Value distribution for holomorphic maps inC n

Abstract

The study of value distribution for holomorphic or meromorphic functions of one complex variable culminated in the very beautiful and deep Nevanlinna theory, which improved upon practically all previously known results. In recent years, value distribution for holomorphic and meromorphic functions in C" and generalizations have been extensively developed, based essentially on a similar theory for one variable (cf. Griffiths [3]), and the results follow at least qualitatively and in general quantitatively the classical one variable theory. As soon as one begins to consider holomorphic maps, however, certain anomalies begin to appear. Perhaps the most perplexing of these is the famous Fatou-Bieberbach map, which is a holomorphic homeomorphism of C 2 into C 2 with Jacobian identically one such that the image omits an open set (cf. Bochner and Martin [2]). In order to develop a coherent theory of value distribution for holomorphic maps, one would like to find sets such that the image of C" under a holomorphic map always intersects these sets. The Fatou-Bieberbach map shows that the image can omit any bounded set, but there are unbounded sets which cannot be omitted. For instance, the theory of holomorphic curves shows that if F : C " ~ C n is a non-degenerate holomorphic map, then the image cannot omit (n + 2) hyperplanes in general position (cf. Vitter [7] for proofs and references and Griffiths [3] for generalizations). These sets are of real dimension ( 2 n 2) in C". Our purpose here is to show that there exist other families of unbounded sets which are in some sense much smaller such that the image of C" under any nondegenerate holomorphic map must intersect these sets. These sets will be finite unions of sets of g~ dimension n in Ir" : {w~C":~eg~(w)= . . .=~eg , , (w)=O, holomorphic in C"}. Since the detailed description of these sets is rather complicated, we will not attempt it here in the introduction. Similar results are obtained for bounded domains D in C ~ which have cg2 boundaries provided that IIF[I grows sufficiently fast near the boundary of D. This leads us to introduce the notion of "essential sets" for families of nondegenerate analytic maps, which we define in the following way: if ~ is a family of non-degenerate holomorphic maps mapping a domain D C C" into C", then a set ECC" will be said to be essential for ~" if for every F ~ and every compact