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Introduction to Value Distribution Theory of Meromorphic Maps

Dedicated to the memory of Aldo Andreotti

  • Conference paper
Complex Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 950))

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.

This work was partially supported by an NSF grant MCS 80-03257.

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  1. University of Notre Dame, Notre Dame, IN, 46556, USA

    Wilhelm Stoll

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Stoll, W. (1982). Introduction to Value Distribution Theory of Meromorphic Maps. In: Complex Analysis. Lecture Notes in Mathematics, vol 950. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0061879

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  • DOI: https://doi.org/10.1007/BFb0061879

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