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Dr. Katz DEq Homework Solutions 14

# Dr. Katz DEq Homework Solutions 14 - is/J"...

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14 N.M. Katz: In fact, it is defined ([35]) as the coboundary map 17 in the long exact sequence of the Rif, and the short exact sequence O-~f* (f2s/r) | f2x/sl ,-l(logD)_~KO/K~-(f2~/~(logD)) (1.4.0.3) -~ t2]/s (log D) --~ 0. This "makes sense", because the coboundary is a mapping 17: R~ f, ((2~c/s (log O)) ~ R q +1 L (f* (Q~S/r) | 12]'~S 1 (log D)) (1.4.0.4) /l ~2~/r | Rq f, (t2]/s (log D)). (The final isomorphism thanks to the local freeness of ~2~/r. ) (1.4.1) The Hodge Filtration. Let us recall that for any complex L ~ the Hodge filtration of L ~ is the filtration by the subcomplexes Fi(L'), where by definition 0 if j--i (1.4.1.1) U(/J)= /j if j>=i. (1.4.1.2) Let us denote by L ~ [n] the complex L ~ +" whose term of degree i
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Unformatted text preview: is/J+", whence (1.4.1.3) F i (L" In]) = (F i +" (L')) [n]. Applying F ~ to the exact sequence (1.4.0.3), we obtain the exact sequence 0 -~ (f* (12~/r) | F i-1 (12],/s (log D))) [ - 1 ] (1.4.1.4) -~ F~(K~ (Q]/r (log D))) V i (f2~c/s (log D))-~ 0. Thus the coboundary maps for the W f, and the exact sequence (1.4.0.3) and (1.4.1.4) "fit together" to form a commutative diagram Rqf,(O]ls(log D))--E-~ v 12Xs/r| (2~/s(log D)) (1.4.1.5) | Rqf, (F *(O]/s (log D))) _~00]/r | Rqf, (U- 1 (O]/s(log D))). Thus we find: (1.4.1.6) PrOlmSition (Griffith's Transversality Theorem). The Gauss- Manin connection respects the Hodge filtration up to a shift of one, i.e. (1.4.1.6.1) 17(FiR~f,(g2]/s(log O))) =f2xs/r| D)))....
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