Algebraic Solutions of Differential Equations
33
HW(a)
na,a
~
n
na
a
\
T
I
/
////pr
\F~J,
(f?~,s))
a"f,(f2x ~)v
//7
F~bs
/
//a
/
/
na
a
/
R
f,(f2x/s)
Let
(,)
denote the cupproduct pairing
(2.3.5.1.3) of De Rham coho
mology. We
have, for each
i,
(2.3.6.10)
0 = (~ (co,), E
V(~j)(co*))
J
and, because
~(coi) is
horizontal,
we have
(2.3.6.11)
0=}" ~j((~ (col), co*.
)]
J /"
J
Because each co* has Hodge filtration
>Na,
the
cupproduct
(~(coi), co*)
depends only on the class
of ~(coi) modulo F "+1R "c q2"
d*~
X/SI,
by (2.3.5.1.4), which is to say on pr. ~ (col)= HW(a) (F~*s
(coi))
Furthermore
denoting
by (,) the cupproduct pairing (2.3.6.1.6) of Hodge cohomology,
we have
(~ (coi), co*)= (pr. ~(coi), co*) = (HW (a)(F,*s(coi)), co*)
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 Fall '11
 NormanKatz
 De Rham cohomology, Algebraic geometry, De Rham, Hodge filtration, Hodge cohomology, Hodge cohomology sheaves

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