II: 91L % be a (not necessarily simultaneous) resolution of T. II has a projective
algebraic representative IIr: 91Lr - 9Gr. Then, off a nowhere dense proper sub-
variety U' of U, II is a weak simultaneous resolution. Then 2£w Ku is constant on
each connected component of U U'. Let Ut be an (analytically) irreducible
component of U. Let y: P1 - Ur be a holomorphic map such that y(0) = 0 and
y(x) E Ut U' for x near 0 but x ¥= 0. y induces a deformation of (V, p) over
P1. By Proposition 4.6, - #
^ -#
#M for u E Ut - U'. Hence - #
-Ku-Kuforu& U'.
Now restrict T to £/', a projective algebraic subvariety of £/. Then, arguing as
before, there exists £/", a nowhere dense subvariety of U', such that -K0 AT0
-ATU Ku for u E U' U". After a finite number of steps, we conclude that
-K0 - K0 -Ku Ku for all u E U,u sufficiently near to 0.
The above argument also shows that S is a subvariety.
5.3. Let A: T- T be the germ of a (flat) deformation of the normal
Gorenstein two-dimensional singularity (V9 p) with T a reduced analytic space. If
Kt - Kt is constant, then also ht is constant.
It suffices to prove the theorem for the special case that X: °V-^ T is
T: % - U, the reduction of the versal deformation of (V, p). Where T has a very
weak simultaneous resolution, ht is locally constant by [52, 42]. Using Proposition
4.7, the result then follows as in the proof of Theorem 5.2.
5.4. Let A: T-* The the germ of a (flat) deformation of the normal
Gorenstein two-dimensional singularity (V, p) with T a reduced analytic space.
Suppose that X has a very weak simultaneous resolution. Then Kt Kt is constant.
This proposition is known; see [46, Proposition 10, p. 10]. Another
proof is as follows. Resolving T, we may assume that Tis Gorenstein. Then there
is a nonzero holomorphic canonical form fionf-
Let II: 91L - T b e
the very weak simultaneous resolution of T. Then II*(£2) has a continuously
varying divisor Kt on each Mt. Since Kt Kt is integer valued, Kt Kt is constant.
Observe the following. Let co: 9H - T be a deformation of a resolution of M
with T reduced. Then [52, 42], co simultaneously blows down if and only if
ht '.= dim
0) is constant. So in the case that (V, p) is Gorenstein, for co,
ht constant implies that Kt Kt is constant. For X of Theorem 5.3, Kt Kt constant
implies that ht is constant.
What about converses to these implications? There are many known examples
(see [9, 21, 54]) of deformations X: T- T where ht = 1 for all / and KrKt
Now look at deformations co: 911- T with (V, p) Gorenstein. With (V, p)
other than rational or minimally elliptic, i.e. with h : = h0 2, there are deforma-
tions co which induce topologically trivial deformations of the exceptional sets in
Mt such that the blow down of Mt is not Gorenstein [29, Theorem 4.3, p. 1282].
Since deformations of a Gorenstein singularity are also Gorenstein, co cannot
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