22 H. B. LAUFER

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, - #

0

• #

0

^ -#

M

• #M for u E Ut - U'. Hence - #

0

• K0

-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.

THEOREM

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.

PROOF.

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.

PROPOSITION

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.

PROOF.

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-

Sing(cV).

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

Hl(Mn

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

changes.

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