14

H. B. LAUFER

For any coherent sheaf 9 on M, H2(M,($) — 0 since M is noncompact of

dimension two [47]. Assuming V Stein, let h — dim Hl(M,&). h is often denoted

0-6(-D) -e-6D-*0

is exact. So if H\M, &(-D)) = 0, then A

1

^ ) = A. So by (4.1),

(4.2) h°(eD) =h-±(DD + DK).

By Proposition 2.1,

(4.3)

A

o

( 0

_

M j f ) = A

_ ^

( f f I

2 _

m ) -

Since H\M, 6(mK)) = 0, all of the obstructions to lifting will vanish. So, as

ideals, with n m ^ 1,

dimSwA = ^(0_

n

J-^(0_

w

J

~2

r—[(A?2 — m2) — (n — w)] .

Now look at the Hilbert function H(v):=

dim(§my/(§m)v+l

of §„. By Theorem

3.5,(Smr = SMr.So

#(* ) = - ~ [ m

2

( 2 , + 1) - m], v 1,

//(0) = / * - ^ ^ ( m

2

- m ) , from (4.3).

We shall now construct the ideal sheaf §m.

We are given the map germ A: T-+ T of Theorem 4.3. In this proof, V will

denote the germ of an analytic space. Recall that V— V0. In this proof, all

representatives of germs will have a subscript r. By blowing up %, away from/?,

we may assume that Xr is a (holomorphic) map, i.e., the zero locus of Xr is disjoint

from the pole locus of \

r

. Let II

r

: 91tr - °Hr be a projective algebraic resolution of

% [18]. Let 911 be the germ of 91tr at U;\p). Let II: 911 - Tbe the germ of II

r

at

911. 9Itr is nonsingular of dimension 3, but II is not necessarily a simultaneous

resolution. We choose 91tr so that the exceptional set S in 911 has normal

crossings. Let p = A°II:91t-^r . We take 0 E T to correspond to 0 in

P1.

Let

M, = p~\t). Observe that M0 need not be reduced. Let Et — Mt n S. Let AT C M0

be the proper transform under IT of V. Let £ denote the restriction of II to AT.

Then we may assume that £: M' — F is a resolution (which is not necessarily the

minimal resolution m\ M - K) and that A/' meets S transversely. M' D & = A' is

the exceptional set in AT. By performing additional blow-ups, we may assume

that A' does not intersect the closure in 911 of S - E0 = U £„ r ^ 0. II: 9 1 - Af0

- T — V0 is a weak simultaneous resolution (see Definition 5.1).

Let Kc^i be the canonical bundle on 9Itr. Let t be a coordinate for T. On M

r

,

define 0(#,

o

): = (9(/^)/( f - ^ © ( A ^ ) . The vector field 3/9/ induces a

nowhere zero section of the normal bundle of Mt in 911, i.e. Mt has a trivial