Foliated Mori Theory
Hyperbolicity of Algebraic Surfaces


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. $ 13\mathrm{c}_1^2>9\mathrm{c}_2$. 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.

Overview of the Proof
Change Log

Michael McQuillan, 15/07/2012