sophisticated arsenal of modifying operations on the special fiber would
be very beneficial. The modest collection presented here is just intended
to be a beginning, and could hopefully prove to be an inspiration for
the future.
In order to be able to use more machinery, more restrictive, but
yet very natural, conditions have to be imposed on tf. This is the
subject of the last two sections of chapter two.
In section 6 we introduce the notion of weak-projectivity. I.e.
the total space K is bimeromorphic to something projectively embeddable.
(This is a concept closely related to being Moisezon.) This allows us
to use Hilbert-scheme methods. (Although Douady (see [Do]) defines the
Hilbert-scheme for any complex manifold, its components are not proper
in general.) The crucial consequence is the principle of "generic
stability" i.e. generically a divisor, say in a general fiber, extends
(i.e. is stable) and specializes to the special fiber (because of the
properness of the components of the Hilbert scheme).
Loosely speaking this makes the special fiber accessible from the
outside. In many ways, this is a condition akin to that of "continuity".
An example of elliptic ruled surfaces degenerating into the union
of two Hopf surfaces, given in chapter three, illustrates the spectacular
failure of this principle in general.
In section 7, finally, we impose a global Kahler condition on H-
This will enable us to use the theory of mixed Hodge structures developed
by Deligne [D],via the Clemens-Schmid exact sequence to obtain very
precise relationships between the Hodge numbers of the components of the
special fiber and those of the general. in particular we can establish
very simple and general formulas for the Betti-numbers (i.e. even
allowing triple points).
As a byproduct obstructions for inserting a divisor ^n as a special
fiber in aKahler manifold are obtained.
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