4 Algebraic Cycles and Singularities of Normal Functions

36 M. Green and Ph. Griffiths
use transcendental arguments, by model theory there will be a purely
algebraic proof of the algebraic statement
sing νζ = ∅ for L ≫ 0 , (4.32)
which is equivalent to the HC;
iii) Finally (and most “heuristically”), because the geometric picture
of the structure of sing νζ is “uniform” for all n — in contrast, for
example, to Jacobi inversion — any purely algebraic proof of (4.32)
that works for n = 1 will work for all n.
4.5 The Poincaré Line Bundle
Given a Hodge class ζ ∈ Hgn (X)prim there is an associated analytic invariant
νζ ∈ H0 (S, JE) and its singular locus
sing νζ ⊂ D.
Although the local behaviour of νζ and subsequent local structure of sing νζ
can perhaps be understood, the direct study of the global behaviour of νζ
and of sing νζ — e.g., is sing νζ = ∅ for L ≫ 0 — seems of course to
be more difficult. In this section we will begin the study of potentially
important global invariants of νζ obtained by pulling back canonical line
bundles (or rather line bundle stacks) that arise from the polarizations on
the intermediate Jacobians J(Xs). We shall do this only in the simplest
non-trivial case and there we shall find, among other things, that
∗
νζ×ζ (Poincaré line bundle) = ζ 2 .
c1
This is perhaps significant since as we have given in section 4.4.1 an heuristic
argument to the effect that any lower bound estimate required for an EHC
will involve ζ 2 .
4.5.1 Polarized Complex Tori and the Associated Poincaré Line
Bundle
The material in this section is rather standard; see for instance [25, Ch. 2].
We shall use the notations
• V is a complex vector space of dimension b,
• Λ ⊂ V is a lattice of rank 2b.
• T = V/Λ is the associated complex torus.