The first sentence on p. 19 seems a bit obscure. Here is a better argument
that the map jstar is injective when i=2m-1:
In the commutative diagram in the middle of page 18 take i=2m-1.
Then r is an isomorphism. Also c and hence b is injective. The
kernel of l has weight 2m-2. Since morphisms in the category of
mixed Hodge structures are strict with respect to the weight filtration
(or since passing to weight graded pieces is an exact functor by
Deligne Hodge II), the intersection of the image of b and the kernel of l
is zero. Thus jstar is injective.
Proof of Proposition 1.9: The surjectivity of the map i_*:H--> H^4(Y) is a
consequence of the Lefschetz hyperplane theorem. This should have been
stated explicitly in the paper, but somehow fell through the cracks.
Here are the details:
Dropping the subscripts the we have an exact sequence,
H^4_V(Y,Z_l(2)) -c-> H^4(Y,Z_l(2)) --> H^4(Y-V,Z_l(2)),
in which the final term is zero, since Y-V is a three dimensional affine
variety [Mi, VI.7.2], and the first term is isomorphic to H^2(V,Z_l(1))
and the map c may be identified with the Gysin map by [Mi, VI.5.4b],
where the references are to Milne's book on etale cohomology.
p. 499 line 7 the S_i in normal font should be in the script font.
The proof of Lemma 7.8 should have explicitly mentioned that the map, $\dot {\rho }_*$,
in the diagram is an isomorphism. Indeed this is already true on the level of
fundamental groups by the proof of 7.7.
Numerous errors were introduced by the editor. Most were caught in the
course of correcting the proofs, but the following two slipped by:
p. 305 statement of Proposition 1.4: Replace "albanese" with "abelian".
p. 309 statement of Proposition 3.5: Delete "We say that".