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Algebraic Solutions of Differential Equations
75
(5.1.1)
Then after the change of base
Spec(C)* T,
Dc
.* D
X c ~
X
(5.1.1.0)
1
Sc~S
Spec(C)
, T
the coherent sheaf with integrable connection on S c
(5.1.I.1)
(R"fc,(~ic/sc(log D)), V
becomes trivial on a finite ~tale covering q~ : S' * S c, i. e., becomes isomorphic
to
(((gs,) ~
d), where b,=the rank of
R"f,(f~/s(logD) )
over the generic
point of S.
Proof.
By (4.3.3), it suffices to prove that the Hodge filtration F on
R"fc,(~}c/sc(log Dc)) is horizontal for the GaussManin connection. For
this it suffices to prove that the Hodge filtration is horizontal over ~c,
since the
obstruction
to the horizontality is the (gsclinear Kodaira
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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