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CURVATURE AND COMPLEX SINGULARITIES 501
The closure F of F* is an analytic subvariety of X x G(n, N), and the projection
7r F-- B
has fibres 7r-(t) Ft the Gaussian images Tt(Vt) for 0, while the fibre Fo is
generally not the closure To(W;). Writing
(5.7) F0 v(V) +
defines A as an analytic subvariety of G(n, N), one possibly having a boundary
corresponding to the points of Vs q 0U, and one whose irreducible components
generally have multiplicities. We shall call A the Pliicker defect (4) associated
to the family {Vt} e B"
It is also possible to define A as a current Ta by the formula
(5.8) Ta(a)= lim I’ Y(a)-I" y’g(a)
0
where a is a C form on G(n, N). Clearly this is just the current associated to
the variety defined by (5.7). With this definition one may prove directly that Ta
is a positive current of type (n, n), which is closed in case V0 has isolated interior
singularities. Moreover, by taking a smaller class of "test forms" a we may
refine the data of the Pliicker defect.
Another possibility is to define A by using resolution of singularities. (5) By
successively blowing up X beginning along the singular locus of V0 we arrive at
2 _B
where 7r-l(t) Vt for # 0 and where 7r-1(0) is a divisor with normal crossings.
We may even assume that the Gauss mappings
Tt: 6-1(t) G(n, N)
are defined for all including O. Writing
7r-’(O) 9o + tXlDa +’’" + t.mDm
where lY0 is the proper transform of V0 (so that 1?0 ---> V0 is a desingularization),
it follows that
Iy0(f’0)
yo(V*o), and
(5.9) !
[.A PtlT0(Ol) +""" + PtmTo(Om).
Each of the characterizations (5.7)-(5.9) of the PRicker defect turns out to be of
use.
Now let a (a l, ", a) be any sequence of integers with
lal-- a n