Chapter three deals with the general principle that the components
of the special fiber should be at least as "special" as the general fiber.
One result of that kind was already established in chapter two,
section 7, where the inequality E p (V) p ( ^ ) was
V comp. of *0
established for Kahler degenerations (i.e. total space Kahler). I believe
it is true without the additional Kahler assumption.
To put one aspect of the above mentioned general principle in pre-
cise form, I propose the following conjecture. The ^-invariant
conjecture. (Recall (cf. chapter I) that the basic measure of "speciality"
is the K-invariant, which subdivides surfaces into four classes, corres-
ponding to K = -1,0,1 and 2.)
The ^-invariant conjecture
Let y - A be a degeneration, then the ^-invariant is upper semi-
continuous in the sense that
K (N, ) max *(V);V component of * .
Chapter three is mainly devoted to an attempt of proving this
The ^-invariant is determined by the asymptotic behavior of the
pluri-genera P (cf. chapter I) and it would be desirable to have formu-
las (or inequalities in the right direction) relating the pluri-genera
of the general fiber with those of the components of the special.
Unfortunately the pluri-genera are very different from the Hodge
invariants, with their strong topological connections, and such formulas
are hard to come by, as a matter of fact it is not clear that they
We have however, using general theory of the lower semicontinuity of
dim H (D ) of a (flatly moving) divisor B, inequalities, but they go in
the wrong direction. Thus we are forced to resort to ad hoc methods. We