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.