Unformatted text preview: 16 N.M. Katz: Then gr~ (M) = F p (M)/F p +a(M) ~ gr~ R p +qf, (f2]/s (log D)) (1.4.1.9.3) l[
Rq f, (~2P/s(log O)). B ecause M is horizontal, it follows from (1.4.1.7) t hat we have a commutative
d iagram a f, (C ,,s(log D)) J ,  (O ,/s(log O)) F V(M) v ~/r grV(M) v __~f2~/r  grP 1(M) (1.4.1.9.4) 1] R ' f , ( ~ / s ( l o g D)) P , ~/r Rq+~f, (~xV})~(logD)) We deduce from it that the Hodge filtration Fi(M) of M is horizontal (i. e.,
each Fi(M) is horizontal) if and only if the restriction to Op gr~(M) o f the
mapping "cupproduct with the KodairaSpencer class" (1.4.1.9.5) p: @ gr~(M), @ f2s~/r
p I(M) p vanishes. (1.4.1.10) R emark. I n practice, the M in (1.4.1.9) will be either all of
R "f,(f2~/s(log D)), or the p rimitive part o f R"f,(f2]/s) in case O is void
a nd X / S is projective and smooth, or the part of R"f, (f2~c/s(logD)) which
t ransforms according to a prechosen irreducible representation of a
finite group (of order prime to all residue characteristics of S) which acts
a s a group of Sautomorphisms of X and preserves D. This last case will
a rise when we discuss Schwartz's list (cf. 6.0). 2. The Cartier Operation and the Conjugate Spectral Sequence
2.0. Throughout this section, we will consider the situation of 1.0, with
t he additional assumption that S is a scheme of characteristic p (a prime
n umber), i.e. that p. ls = 0 in ~Ys.
(2.0.1) Recall that for any Sscheme n: Y, S, the Sscheme y~v) is by
d efinition the fibre product of 7~: Y,S and the a bsolute F robenius ...
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 Fall '11
 NormanKatz
 Hodge filtration Fi, prechosen irreducible representation, Conjugate Spectral Sequence

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