Introduction to Value Distribution Theory of Meromorphic Maps

Abstract

Value distribution of functions of one complex variable has grown into a huge theory. In several complex variables the theory has not been developed to such an extent, but considerable progress has been made. In these notes an introduction to some aspects of the several variable theory is provided. The two Main Theorems are proved for meromorphic maps f : W → ℙ(V) from a complex vector space W into complex projective space, where the intersection of f(W) with hyperplanes is studied. The First Main Theorem is established along standard lines. Vitter’s Lemma of the Logarithmic Derivative is proved by a method of Biancofiore and Stoll. The Second Main Theorere and the Defect Relation is obtained by the original method of H. Cart an following ideas of Vitter. The value distribution theory of Carlson, Griffiths and King for holomorphic maps from parabolic manifolds into projective algebraic varieties is outlined without proof. These results are specialized to holomorphic maps f : W → ℙ(V) where the intersection of f(W) with hypersurfaces of degree p are studied. Here the Carlson, Griffiths, King theory has to require that rank f = dim ℙ(V). It is a difficult open question of this restriction can be removed. We formulate a recent result of A. Biancofiore which gives a partial answer and which is the best answer available at present.KeywordsVector SpaceLine BundleHolomorphic FunctionComplex Vector SpaceHermitian Vector SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.