4 Algebraic Cycles and Singularities of Normal Functions

40 M. Green and Ph. Griffiths
Proof that (4.37) is well defined: Denote by PM the RHS of (4.37) . Since
PM⊗R = PM ⊗ PR
we have to show that for a line bundle R → T
PR =0 if c1(R) =0. (4.38)
In fact we will show that PR is canonically trivial. We will check that
Assuming this we have
Then, by definition, for a, a ′ ∈ T
c1(R) =0⇒ c1(PR) =0. (4.39)
PR ∈ Pic 0 (T × T ) ∼ = Pic 0 (T ) ⊕ Pic 0 (T ) .
(PR) (a,e) = R ∗ e
(PR) (e,a ′ ) = R ∗ e
so that the two “coordinates” of PR are zero, hence PR is trivial. To make
the trivialization canonical we need to show independence of scaling, and
this is the role of the Me factor.
To verify (4.39), in general we may choose coordinates x i ,y i ∈ Λ ∗ Z
any line bundle R has c1(R) represented by
i
λidx i ∧ dy i .
Using coordinates (x i ,y i ,x ′j ,y ′j )onΛR ⊕ ΛR and using that
µ i ((x, y) ˙+(x ′ ,y ′ )) = (x i + x ′i ,y i + y ′i )
where ˙+ is the group law on T , it follows that c1(PR) is represented by
i
λi(dx i ∧ dy ′i + dx ′i ∧ dy i ) .
In particular, if the λi = 0 then (4.39) follows.
Remark. For later use we note for Q as above
c1(PQ) =
In particular
i
dx i ∧ dy ′i + dx ′i ∧ dy i .
so that
c1(PQ) 2b [T × T ]=2 b . (4.40)