On McQuillan's "tautological inequality" and the Weyl-Ahlfors theory of associated curves

TL;DR

This paper presents a new proof of a classical theorem on holomorphic curves in projective space, utilizing McQuillan's tautological inequality and algebraic geometry techniques, with implications for diophantine conjectures.

Contribution

It introduces a novel proof approach based on successive minima and McQuillan's inequality, emphasizing algebraic geometry over differential geometric methods.

Findings

New proof of Cartan's theorem using algebraic geometry techniques.

Encapsulation of analysis within a modified McQuillan-like inequality.

Proposal of a diophantine conjecture related to McQuillan's inequality.

Abstract

In 1941, L. Ahlfors gave another proof of a 1933 theorem of H. Cartan on approximation to hyperplanes of holomorphic curves in P^n. Ahlfors' proof built on earlier work of H. and J. Weyl (1938), and proved Cartan's theorem by studying the associated curves of the holomorphic curve. This work has subsequently been reworked by H.-H. Wu in 1970, using differential geometry, M. Cowen and P. A. Griffiths in 1976, further emphasizing curvature, and by Y.-T. Siu in 1987 and 1990, emphasizing meromorphic connections. This paper gives another variation of the proof, motivated by successive minima as in the proof of Schmidt's Subspace Theorem, and using McQuillan's "tautological inequality." In this proof, essentially all of the analysis is encapsulated within a modified McQuillan-like inequality, so that most of the proof primarily uses methods of algebraic geometry, in particular flag varieties. A diophantine conjecture based on McQuillan's inequality is also posed.