Dr. Katz DEq Homework Solutions 13

Dr. Katz DEq Homework Solutions 13 - (log D associated...

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Algebraic Solutions of Differential Equations 13 The coboundary mapping associated to (1.3.1.2) (1.3.1.3) ~: Hq(X, f2~/s(log D))-~ Hq+'(X, f*(f2~s/r)| D)) is the cup-product with the Kodaira-Spencer class (1.1.2) p ~ Ext,,, (f21XlS (log D), f* (f21Slr)) n'(x, Hom(f2'x/s(log D), f*(f2's/r) ) ~ nl(x,f*(f2's/r)| Dero(X/S)). Proof. This is just (1.2.2.4.5) applied to the exact sequence (1.3.1.1), in which case the element ~ of (1.2.2.4.5) is the Kodaira-Spencer class. Localizing on S, we have (1.3.2) Corollary. Hypotheses as in (1.3.1), the coboundary mapping (3 Rq f, (aVx/s (log O)) -, R q +' f, (f* (fJ~S/T) | O~-fS 1 (log O)) (1.3.2.1) /
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Unformatted text preview: (log D)) associated to (1.3.1.1) is given by cup-product with the Kodaira-Spencer mapping (1.1.3), viewed as an element p ~ Hom~ (Der (S/T)), R' f, (Dero (X/S)) (1.3.2.2) t H ~ (S, Y2~S/T) | R~ f, (Dero (X/S)). 1.4. Application to the Gauss-Manin Connection (1.4.0) The construction of the Gauss-Manin connection on HoR (X/S (log O)) = R f, (f2x/s (log O)) is based on the fact that the Koszul filtration of f2~/r(1Og D) arising from the exact sequence (1,1.1) (1.4.0.1) 0 -,f* (f2~S/T) --, fJ~/T (log D) -, f2~/s(1Og D) -, 0 is a filtration by subcomplexes, and that the associated graded complexes are given by (1.4.0.2) gr~(f2]/T(lOgD))'~f * ~2~/T | 12jc~i (log D)....
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