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Algebraic Solutions of Differential Equations
71
mixed Hodge structures on S an
(4.3.5.0)
(n n (
O an, Z))san ~ R" 7~, n (Z)
has as image a constant subfamily of mixed Hodge structures. Its image is
the largest constant sublocal system of R" ft. n
(Z).
Proof.
The first assertion results from the fact that the category of
families of mixed Hodge structures is an abelian subcategory of the cate
gory of local systems, and that the formation of images (as well as of
kernels and cokernels) commutes with the inclusion of the category of
families of mixed Hodge structures into that of local systems. The second
assertion will result from the equality a.,~itTO'n
~'O,n__
x., 2
in the usual Leray
spectral sequence in integral cohomology of rt~": Ua"~ S an, which itself
follows from the following proposition.
(4.3.6)
Proposition.
Hypotheses as in
(4.3.5),
the Leray spectral sequence
Rq
an
(4.3.6.0)
E~'q=HP(S an,
~z, (Z)) ~
HP+q(U an,
Z)
is degenerate at E 2.
It suffices to show that E g'q =0 unless p = 0 or p = 1. This is
true, because the R q n."(Z) are local systems on S"", and because S a" is
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 Fall '11
 NormanKatz

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