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Algebraic Solutions of Differential Equations
21
is endowed with the GaussManin connection (cf. [24], 3.5). For reasons
which will soon become apparent (cf. (2.3.3)below), we call this the
conjugate spectral
sequence, and
the corresponding filtration of
R"f,(f2~/s(log D)) the conjugate filtration, noted
i
F~'on. This filtration is
horizontal,
and is every bit as interesting as the Hodge filtration.
(2.3.1)
Let us explicate the
Ez
term of the conjugate spectral sequence
with the aid of Cartier's isomorphism (2.1.1).
(2.3.1.1)
Lemma. The
Cartier isomorphism ~ defines an isomorphism of
(gsmodules for each pair a, b of nonnegative integers
(2.3.1.1.1)
Ea,~_
~
b
co, 2  R f, (Jr ~ (f2x/s(log D)))
,
R"f~,P)(tr*(Qbxls(log D))).
Proof.
From the commutativity of the diagram
X
r
~ X~p)
S
and the fact that F is a homeomorphism, we have isomorphisms
(2.3.1.1.2) ~o,E3
'b =
Raf, (~'r ~b (t2~,./s
(log D))) ~
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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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