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6
N.M. Katz:
1. Generalities on the KodairaSpencer Class
and the GaussManin Connection
1.0. The Geometric Setting
Throughout this Section 1, we will consider the situation
D ~
i ~X,
J
~
U=XD
(1.0.1)
s
T
in which T is an arbitrary base scheme, S is a smooth Tscheme (via g),
which will play the role of a parameter space, X is a smooth Sscheme
(via f), whose fibres over S are "parameterized" by S, and D is (via /)
a union of divisors Di in X, each of which is smooth over S (hence also
over T), and which have normal crossings relative to S (hence also relative
to T). This situation persists after arbitrary change of base
T' ~ T.
In
practice, X
is usually proper over S, and should be thought of as
a particularly nice compactification of the smooth "open" Sscheme
U =XD,
which is psychologically prior to X. We allow D to be the
empty divisor, corresponding to U being proper over S. We do
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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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