On the Inverse of Monoidal Transformation

Abstract

When we have a complex analytic manifold and a complex analytic submanifold of codimension r^2 of Xy we can form the monoidal transform of with centre M. (By a manifold, we shall understand a paracompact connected one through this paper.) is a complex analytic manifold with the same dimension n as X, there exists a holomorphic mapping n from onto X, and n is an analytic homeomorphisrn between X—S and X—M, where S=n~(M^). (More properly, we should say (X, n) is the monoidal transform of X) S is an analytic submanifold of of codimension 1, and is in a peculiar position in X: The restriction of n to 5 makes x:S-*M an analytic fibre bundle with projective (r — 1)-space as the standard fibre. (More specifically, 5 is the normal bundle of in X, with the zero cross section deleted and divided by the group C* operating as multiplication by constants on each fibre.) If we denote the fibre 71^(0) by La(a^M), then we have [S]za= M , where [5] and [e] denote the complex line bundles defined by the divisor 5 of and the hyperplane e of P~ = La respectively, and [S]La denotes tne restriction of [5] to La. Now the inverse problem of the monoidal transformation is the following: Suppose we have a complex analytic manifold X and a submanifold 5 of of codimension 1. Let S have a structure of a holomorphic fibre bundle over an analytic manifold with projective (r— l)-space as a standard fibre (m + r = ri). Then under what condi-