WEAK SIMULTANEOUS RESOLUTION 5
Now consider the remaining case, K-Aj — 0. Then (L — Yk_x)-Aj - 1 . If
(L — Yk_x)'AJk 0, then we are done, as above. For the remaining subcase,
(L — Yk_x)-Ajk — - 1 , we have that LAj—K- Ajk — 0. Then L restricted to AJk,
denoted by Lk, is the trivial bundle. By the induction hypothesis, since h 2, L
has a section which is nonzero at a point in supp(YA:_1) n Aj . Since LA; is the
trivial bundle on Aj , this section is nonzero at all points of Aj . This completes
the proof of Theorem 3.1.
Theorem 3.2 below is somewhat stronger than [46, Corollary 5(i), p. 5].
Retain the notation of the proof of Theorem 3.1. Let ^ b e a sheaf on M. Recall
that T(A, §) := dirlim T(U, ¥)9 where U varies over all open neighborhoods of
If Lx and L2 are line bundles, we let Lj + L2 denote Lx 8 L2.
3.2. Let M be a strictly pseudoconvex two-manifold with connected
exceptional set A. Suppose that M is the minimal resolution of its blow-down, i.e.
Ai-K§ for all irreducible components of A. Let Lx and L2 be line bundles on M
such that Lx-Ai^2K'Ai and L2At 3K • At for all i. Then the canonical map
T(A, 0(L,)) ®
)) - T(A, 0(L, + L2))
We first do a formal computation, using essentially infinitesimal
neighborhoods of A. We then show that the formal computation suffices.
Recall the Yk of (2.4). Let Cs = nZ + Yk for n ^ 0, 0 k /; s = nl + £. We
shall show that, for all s, the map
T: r(^,e(L,)) ®
)) -+T(A,e(L, + L2))/T(A9e(Lx + L2 - CS))
is onto. By taking successive quotients
r ( ^ , 0(L, + L2))/T(A, 0(L, + L2 - C,))
r(^, 0(L, + L2 - c,))/r(^, G(L, + L2 - c2))
T(^ , 0(L, + L2 - C,_,))/r(^ , 0(L, + L2 - C,)),
it suffices to show that
(3.1) imr D T(A, 6(LX + L2 - C,_,))/r(,4, ©(L, + L2 - C j ) .
By Proposition 2.1, T(A9 6(L, + L2 - C ^ ) ) / ^ , 0(L, + L2 - Q ) may be
identified with T(A, e(Lx + L2- CS_X)/&(LX + L2 - Cs)).
Let ^4, be such that Cs = Cs_x + Ar By Theorem 3.1, there exists / G
T(^4, 0(Lj)) such that/is nonzero at all singular points of A. Multiplication b y /
gives an injective map
- CS_,)/6(L2 - C,))
- r ( ^ „ 0(L, + L2 - €,_,)/©(£, + £
- CJ) .