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Unformatted text preview: cohomology sheaf of the complex (4.1.1.10) E~o(an) v dv~ ' E,o(an) v dr~ E~o(an) v. By induction, this complex receives isomorphically the complex (4.1.1.11) Ero(abs ) ar~ ,E,o(abs) a*~ whose first cohomology sheaf is E,o+l(abs ). This implies that Ero+l(abs)  ,Ero+l(an) v, and proves the lemma. We may now conclude the proof of (4.1.1) by noting the filtered quasi isomorphisms (cf. (2.2.2.1) and [8], (3.1.7.1) and Prop. (3.1.8)) ((Qi/c (log D)) "n, W), ' ((Q]/c (log D)) "n, z <__ .) (4.1.1.12) l (j, (t]~/~), z =< .). Q.E.D. In the course of the above, we made use of the following fact, applied to X a" and the intersections/)anc~ c~ D an S an, iL "" i, over in order to prove that (4.1.1.9) was an isomorphism for r= 1. (4.1.2) Proposition. Let f: Ys 5 r be a proper and smooth morphism of complex manifolds. The canonical morphism of sheaves on 5 ~ (4.1.2.0) Rqf,(C) ~, Rqf,(t2~/c) ,Rqf,(a~/~)...
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 Fall '11
 NormanKatz

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