/pdf/another-proof-of-the-defect-relation-for-moving-targets-56qdkrgts7.pdf

Another proof of the defect relation for moving targets

Basic Algebraic Geometry

Characteristics of the behavior of a meromorphic function and the first fundamental theorem Meromorphic functions of finite order The second fundamental theorem Deficient values Asymptotic properties of meromorphic functions and deficiencies Value distribution with respect to the arguments Applications of Riemann surfaces to value distribution On the magnitude of an entire function Notes A survey of some results after 1970 Bibliography References added to the English edition Author index Subject index Notation index.

Value Distribution of Meromorphic Functions

It is well known that the integrability (solvability) of a differential equation is related to the singularity structure of its solutions in the complex domain - an observation that lies behind the Painleve test. A number of ways of extending this philosophy to discrete equations are explored. First, following the classical work of Julia, Birkhoff and others, a natural interpretation of these equations in the complex domain as difference or delay equations is described and it is noted that arbitrary periodic functions play an analogous role for difference equations to that played by arbitrary constants in the solution of differential equations. These periodic functions can produce spurious branching in solutions and are factored out of the analysis which concentrates on branching from other sources. Second, examples and theorems from the theory of difference equations are presented which show that, modulo these periodic functions, solutions of a large class of difference equations are meromorphic, regardless of their integrability. It is argued that the integrability of many difference equations is related to the structure of their solutions at infinity in the complex plane and that Nevanlinna theory provides many of the concepts necessary to detect integrability in a large class of equations. A perturbative method is then constructed and used to develop series in z and the derivative of log (z ), where z is the independent variable of the difference equation. This method provides an analogue of the series developed in the Painleve test for differential equations. Finally, the implications of these observations are discussed for two tests which have been studied in the literature regarding the integrability of discrete equations.

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On the extension of the Painlevé property to difference equations

These notes are an expanded version of lectures delivered at the AMS Summer School on Algebraic Geometry, held at Santa Cruz in July 1995 The main goal of the notes is to study complex varieties (mostly compact or projective algebraic ones), through a few geometric questions related to hyperbolicity in the sense of Kobayashi A convenient framework for this is the category of “directed manifolds”, that is, the category of pairs (X, V ) whereX is a complex manifold and V a holomorphic subbundle of TX  If X is compact, the pair (X, V ) is hyperbolic if and only if there are no nonconstant entire holomorphic curves f : C → X tangent to V (Brody’s criterion) We describe a construction of projectivized kjet bundles PkV , which generalizes a construction made by Semple in 1954 and allows to analyze hyperbolicity in terms of negativity properties of the curvature More precisely, πk : PkV → X is a tower of projective bundles over X and carries a canonical line bundle OPkV (1) ; the hyperbolicity of X is then conjecturally equivalent to the existence of suitable singular hermitian metrics of negative curvature on OPkV (−1) for k large enough The direct images (πk)⋆OPkV (m) can be viewed as bundles of algebraic differential operators of order k and degree m, acting on germs of curves and invariant under reparametrization Following an approach initiated by Green and Griffiths, we establish a basic Ahlfors-Schwarz lemma in the situation when OPkV (−1) has a (possibly singular) metric of negative curvature, and we infer that every nonconstant entire curve f : C → V tangent to V must be contained in the base locus of the metric This basic result is then used to obtain a proof of the Bloch theorem, according to which the Zariski closure of an entire curve in a complex torus is a translate of a subtorus Our hope, supported by explicit Riemann-Roch calculations and other geometric considerations, is that the Semple bundle construction should be an efficient tool to calculate the base locus Necessary or sufficient algebraic criteria for hyperbolicity are then obtained in terms of inequalities relating genera of algebraic curves drawn on the variety, and singularities of such curves We finally describe some techniques introduced by Siu in value distribution theory, based on a use of meromorphic connections These techniques have been developped later by Nadel to produce elegant examples of hyperbolic surfaces of low degree in projective 3-space; thanks to a suitable concept of “partial projective projection” and the associated Wronskian operators, substantial improvements on Nadel’s degree estimate will be achieved here 2 J-P Demailly, Kobayashi hyperbolic projective varieties and jet differentials

Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials

Nevanlinna Theory for Meromorphic Functions and Roth's Theorem Holomorphic Curves into Compact Riemann Surfaces and Theorems of Siegel, Roth, and Faltings Holomorphic Curves in Pn(C) and Schmidt's Sub-Space Theorem The Moving Target Problems Equi-Dimensional Nevanlinna Theory and Vojta's Conjecture Holomorphic Curves in Abelian Varieties and the Theorem of Faltings Complex Hyperbolic Manifolds and Lang's Conjecture.

Nevanlinna Theory and Its Relation to Diophantine Approximation

In 1933, H. Cartan proved a defect relation Σ q j =1 δ f ( H j ) ≤ n + 1 for a linearly nondegenerate holomorphic curve f : [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] and hyperplanes H j , 1 ≤ j ≤ q , in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] in general position. This paper extends it to holomorphic curves intersecting hypersurfaces. In 1979, B. Shiffman conjectured that if f : [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] is an algebraically non-degenerate holomorphic map, and D 1 , . . . , D q are hypersurfaces in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /] in general position, then Σ q j =1 δ f ( D j ) ≤ n + 1. This paper proves this conjecture.

A defect relation for holomorphic curves intersecting hypersurfaces

This paper establishes a defect relation for algebraically nondegenerate holomorphic mappings into an arbitrary nonsingular complex projective variety V (rather than just the projective space) intersecting possible nonlinear hypersurfaces, extending the result of H. Cartan.

Holomorphic curves into algebraic varieties

The defect relation for holomorphic maps
 
$$f:\mathbb{C} \to \mathbb{P}_n $$

 in respect to slowly moving target hyperplanes in
 
$$\mathbb{P}_n $$

 is proved. The sharp defect boundn+1 is obtained.

The second main theorem for moving targets

In this paper, we prove a uniqueness theorem for holomorphic curves with moving targets without counting multiplicity.

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Min Ru

Papers

Nevanlinna Theory and Its Relation to Diophantine Approximation

A defect relation for holomorphic curves intersecting hypersurfaces

Holomorphic curves into algebraic varieties

The second main theorem for moving targets

A uniqueness theorem with moving targets without counting multiplicity