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Dr. Katz DEq Homework Solutions 68

Dr. Katz DEq Homework Solutions 68 - 68 N.M Katz W hen...

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