The lemma of the logarithmic derivative in several complex variables

Vitter, Al

doi:10.1215/s0012-7094-77-04404-0

Cited by 112 publications

(50 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Differentiation of (8) (write the integral over rB in polar coordinates) gives (11) Q (r) = 2nr 2n−1 P (r).…”

Section: 2mentioning

confidence: 99%

“…Friedland and Milnor [5] proved this, in the polynomial context, by using degrees of polynomials as a measure of growth. As was done in [1], we instead use some properties of the Nevanlinna characteristic of functions f : C n → C. One of these, namely that the characteristic of the gradient of an entire function f is dominated by that of f , has previously been proved via the much more difficult so-called "lemma of the logarithmic derivative" in several variables [1], [11]. We give a very simple proof of the gradient inequality in an Appendix.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Periodic automorphisms of C^n

Ahern¹,

Rudin²

1995

Indiana Univ. Math. J.

View full text Add to dashboard Cite

show abstract

“…Differentiation of (8) (write the integral over rB in polar coordinates) gives (11) Q (r) = 2nr 2n−1 P (r).…”

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

Periodic automorphisms of C^n

Ahern¹,

Rudin²

1995

Indiana Univ. Math. J.

View full text Add to dashboard Cite

show abstract

“…Next Theorem 2.5.1, called Lemma on logarithmic derivatives, is most basic in our paper. Lemma on logarithmic derivatives is first discussed by Nevanlinna [Nev39] for a meromorphic function on C, and then by Vitter [Vi77] for a meromorphic function on C m . Following general form is obtained by Noguchi [No85].…”

Section: 4mentioning

confidence: 99%

“…After Nevanlinna's work, A. L. Vitter [Vi77] generalized this lemma to a meromorphic function on the m-dimensional complex affine space C m .…”

Section: §1 Introductionmentioning

confidence: 99%

Algebro-geometric version of Nevanlinna’s lemma on logarithmic derivative and applications

Yamanoi¹

2004

Nagoya Mathematical Journal

View full text Add to dashboard Cite

Abstract. In this paper we shall establish some generalization of Nevanlinna's Lemma on Logarithmic Derivative to the case of meromorphic maps from a finite analytic covering space over the m-dimensional complex affine space C m to a smooth complex projective variety. Then we shall apply this to "the Second Main Theorem" in Nevanlinna theory in several complex variables. §1. Introduction Since a meromorphic function f on C m can be considered as a meromorphic map f : C m P 1 , it is a natural problem to generalize the lemma of the logarithmic derivative to the case of a meromorphic map f : C m X for a complex projective manifold X. In this direction, J. Noguchi [No77], [No85] established the lemma of the logarithmic derivative for a meromorphic map f : Y X and global logarithmic 1-forms on X, where Y π → C m is a finite analytic covering space over C m . And he established the second main theorem type inequality for a meromorphic map f : Y X and a divisor D ⊂ X on X such that there are sufficiently many global logarithmic 1-forms on X with only logarithmic poles on D.The main purpose of this paper is to establish another geometric formulation of the lemma of the logarithmic derivative for a meromorphic map

show abstract

“…In the case that V is an ^-dimensional complex projective space P n (C), Stoll [7] and Vitter [8] proved the Nevanlinna's second main theorem for meromorphic mappings of C m into P n (C) in the following form.…”

Section: For a Divisor Z)( $ 0) In C M We Write N(d T) = F 9mm Andmentioning

confidence: 99%

Algebraic degeneracy theorem for holomorphic mappings into smooth projective algebraic varieties

Aihara

Mori

1981

Nagoya Mathematical Journal

View full text Add to dashboard Cite

The famous Picard theorem states that a holomorphic mapping /: C -• P\C) omitting distinct three points must be constant. Borel [1] showed that a non-degenerate holomorphic curve can miss at most n + 1 hyperplanes in P n (C) in general position, thus extending Picard's theorem (n = 1 [4], they proved that a holomorphic mapping /: C m -> P n (C) omitting any n + 2 hyperplanes in general position must have the image lying in a hyperplane, especially Green showed that the same result holds under the condition that hyperplanes are distinct. Furthermore, in [5] he proved that a holomorphic mapping / of C m into a projective algebraic variety V of dimension n omitting n + 2 non-redundant hypersurface sections must be algebraically degenerate. On the other hand, in the equidimensional case, Carlson and Griffiths [2] obtained a generalization of Nevanlinna's defect relation for holomorphic mappings of C n into an ra-dimensional smooth projective algebraic variety V. By their results, a holomorphic mapping /: C n -> P n (C) having the Nevanlinna's deficiency δ(D) = 1 for a hypersurface D C P n (C) of degree ^ n + 2 with simple normal crossings, must be degenerate in the sence that J f = 0 on C\ While, Noguchi [6] obtained an inequality of the second main theorem type for holomorphic curves in algebraic varieties, thus a holomorphic curve / in an algebraic variety V which has the Nevanlinna's deficiency δ(Σ) = 1 for hypersurfaces Σ with some conditions in V must be algebraically degenerate. In this paper, we shall show that for n+2 ample divisors {D^H with normal crossings, any holomorphic mapping of C m into an n-dimensional smooth projective algebraic variety

show abstract

The lemma of the logarithmic derivative in several complex variables

Cited by 112 publications

References 0 publications

Periodic automorphisms of C^n

Periodic automorphisms of C^n

Algebro-geometric version of Nevanlinna’s lemma on logarithmic derivative and applications

Algebraic degeneracy theorem for holomorphic mappings into smooth projective algebraic varieties

Contact Info

Product

Resources

About