10 S. BERHANU In thew coordinates, the defining function of Misgiven by p(w', wn) = p(w', Wn + 2J=l w i). We will show that if Wn = Un + J=T Vn is sufficiently small, p(O', Wn) :: 0 whenever Vn 2:: -u i. Indeed, p(O', Wn) p(O', Wn + 2J=lw ) q?(O',~(wn + 2J=Tw ))- ' S(wn + 2J=Tw ) q?(O', Un- 4unvn)- Vn- 2u + 2v 4 ( 1 4 )4 2 2 2 2 Un - Vn - Vn - Un + Vn. Suppose now Vn 2:: -u i. Then for Wn small, if Vn 2:: 0, then -vn + 2v i 0 and so p(O', wn):: 0 if Vn :: 0, since Vn 2:: -u , we have v :: u~ and so again for Wn small, p(O', wn):: 0. We now let p = p and we have new coordinates z satisfying: Yn 2:: -x = p(O', Zn):: 0. The biholomorphic map G also leaves the form off unchanged, therefore in the new coordinates we may still write J(z', Zn) = b(z', Zn)(z + a!(z')zn + ao(z')), b(O) -f. 0 and aj (0') = 0. We may also assume that b(O) = 1. Let h(z) = b(O, z)z 2 . We claim that there is a holomorphic function a(z) such that a(O) = -1 and h(z) = h(a(z)z). To see this, consider the holomorphic function F(z, a)= h(az) ~ h(z) z which is defined near (0, -1). F(O, -1) = 0 and ~~ (0, -1) = -2. Therefore, by the implicit function theorem, there is t 0 and a holomorphic function a(z) defined on lzl t such that a(O) = -1 and F(z,a(z)) = 0. Hence h(z) = h(a(z)z). From the equation F(z, a(z)) = 0, we have a'(O) =- ~ab (0). UZn Let W, = {(O',zn): Yn 2:: -x }n{(O',zn): lznl t}. We know that since f(W,) ~ f(rJ), 0, the neighborhood of 0 in M, f is open at 0 whenever the following holds: if (0', z) rf. W,, lzl t, then a(z)z E W,. Suppose lzl t and (0', z) rf. W,. If z = x + J=Ty, y -x 2 . Notice that {jb 2 3 a(z)z = -z- -(O)z + O(z ). OZn Letting "'ab (0) = s + J=Tt, we have: UZn ' S(a(z)z) -y- 2sxy- t(x 2 - y2 ) + O(z3 ) _!!_ + ~x 2 - 2sxy- t(x 2 - y2 ) + O(z3 ) 2 2 -~ + (ty- 2sx)y + (~- t)x 2 + O(z3 ). Now :::.JL 0 since y -x 2 and the term (ty- 2sx)y can be absorbed in :::.jL if tis small enough. Hence if {jb 1 It I= P~(O)I :: - , u---n then ' S(a(z)z) 0, ie. a(z)z E W,, implying that f is open at 0.
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