68
N.M. Katz:
When tensored with Q, it degenerates at E3 and defines (a renumbering
of) the filtration W on R"n,n(Q). Thus the assosciated graded families
of Hodge structures gr w of our family of mixed Hodge structures are
isogenous
to various of the E3 terms of the Leray spectral sequence
(4.3.1.0) over Z.
As these latter are subquotients (i.e., quotients of subobjects) of
the E2 terms of (4.3.1.0), it remains to polarize the
E2
terms.
In the notation of (4.0.1.4), we have ((q) denoting 
cf, (4.2.0)).
Ev, q=Si,<.G.
.<iqRpfa"(i,
)
.....
iq).(Z)(q)
if q=~0
(4.3.1.2)
lR~'f,"
(Z)
if q = 0.
Because finite direct sums of polarizable families are polarizable,
and Tate twists H( q) of polarizable families H are polarizable, we need
only remark the' following proposition, applied to X and to all inter
sections
Dii n.
.. n Diq.
(4.3.1.3)
Proposition (Hodge's Index Theorem).
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 Fall '11
 NormanKatz
 Algebraic geometry, Hodge, spectral sequence, mixed Hodge structures, polarizable families, Hodge structures grw

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