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The concepts of general position and a second main theorem for non-linear divisorsy

H.F. Law
P.P.W. Wong
P.P.W. Wong
Journal / Anthology

Complex Variables and Elliptic Equations
Year: 2011
Volume: 56
Issue: 1-4
Page range: 375-398

The recent works of Evertse–Ferretti (Evertse and Ferretti, A generalization of the subspace theorem with polynomials of high degree, Dev. Math. (2008), pp.175–198) and Corvaja–Zannier (Corvaja and Zannier, On a general Theu’s equation, Ann. Math. 160 (2004), pp. 705–726; Corvaja and Zannier, On integral points on surfaces, Amer. J. Math. 126 (2004), pp. 1033–1055) in diophantine approximations (Ru, A defect relation for holomorphic curves intersecting hypersurfaces, Amer. J. Math. 126 (2004), pp. 215–266; An, A defect relation for non-Archimedean analytic curves in arbitrary projective varieties, Proc. Amer. Math. Soc. 135 (2007), pp. 1255–1261) in complex and p-adic Nevanlinna theory extend the classical subspace theorem and the classical second main theorem to the case of non-linear divisors in general position. These had been long standing problems in diophantine approximation and Nevanlinna theory. However, their results when specialized to the case of hyperplanes are weaker than the classical results. In this article, we refine the concept of general position to the concepts of p-jet general position. These concepts of general position involve jets of order p and coincide with the usual concept of general position for hyperplanes, but are different for hypersurfaces of higher degrees. With the assumption that the hypersurfaces are in n-jet general position, a second main theorem, with ramification term, for non-linear divisors and d-non-degenerate map f :C!Pn is obtained. The result when specialized to hyperplanes is precisely the classical result of Ahlfors (The theory of meromorphic curves, Acta Soc. Sci. Fenn. 3 (1941), pp. 1–31) (see also Cartan, Sur les zeros des combinations lineares de p fonctions holomorphes donees, Mathematica 7 (1933), pp. 5–31; Stoll, About the value distribution of holomorphic maps into the projective space, Acta Math. 123 (1969), pp. 83–114; Cowen and Griffiths, Holomorphic curves and metrics of nonnegative curvature, J. Anal. Math. 29 (1976), pp. 93–153; for small functions, see Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192 (2004), pp. 225–294). In fact the proof is a modification of Ahlfors proof. There are a number of variations of Ahlfors proof, we choose the ‘approximate negatively curved’ approach used in Cowen and Griffiths (Holomorphic curves and metrics of non-negativecurvature, J. Anal. Math. 29 (1976), pp. 93–153), Wong (Defect relation for meromorphic maps on parabolic manifolds and Kobayashi metrics on Pn omitting hyperplanes, Ph.D. thesis, University of Notre Dame (1976)), Wong (Holomorphic curves in spaces of constant curvature, in Complex Geometry (Proceedings of the Osaka Conference, 1990), Lecture Notes in Pure and Applied Mathematics, Vol. 143, Marcel Dekker, New York, Basel, Hong Kong, 1993, pp. 201–223), Wong (On the second main theorem of Nevanlinna theory, Amer. J. of Math. 111 (1989), pp. 549–583) and Cowen (The Kobayashi metric on Pn n (2nþ1) hyperplanes, in Value Distribution Theory, Part A, R.O. Kujala and A.L. Vitter III, eds., Marcel Dekker, New York, 1974, pp. 205–223).