On Mcquillan’s “tautological Inequality” and the Weyl-ahlfors Theory of Associated Curves

In 1941, L. Ahlfors gave another proof of a 1933 theorem of H. Cartan on approximation to hyperplanes of holomorphic curves in Pn . 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. Cartan’s theorem on value distribution with respect to hyperplanes in P in general position [Ca] remains to this day as one of the pillars of value distribution of holomorphic curves in higher-dimensional spaces. Theorem 0.1 (Cartan). Let n ∈ Z>0 , and let H1, . . . ,Hq be hyperplanes in P n C in general position (i.e., all collections of up to n + 1 of the corresponding linear forms on C are linearly independent). Let f : C → P be a holomorphic curve whose image is not contained in any hyperplane. Then