On Nevanlinna's second main theorem in projective space Article DOI:
Cite this article as: Ye, Z. Invent Math (1995) 122: 475. doi:10.1007/BF01231453 Summary
We first prove a theorem concerning higher order logarithmic partial derivatives for meromorphic functions of several complex variables. Then we show the best nature of the second main theorem in Nevanlinna theory under two different assumptions of non-degeneracy of meromorphic mappings
f : ℂ → ℙ n for arbitrary positive integers n n and m. Moreover, we derive a upper bound of the error term in the second main theorem for meromorphic mappings of finite order. Finally, we demonstrate the sharpness of all upper bounds in our main theorems.
Oblatum 28-IX-1994 & 29-V-1995
A. Biancofiore, W. Stoll: Another proof of the lemma of the logarithmic derivative in several complex variables. In: J. Fornaess (ed.) Recent developments in several complex variables, pp. 29–45, 1981
J. Carlson, P. Griffiths: A defect relation for equidimensional holomorphic mappings between algebraic varieties. Ann. of Math.,
W. Cherry: The Nevanlinna error term for coverings generically surjective case. In: W. Stoll, editor, Proceedings symposium on value distribution theory in several complex variables, pp. 37–54. University of Notre Dame Press, 1992
A. Gol'dberg, A. Grinshtein: The logarithmic derivative of a meromorphic function. Matematicheskie Zametki,
W. Hayman: Meromorphic Functions. Clarendon Press, 1975
A. Hinkkanen: A sharp form of Nevanlinna's second fundamental theorem. Invent. math.
A. S. Kolokolnikov: On the logarithmic derivative of a meromorphic function. Matematicheskie Zametki,
S. Lang: Introduction to complex hyperbolic spaces. Springer, Berlin Heidelberg New York 1987
S. Lang: The error term in Nevanlinna theory. Duke Math. J.,
S. Lang: The error term in Nevanlinna theory. II. Bull. of the AMS,
S. Lang, W. Cherry: Topics in Nevanlinna Theory. volume 1433 of Lecture Notes in Math. Springer Berlin Heidelberg New York 1990
J. Miles: A sharp form of the lemma on the logarithmic derivative. J. London Math. Soc.
B. Shiffman: Holomorphic curves in algebraic manifolds. Bull. Am. Math. Soc.
B. Shiffman: On holomorphic curves and meromorphic maps in projective space. Indiana Univ. Math. J.
L. R. Sons, Z. Ye: The best error terms of classical functions (to appear in Complex Variables)
A. Vitter: The lemma of the logarithmic derivative in several complex variables. Duke Math J.
P. Vojta: Diophantine Approximations and Value Distribution Theory. volume 1239 of Lecture Notes in Math. Springer-Verlag, 1987
P. M. Wong: On the second main theorem of Nevanlinna theory. Am. J. Math.
P. M. Wong, W. Stoll: Second main theorem of Nevanlinna theory for non-equidimensional meromorphic maps. Am. J. Math.
Z. Ye: The error term of holomorphic mappings in Nevanlinna theory. Proc. AMS
Z. Ye: The error terms of most meromorphic functions in value distribution theory. (preprint)
Z. Ye: A sharp form of Nevanlinna's second main theorem of several complex variables (to appear in Math. Z)
Z. Ye: On Nevanlinna's error terms. Duke Math. J.