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Algebraic Solutions of Differential Equations
105
integrable Tconnection V,~ on (gs is a plinear mapping
(7.1.0)
0,~:
Der(S/T) ~
Endr
(9 s
which is
additive
in the variable closed oneform o9.
(7.1.1)
Recall from (6.0.2.5) the commutative diagram (in which g: S ~ T
is the structural morphism)
S.,
F
~ S (p)
~
~, S
v
) S(p)
T
Fabs ~T
the middle square of which is cartesian.
(7.1.2)
Proposition.
The
pcurvature 0o, of Vo is given by the formula
(7.1.2.0)
0o (D) = F* ((a* (o9) cg(o9), r
(D)))
where ~ is Cartier's isomorphism
(2.2.1) jr
(F, f2~/r)
~ ' ~s'(,,/r
(applied
to the class of the closed form o9), ~* (D) is the Tderivation of S (p) defined by
(7.1.2.1)
a*(D)(a*(f))=a*(D(f)),
f a local section of (gs,
and (,) is the canonical pairing of Os~prmodules
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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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