4 Algebraic Cycles and Singularities of Normal Functions

34 M. Green and Ph. Griffiths
Then it may be shown that (this is essentially ρ(iii) = ρ(v) in section 4.2.4
above)
From (4.24) a consequence is that
HC ⇒ α is injective on Hg n (X)prim . (4.26)
a class ζ ∈ H n (Ω n X )prim is integral ⇔ the residues of ζ are integral (4.27)
The spectral sequence argument also gives
H 1 R 0 πI∆ ⊗ (KX ⊗ L n ⊗ H n ) =0 for L ≫ 0
⇒ α is injective
⇒ HC.
(4.28)
The statements (4.26) and (4.28) give precise meaning to the general principle:
The HC may be reduced to (in fact, is equivalent to) a statement about the
global geometry of
We thus have:
∆ ⊂ X × S. (4.29)
HC ⇔ geometric property of (4.29) when L ≫ 0.
Above we have discussed the question: Can we a priori estimate how positive
L must be? The condition L ≫ 0 in this section requires sufficient
positivity to have vanishing of cohomology plus Castelnuovo-Mumford regularity.
Above, we gave a heuristic argument to the effect that for each ζ
the condition L ≫ 0 must also involve ζ 2 .
Discussion: Denote by ˇ Xk the dual variety to the image of
One may ask the question
X → P ˇ H 0 (OX(L k )) .
What are the properties of the singular set ˇ Xk,sing of ˇ Xk for k ≫ 0?
Although we shall not try to make it precise, one may imagine two types
of singularities: (i) Ones that are present for a general X ⊂ P ˇ H 0 (OX(L))
having the same numerical characters as X; in particular, they should be
invariant as X varies in moduli. (In this regard, one may assume that L → X
is already sufficiently ample so as to have those vanishing theorems that will
ensure that dim ˇ Xk can be computed from the numerical characters of X1).
(ii) Ones that are only present for special X. What our study shows is that: