Algebraic Solutions of Differential Equations 33 H-W(a) n-a,a ~ n n-a a \ T I / ////pr \F~J, (f?~,s)) a"f,(f2x ~)v //7 F~bs / //a / / n--a a / R f,(f2x/s) Let (,) denote the cup-product pairing (18.104.22.168.3) of De Rham coho- mology. We have, for each i, (22.214.171.124) 0 = (~ (co,), E V(~j)(co*)) J and, because ~(coi) is horizontal, we have (126.96.36.199) 0=}-" ~j((~ (col), co*. )] J /" J Because each co* has Hodge filtration >N-a, the cup-product (~(coi), co*) depends only on the class of ~(coi) modulo F "+1R "c q2" d*~ X/SI, by (188.8.131.52.4), which is to say on pr. ~ (col)= H-W(a) (F~*s (coi))- Furthermore denoting by (,) the cup-product pairing (184.108.40.206.6) of Hodge cohomology, we have (~ (coi), co*)= (pr. ~(coi), co*) = (H-W (a)(F,*s(coi)), co*)
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De Rham cohomology, Algebraic geometry, De Rham, Hodge filtration, Hodge cohomology, Hodge cohomology sheaves