Dr. Katz DEq Homework Solutions 62

Dr. Katz DEq Homework Solutions 62 - 62 N.M. Katz: is an...

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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 Gauss-Manin connection) of R q/, (f2~./~). P roof T he proof is an exercise in the definition o f the Gauss-Manin 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 , ( E f'q=RV+q f,(gr~ ~2~c/c)=~.RP+qf,((2~r/c). B y (the analytic version of) (, we have ( Fp, ~--- ov~ / C | Rqf, (f2}/y). L'I ~5 B y d efinition o f the Gauss-Manin connection (cf. [35]), the differential ( dr, q: E f ' q ~ E f + l ' q is the mapping d| ( plQV d educed from the Gauss-Manin 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 ( (_9~| c Rq f , (Q~/~)v__~ Rq f , (~2~:/~) is an isomorphism. Thus we have an isomorphism ( 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 C-spaces, it is automatically f iat o ver C, and h ence we have ( v.q,,~ 9 9 E z - ~ v (f2~/c) @cR qf , (f~r/~) v9 B y the Poincar6 lemma, ( ~f'v (f2~/c) = if p:60. T hus we have ( E V.q_fRqf,(f2~/~) v i f p = 0 2-)0 if p#:0. ...
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