THE RISE OF FUNCTIONS
19
We remark that a quotient Ofholomorphic functions at P0 can be similarly
extended from a neighborhood of P0 to a neighborhood of Pn~l by extending
numerator and denominator separately, so that it was not even necessary
to exclude from the class {∕} meromorphic functions on Vn that are non
regular at P0.
If now we couple this remark with the observation that the Hopf blow-up
of an algebraic variety is again algebraic [17], we arrive at the following
insight which ought to have a sobering effect on any devotee of functional
analysis of our times. A compact complex manifold Vn, of complex dimension
n ≥ 2, even when algebraic, cannot be “completely characterized” by the
assemblage of [holomorphic and) meromorphic functions on it, even if to the
scalar functions all possible tensorial functions [that is, tangential vector
bundles) be added. For n ≥ 3, even non tangential holomorphic vector bundles
may be added (see [IS, pp. 192-195]).
A Parting Thought. The statement just made and its rationale exemplify
a developing trend that bids fair to take over and prevail in geometrically
oriented analysis for decades to come. During nearly a hundred years, since
after 1870, geometrically controlled analysis was searching for and striving
to articulate “harmonies,” “symmetries,” “homogeneities”; and among
cognoscenti, the credal inspiration for this mathematical state of mind drew
from something called the Erlanger Program, whatever that was. Mighty
achievements ensued: the theories of Lie groups, Lie algebras, symmetric
spaces, even, in part, of automorphic functions among such. And yet, all
along, something new and different was burgeoning, something that tried
to overcome, or at least to make itself independent of, the “retrogressive”
seeking of bigger and better symmetries and homogeneities. All truly
exciting achievements in analytical and differential topology of recent years,
beginning with the pioneering efforts of Marston Morse decades ago, have
been of this novel kind, and they seem to be a truer fulfillment of the general
aspirations of our century than that which had preceded. And there is
wisdom to such aspirations. For instance, in the realm of several complex
variables, nothing is less accessible to present-day analysis than a compact
complex manifold that is simply connected and does not have a single
complex automorphism acting upon it. Yet a “random” compact complex
manifold is probably of this kind, and it is crying out for something to be
done about it.
In the second half of the 20th century, our universe of thought, feeling,
perception, and physical and cosmological reality somehow refuses to be
placidly “symmetrical,” and if it is symmetrical, then only in a crude surface