The name is absent

holomorphic mappings into unspecified complex spaces. This suggests that
it ought to be possible to dispense with the “special” unbranched mapping
g into an equidimensional space S' and to put some “general” restriction
in its place. Now, we have found in [16] that it suffices instead to demand
that the totality of objects {∕} separates points on D in the following sense.

We assume that in any holomorphic coordinate system around any point
of D, every object /is characterized by a set of components each of which is
a holomorphic function in the given coordinates. Now, take any two points
P, P' on D, different or not, a coordinate neighborhood N : {z} of P, and a
coordinate neighborhood N' : {z'} of P', such that there exists a one-one
holomorphic mapping z' = φ(z) from N to N'. Take all functional elements
{f(z')} in N', and form their transforms {g(z)} ≡ {∕(≠(z))} in TV. Now, our
hypothesis is that whenever the totality of elements {g(z)} is the same,
object by object, as the totality of elements {∕(z)}, then the two points P, P'
are identical, the coordinate systems {z}, {z'} are identical, and <∕>(z) is the
identity mapping.

With this definition we proved in [16] the following proposition.

It follows from mere analyticity of the data that any ensemble {D,f}
whatsoever always has maximal extensions {D,f}, but there are usually
many such. If however the ensemble {D,f} has the separation property
just described, and if we consider extensions {D,f} with the same separation
property, then the maximal extension is unique.

For complex dimension n ≥ 2 an interesting case of non uniqueness
can be exhibited by use of the Hopf blow-up as follows. Let V" be a compact
complex manifold, say algebraic, and let P⁰ be a point on it. Let D be the
manifold V" — P⁰ (that is, V" minus the one point P⁰), and let f represent
all meromorρhic functions on V" which are non singular at P⁰. The “natural”
maximal extension is D = V", and the resulting {D, f} is indeed maximal
because D, being compact, is non continuable. However, instead of adding
merely the point P⁰ we can also add, by performing a Hopf blow-up, a
projective space P"~^l of n — 1 complex dimensions. The resulting complex
manifold

D' = {y>< _ po} ↑jpn-ι

is again compact, and there is a corresponding maximal extension

{D',f'},

in which f∖P) arises from/(P) by assigning to all points P of Pⁿ~^l the
constant value/(P⁰). In this new maximal extension, {∕^,} no longer separates
points.

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