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Unformatted text preview: 62 N.M. Katz: is an isomorphism between its source and the subsheaf Rq f,(f2~c/~) v of
g erms of horizontal sections (for the GaussManin connection) of R q/, (f2~./~).
P roof T he proof is an exercise in the definition o f the GaussManin
c onnection (cf. (1.4) and [35]). We consider the Koszul filtration K
(of. (1.4)) of the complex s2]~/c, and the corresponding spectral sequence
f or the functors R f ,
(4.1.2.1) E f'q=RV+q f,(gr~ ~2~c/c)=~.RP+qf,((2~r/c). B y (the analytic version of) (1.4.0.2), we have
(4.1.2.2) Fp, ~ ov~ / C  Rqf, (f2}/y).
L'I
~5 B y d efinition o f the GaussManin connection (cf. [35]), the differential (4.1.2.3) dr, q: E f ' q ~ E f + l ' q is the mapping
d (4.1.2.4) plQV d educed from the GaussManin connection V= d ~
B ecause the ROf, (f2]./~) are coherent sheaves on the complex manifold
t hey are locally free of finite rank, and the canonical mapping
(4.1.2.5) (_9~ c Rq f , (Q~/~)v__~ Rq f , (~2~:/~) is an isomorphism. Thus we have an isomorphism
(4.1.2.6) El" ~~ f2)/c  Rq f , (f2~c/~)v in terms of which d ('q=d ( 4.l.2.7) A s R qf, (fl;c/s~)v is a sheaf of Cspaces, it is automatically f iat o ver C, and
h ence we have
(4.1.2.8) v.q,,~
9
9
E z  ~ v (f2~/c) @cR qf , (f~r/~) v9 B y the Poincar6 lemma,
(4.1.2.9) ~f'v (f2~/c) = if p:60. T hus we have
(4.1.2.10) E V.q_fRqf,(f2~/~) v i f p = 0
2)0 if p#:0. ...
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 Fall '11
 NormanKatz

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