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8
N.M. Katz:
which gives rise to an element
p e Ext,,,
(f21x/s
(log D), f * (t2~/r))
(1.1.2)
~ H t (X, Dero (X/S) @r f*
(I2~/r)).
This element is called the
KodairaSpencer class
of the situation 1.0. It's
image in
U ~ (S, R'f, (Der o (X/S) 
f* (f2~/r)))
(1.1.3)
~ U ~ (S, n I f. (Der o (X/S)) 
(2~/r)
Home~(Der(S/T)), R~
f. (Dero(X/S))
is the
KodairaSpencer mapping
(but still denoted p !), which may be
explicated as follows. Let D be a section of
Der(S/T)
over an affine open
set ~ c S. Then p (D) is an element of n I (f
1
(q[),
Dero (f  1
(q/)/q/)) which
may be given explicitly as follows. Let {Vii} be an affine open cover of
fl(q/). By the exactness of the dual of (1.1.1) over each Vii (or, more
"directly", by the explicit description (1.0.3) via local coordinates), we
may choose, for each i, a derivation
Di~H~
Dero(X/T))
which
extends
the given derivation
D ~H ~ (ql, Der(S/T)).
Because
D i
and D~ extend the
same
derivation D of q/, the difference D i  Dj lies in
H ~ (Vii n Vj, Dero(X/S)).
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 Fall '11
 NormanKatz

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