Ja n 20 07 Almost periodic divisors , holomorphic functions , and holomorphic mappings ∗ Favorov

Almost periodic divisors, holomorphic functions, and holomorphic mappings * Favorov S.Ju. Abstract We prove that to each almost periodic, in the sense of distributions, divisor d in a tube T Ω ⊂ C m one can assign a cohomology class from H 2 (K, Z) (actually, the first Chern class of a special line bundle over K generated by d) such that the trivial cohomology class represents the divisors of all almost periodic holomorphic functions on T Ω ; here K is the Bohr compactification of R m. This description yields various geometric conditions for an almost periodic divisor to be the divisor of a holomorphic almost periodic function. We also give a complete description for the divisors of homogeneous coordinates for holomorphic almost periodic curves; in particular, we obtain a description for the divisors of meromorphic almost periodic functions. The classical theory of almost periodic functions has found a lot of applications in various branches of mathematics, from differential equations (cf. [37], [7], [25]) to number theory (cf. [3], [38]). In spite of the fact that the whole theory was originally motivated by problems in complex analysis (H.Bohr [5], p.3), analytic aspects of the theory are less known. Holomorphic almost periodic functions have certain specific properties, mainly because almost periodicity of a holomorphic function causes strong restrictions on the distribution of its values (for example, of its zeroes). The main contributions to the classical theory of holomorphic almost periodic functions of one variable are due to for a detailed presentation of the subject, see [20] and [24].