4 Algebraic Cycles and Singularities of Normal Functions

28 M. Green and Ph. Griffiths
Returning to (4.11) which only gives information for odd n, for the first
even case n = 2 for purposes of illustration we assume that there exists a
rank-two vector bundle E → X with
c2(E) =ζ
c1(E) =aλ .
Setting as usual E(m) =E⊗L m , let m0 be such that there is σ ∈ H 0 (OX(E(m0)))
with (σ) =W where N ∨ W/X ∼ = E(m0)|W is ample. Then we claim that
m 2 0 + bm0 >c ζ 2 (4.15)
where b, c are constants depending only on X, E and L.
Proof By another result due to Fulton-Lazarsfeld (loc. cit.), since N W/X is
ample we have
from which(4.15) follows.
c1(N W/X) 2 >c2(N W/X) ,
This again suggests the possibility of there being, in general, a lower bound
on m0 for which (4.8) holds in terms of |ζ| 2 .
This possibility is reinforced by the following considerations: Let M be a
quasi-projective algebraic variety parametrizing a family of smooth projective
X’s with reference variety X0 ∈ M. For example, M could be a moduli
space if such exists. Letting U be a sufficiently small neighborhood of X0
and ζ ∈ Hg n (Xc)prim, the locus
Uζ = U ∩{X ∈ U : ζ ∈ Hg n (X)prim}
of nearby points where ζ remains a Hodge class is an analytic variety. By a
theorem of Cattani-Deligne-Kaplan [8] it is part of an algebraic subvariety
Mζ ⊂ M .
We shall write points of Mζ as (X, ζ) to signify that there is a Hodge class
ζ extending the one defined over Uζ, where we may have to go to a finite
covering to make ζ single-valued. For each k, m with m>0, k = 0we
consider the subvarieties
Mk,m = {(X, ζ) ∈ Mζ : kζ + mλ n =[Z]}
where Z is an effective algebraic cycle.