := S/J is the Jacobian ideal o, and R k denotes its degree k component. The map identiies then, up to a coeicient, to the map given by multiplication R d 2+N(p+1) Hom(T M,,R d 2+N(p+2) ). It ollows rom this that the sequence (1.6) identiies to the ollowing piece o the Koszul complex o the Jacobian ring R with respect to the action o T M, on it by multiplication: R d 2+Np s+1 δ TM, R d 2+N(p+1) s TM, s 1 δ R d 2+N(p+2) TM,.(1.7) Now, byassumption, iw istheimageoρ, W S N isabase-pointreelinear system, because it contains the jacobian ideal J N, and it satisies codimw = C. One veriies that it suices to check exactness at the middle o the exact sequences (1.7) in the considered range, with T M, replaced with W. This last act is then a consequence o the ollowing theorem due to M. Green : Theorem 5 [9] Let W S N be a base-point ree linear system. Then the ollowing sequence, where the dierentials are the Koszul dierentials S d 2+Np s+1 δ W S d 2+N(p+1) s s 1 δ W S d 2+N(p+2) W (1.8) is exact or d 2+Np s+codimw. Using the act that the Jacobian ideal is generated by a regular sequence in degree N 1, one then shows that the same is true when S i is replaced with R i in (1.8), at least i d 2+N(p+1) N 1. We now conclude the proo o i). We have just proved that (1.5) is exact at the middle i d 2+Np s+c, d 2+N(p+1) N 1. Since we assumed N d + 2, the second inequality is satisied when p 1. Next, i q d r, s d r, we have p r 1, s d r. 6

7 Hence, the exactness o (1.5) in the range q d r, s d r will ollow rom the inequality d 2+Nr d r +C, that is (1.2). ii) The proo is exactly similar, and we just sketch it in order to see where the numerical assumption is used. We irst observe that by a mixed Hodge structure argument (c [13]), it suices, in order to get the surjectivity o the restriction map : j : H 2d r i (M P d+1,c) H 2d r i (X M,C), to show the surjectivity o the restriction map : j : F d r+γ i H 2d r i (M P d+1,c) F d r+γ i H 2d r i (X M,C), where γ i is the round-up o r i 2d r i. (This is because the round-up o is 2 2 d r +γ i.) We reduce then this last act to showing : The restriction map j : R l π P (Ω k M P d+1 ) R l π X (Ω k X M ) is bijective (1.9) or l d i γ i, k +l 2d r i. Expressing the cohomology groups above with the help o the IVHS on the primitive cohomology o the ibers o π X, this is reduced to proving : The sequence (1.5) is exact at the middle or q d i γ i, p + q = d, p+s+q 2d r i. Using the Carlson-Griiths theory, we are now reduced to prove : the ollowing sequence : R d 2+Np s+1 δ TM, R d 2+N(p+1) s TM, s 1 δ R d 2+N(p+2) TM, (1.10). is exact or p γ i +i, s d r i. As in the previous proo, we now apply the theorem 5 and conclude that the last statement is true i d 2+N(γ i +i) C +d r i. (1.11) Now it is clear that the γ i +i are increasing with i, while the C+d r i are decreasing with i. Hence it suices to have (1.11) satisied or i = 1, which is exactly inequality (1.3). 7

8 Denoting by H 2d r (X M ) prim the quotient H 2d r (X M ) prim /j (H 2d r (M P d+1 )), we shall only be interested with the pure part W 2d r H 2d r (X M ) prim, which is the part o the cohomology which comes rom any smooth projective compactiication o X M. It carries a pure Hodge structure o weight 2d r. Corollary 2 Under the assumptions o theorem 4, the Hodge structure on W 2d r H 2d r (X M ) prim is o Hodge level r 2. Proo. Recall that the Hodge level o a Hodge structure H, H C = H p,q is Max{p q, H p,q 0}. Since we know that F d W 2d r H 2d r (X M ) prim = 0, we have H p,q (W 2d r H 2d r (X M ) prim ) = 0, or p d. Since H p,q = 0 or p + q 2d r, it ollows that the Hodge level is d 1 (2d r (d 1)) = r 2. 2 Proo o theorem 1 We prove theorem 1 by contradiction. Using Chow varieties, or relative Hilbert schemes, we see that there exist countably many quasi-projective varieties B parameterizing triples (t,,φ s ), where t S, U, and φ s is a rational map φ s : Y s X which is generically inite onto its image. For ixed generic, our assumption is that the images o such φ s ill-in X, and a countability argument then shows that there exists one B, which dominates U via the second projection, and which is such that the universal rational map is dominating, where Φ : K X U π : K B is thepull-back via the irst projection Ψ : B S othe amilyy S, and, as in the previous section, X U is the universal hypersurace parameterized by U. We shall denote by B the (generic) iber o the second projection q : B U and π : K B the induced amily. By taking desingularizations, we may assume that B hence K are smooth, and by assumption the map π is smooth. Since is generic, B and K are then also smooth. Finally, we may, up to replacing B by a closed subvariety, assume that the restriction φ : K X o Φ to K is generically inite and dominating. In particular, dimb = d r. 8

9 Now we make the ollowing construction : denote by B the space B S B. Restricting tozariskiopensets ob andb, wemayassumethatb issmooth. The generic point o B parameterizes, via the second projection p : B B, a variety Y t together with a rational map φ t, : Y t X, and, via the second projection, a rational map φ t, : Y t X. Let ρ : B U be the composition o the irst projection and the map q : B U and let m : B S be the natural map. We shall also use the notation K = K B B = t B Y m(t), and X := X U B = t B X ρ(t). The map Φ induces a rational map Φ : K X which is compatible with the maps K B and X B. It ollows that the graph Γ o Φ is contained in Y := K B X = K B X and is o codimension d in Y. Note that Y contains K X and that Γ K X is nothing but the graph o φ. Now, the class o this last graph in H 2d (K X,Q) does not vanish in H 2d (K X,Q)/H 2d (K P d+1,q). Indeed, its Künneth component φ in Hom(H d (X,Q) prim,h d (K,Q)) doesnotvanish, becausen d+2,sothatthetranscendentpartoh d (X,Q) prim, that is the orthogonal o all sub-hodge structures which are o level < d, is non-zero, so that it cannot be annihilated by φ, because φ is dominating. It ollows that the class γ o Γ in H 2d (Y,Q) is non zero modulo H 2d (K P d+1,q). Recall next that with the help o a polarization, that is a choice o a relatively ample line bundle on Y π X, the cohomology H 2d (Y,Q) splits into a direct sum H 2d (Y,Q) = H 2d l (X,R l π Q). l It is easy to check by similar reasons as above that the component γ r o γ in H 2d r (X,R r π Q)/H 2d r (B P d+1,r r π Q), 9

10 where in the second term π is the natural map K P d+1 B P d+1, is non zero. We may inally or the same reason reine this, replacing R r π Q by its quotient R r π Q tr, that is its quotient by the maximal sub-hodge structure which exists generically on B and is not o maximal Hodge level r. Note that γ, γ r and γ r,tr are Hodge classes, that is belong to the F d W 2d -level o the considered cohomology groups, which all have mixed Hodge structures. Next we denote by g : B B, g := g π X : X B, the natural maps. We observe that shrinking B i necessary, the ibers B,t o g are smooth and the map ρ B,t has constant corank C = dims, because ρ : B U is submersive near B, and the ibers o g identiy to the ibers o B S. Hence we may apply theorem 4 and its corollary. It says that under the assumtions 0.1, the ibers X,t o g satisy H 2d r i (X,t,Q) = 0, i 1. On the other hand, we observe that because π : Y X is a iber product K B X, the local system R r π Q is a pull-back rom B : The vanishing above implies then that R r π Q = g 1 (R r π Q). (R 2d r i g (R r π Q tr )) prim = 0, i 1, where prim here denotes the quotient by the ambiant cohomology (while beore it was used to mean the orthogonal to ambiant cohomology). It ollows by Leray spectral sequence that the class γ r,tr does not vanish along the ibers o g, that is in H 0 (R 2d r g (R r π Q tr ) and by restriction to the general iber, it does not vanish in H r (Y t,q) tr H 2d r (X,t,Q)/H 2d r (B,t P d+1,q). Note that it is a Hodge class in H r (Y t,q) tr ) H 2d r (X,t,Q)/H 2d r (B,t P d+1,q)) == Hom(H r (Y t,q) tr,h 2d r (X,t,Q)/H 2d r (B,t P d+1,q)). Since the mixed Hodge structure on the let is pure, this implies that it belongs to Hom MHS (H r (Y t,q) tr,h 2d r (X,t,Q)/H 2d r (B,t P d+1,q)) = HomHS (H r (Y t,q) tr,w 2d r H 2d r (X,t,Q)/H 2d r (B,t P d+1,q)), where MHS and HS mean morphisms o mixed (respectively pure) Hodge structures. Since by deinition H r (Y t,q) tr has no quotient Hodge structure which is o Hodge level < r, we get now a contradiction with corollary 2, which says that the Hodge structure on W 2d r H 2d r (X,t,Q)/H 2d r (B,t P d+1,q) has Hodge level < r. 10

11 3 Rational maps rom abelian varieties to Calabi- Yau hypersuraces and other applications Let us apply theorem 1 to the case o Calabi-Yau hypersuraces, that is hypersuraces o degree N = d+2 in P d+1. The moduli space o r-dimensional abelian varieties with given polarization type is o dimension r(r+1). Hence the 2 conditions o theorem 1 become : (d+3)r 2d+ r(r +1) 2 +2, (γ +1)(d+2) 2d r +1+ r(r+1) 2. It is not hard to check that this is satisied or 2 r d. Hence we get in this case theorem 2. When r = 1, the inequality (0.1) is never satisied so that our argument deinitely does not apply to the study o elliptic curves in Calabi-Yau hypersuraces. In act we could adapt our proo o theorem 2 to work as well or Calabi-Yau hypersuraces in a product o projective spaces. On the other hand, certain generic Calabi-Yau hypersuraces in a product o projective spaces are swept out by elliptic curves, eg the hypersurace o bidegree (3,3) in P 2 P 2. This shows that or r = 1, a dierent argument has to be ound. We can however prove the ollowing : Lemma 2 I X is a Calabi-Yau hypersurace o dimension 2, X is not rationally swept out by elliptic curves o ixed modulus. Proo. Indeed, ixing otherwise the modulus o the elliptic curve, we would get, or at least one elliptic curve E, an hypersurace M E in the moduli space M o X consisting o X s which are rationally dominated by some E B. For such an X, there must be an inclusion o rational Hodge structures induced by the dominant rational map φ : E B X : φ : H d (X ) prim H 1 (E) H d 1 (B), (3.12) because the Hodge structure on H d (X) prim is simple. I we now let vary in M E, only B deorms with, not E, and it ollows that the ininitesimal variation o Hodge structure on H 1 (E) H d 1 (B) : H p,q (H 1 (E) H d 1 (B)) H p 1,q+1 (H 1 (E) H d 1 (B)) Ω ME has the ollowing orm at the point M E : (α β) = α B (β) or α H r,s (E), β H p r,q s (B), where B is the ininitesimal variation o Hodge structure on H d 1 (B). Hence the Yukawa couplings o the IVHS on H 1 (E) H d 1 (B), that is the iterations o, have the ollowing property: 11

12 η H d,0 (H 1 (E) H d 1 (B)), the map vanishes. d (η) : S d T ME, H 0,d (H 1 (E) H d 1 (B)) I there is along M E an injective morphism o Hodge structure (3.12), it ollows that the same property is true or the IVHS o the amily o X s parameterized by M E, namely the Yukawa couplings o X vanish on the hyperplane K := T ME, S d+2. The Carlson-Griiths theory [10], [2] shows easily that this is not the case. Indeed, these Yukawa couplings identiy to the multiplication map S d (S d+2 ) R d(d+2). Assume they vanish on K. Since K is a hyperplane in S d+2, the subspace K := [K : S 1 ] S d+1 has codimension d + 2. It is without base-point, since T ME, contains J. It ollows then rom [8], that S d+3 K = S 2d+4. But K 2 contains S 1 K K = K S 1 K = W S d+3 = S 2d+4. Hence K 2 = S 2d+4 and similarly K d = S d(d+2) contradicting the act that K d J d(d+2). Remark 2 In the case o odd dimensional varieties, one can also use the Mumord-Tate group argument due to Deligne ([6], p 224) to get this result. Corollary 3 I Lang s conjecture is true, any Calabi-Yau hypersurace X o dimension 2 has a divisor which is uniruled. Proo. Indeed, we know that X is rationally swept out by elliptic curves, but not by elliptic curves with constant modulus. Hence there is a diagram K φ X π B, where we may assume that K and B are smooth, projective, where K is a smooth projective model o the amily K B on which φ is deined, and that the map j : B P 1 is deined, Now, since φ is generically inite, or generic t P 1, the divisor K t := (j π) 1 (t) must be sent by φ onto a divisor o X, and it ollows that or any t the image by φ o the divisor K t := (j π) 1 (t) must contain a divisor o X. Taking t =, and noting that any component o K is uniruled, gives the result. 12

13 Finally, we turn to Theorem 3. We simply note or this that in the proo o Theorem 1, we used the dominating rational map φ : K X only to deduce that there is a corresponding inclusion φ : Hd (X ) prim H d (K ) o Hodge structures. The Chow argument we used would work equally with graphs o rational maps replaced with cycles in the product Y t X. Hence we conclude that everywhere in the paper, we could replace dominating rational maps by cycles in CH d (K X) inducing a non-zero morphism o Hodge structure H d (X) prim H d (K). (Note that such a non-zero morphism should be in act injective because the Hodge structure on H d (X) prim is simple or generic X.) Using the act that the moduli space o curves o genus g 1 has dimension 3g 3, g 2 or 1, g = 1, we see that theorem 3 is then a consequence o theorem 1, where we replace dominating rational maps with correspondences inducing a non zero morphism o Hodge structure. Reerences [1] M. Asakura, S. Saito. Noether-Leschetz locus or Beilinson-Hodge cycles on open complete intersections, preprint [2] J. Carlson, Ph. Griiths. Ininitesimal variations o Hodge structures and the global Torelli problem, in Journées de géométrie algébrique, (A. Beauville Ed.), Sijtho-Nordho, (1980), [3] L. Chiantini, A.-F. Lopez, Z. Ran. Subvarieties o generic hypersuraces in any variety, Math. Proc. Cambr. Phil. Soc [4] H. Clemens. Curves in generic hypersuraces, Ann. Sci. École Norm. Sup. 19 (1986), [5] H. Clemens, Z. Ran. Twisted genus bounds or subvarieties o generic hypersuraces, to appear Amer. J. Math. [6] P. Deligne. La conjecture de Weil pour les suraces K3, Inventiones Math. 15, (1972). 13

14 [7] L. Ein. Subvarieties o generic complete intersections, Invent. Math. 94 (1988), [8] M. Green. Restrictions o linear series to hyperplanes, and some results o Macaulay and Gotzmann, in Algebraic curves and projective geometry, (E. Ballico, C. Ciliberto Eds), Lecture Notes in Mathematics 1389, Springer- Verlag 1989, [9] M. Green. A new proo o the explicit Noether-Leschetz theorem, J. Di. geom. [10] Ph. Griiths. Periods o certain rational integrals, I, II, Ann. o Math. 90 (1969) [11] S. Lang. Hyperbolic and diophantine Analysis, Bulletin o the American Math. Soc. Vol. 14, 2, (1986), [12] D. Mumord. Rational equivalence o zero-cycles on suraces, J. Math. Kyoto Univ. 9 (1968), [13] M. Nori. Algebraic cycles and Hodge theoretic connectivity, Invent. Math. 111 (1993) [14] A. Otwinowska. Asymptotic bounds or Nori s connectivity theorem, preprint [15] C. Voisin. On a conjecture o Clemens on rational curves on hypersuraces, Journal o Dierential Geometry 44 (1996) ( + correction J. Dierential Geometry 49 (1998),