# A Defect Relation for Equidimensional Holomorphic Mappings Between Algebraic Varieties

@article{Carlson1972ADR, title={A Defect Relation for Equidimensional Holomorphic Mappings Between Algebraic Varieties}, author={James A. Carlson and Phillip A. Griffiths}, journal={Annals of Mathematics}, year={1972}, volume={95}, pages={557} }

0. Introduction 1. Notations, terminology, and sign conventions (a) Line bundles and Chern classes (b) Currents and forms in C0 2. Construction of a volume form 3. A second main theorem for non-degenerate maps 4. The defect relation (preliminary form) 5. The first main theorem and defect relation 6. Variants and applications (a) Schottky-Landau theorems (b) Remarks on the case ci(L) + ci(Kv) ? 0 (c) Holomorphic mappings with growth conditions Bibliography

#### 140 Citations

The second main theorem for holomorphic curves into semiabelian varieties II

- Mathematics
- 2008

Abstract We establish the second main theorem with the best truncation level one for the k-jet lift Jk (ƒ) : → Jk (A) of an algebraically non-degenerate entire holomorphic curve ƒ : → A into a… Expand

Two Results in the Global Theory of Holomorphic Mappings

- Mathematics
- 1974

Publisher Summary
The substantial progress in the global study of general holomorphic mappings depends on understanding the following problems: (A) the extent to which the basic results of… Expand

Algebraic degeneracy theorem for holomorphic mappings into smooth projective algebraic varieties

- Mathematics
- 1981

The famous Picard theorem states that a holomorphic mapping f : C → P 1 ( C ) omitting distinct three points must be constant. Borel [1] showed that a non-degenerate holomorphic curve can miss at… Expand

A DEFECT RELATION FOR HOLOMORPHIC CURVES INTERSECTING HYPERSURFACES By MIN RU

- 2004

In 1933, H. Cartan proved a defect relation ∑q j=1 δf (Hj) ≤ n + 1 for a linearly nondegenerate holomorphic curve f : C → Pn(C) and hyperplanes Hj, 1 ≤ j ≤ q, in Pn(C) in general position. This paper… Expand

SOME GENERALIZATIONS OF THE SECOND MAIN THEOREM

- 2014

In [18], the author established a defect relation for holomorphic mappings from C into the projective variety X ⊂ P (C) intersecting divisors of X coming from hypersurfaces in P . The purpose of this… Expand

The second main theorem for holomorphic curves into semi-Abelian varieties

- Mathematics
- 2002

We establish the second main theorem with the best truncation level one Tf (r;L(D)) N1(r; f∗D) + Tf (r)|| for an entire holomorphic curve f : C → A into a semi-abelian variety A and an arbitrary… Expand

Nevanlinna theory of holomorphic mappings for singular divisors

- Mathematics
- 2020

We establish a defect relation for holomorphic mappings for singular divisors, from a non-positively curved complete Kähler manifold into a complex projective algebraic manifold. Let M be a… Expand

The value distribution of holomorphic curves into semi-Abelian varieties

- Mathematics
- 2000

Abstract Let f: C →A be an entire holomorphic curve into a semi-Abelian variety A . Then the Zariski closure of f( C ) is a translate of a semi-Abelian subvariety of A (logarithmic Bloch–Ochiai's… Expand

Holomorphic curves and metrics of negative curvature

- Mathematics
- 1976

In this paper we shall prove some of the basic results in the theory of holomorphic curves using the method of negative curvature which has recently been fruitful in the study of equidimensional… Expand

AN EXTENSION OF THE CARTAN-NOCHKA SECOND MAIN THEOREM FOR HYPERSURFACES

- Mathematics
- 2009

In 1983, Nochka proved a conjecture of Cartan on defects of holomorphic curves in ℂPn relative to a possibly degenerate set of hyperplanes. In this paper, we generalize Nochka's theorem to the case… Expand

#### References

SHOWING 1-10 OF 20 REFERENCES

Some degeneracy theorems for entire functions with values in an algebraic variety

- Mathematics
- 1972

In the first part of this paper we prove the following extension theorem. Let P* be a q-dimensional punctured polycylinder, i.e. a product of disks and punctured disks. Let Wn be a compact complex… Expand

On a Differential-Geometric Method in the Theory of Analytic Stacks.

- Mathematics, Medicine
- Proceedings of the National Academy of Sciences of the United States of America
- 1953

Results to appear in the Acta Mathematica 1954, or in the Report5 of the Michigan Conference of June 1953, together with three earlier papers of this series and papers by Kaplan6 are listed below. Expand

Analytic Functions

- Nature
- 1962

Analytic FunctionsBy R. Nevanlinna, H. Behnke and H. Grauert, L. V. Ahlfors, D. C. Spencer, L. Bers, K. Kodaira, M. Heins, J. A. Jenkins. (Princeton Mathematical Series, No. 24.) Pp. vii + 197.… Expand

Fonctions entières (n variables) et fonctions plurisousharmoniques d'ordre fini dansCn

- Mathematics
- 1964

Die beiden Hauptsätze der Wertverteilungstheorie bei Funktionen mehrerer komplexer Veränderlichen (II)

- Mathematics
- 1953

On a differential-geometric method

- Proc. Nat. Acad. Sci. U.S.A
- 1953

Df Dfr] for all r, and thus N(D, r) < d log r + 0(1). Using (5.12) we have T(L, r) ! d log r

- Df Dfr] for all r, and thus N(D, r) < d log r + 0(1). Using (5.12) we have T(L, r) ! d log r

Differential geometry-its past and future

- Proc. Nice Con- gress