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
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 NormanKatz
 Derivative, Algebraic geometry, locally free sheaf, arbitrary base scheme, KodairaSpencer Class, 1.0.2.1

Click to edit the document details