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 (126.96.36.199) over Z. As these latter are sub-quotients (i.e., quotients of sub-objects) of the E2 terms of (188.8.131.52), it remains to polarize the E2 terms. In the notation of (184.108.40.206), we have ((-q) denoting | cf, (4.2.0)). Ev, q=Si,<.G..<iqRpfa"(i, ) .....iq).(Z)(-q) if q=~0 (220.127.116.11) 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. (18.104.22.168) Proposition (Hodge's Index Theorem).
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