Dr. Katz DEq Homework Solutions 61

Dr. Katz DEq Homework Solutions 61 - cohomology sheaf of...

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Algebraic Solutions of Differential Equations 61 (4.1.1.9) Lemma. The morphism (4.1.1.8) of spectral sequences induces an isomorphism between (4.1.1.6) and the sub-spectral sequence in local systems of (4.1.1.2) obtained by taking germs of local horizontal sections. Proof Let's denote by E~(an) the terms of (4.1.1.2) by E,(an) v their sheaves of germs of horizontal sections, and by E,(abs) the terms of (4.1.1.6). It follows from the definition of the Gauss-Manin connection in terms of the Koszul filtration (cf. (1.4)) that the canonical morphisms E,(abs)-* E,(an) factor through E~(an) v. It remains only to prove induc- tively that the mappings E, (abs) -* E, (an) v are isomorphisms. For r = 1, this is proven in [6]. Below we will indicate another con- ceptual proof (cf. (4.1.2)). Suppose the result for all r<r o. Because the functor "germs of horizontal sections" is exact, E,o+~(an) v is the first
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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)) &quot;n, W), ' ((Q]/c (log D)) &quot;n, z &lt;__ .) (4.1.1.12) l (j, (t]~/~), z =&lt; .). Q.E.D. In the course of the above, we made use of the following fact, applied to X a&quot; and the intersections/)anc~ c~ D an S an, -iL &quot;&quot; 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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