24 H. B. LAUFER
THEOREM
5.6. Let f:Ta-+ U be as above. Let S = {u G U\ Ku-Ku = K0-K0).
Thenf(Ta) = S.
PROOF.
f(Ta) is a subvariety of S by [13] and Proposition 5.4. We shall assume
that f(Ta) 7^ S and reach a contradiction. Let Sx be an irreducible component of
S such that Sx £ f(Ta). Then dim [Sx n f(Ta)] dim Sx. Temporarily blow up S
at 0, j: S" S. Let E = J_1(0). Let primes' denote proper transforms via f. For
any irreducible subvariety Z of S, dim (Is Pi Z') + 1 = dim Z' = dim Z. So there
exists^ E E n S{ such that & E C\ (Sx n f(Ta))\
By Theorem 5.2, S is a subvariety of U. Let S = US,- be the decomposition of
S into irreducible components. Observe as follows that for each Si9 there is a
(projective algebraic) subvariety Sir of Ur such that St is an (analytically)
irreducible component of the germ of Sir at 0: Let II
r
:
GJ\ir
%r be a not
necessarily simultaneous resolution of %r. Then off a nowhere dense subvariety
of U'r of Ur9 Hr is a weak simultaneous resolution. Take U'r D Sing(£/r). So AT
W
ATM
is constant on each connected component of Ur U'r. If St is an irreducible
component of U9 then let Sir = Ur9 and we are done. If not, St is contained in the
germ of U'r at 0. Now repeat the argument using U'r.
Then there is a holomorphic map yr': P1 -* 5{
r
such that y/(0) = y9 yr: = f
r
° yr'
is not constant, and for t near 0 in P1, t ^ 0, yr(f) E Sx - f(Ta) - U S,., i ¥* 1. yr
induces a deformation A: F- JP. By Theorem 4.3 and [8], there is a finite base
change a: Tx -* T such that the induced deformation \x\ °VX - 7^ has a very weak
simultaneous resolution, II: 9IL, - %. A^ % r, is induced from T.%-* [/via
y ° a: Tx U.
Let p
1
:=X
1
°II: ( D1l
1
-»7'
1
. Then the deformation p1 may be induced from
a: 911 Q [33, Theorem 5, p. 515]. Let us recall part of the proof of [33, Theorem
5, p. 515]: B : = Q X Tx. We form a deformation
T.^-^B
such that T|
0xr i
^ px
and T|£XO « co. There is a submersion a: B ^ Q which induces r from co. We may
take T
j e x o
: Q -^ Q to be the identity map. R : = a(9 X r,) C Ta.
Since P! is not the trivial deformation and Tx is one-dimensional, a: 0 X Tx Ta
is proper and R is (the germ of) an irreducible subvariety. Let Bx :=
a_1(i^),
an
irreducible subvariety of B. In BX9 R may be identified with RX O, which is
transverse to 0 X Tx. KhKh is constant on Yf
e
for b G i^. Since A, is constant on
MlntGTl9hbis also constant on Ybfor b E Bx. Hence we may simultaneously
blow down r\ ^ ^ B over BX9 yielding T,: % - BX9 a (flat) deformation of (V9 p).
On O X Tx C BX9 rx restricts to a deformation isomorphic to Xx: °VX Tx. On
i? X O C 5,, T, restricts to a deformation isomorphic to ma restricted to R. Let
g: (O X Tj) U (RX O) -* Ubz given by g
) o x r i
= y ° « and gj/?xo be the restric-
tion of /: Ta - U to R. Since O X T j and i? X O are transverse in BX9 g is
holomorphic and induces T, restricted to (OX Tx) and ( # X O). By [40, Theoreme
(b), pp. 162-163] g may be extended to a holomorphic map gx\ Bx £/. Neces-
sarily, g(£,) C 5.
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