THE ENVELOPE OF HOLOMORPHY OF A TWO-MANIFOLD IN C* 57
Lr(∕ι,p). Given a singularity manifold of order r, ScJr(n,p), we can
define a submanifold S(V,M) of J'(V,M), and the codimension of S in
Jr(n,p) equals the codimension of S(V,M) in Jr(V,M).
Let S(f) = (Jr(∕)^*(S,(KM))), where /: V → M is of class Vr. Assume
the codimension of S(V,M) in Jr(V,M) is q.
Thom’s Transversality Theorem gives us the following results (assume
(s — r) > max(∕ι-q,0)):
(i) If q > h , the set of maps f in L(V,M,s) such that S(J) = 0 is dense
in L(V,M,s).
(ii) If q ≤ n, the set of maps f in L(V,M,s) such that S(J) = 0 or
S(f) is a submanifold of V of codimension q is dense in L(V,M,s).
Ws want to apply the Iransversality theory to the case V = M2 (M2 is
compact, real two-dimensional), M = C2, and f∙.M2→C2 is an em-
bedding.
If p is a point in M2, Jl(2,2) can be identified with four-dimensional
complex euclidean space. If p is an exceptional point of M2 under a ttfo
map f, then the complex Jacobian off has a vanishing determinant at p.
Since the singularity S1 of J1(2,2), which is defined by the vanishing of
the Jacobian determinant, is invariant under coordinate changes on M2
and C2, it is a singularity manifold of order 1, and we can thus define
Sl(M2,C2) in Jl(M2,C2).
Consider the set L(M2,C2,∞). Since the real codimension of Sl(M2,C2)
in Jl(M2,C2) is 2, we have that the set of maps f in L(M2,C2,∞) such
that Sl(f) = 0 or S1(∕ ) is a submanifold of dimension O in M2 is open
and dense in L(M2, C2, co).
Because the embeddings are an open set in L(M2,C2, co), we find that
the set of embeddings f such that S1(f) = 0 or Sl(f) contains a finite
number of points is open and dense in the set of all ⅞f°0 embeddings. If
sɪ(ʃ) = 0 and M2 is orientable, then M2 is totally real under the em-
bedding f and is thus a torus, as mentioned earlier.
If p is an exceptional point in an embedded two-manifold in C2, the
equations for a neighborhood of p are
∑1 = и + iυ = w
z2 = αw2 + βiv2 + γww + Λ(w).
If y ≠ O and I β I ≠ )J y I these equations can be put in the form
z1 = u + iv = w
z2 = β(w2 + w2) + ww + Λ(w)