WEAK SIMULTANEOUS RESOLUTION 15
normal bundle. So Kt coincides with the usual canonical bundle on Mt at the
regular points of Mt. To restrict a 3-form on 911 to a 2-form on Mt, contract with
3/3/ and then restrict the coefficients to Mt. There is the same construction for
mKlo i.e. e(mKto) := 6(mK^)/(t - t0)e(mK^).
Since V is Gorenstein, so is T. There exists Q on T, a holomorphic nonzero
3-form on T Sing(T). The restriction of £2 to Vn i.e. the contraction with 3/3/
followed by the restriction of the coefficients, gives w„ a holomorphic nonzero
2-form on Vt - Sing(F,). On % 0 S
0 m
~ 0 can be thought of as meromorphic
sections of mK^ which are holomorphic o n T -
Sing(c\T).
We now define §m. Our definition has been chosen to be independent of any
resolution of T. Let im be the sheaf of ideals in 0 fi8wgiven by those sections of
0
08m
which for t 0 restrict to elements of Sw/ on F
r
Thus the restriction of
elements of imp to F p consists of (the germs at p of) those meromorphic
sections of mKv which are holomorphic on V p and have extensions to 0
fi2m
such that on each Vn t ^ 0, the extension extends holomorphically to Mr We first
need that §m is coherent. Recall IT. 91L - T. Since we may choose small repre-
sentatives for II which are proper maps, II ^©(mA^)) is coherent. II ^(©(mA^))
C$
m
. In fact, on
CV—
VQ, where II is a weak simultaneous resolution,
ri5|{(0(m^g1L)) = hm because: Look at a, an element of 0-S®™, and suppose
a £ U^eimK^)). We need that a £ f
OT
. a £ I I ^ / n A ^ ) means that II*(a) is
not holomorphic on 91L, i.e. that 11(a) has poles P on S, the exceptional set in
911. P has dimension 2. Let Z be the zero set of II*(a). Then P D Z has
dimension 1. So £, := S D Mt, the exceptional set in Mn can be contained in
P D Z for only a discrete subset S of T - {0}. Then £, n (P - (P n Z)) ^ 0
for / E r {0} S. Then the restriction of a to Mr is not in Sm
t
for / E T
{0} - 5.
Since ©cy is Noetherian, $ , the stalk of ffm at the initial singular point /?, is
finitely generated over 0K;7. II ^(©(raA^)), which is contained in 5m and which
equals 5m on T - V09 is finitely generated near/?. Hence $m is finitely generated
near p and hence is coherent.
We define imt := im/{t t0)$m. Temporarily replace II over T {0} by the
simultaneous minimal resolution. Then H\Mn &(mKt)) = 0 for t ^ 0. Then by
[42, Bemerkung, pp. 94-95], imJ = Sw
f
, / * 0.
Observe that the natural map $
0
- 0
F
- co®m is an injection. Indeed, if a is an
element of im which vanishes upon restriction to V p, then a/t has removable
singularities on V p and so defines an element /? of im. Then a = r/?, i.e.
Let Sm : = 0 Q®m/3m. Then Sm is supported on Sing(T). A*(SJ can have no
torsion. For if an element a of 0 S20m maps to 0 in Sm above f ¥^ 0, then a
restricts to an element of Sw
t
for r ^ 0 and so a is an element of $m, by the
definition of $m. So A+(Sm) is a locally free sheaf on the one-dimensional space T,
including t = 0, of rank dim 0
V(

cofm
/ S
m p
f ^ 0. By (4.3),
K K
(4.4) RankX,(2
m
) = A,
'-^(m2
- m), t¥=0.
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