20 H. B. LAUFER
supported on E0 near M' such that near AT, on 9tt,
(4.12) ? T = 6(mK^ + + M^ - Hm)
and such that Hm C) M' = mJ + D'.
So 3"is locally free near M'. By [41, §2, pp. 268-272 or 19, §1, pp. 314-320], we
may modify 9Hr so that ^becomes locally free on 91tr. This modification does not
change
(DILr
near M' since ? T is already locally free near M'. We may extend our
definition of Hm above and say that there is a divisor Hm on 91Lr such that
!T = QimKa^ + F* + M^- Hm) on 91tr. Near AT, #
w
is supported on £0 and
Hm D M' = mJ + D'. Then observe that p is an isolated point of IIr(Mr' Pi
supp#J.
Since 0((m + l)iC - D) has no base points on M, 0((m + 1)^ - (m + 1)/ -
D') has no base points on M'. Then we may choose initial meromorphic sections
ofkr
of 0((m + \)K - (m + 1)7 - £') on M; such that (a), (b) and (c) are
satisfied for the germs
(a*)~l(ok)
of the okr at A'. %r(°k,r)
*s a
meromorphic
pluricanonical form on Vr which is not necessarily holomorphic at p and at some
other subvarieties of Vr. By multiplying £r{°k,r) ^Y
a
meromorphic function, we
may assume that ir{ok r) has a zero of any desired order at n
r
( Afr' D supp Hm)
p. Recall that a^ is the extension via (4.7) and (4.8) of ok from
Mr
to M0. So there
is an effective divisor Gm+l on %, which we also choose to be ample, such that all
of the ok have representatives ok
r
on M0r such that, letting S := supp(M0r)
supp(£0),
(4.13) a
M s
G r(S, 0((m + l)tf0) 0 0(G*+1 - tfj).
0(G£+I) is a quasipositive sheaf. So also 5'® 0(G*+1) is quasipositive. Then [16,
Satz 2.2, p. 273], tf'(M„ 0 ( ^ + G*+1 + Mx - M0r) ® J ) = 0.
Letirm+1
= Gm+i + Fm + 2A-'(oo). Using (4.12), we see that
0 - T(Mr,e((m + \)K^+F*+l - M0,r - Hm))
- T(Mr, 0((m + 1 ) ^ + F*+i - Hm))
- r(A/0r, 6((#n + 1)AT0) 0 0(F*+1 - / f j )
is an exact sequence.
In order to complete the proof of the lemma, we need that
okt, G r(M0?r, 0((m + \)K0) ® 0(/£
+ 1
- Hj).
(4.13) says that this is true on supp(M0 r) supp(£0). So we need to examine ok
near E0. Recall that the divisor of ok is at least (m + 1)/ + D' mJ + D'. Then,
as in (4.10) and (4.11), any local lifting dk of ok has a divisor C ^
(JC(«+D«+c),(m+i)H^i Locally, H =
(xma+cymb+d)
and £0 =
(xayb).
So ^ G
0((m + 1 ) ^ - Hm) and in fact
dkE6((m+\)K^-Hm-M0)
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