Dr. Katz DEq Homework Solutions 62

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 , (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 Gauss-Manin 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 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 (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 C-spaces, 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. ...
View Full Document

Ask a homework question - tutors are online