4 Algebraic Cycles and Singularities of Normal Functions

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24 M. Green and Ph. Griffiths so that from the complex (∗) HZ,e;s0 →⊕(Ti − I)HZ,e;s0 we obtain Now ˜ν is well-defined up to → ⊕(Ti− I)(Tj − I)HZ,e;s0 i
Algebraic Cycles and Singularities of Normal Functions 25 4.4 Issues and Deeper Structures 4.4.1 The Effective Hodge Conjecture (EHC) Let X be a smooth variety, of any dimension for the moment, L → X a very ample line bundle with Chern class λ = c1(L), and ζ ∈ Hg p (X) . Since torsion consideration will be important in this section we shall use Z coefficients. The (HC) is equivalent to the statement (1) There exist integers k0, m0 such that for m ≧ m0 where Z is an effective, integral algebraic cycle. Indeed, if k0ζ + m(λ) p =[Z] (4.8) k0ζ =[Z ′ − Z ′′ ] where Z ′ ,Z ′′ are effective cycles, then writing Z ′′ = niZi, ni ∈ Z , where the Zi are irreducible of codimension p, by passing hypersurfaces of high degree through each Zi we will have for a subvariety Wi, which then gives Zi + Wi = H1 ∩···∩Hp [−Zi] =lλ p +[Wi] for some integer l, from which (1) follows. We note that if (1) holds for m = m0, then it also holds for m ≧ m0. The (EHC), in various forms to be discussed below, asks for (1) with estimates on k0,m0. We are grateful to the referee for pointing out to us the following result of Kollár, which illustrates the care that must be taken in consideration of the torsion coefficient k0. We consider the space M of smooth hypersurfaces of degree d ≧ n + 1. Then Denoting by X ⊂ P n+1 , n ≧ 3 Hg n−1 (X) ∼ = H 2n−2 (X, Z) ∼ = Z . Mk ⊂ M
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