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X , with coefficients in k, and let X denote, as before, the affine n-space defined over k ; then for every 0, in Y(X,) the following integral: defines a holomorphic function Z,,, on the subset of Q(k2) defined by a(w) > 0 ; and it has a meromorphic continuation to the whole Q(k,"). Furthermore the function F$u is in L1(k,) if and only if I FiU(i*) 5 const. max (1, /i*j,)-" Theorem 2. Let b denote a point of Y , and @, an element of 9 ( X v ) satisfying 0, 2 0, 0,(h(b)) > 0 ; then the existence of F,,,(O) implies that the numerical data of h have the property (P,) at b ; and the stronger assumption: implies that the numerical data of h have the property (P) at b.

It is my pleasure to express my gratitude to E. Horikawa who guided me to his theory of deformations of varieties carrying divisors ([3]) which was very helpful in the first part of this study. I am also grateful to T. Shioda and S. Iitaka for their valuable conversations with me in connection with this problem. Notations and conventions In addition to the basic notation introduced in use the following notations and conventions. 1) Available at the Univ. of Tokyo. 5 2, we shall also frequently FROBESICS CORRESPONDENCES For each 0 5 in 5 n and an R,-algebra A, (resp.

9. In €j 9, it is assumed that n 2 1. Let C, be a smooth R,-scheme that extends C and denote by 9, the local ring at the generic point of C,. Then, as an R,-algebra, R, is determined uniquely by C and n. In fact, if Uo is any affine open set of C, the smooth R,-scheme U, that extends Uo is unique up to R,-isomorphisms and 9, is the inductive limit of the rings of sections of open sets of U,. Clearly, 9,is a local ring which is a flat R,-algebra such that 9 , / z = 9, is the function field of C.