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
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