16

H. B. LAUFER

Of course, in (4.4) Kt • Kt is the same for all t ¥=0 because Kt • Kt is defined

topologically and n is a weak simultaneous resolution. ht is also independent of /

for t ^ 0; let m = 0 in (4.4).

The proofs of Lemmas 4.4 and 4.5 below are postponed until later in this

section. Observe that the hypotheses of Lemmas 4.4 and 4.5 do not assume that

Kt • Kt is constant. Sm : = Sw 0.

LEMMA

4.4. Let X: T- T be a germ of a flat deformation of the germ (V, p) ofa

normal Gorenstein two-dimensional singularity with T smooth of dimensiion 1. If

$m o £ §m, then there exists c 0 such that for all large v,

dimimvfi/(%mvf\imv0)cv2.

LEMMA

4.5. Let X: T- T be a germ of a flat deformation of the germ (F, p) ofa

normal Gorenstein two-dimensional singularity. Suppose that the map germ X: °\f-* T

has a representative Xr: °V ^ -»

P1

such that ^ is a projective algebraic variety and Xr

is a meromorphic function on %. Then, as a function of m, dimS

m

/(S

w

0 $m0) is

bounded by a linear function of m.

Consider the inclusions given in (4.5). Think of each inclusion as an injection.

(4.5) is commutative. All sheaves are over V.

©•"0m"

= t,,o

(4-5) fi\j Ur

o

The codimension of im a is given by (4.4). The codimension of im /? is given by

(4.3) with h — h0 and K- K — K0- K0. Lemma 4.5 estimates the codimension of

im 8. Lemma 4.4 is about the codimension of im y.

Fix m. In (4.5), look at the rate of quadratic growth of the codimensions of the

inclusions as v - oo. Codim(/? ° S) is asymptotic to (K0 • K0)(m2v2). im(/? °8) =

im(a o y). Since we have not yet used that Kt • Kt is constant, we have proved the

PROPOSITION

4.6. Let X: T- Tbe as in Lemma 4.5. Then -K0-K0 -Kt• Kn

We now use the hypothesis that

K^-KQ-

Kt-Kr Then (4.5) and Lemma 4.4

imply that im

0

C Sm. With v = 1, (4.5) becomes:

0 ' ^ m = 4m,o

(4.6)

fi

u

Then A0 ft,. But A0 ht [11, Theoreme 1, p. 144]. So,

PROPOSITION

4.7. Let X: T-^ r be as in Theorem 4.3. 77ie« ht is constant.

Then from (4.6), Sw = Sm n t , o = W