ASYMPTOTIC BOUNDS FOR NORI’S CONNECTIVITY THEOREM

ANIA OTWINOWSKA

Abstract. Let Y be a smooth complex projective variety. I study the cohomology of smooth families of hypersurfaces X → B for B ⊂ ℙH^{0}(Y, O(d)) a codimension c subvariety. I give an asymptotically optimal bound on c and k when d → ∞ for the space H^{k}(Y × B, X , ℚ) to vanish, thus extending the validity of Lefschetz Hyperplane section Theorem and Nori’s Connectivity Theorem [No]. Next, I construct in the limit case explicit families of higher Chow groups whose class does not vanish in H^{k}(Y × B, X , ℚ). Some of them are indecomposable. This suggests that in the limit case the space H^{k}(Y × B, X , ℚ) should be spanned by higher Chow groups, containing [No] and [O] as special cases (cf. conjecture 2).

0. Introduction

0.1. Nori’s connectivity theorem. Let Y be a smooth complex proper algebraic variety of dimension N +1 with an ample invertible sheaf L and π: X → ℙH^{0}(Y, L) the universal family of hypersurfaces of class c_{1}(L). For any b ∈ ℙH^{0}(Y, L), let X_{b} = π^{−1}(b) be the corresponding hypersurface of Y (throughout the paper, the notation ℙV = (V \ {0}) /ℂ^{∗} denotes the projectivisation of a vector space V).

The Lefschetz hyperplane section theorem asserts that for k + 1 ∈ {0, …, N},

H^{k+1}(Y, X_{b}, ℤ) = 0.

Nori’s connectivity theorem extends this result to generically defined cohomology classes of degree all the way up to 2N = dim_{ℝ} X. More precisely, let T be a smooth variety and ϕ: T → ℙH^{0}(Y, L) a differentiable map. Let X_{T} = X ×_{ℙH^{0}(Y,L)} T be the universal hypersurface over T and let Y_{T} = Y × T. By the Leray spectral sequence, the Lefschetz Hyperplane Section Theorem implies that for k + 1 ∈ {0, …, N},

H^{k+1}(Y_{T} , X_{T} , ℤ) = 0.

(cf. section 0.2.1).

With stronger assumptions one can say more. Assume that T is a smooth algebraic variety and that ϕ: T → ℙH^{0}(Y, L) is a smooth (hence dominant) morphism. Let O(1) be a fixed very ample invertible sheaf on Y and assume L = O(d) for some positive integer d.

Theorem (Nori’s connectivity theorem [No]). For d large enough and k+1 ∈ {0, …, 2N},

H^{k+1}(Y_{T} , X_{T} , ℚ) = 0.

Note that Nori’s connectivity theorem holds more generally for families of complete intersections in Y (instead of only hypersurfaces). The more general setting is motivated by the construction of algebraic cycles which are homologically but not algebraically equivalent to zero. Since in this paper I do not consider applications to algebraic equivalence, I restrict myself to the case of hypersurfaces, which involves all the ideas and simplifies notations.

Despite its topological nature, the proof of Nori’s connectivity theorem relies on Hodge theory (cf. [No], section 1). Namely, let Ω_{Y_{T} ,X_{T}}^{p} be the coherent sheaf of O_{Y_{T}} -modules defined by the short exact sequence

0 → Ω_{Y_{T} ,X_{T}}^{p} → Ω_{Y_{T}}^{p} → j_{T ∗}Ω_{X_{T}}^{p} → 0,

where j_{T} : X_{T} → Y_{T} is the closed immersion. Then H^{p+q+1}(Y_{T} , X_{T} , ℂ) = ℍ^{p+q+1}(Y_{T} , Ω_{Y_{T} ,X_{T}}^{•}). Following Nori, let

G^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ) = Im(ℍ^{p+q+1}(Y_{T} , Ω_{Y_{T} ,X_{T}}^{•≥p}) → H^{p+q+1}(Y_{T} , X_{T} , ℂ))

denote the Hodge-Grothendick filtration. Nori shows that there is a natural inclusion

F^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ) ⊂ G^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ),

where F^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ) denotes the Hodge filtration of the mixed Hodge structure on H^{p+q+1}(Y_{T} , X_{T}). Hence if for some positive integer k one has H^{q+1}(Y_{T} , Ω_{Y_{T} ,X_{T}}^{p}) = 0 for all p + q + 1 = k + 1 and q + 1 ≤ k/2 + 1, then by the Hodge–Fröhlicher spectral sequence G^{k−[k/2]}H^{k+1}(Y_{T} , X_{T} , ℂ) = 0, hence F^{k−[k/2]}H^{k+1}(Y_{T} , X_{T} , ℂ) = 0 and a mixed Hodge structures argument implies H^{k+1}(Y_{T} , X_{T} , ℚ) = 0. Hence the connectivity theorem follows from the following more precise Hogde-theoretical statement

Theorem (Nori). For d large enough, p + q + 1 ≤ 2N and q + 1 ≤ N,

H^{q+1}(Y_{T} , Ω_{Y_{T} ,X_{T}}^{p}) = 0.

0.2. Vanishing theorems. The first goal of this paper is to investigate the minimal assumptions for Nori’s connectivity theorem to extend. Natural questions are:

− how large does d need to be?

− what happens if the map ϕ: T → ℙH^{0}(Y, O(d)) fails to be dominant, i.e if one considers families of hypersurfaces which are not general, but close to being general?

− can the above assertions be made more precise for a fixed integer k (in Nori’s connectivity theorem), and for fixed integers p and q (in the Hodge-theoretical version)?

The first question was studied in Nagel [N] and Voisin [V2]. In this paper I give the following asymptotic answer to the above questions.

Let T be a smooth algebraic variety with an algebraic morphism ϕ: T → ℙH^{0}(Y, O(d)), such that X_{T} is smooth. I assume that for all t ∈ T the cokernel of the differential map dϕ(t): T_{T,t} → T_{ℙH^{0}(Y,O(d)),ϕ(t)} is of dimension at most an integer c.

The notation ℕ stands for nonnegative integers (hence 0 ∈ ℕ), and ]0, 1[⊂ ℝ denotes the open interval.

Theorem 1. For all ε ∈]0, 1[ and for all p ∈ ℕ and for q + 1 ∈ {0, …, N}, there is an integer D depending only on ε, p, Y and O(1), such that for all d ≥ D,

H^{q+1}(Y_{T} , Ω_{Y_{T} ,X_{T}}^{p}) = 0

provided that

c ≤ (1 − ε)\frac{d^{N−q}}{(N − q)!}.

A straightforward adaptation of Nori’s argument sketched above shows that in the numerical situation of Theorem 1 G^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ) = 0, hence F^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ) = 0, and the following topological statement holds.

Corollary . For all ε ∈]0, 1[ and k + 1 ∈ {0, …, 2N} there is an integer D depending only on ε, Y and O(1) such that for all d ≥ D,

H^{k+1}(Y_{T} , X_{T} , ℚ) = 0

provided that

c ≤ (1 − ε)\frac{d^{N−[k/2]}}{(N − [k/2])!}.

The following remarks show that the bounds given by Theorem 1 and the Corollary are close to being asymptotically optimal:

0.2.1. Case k + 1 ∈ {0, …, N}. By the Lefschetz hyperplane section theorem one has H^{k′+1}(Y, X_{t}, ℚ) = 0 for all k′ ∈ {0, …, k} and t ∈ T, hence R^{k′+1}π_{∗}(ℚ → j_{T ∗}ℚ) = 0 and the Leray spectral sequence implies H^{k+1}(Y_{T} , X_{T} , ℚ) = 0 with no assumption on c. A similar argument for p + q + 1 ∈ {0, …, N} gives a more precise result: since F^{p′}H^{p′+q′+1}(Y, X_{t}, ℚ) = 0 for all p′ ∈ {0, …, p}, q′ ∈ {0, …, q} and t ∈ T, one has ℍ^{q′+1}(Y, Ω_{Y,X_{t}}^{•≥p′}) = 0, hence R^{q′+1}π_{∗}Ω_{Y_{T} ,X_{T} /T}^{•≥p′} ⊗ Ω_{T}^{p−p′} = 0; the spectral sequence induced by the Leray filtration on Ω_{Y_{T} ,X_{T}}^{•≥p} implies R^{q′+1}π_{∗}Ω_{Y_{T} ,X_{T}}^{•≥p} = 0, then the holomorphic Leray spectral sequence implies H^{q+1}(Y_{T} , Ω_{Y_{T} ,X_{T}}^{p}) = 0 with no assumption on c.

0.2.2. Case k + 1 ∈ {N + 1, …, 2N}. I show in the next section (see Theorem 3 below) that under a weak assumption on Y, the bound on c is asymptotically optimal for any couple of fixed integers p and q such that N/2 ≤ q ≤ N − 1 and p ≥ q. This implies that the bound given by the Corollary is asymptotically optimal.

0.2.3. Case k + 1 ≥ 2N + 1. One has H^{2N+1}(Y_{T} , X_{T} , ℚ) ≠ 0 for any positive integer c, d > N and T = X_{B}, where B ⊂ ℙH^{0}(Y, O(1)) is any smooth codimension c subvariety contained in the locus of smooth hypersurfaces. Indeed, the condition d > N implies H^{0,N}(X_{b}, ℚ) ≠ 0 for all b ∈ B, hence (cf. [No], section 0.2) the cohomology class of the diagonal embedding of X_{B} in X_{T} = X_{B} ×_{B} X_{B} is an element of H^{2N}(X_{T} , ℚ) which does not belong to j_{T}^{∗} H^{2N}(Y_{T} , ℚ), hence defines a non-zero element of H^{2N+1}(Y_{T} , X_{T} , ℚ).

0.3. Some examples of algebraic classes. The aim of this section is to give explicit constructions of algebraic classes in F^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ) for q + 1 ∈ {0, … N}, p ≥ q and c close to the bound given by theorem 1.

0.3.1. The regulator map. For any smooth algebraic variety X and positive integers p ≥ q the higher Chow groups CH^{p}(X, p − q) are defined in [B]. In this paper they are always implicitly assumed to be with ℚ coefficients. They come with the Deligne class map

cl_{p,p−q}(D): CH^{p}(X, p − q) → H_{D}^{p+q}(X, ℚ(p)),

where H_{D}^{p+q}(X, ℚ(p)) denotes Deligne cohomology. There is a natural map

r_{p,p−q}: H_{D}^{p+q}(X, ℚ(p)) → F^{p}H^{p+q}(X, ℂ),

where F^{p}H^{p+q}(X, ℂ) denotes the Hodge filtration. Composing cl_{p,p−q}(D) with r_{p,p−q} one gets the regulator map

reg_{p,p−q}: CH^{p}(X, p − q) → F^{p}H^{p+q}(X, ℂ).

Note that this map is identically zero for p > q whenever X is proper.

Consider the case X = X_{T} (note that X_{T} is not proper). Let

H^{p+q}(X_{T} , ℂ)_{v} = coker (j_{T}^{∗} : H^{p+q}(Y_{T} , ℂ) → H^{p+q}(X_{T} , ℂ))

denote the vanishing cohomology, let F^{p}H^{p+q}(X_{T} , ℂ)_{v} denote the image of F^{p}H^{p+q}(X_{T} , ℂ) in H^{p+q}(X_{T} , ℂ)_{v} and let

\overline{reg}_{p,p−q}: CH^{p}(X_{T} , p − q) → F^{p}H^{p+q}(X_{T} , ℂ)_{v} ⊂ F^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ)

be the map deduced from reg_{p,p−q} by projection on H^{p+q}(X_{T} , ℂ)_{v}.

An element of F^{p}H^{p+q}(X_{T} , ℂ)_{v} (or F^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ)) will be called algebraic if it lies in the image of \overline{reg}_{p,p−q}.

0.3.2. Chow groups. In the case p = q higher Chow groups are Chow groups and the regulator is the class map. I then write CH^{p}(X_{T}) for CH^{p}(X_{T} , 0), cl_{p} for reg_{p,0} and \overline{cl}_{p} for \overline{reg}_{p,0}

The following is a variation of my result in [O].

Theorem 2. Let p, q, b and c be positive integers such that p + q = N, q ∈ {[(N + 1)/2], …, N − 1\}, b ≥ 1 and c ≤ b\frac{d^{N−q}}{(N−q)!}. Assume ϕ(T) ⊂ ℙH^{0}(Y, O(d)) lies in the locus parametrizing smooth hypersurfaces. Then there is an integer D depending only on b, Y and O(1) such that for all d ≥ D the following holds.

If N is even and p = q = N/2 then the space F^{p}H^{p+q}(X_{T} , ℂ)_{v} is spanned via \overline{cl}_{p} by classes of flat families of algebraic cycles in X_{T} of codimension p and degree at most b.

If q > N/2 then F^{p}H^{p+q}(X_{T} , ℂ)_{v} = 0.

0.3.3. Higher Chow groups. I now fulfill the promise made in section 0.2.2. The notation ∼_{d→∞} denotes asymptotic equivalence for d → ∞.

Theorem 3. Let p and q be positive integers such that p ≥ q and q ∈ {[(N +1)/2], …, N − 1}. Assume that Y ⊂ ℙH^{0}(Y, O(1))^{∨} contains a linear subvariety V ⊂ ℙH^{0}(Y, O(1))^{∨} of codimension q + 1 in Y and let T ⊂ ℙH^{0}(Y, O(d)) be the subvariety parametrizing smooth hypersurfaces containing V. Then for d ≫ 0 the variety T is of codimension c ∼_{d→∞} \frac{d^{N−q}}{(N−q)!}, and there is a higher cycle Z ∈ CH^{p}(X_{T} , p−q) such that \overline{reg}_{p,p−q}(Z) ≠ 0.

Hence in this situation the bound on c in Theorem 1 and its corollary are asymptotically optimal.

Note that the assumption on Y holds for Y = ℙ_{ℂ}^{N+1}, which should be regarded as the main example.

0.3.4. Indecomposable Higher Chow groups. For all i ∈ {0, …, p} and j ∈ {0, …, p − q} there are natural product maps

ψ_{i,j}: CH^{p−i}(X_{T} , p − q − j) ⊗ CH^{i}(X_{T} , j) → CH^{p}(X_{T} , p − q).

I say that a higher cycle in CH^{p}(X_{T} , p − q) is decomposable if it belongs to the subgroup spanned by the images of the maps ψ_{i,j} for (i, j) ∉ {(0, 0), (p, p − q)}, and that it is indecomposable otherwise.

The proof of Theorem 3 relies on an explicit construction of higher cycles in CH^{p}(X_{T} , p− q). More precisely, since CH^{1}(X_{T} , 1) ≃ H^{0}(X_{T} , O_{X_{T}}^{∗}) ≃ H^{0}(T, O_{T}^{∗}), iterating the above map p − q times I obtain the map

ψ_{1,1}^{p−q}: CH^{q}(X_{T}) ⊗ H^{0}(T, O_{T}^{∗})^{p−q} → CH^{p}(X_{T} , p − q).

For p > q the higher cycles in CH^{p}(X_{T} , p − q) I construct are in the image of ψ_{1,1}^{p−q}. Since ψ_{1,1}^{p−q} commutes to the restriction to a fiber, it follows that the restriction of these higher cycles to any fiber X_{t}, t ∈ T, is decomposable.

My result should be compared to Voisin’s [V2], who studies the case Y = ℙ_{ℂ}^{N+1}, p = N, q = N − 1, d = 2N and T ⊂ ℙH^{0}(Y, O(d)) an open subset (this implies c = 0). She constructs a family of higher cycles in CH^{N}(X_{T} , 1) that limits the validity of Nori’s connectivity theorem and whose restriction to a very general fiber X_{t} for t ∈ T is indecomposable. This suggests that for all p, q such that p + q ≥ N and p ≥ q there might be higher cycles in CH^{p}(X_{T} , p − q) for c ∼_{d→∞} \frac{d^{N−q}}{(N−q)!} whose restriction to a very general fiber X_{t}, t ∈ T, is indecomposable. Supporting this intuition, I prove the following slightly weaker result for Y = ℙ^{N+1}, N even, q = N/2 and p = q + 1.

Theorem 4. Assume N even, q = N/2 and that Y ⊂ ℙH^{0}(Y, O(1))^{∨} contains a linear subvariety of ℙH^{0}(Y, O(1))^{∨} of codimension q in Y. Then for d ≫ 0 there is a subvariety U ⊂ ℙH^{0}(Y, O(d)) contained in the locus parametrizing smooth hypersurfaces, of codimension c ∼_{d→∞} 2\frac{d^{N−q}}{(N−q)!}, and a higher cycle Z_{U} ∈ CH^{q+1}(X_{U}, 1) such that

− \overline{reg}_{q+1,1}(Z_{U}) ≠ 0, and

− the restriction of Z_{U} to a very general fiber X_{u} for u ∈ U is indecomposable.

The cycle Z_{U} is constructed explicitly and is much simpler than Voisin’s example. The proof of indecomposability relies on Theorems 1 and 2.

I believe that Theorem 4 should generalize as follows.

Hope. Let p and q be positive integers such that p > q and q ∈ {[(N + 1)/2], …, N − 1}. Assume Y = ℙ^{N+1}. Then for d ≫ 0 there is a subvariety U ⊂ ℙH^{0}(Y, O(d)) of codimension c ∼_{d→∞} (p − q + 1)\frac{d^{N−q}}{(N−q)!} and a higher cycle Z_{U} ∈ CH^{p}(X_{U}, p − q) such that

− \overline{reg}_{p,p−q}(Z_{U}) ≠ 0, and

− the restriction of Z_{U} to a very general fiber X_{u}, u ∈ U, is indecomposable.

Theorem 4 is the case p = N/2 + 1, q = N/2.

Note that c ∼_{d→∞} 2\frac{d^{N−q}}{(N−q)!} in Theorem 4 (and c ∼_{d→∞} (p−q+1)\frac{d^{N−q}}{(N−q)!} in its conjectural generalization) while Theorem 3 gives examples of decomposable cycles in CH^{p}(X_{T} , p−q) with a non zero vanishing cohomology class for c ∼_{d→∞} \frac{d^{N−q}}{(N−q)!}. This suggests that the bound on c in Theorem 4 may not be optimal.

Question . Is there an indecomposable higher cycle Z_{U} ∈ CH^{p}(X_{U}, p − q) such that \overline{reg}_{p,p−q}(Z_{U}) ≠ 0 for c ∼_{d→∞} \frac{d^{N−q}}{(N−q)!}?

0.4. Algebraicity conjecture. The results of Section 0.3 suggest that for c slightly

bigger than the bound \frac{d^{N−q}}{(N − q)!} given by Theorem 1 the space F^{p}H^{p+q}(X_{T} , ℚ)_{v} should

be algebraic.

More precisely, I believe the following should hold

Conjecture 1. For all integers b ∈ ℕ^{∗} and p ∈ ℕ there is an integer D depending only on b, p, Y and on O(1), such that for all d ≥ D and 0 ≤ q ≤ N − 1, if

c ≤ b\frac{d^{N−q}}{(N − q)!}

then the map \overline{reg}_{p,p−q} : CH^{p}(X_{T} , p − q) ⊗ ℂ → F^{p}H^{p+q}(X_{T} , ℂ)_{v} is surjective. By convention, CH^{p}(X_{T} , p − q) = 0 for p < q.

0.4.1. Case p + q + 1 ∈ {0, …, N}. Conjecture 1 is true since the target cohomology space vanishes (cf. section 0.2.1).

0.4.2. Case p + q = N, q ≥ N/2. Conjecture 1 holds by Theorem 2 provided that ϕ(T) ⊂ ℙH^{0}(Y, O(d)) lies in the locus parametrizing smooth hypersurfaces.

Mimicking Nori’s argument, it is easy to see that Conjecture 1 implies the following purely topological statement.

Conjecture 2. For every positive integer b there is an integer D depending only on b, Y and O(1) such that for all d ≥ D and k ∈ {N, …, 2N − 1} and for

c ≤ b\frac{d^{N−[k/2]}}{(N − [k/2])!},

− if k is even then the map \overline{cl}_{k/2} : CH^{k/2}(X_{T}) ⊗ ℂ → F^{k/2}H^{k}(X_{T} , ℂ)_{v} is surjective; − if k is odd then the maps

\overline{reg}_{(k+1)/2,1} : CH^{(k+1)/2}(X_{T} , 1) ⊗ ℂ → H^{k}(X_{T} , ℂ)_{v}, and
ψ ∘ \overline{reg}_{(k+1)/2,1} : CH^{(k+1)/2}(\overline{X_{T}} , 1) ⊗ ℂ → H^{k}(\overline{X_{T}} , ℂ)_{v} → H^{k}(X_{T} , ℂ)_{v}

generate H^{k+1}(Y_{T} , X_{T} , ℂ)_{v}.

Here \overline{X_{T}} is the variety isomorphic to X_{T} as a real manifold and endowed with the opposite complex structure and ψ is the complex conjugation.

0.4.3. Case k ∈ {0, …, N}. Conjecture 2 is true since the target cohomology spaces vanishes (cf. section 0.2.1).

0.4.4. Case k even. Theorem 1 implies that in the numerical situation of Conjecture 2, one has F^{k/2+1}H^{k+1}(Y_{T} , X_{T} , ℂ) = 0, hence by a mixed Hodge structures argument the space H^{k+1}(Y_{T} , X_{T} , ℂ) is of pure Hodge type (k/2, k/2) (like in [No], section 0.2), hence the subspace H^{k}(X_{T} , ℂ)_{v} is a sub–Hodge structure of pure Hodge type (k/2, k/2). Hence for k even Conjecture 2 is implied by Hodge Conjecture. 0.5. Summary. The figures summarize the results. Assume c ≤ b\frac{d^{N−q}}{(N−q)!} and that ϕ(T) ⊂ ℙH^{0}(Y, O(d)) lies in the locus parametrize smooth hypersurfaces. a : F^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ) = 0 for any c by section 0.2.1. b : F^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ) = 0 for c ≤ (1 − ε)\frac{d^{N−q}}{(N−q)!} by Theorem 1 and F^{p}H^{p+q}(X_{T} , ℂ)_{v} is spanned by classes of flat families of algebraic cycles of codimension N/2 and degree at most b for c ≤ b\frac{d^{N−q}}{(N−q)!} by Theorem 2, as predicted by Conjecture 1. c : F^{p}H^{p+q}(X_{T} , ℂ)_{v} = 0 for c ≤ b\frac{d^{N−q}}{(N−q)!} by Theorem 2. d : F^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ) = 0 for c ≤ (1 − ε)\frac{d^{N−q}}{(N−q)!} by Theorem 1 and one should still have F^{p}H^{p+q}(X_{T} , ℂ)_{v} = 0 for c ≤ b\frac{d^{N−q}}{(N−q)!} by Conjecture 1. e : F^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ) = 0 for c ≤ (1 − ε)\frac{d^{N−q}}{(N−q)!} by Theorem 1 and F^{p}H^{p+q}(X_{T} , ℂ)_{v} should be spanned by \overline{cl}_{p}(CH^{p}(X_{T})) for c ≤ b\frac{d^{N−q}}{(N−q)!} by Conjecture 1; by Theorem 3 there is a variety Y such that \overline{cl}_{p}(CH^{p}(X_{T})) ≠ 0 for c ∼ \frac{d^{N−q}}{(N−q)!}. f : F^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ) = 0 for c ≤ (1 − ε)\frac{d^{N−q}}{(N−q)!} by Theorem 1 and F^{p}H^{p+q}(X_{T} , ℂ)_{v} should be spanned by \overline{reg}_{p,p−q}CH^{p}(X_{T} , p − q) for c ≤ b\frac{d^{N−q}}{(N−q)!} by Conjecture 1; by Theorem 3 there is a variety Y such that \overline{reg}_{p,p−q}CH^{p}(X_{T} , p − q) ≠ 0 for c ∼ \frac{d^{N−q}}{(N−q)!} and by Conjecture there should be an indecomposable higher cycle in CH^{p}(X_{T} , p − q) with a non zero image under \overline{reg}_{p,p−q} for c ∼ (p − q + 1)\frac{d^{N−q}}{(N−q)!}. g : as in f; by Theorem 4 there is a variety Y and an indecomposable higher cycle in CH^{p}(X_{T} , p − q) with a non zero image under \overline{reg}_{p,p−q} for c ∼ 2\frac{d^{N−q}}{(N−q)!}. h : F^{p}H^{p+q+1}(Y_{T} , X_{T} , ℂ) = 0 for c ≤ (1 − ε)\frac{d^{N−q}}{(N−q)!} by Theorem 1 and F^{p}H^{p+q}(X_{T} , ℂ)_{v} should be spanned by \overline{reg}_{p,p−q}(CH^{p}(X_{T} , p − q)) for c ≤ b\frac{d^{N−q}}{(N−q)!} by Conjecture 1. More precisely, I believe that F^{p}H^{p+q}(X_{T} , ℂ)_{v} = 0 for c ≤ b\frac{d^{N−q}}{(N−q)!} in this case. If this is true one may further ask what is the (asymptotically) smallest c such that \overline{reg}_{p,p−q}(CH^{p}(X_{T} , p − q)) ≠ 0? This problem is linked to the existence of non trivial Hodge substructures in H^{p+q}(X_{T} , ℂ)_{v}. Note that I do not have any example of a non-zero element in CH^{p}(X_{T} , p− q)/j_{T}^{∗} (CH^{p}(Y_{T} , p − q)), for any c.

0.6. Remark. For Y = ℙ^{N+1} the proofs below can easily be modified (and even simplified, since one avoids the technical Lemma 1.3.1) to give explicit bounds on the codimension c in Theorems 1, 3 and 4, which are very close to being optimal.

1. Proof of Theorem 1 and of its Corollary

1.1. Reduction to the local case. A straightforward adaptation of Nori’s argument shows that if H^{q+1}(Y_{t}, Ω_{Y_{T} ,X_{T} |Y_{t}}^{p}) = 0 for all t ∈ T then H^{q+1}(Y_{T} , Ω_{Y_{T} ,X_{T}}^{p}) = 0 ([No], Lemma 2.1). This shows that Theorem 1 is implied by the following purely local-analytic statement.

Proposition 1.2. For all ε ∈]0, 1[ there is an integer D depending only on p, ε, Y and on O(1) such that

H^{q+1}(Y_{t}, Ω_{Y_{T} ,X_{T} |Y_{t}}^{p}) = 0

for all integers d ≥ D, q ∈ {−1, …, N − 1}, for all smooth analytic varieties T endowed with a morphism ϕ: T → H^{0}(Y, O(d)) such that

− for all t ∈ T the cokernel of dϕ(t) is of dimension at most (1 − ε)\frac{d^{N−q}}{(N−q)!} and − X_{T} is smooth.

1.3. Proof of proposition 1.2. Since the proof of proposition 1.2 proceeds by induction on several parameters, it is convenient to introduce the following notation. For a fixed positive integer d and a projective space ℙ_{ℂ}^{A} I say that assertion H(Y, p, q, c, a) holds for the smooth projective variety Y ⊂ ℙ_{ℂ}^{A} of dimension N + 1 > 0 and for the integers (p, q, c, a) ∈ ℕ × {−1, …, N − 1} × ℕ × ℤ if

H^{q+1}(Y_{t}, Ω_{Y_{T} ,X_{T} |Y_{t}}^{p} ⊗ O(a′)) = 0

for all a′ ≤ a, for all smooth analytic varieties T endowed with a morphism ϕ: T → H^{0}(Y, O(d)) such that X_{T} is smooth, and for all t ∈ T such that the cokernel of dϕ(t) is of dimension at most c.

The proof relies on the following four lemmas.

Recall (cf. [G1]) that for any positive integer c, a d-decomposition of c is the unique series (c_{d}, …, c_{r}) such that r ∈ {1, …, d}, c_{d} > ⋯ > c_{r} ≥ r and

c = \binom{c_{d}}{d} + ⋯ + \binom{c_{r}}{r}.

Let

c_{<d>} = \binom{c_{d} − 1}{d} + ⋯ + \binom{c_{r} − 1}{r},

with the convention \binom{α}{β} = 0 for α < β.

The asymptotic behavior of the function c ↦ c_{<d>} is described by the following lemma.

Lemma 1.3.1. For all integers M ≥ 2 there exists a function f_{M}: ]0, 1[×ℕ → ℝ, (ε, d) ↦ ε′ such that for all ε ∈]0, 1[ ε′ → ε^{\frac{M−1}{M}} for d → ∞, and such that for all nonnegative integers

c ≤ (1 − ε)\frac{d^{M}}{M!}

one has

c_{<d>} ≤ (1 − ε′)\frac{d^{M−1}}{(M − 1)!}.

The following lemma is the key ingredient of my proof. It relies on Green’s hyperplane section theorem [G1].

Lemma 1.3.2. Let Y′ be a generic hyperplane section of Y. Then for q ≤ N − 2

\begin{matrix}H(Y′, p, q, c_{<d>}, a) \\ H(Y′, p − 1, q, c_{<d>}, a − 1) \end{matrix} \} ⇒ H(Y, p, q, c, a).

By convention, assumption H(Y′, −1, q, c_{<d>}, a − 1) holds.

The last two lemmas will only be used for q = N − 1. Both rely on an idea of Green ([G2], p. 193-194).

Lemma 1.3.3. Assume c > 0. Then

\begin{matrix}H(Y, p + 1, q, c − 1, a) \\ H(Y, p + 1, q − 1, c, a) \end{matrix} \} ⇒ H(Y, p, q, c, a).

By convention, assumption H(Y, p + 1, −2, c, a) holds.

The last lemma has been proved by Nori for a = 0 ([No], remark 3.10); the general case involves no new ideas. See also [N] for computations of an explicit bound for s.

Lemma 1.3.4. There is an integer s depending only on Y and O(1) such that for all d ≥ p + a + s the assertion H(Y, p, q, 0, a) holds for all q ∈ {−1, …, N − 1}.

Lemma 1.3.4 will only be used for q = N − 1.

1.3.5. Proof of proposition 1.2. Postponing the proofs of the above Lemmas, I now show how they imply proposition 1.2.

I show more generally that for all ε ∈]0, 1[ there is an integer D depending only on ε, p, a and Y such that assertion H(Y, p, q, c, a) holds for all d ≥ D, q ∈ {−1, …, N − 1} and c ≤ (1 − ε)\frac{d^{N−q}}{(N−q)!}. The case a = 0 is the proposition 1.2.

I proceed by induction on N − q, c and N.

Assume N − q = 1. The condition on c simplifies to c ≤ (1 − ε)d. Choose D such that εD ≥ p + a + s. Thus for d ≥ D one has d ≥ (1 − ε)d + εD ≥ c + p + a + s.

The case c = 0, N − q = 1 follows from Lemma 1.3.4.

The case c > 0, N = 0 and q = −1 follows from Lemma 1.3.3 by increasing induction on c. For the case c > 0, N − q = 1 let Y′ be a generic hyperplane section of Y. Lemmas 1.3.2 and 1.3.3 provide successive implications

\begin{matrix}H(Y, p + 1, q, c − 1, a) \\ \begin{matrix}H(Y′, p + 1, q − 1, c_{<d>}, a) \\ H(Y′, p, q − 1, c_{<d>}, a − 1) \end{matrix} \} ⇒ H(Y, p + 1, q − 1, c, a) \end{matrix} \} ⇒ H(Y, p, q, c, a).

Assertion H(Y, p, q, c, a) follows by increasing induction on both c and N.

Assume N − q > 1 and let Y′ be a generic hyperplane section of Y. By induction on N and by Lemma 1.3.1 applied to M = N − q, assertions H(Y′, p, q, c_{<d>}, a) and H(Y′, p − 1, q, c_{<d>}, a − 1) hold. Hence by Lemma 1.3.2 assertion H(Y, p, q, c, a) holds. □

1.4. Proof of the Lemmas.

1.4.1. Proof of Lemma 1.3.1. Let e be the largest positive integer such that

(1 − ε)\frac{d^{M}}{M!} ≤ \binom{M + d}{d} − \binom{M + e}{e}.

The identities

\binom{M + d}{d} − \binom{M + e}{e} = ∑_{i=1}^{d−e} \binom{M + d − i}{d − i + 1}

\binom{M − 1 + d}{d} − \binom{M − 1 + e}{e} = ∑_{i=1}^{d−e} \binom{M − 1 + d − i}{d − i + 1}

imply that one has

[\binom{M + d}{d} − \binom{M + e}{e}]_{<d>} = \binom{M − 1 + d}{d} − \binom{M − 1 + e}{e}.

Since the map c ↦ c_{<d>} is increasing (cf. [G1]), and since c ≤ (1 − ε)\frac{d^{M}}{M!} , this implies

c_{<d>} ≤ \binom{M − 1 + d}{d} − \binom{M − 1 + e}{e}.

Fix ε ∈]0, 1[ and consider e as a function of d. Then

\binom{M − 1 + d}{d} − \binom{M − 1 + e}{e} ∼_{d→∞} (1 − ε^{\frac{M−1}{M}})\frac{d^{M−1}}{(M − 1)!},

Indeed, since \binom{M+d}{d} ∼_{d→∞} \frac{d^{M}}{M!} , the definition of e implies \binom{M+e}{e} ∼_{d→∞} ε\frac{d^{M}}{M!} , hence \frac{e^{M}}{M!} ∼_{d→∞} ε\frac{d^{M}}{M!} and e ∼_{d→∞} ε^{1/M}d. The assertion then follows from the equivalences \binom{M−1+d}{d} ∼_{d→∞} \frac{d^{M−1}}{(M−1)!} and \binom{M−1+e}{e} ∼_{d→∞} \frac{e^{M−1}}{(M−1)!} ∼_{d→∞} ε^{(M−1)/M} \frac{d^{M−1}}{(M−1)!}.

Define ε′ ∈ ℝ by the formula

\binom{M − 1 + d}{d} − \binom{M − 1 + e}{e} = (1 − ε′)\frac{d^{M−1}}{(M − 1)!}.

Then c_{<d>} ≤ (1 − ε′)\frac{d^{M−1}}{(M−1)!} and ε′ ∼_{d→∞} ε^{\frac{M−1}{M}}. □

1.4.2. Proof of Lemma 1.3.2. Let i: Y′ → Y be the projective embedding, let Y′_{T} = Y′×T and let X′_{T} = X_{T} ×_{Y_{T}} Y′_{T}. Fix t ∈ T. Since Y′ is generic, one may assume that X′_{T} is smooth and that X_{t} meets Y′_{t} transversally in Y_{t}. Shrinking T if necessary, one may further assume that for all u ∈ T, X_{u} meets Y′_{u} transversally in Y_{u}.Then the image of the map ϕ: T → ℙH^{0}(Y, O(d)) does not meet the base locus of the natural rational map ℙH^{0}(Y, O(d)) → ℙH^{0}(Y′, O(d)), hence there is a well-defined composed map ϕ′: T → ℙH^{0}(Y′, O(d)).

Let – p_{Y} : H^{0}(Y, O(d)) \ {0} → ℙH^{0}(Y, O(d)) be the natural projection,

– E = p_{Y}^{∗} Im (dϕ(t)) and c = dim coker (dϕ(t)) = codim (E, H^{0}(Y, O(d))),

– S = ⨁_{δ∈ℕ} S_{δ}, S_{δ} = Sym^{δ}H^{0}(Y, O(1)) be the homogeneous polynomial ring of the projective embedding Y → ℙH^{0}(Y, O(1))^{∨} and I ⊂ S be the homogeneous ideal of Y. Define the data p_{Y′}, E′, c′, S′, I′ for Y′ as p_{Y} , E, c, S, I for Y.

One has c = codim (E + I_{d}, S_{d}) and c′ = codim (E′ + I′_{d}, S′_{d}). Since Y′ ⊂ Y is generic, Green’s hyperplane theorem [G1] asserts that

c′ ≤ c_{<d>}.

Hence the cokernel of dϕ′(t) is of dimension at most c_{<d>}.

The sheaf Ω_{Y_{T} ,X_{T} |Y′_{T}}^{p} fits into the short exact sequence

0 ⟶ Ω_{Y′_{T} ,X′_{T}}^{p−1} ⊗ O′_{Y} (−1) ⟶^{dH∧} Ω_{Y_{T} ,X_{T} |Y′_{T}}^{p} ⟶ Ω_{Y′_{T} ,X′_{T}}^{p} ⟶ 0

where H ∈ H^{0}(Y, O(1)) is an equation of Y′. Restricting this sequence to Y′_{t} and tensorising it by O′_{Y} (a) one gets the associated long exact sequence

H^{q+1}(Y′_{t}, Ω_{Y′_{T} ,X′_{T} |Y′_{t}}^{p−1} ⊗ O′_{Y} (a − 1)) ⟶^{dH∧} H^{q+1}(Y′_{t}, Ω_{Y_{T} ,X_{T} |Y′_{t}}^{p} ⊗ O′_{Y} (a)) ⟶

⟶ H^{q+1}(Y′_{t}, Ω_{Y′_{T} ,X′_{T} |Y′_{t}}^{p} ⊗ O′_{Y} (a)).

The term on the left vanishes by assumption H(Y′, p − 1, q, c_{<d>}, a − 1) and the term on the right vanishes by assumption H(Y′, p, q, c_{<d>}, a). Hence the middle term vanishes as well: one has

H^{q+1}(Y′_{t}, Ω_{Y_{T} ,X_{T} |Y′_{t}}^{p} ⊗ O′_{Y} (a)) = 0. (1)

For any locally free sheaf F on Y_{t} there is a short exact sequence

0 ⟶ F ⊗ O(−1) ⟶^{H} F ⟶ i_{∗}i^{∗}F ⟶ 0,

Take F = Ω_{Y_{T} ,X_{T} |Y_{t}}^{p} ⊗ O_{Y} (a) and consider the associated long exact sequence

H^{q+1}(Y_{t}, Ω_{Y_{T} ,X_{T} |Y_{t}}^{p} ⊗ O_{Y} (a − 1)) ⟶^{H} H^{q+1}(Y_{t}, Ω_{Y_{T} ,X_{T} |Y_{t}}^{p} ⊗ O_{Y} (a))

⟶ H^{q+1}(Y′_{t}, Ω_{Y_{T} ,X_{T} |Y′_{t}}^{p} ⊗ O′_{Y} (a)).

The term on the right vanishes by (1). Since q ∈ {0, …, N − 1}, by increasing induction on a the term on the left vanishes as well, the case a ≪ 0 following from Serre’s vanishing theorem. Hence the term in the middle vanishes. □

1.4.3. Proof of Lemma 1.3.3. Since the assertion depends only on the tangent space of T at t, one may replace T by a small open neighborhood of t in the tangent space of T at t; hence one may assume T = B × S, where B ⊂ ℙH^{0}(Y, O(d)) is an open neighborhood of ϕ(t) of codimension at most c and S is an open ball.

Let B′ ⊂ ℙH^{0}(Y, O(d)) be a smooth analytic variety containing B as a codimension 1 subvariety, and let T′ = B′ × S. Then Y_{T} ⊂ Y_{T′} is a smooth codimension 1 subvariety and one has the following short exact sequence of sheaves on Y_{T}:

0 ⟶ Ω_{Y_{T} ,X_{T}}^{p} ⟶ Ω_{Y_{T′},X_{T′}|Y_{T}}^{p+1} ⟶ Ω_{Y_{T} ,X_{T}}^{p+1} ⟶ 0.

Restrict this sequence to Y_{t}, tensorise it with O(a) and consider the associated long exact sequence

H^{q}(Y_{t}, Ω_{Y_{T} ,X_{T} |Y_{t}}^{p+1} ⊗O(a)) ⟶ H^{q+1}(Y_{t}, Ω_{Y_{T} ,X_{T} |Y_{t}}^{p} ⊗O(a)) ⟶ H^{q+1}(Y_{t}, Ω_{Y_{T′},X_{T′}|Y_{t}}^{p+1} ⊗O(a)).

The terms on the right and on the left vanish by assumption, hence the term in the middle vanishes as well. □

2. Proof of Theorem 2

Consider the exact sequence

R^{k}π_{∗}ℂ → R^{k}π_{∗}(j_{T ∗}ℂ) → R^{k}π_{∗}(ℂ → j_{T ∗}ℂ) → R^{k+1}π_{∗}ℂ → R^{k+1}π_{∗}(j_{T ∗}ℂ)

and let H_{X_{T} ,v}^{k} = coker (R^{k}π_{∗}ℂ → R^{k}π_{∗}(j_{T ∗}ℂ)) denote the locally trivial subsheaf of R^{k}π_{Y ∗}(ℂ → j_{T ∗}ℂ) whose fiber at any point t ∈ T is canonically isomorphic to H^{k}(X_{t}, ℂ)_{v}. Let F^{p}H^{0} (T, H_{X_{T} ,v}^{k}) denote the space of sections of H_{X_{T} ,v}^{k} whose fiber at any point t ∈ T belongs to F^{p}H^{k}(X_{t}, ℂ)_{v}.

In [O] I have shown

Theorem ([O]). For all integers b ∈ ℕ^{∗} there is an integer D depending only on b, Y and on O(1), such that for all d ≥ D, for all q ∈ {[(N + 1)/2], …, N − 1}, for all

c ≤ b\frac{d^{N−q}}{(N − q)!}

for all codimension c analytic varieties T ⊂ ℙH^{0}(Y, O(d)) lying in the locus parametrizing smooth hypersurfaces and for all t ∈ T, the following holds.

If N is even and q = N/2 then the space F^{N−q}H^{0} (T, H_{X_{T} ,v}^{N}) is spanned by families of algebraic cycles of codimension q and degree at most b.

If q > N/2 then F^{N−q}H^{0} (T, H_{X_{T} ,v}^{N}) = 0.

Since the statement is local-analytic, one can replace the analytic space T ⊂ ℙH^{0}(Y, O(d)) by any algebraic variety T endowed with a morphism ϕ: T → ℙH^{0}(Y, O(d)) such that

− ϕ(T) lies in the locus parametrizing smooth hypersurfaces and

− for all t ∈ T the cokernel of the differential dϕ(t): T_{T,t} → T_{ℙH^{0}(Y,O(d)),ϕ(t)} is of

dimension at most c; This stronger version of the theorem of [O] is equivalent to Theorem 2. Indeed, one

only needs to show

F^{N−q}H^{N}(X_{T} , ℂ)_{v} ≃ F^{N−q}H^{0} (T, H_{X_{T} ,v}^{N}) .

But the Lefschetz hyperplane Theorem and the Leray spectral sequence give a canonical isomorphism

H^{N}(X_{T} , ℂ)_{v} ≃ H^{0} (T, H_{X_{T} ,v}^{N}) .

Since for any t ∈ T, the restriction map H^{N}(X_{T} , ℂ)_{v} → H^{N}(X_{t}, ℂ)_{v} is a homomorphism of Hodge structures, hence is strict, the result follows. □

3. Proof of Theorem 3

Let T be an open affine dense subset of the variety all smooth hypersurfaces containing V. The condition q ≥ N/2 implies that for d large enough T is non–empty (cf. for example [K-A]). Hence its Zariski closure \overline{T} in ℙH^{0}(Y, O(d)) is the linear subspace of all hypersurfaces containing V, and is of codimension h^{0}(V, O(d)) ∼_{d→∞} \frac{d^{N−q}}{(N−q)!}. Hence c ∼_{d→∞} \frac{d^{N−q}}{(N−q)!}.

For all i ∈ {1, …, p − q} choose hyperplane sections H_{i} ⊂ T in general position and and let f_{i} ∈ H^{0}(T, O_{T}) be the equations of H_{i}; Let P_{i} = ⋂_{j>i} H_{j}, Q_{i} = ⋃_{j≤i} H_{j} and T_{i} = P_{i} \ (P_{i} ∩ Q_{i}). For j ≤ i the functions f_{j} do not vanish on T_{i}, hence belong to H^{0}(T_{i}, O_{T_{i}}^{∗}) and there are well-defined higher cycles in CH^{q+i}(X_{T_{i}}, i)

Z_{i} = ψ_{1,1}^{i}([V × T_{i}], f_{1}, …, f_{i}).

I show by increasing induction on i that \overline{reg}_{q+i,i}(Z_{i}) ≠ 0. Case i = p−q implies Theorem 3.

Case i = 0 a straightforward adaptation of [V2], Proposition 4.3: for d ≫ 0 and for any open subset U_{0} ⊂ T_{0} the cohomology class of the cycle [V × U_{0}] ∈ CH^{q}(X_{U_{0}}) in H^{2q}(X_{U_{0}}, ℂ) does not belong to j_{U_{0}}^{∗}(H^{2q}(Y_{U_{0}}, ℂ)).

Assume i > 0. Then T_{i−1} ∪ T_{i} = P_{i} \ (P_{i} ∩ Q_{i−1}), hence T_{i−1} is a hyperplane section of T_{i−1} ∪ T_{i} and T_{i} is the complementary open set. Hence there is a commutative diagram

\xymatrix{CH^{q+i}(X_{T_{i}}, i) \ar[r]^{l_{q+i,i}} \ar[d]^{_{\overline{reg}_{q+i,i}}} & CH^{q+i−1}(X_{T_{i−1}}, i − 1) \ar[d]^{_{\overline{reg}_{q+i−1,i−1}}} \\ F^{q+i}H^{2q+i+1}(Y_{T_{i}}, X_{T_{i}}, ℂ) \ar[r]^{res} & F^{q+i−1}H^{2q+i}(Y_{T_{i}}, X_{T_{i−1}}, ℂ)}

where res is the residue map and l_{q+i,i} is the linking map for the long exact sequence of higher Chow groups. One has

l_{q+i,i}(Z_{i}) = (l_{q,0} ∘ ψ_{1,1}^{i})([V × T_{i}], f_{1}, …, f_{i}) = ψ_{1,1}^{i−1}([V × T_{i−1}], f_{1}, …, f_{i−1}) = Z_{i−1}.

The induction assumption \overline{reg}_{q+i−1,i−1}(Z_{i−1}) ≠ 0, and the commutativity of the diagram implies \overline{reg}_{q+i,i}(Z_{i}) ≠ 0. □

4. Proof of Theorem 4

4.1. Construction of the cycle Z_{U}. Let P ⊂ Y be a codimension q linear subspace, let R ⊂ Y be a codimension q + 2 linear subspace such that R ⊂ P and let S be the blow-up of P along R. There are natural maps ψ: S → ℙ_{ℂ}^{1} (where ℙ_{ℂ}^{1} parametrizes hyperplanes of P containing R) and s: S → Y. For any x ∈ ℙ_{ℂ}^{1} let L_{x} = s(ψ^{−1}(x)) denote the codimension q + 1 linear subspace of Y. For any subvariety B ⊂ ℙH^{0}(Y, O(d)) let C_{B} = S ×_{Y} X_{B} and let ψ_{B}: C_{B} → ℙ_{ℂ}^{1} and s_{B}: C_{B} → X_{B} be the morphisms deduced from ψ and s by base change.

Let U, T, T′ and T′′ denote subvarieties of the locus of smooth hypersurfaces in ℙH^{0}(Y, O(d)) defined as follows:

− t ∈ U ⟺ [L_{0} ∩ X_{t}] = dR and [L_{∞} ∩ X_{t}] = dR;

− t ∈ T ⟺ L_{0} ⊂ X_{t} and [L_{∞} ∩ X_{t}] = dR;

− t ∈ T′ ⟺ L_{∞} ⊂ X_{t} and [L_{0} ∩ X_{t}] = dR;
− t ∈ T′′ ⟺ L_{0} ∪ L_{∞} ⊂ X_{t}.

The condition q ≥ N/2 implies that for d large enough these varieties are non–empty (cf. for example [K-A]). Let \overline{U}, \overline{T}, \overline{T′} and \overline{T′′} denote their Zariski closures in ℙH^{0}(Y, O(d)). Then \overline{T} and \overline{T′} are hyperplane sections of the projective space \overline{U}, and \overline{T′′} = \overline{T} ∩ \overline{T′} is a linear subspace of codimension 2 in \overline{U}. Since codim (\overline{T′′}, ℙH^{0}(Y, O(d))) ∼_{d→∞} 2\frac{d^{N−q}}{(N−q)!} as in section 3, one has codim (U, ℙH^{0}(Y, O(d))) ∼_{d→∞} 2\frac{d^{N−q}}{(N−q)!}.

By construction, s_{U}(ψ_{U}^{−1}(0)) = R × U = s_{U}(ψ_{U}^{−1}(∞)), hence the couple (C_{U}, ψ_{U}) defines an element Z_{U} ∈ CH^{q+1}(X_{U}, 1).

4.2. Proof of \overline{reg}_{q+1,1}(Z_{U}) ≠ 0. There is a natural linking homomorphism

l_{q+1,1} : CH^{q+1}(X_{U}, 1) → CH^{q}(X_{T}).

Since C_{T} is the union of a variety dominating ℙ_{ℂ}^{1} and of the variety ψ_{T}^{−1}(0) ≃ L_{0} × T, one has

l_{q+1,1}(Z_{U}) = d[s_{T} (ψ_{T}^{−1}(0))] = d[L_{0} × T].

I now proceed as in section 3. A straightforward adaptation of the argument of [V2], Proposition 4.2 shows that for d ≫ 0 the class cl_{q}(d[L_{0} × T]) ∈ H^{2q}(X_{T} , ℂ) does not belong to j_{T}^{∗} H^{2q}(Y_{T} , ℂ), hence \overline{cl}_{q}(d[L_{0} × T]) ∈ H^{2q+1}(Y_{T} , X_{T} , ℂ) is non-zero. Since the diagram

\xymatrix{CH^{q+1}(X_{U}, 1) \ar[r]^{l_{q+1,1}} \ar[d]^{_{\overline{reg}_{q+1,1}}} & CH^{q}(X_{T}) \ar[d]^{_{\overline{cl}_{q}}} \\ H^{2q+2}(Y_{U}, X_{U}, ℂ) \ar[r]^{res} & H^{2q+1}(Y_{T} , X_{T} , ℂ)}

is commutative, it follows that \overline{reg}_{q+1,1}(Z_{U}) ≠ 0.

4.3. Indecomposability of the cycle Z_{U}. Since \overline{reg}_{q+1,1}(Z_{U}) ≠ 0, it is enough to show that the space of decomposable cycles is contained in the kernel of \overline{reg}_{q+1,1}.

Let i ∈ {1, …, q}. One has to show

reg_{q+1,1} ∘ ϕ_{i,0}(CH^{i}(X_{U}) ⊗ CH^{q+1−i}(X_{U}, 1)) ⊂ j_{U}^{∗}F^{q+1}H^{2q+1}(Y_{U}, ℂ).

Since the diagram

\xymatrix{CH^{i}(X_{U}) ⊗ CH^{q+1−i}(X_{U}, 1) \ar[r]^{ϕ_{i,0}} \ar[d]^{cl_{i}⊗reg_{q+1−i,1}} & CH^{q+1}(X_{U}, 1) \ar[d]^{_{reg_{q+1,1}}} \\ F^{i}H^{2i}(X_{U}, ℂ) ⊗ F^{q−i+1}H^{2q−2i+1}(X_{U}, ℂ) \ar[r]^{⌣} & F^{q+1}H^{2q+1}(X_{U}, ℂ)}

is commutative, it is enough to show

F^{i}H^{2i}(X_{U}, ℂ) ≃ F^{i}H^{2i}(Y_{U}, ℂ) and
F^{q−i+1}H^{2q−2i+1}(X_{U}, ℂ) ≃ F^{q−i+1}H^{2q−2i+1}(Y_{U}, ℂ).

If i < q these isomorphisms follow from Theorem 1.

If i = q the second isomorphism is straightforward: one has

CH^{1}(X_{U}, 1) ≃ H^{0}(U, O^{∗}(U)) ≃ CH^{1}(Y_{U}, 1).

It remains to show F^{q}H^{2q}(X_{U}, ℂ) ≃ F^{q}H^{2q}(Y_{U}, ℂ). By Theorem 2 it is enough to show that X_{U} does not contain any flat family of subvarieties of codimension q and degree 1 or 2.

Assume by contradiction that X_{U} contains such a family. Let H denote the space of all subvarieties of Y of pure codimension q + 1 and degree 1 or 2 and, for a subvariety W of Y, let T_{W} ⊂ ℙH^{0}(Y, O(d)) denote the family of degree d hypersurfaces of Y containing W. By assumption one has U ⊂ ⋃_{W ∈H} T_{W}.

Choose W ∈ H such that T_{W} meets U and codim (T_{W} ∩ U, U) ≤ dim H. One has codim (T_{W} ∩ \overline{U}, \overline{U}) ≤ dim H, hence codim (T_{W} ∩ \overline{T′′}, \overline{U}) ≤ dim H + 2. But \overline{T′′} = T_{L_{0}∪L_{∞}}, hence T_{W} ∩ \overline{T′′} = T_{W ∪L_{0}∪L_{∞}} and codim (T_{W ∪L_{0}∪L_{∞}}, \overline{U}) ≤ dim H + 2. Since dim H + 2 does not depend on d, this implies codim (T_{W ∪L_{0}∪L_{∞}}, ℙH^{0}(Y, O(d))) ∼_{d→∞} 2\frac{d^{N−q}}{(N−q)!}.

For d large enough this forces deg(W ∪ L_{0} ∪ L_{∞}) ≤ 2, hence W ⊂ L_{0} ∪ L_{∞}. But the hypersurfaces parametrized by T_{W} ∩ U contain W and meet L_{0} ∪ L_{∞} at R = L_{0} ∩ L_{∞}, hence W ⊂ R, contradicting codim R = q + 2. □

4.4. Indecomposability of the restriction of Z_{U} to a very general fiber X_{u}, u ∈ U. This follows from the indecomposability of Z_{U} by Proposition 4.5 of [V2].

Acknowledgments. I would like to thank Stefan Müller-Stach for explaining to me what a higher Chow group is, and for a useful remark on conjecture 1.

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