The purpose of these web pages is to serve as a reference point for the author's programme to establish that algebraic surfaces of general type are, modulo a finite number of rational and elliptic curves, hyperbolic, so, inter alia the Green-Griffiths conjecture for the same by way of what is functorially, with respect to the ideas, the Mori theory of foliation by curves. Although the motivating problem is 2 dimensional, it requires, in principle, the study of foliations by curves up to ambient dimension a few hundred, so, to all intents and purposes, arbitrary. The motivating problem may be viewed as a highly structured, and strictly easier, sub-case of the classification problem for foliations by curves in numerical Kodaira co-dimension 1.
That a proof should require its own web site
is somewhat unusual. It is, however, unusually
long. The
current situation with the release of
the residue lemma in dimension 3
is that all pre-requisites for classification
in numerical Kodaira co-dimension 1 and ambient
dimension 3 have been achieved. In particular,
as regards algebraic surfaces,
the programme works perfectly whenever
the surface has enough two jets, e.g.
.
Nevertheless, the residue lemma is
a lemma within a larger whole, and
it did not appear appropriate to
explain the whole in the introduction
to the specific. This, however, creates
a certain lacuna since one must know
how to put the jigsaw together, and
the pieces are somewhat scattered,
albeit Uniform Uniformisation,
modulo ignoring the chapter on residue
theory, is a logically sufficient reference.
As such, these pages are essentially
an appendix to the residue lemma whose
aim is to permit an understanding of
the whole in a matter of hours, as
opposed to the days or weeks that
might be required without them.
