LOCAL HOLOMORPHIC CONVEXITY OF A TWO-MANIFOLD IN C2 177
very many, and in fact the set of manifolds which do not have this property
is dense, in a special sense.
It will be shown how to construct a function f0 of the form (1.2) for
which there exists no non constant function φ, holomorphic in a neigh-
borhood of 0 in C2 and such that
(3.1) z → ≠(z,∕0(z))
is real for all z near 0. A function ∕0 with this property can be found with
any prescribed order of contact at 0 to a given infinitely differentiable
representation of the form (1.2). In particular, there is no prospect for
finding a biholomorphic coordinate change to convert an arbitrary repre-
sentation (1.2) into one of the same form where f takes real values, for one
of the coordinate functions of such a transformation would have to take
real values on M.
These examples probably permit no non constant holomorphic functions
for which (3.1) has rank ≤ 1 near 0, but here it is only shown that (3.1)
cannot be real.
An example can be found easily if it is not required to satisfy (1.2):
/(z) = z2z is a function whose graph permits no non Constantholomorphic
function φ which takes real values on it. This is seen by equating the co-
efficients of the general term zpz^, in the power series relation
φ(z, z2z) = fl5(z,z2z).
To find this behavior in a function f of the form (1.2) seems to require
more effort. The construction below produces one whose graph M is in fact
a determining set for real-analytic functions, meaning that any real-analytic
function in a connected neighborhood U of zero which is zero on M vanishes
identically on U. Once this is established and φ is holomorphic on U and
real on M, then Im φ vanishes on U, which implies that φ is constant.
The idea of the construction is roughly that M should be a determining
set if it has infinite order of contact with the graph of Q at O, but is not
contained in any of the obvious real-analytic varieties through the graph
of Q. It is convenient to make the construction in several steps.
Lemma 3.1. Suppose {(xn, JF,,)} is a sequence in R2 such that xll ≠ O
and y„ ≠ O for any n, xn→ O and for any integer p ≥ O,
yn∣χpn → θ∙
Ifh is real-analytic in a connected neighborhood of O in R2 and h(xn,yn) = O
for all n, then h = O.