On some fields of meromorphic functions on fibers

Abstract

1.1. In this paper we consider the extension problem of meromorphic functions on fibers of complex analytic fiber spaces to neighborhoods of the fibers. Let X-> Y be a complex analytic fiber space, where X and Y are normal and connected complex spaces and n is a proper holomorphic mapping of X onto Y with irreducible fibers. We denote by Kt the meromorphic function field of a fiber =n~' (f)9 and by K't the subfield of Kt consisting of all elements of Kt which can be extended to some neighborhoods of Xt. By [6] or [9], the field Kt is isomorphic to a finite algebraic extension of a rational function field. We discuss here the following problem. Let /!, •••,// be meromorphic functions on X and g be a meromorphic function on a fiber Xt which is dependent on flit, •••//,*, where fitt(i = l, • • • , / ) is the analytic restriction of /,to Xt. Then, can we extend the function g to a meromorphic function on some neighborhood of Xtl We can answer this problem as follows. (I) The complement of the set {t^Y any meromorphic function on Xt which is dependent on f l i t , ••, / / , / can be extended to some neighborhoods of Xt} is nowhere dense in Y. The proof of this theorem is essentially due to the Stein factorization of a proper holomorphic mapping. This notion (or the