Supplement to “On the Inverse of Monoidal Transformation”

Abstract

In this note we shall fill a gap in the original proof of the Main Theorem in the previous paper ^4], and sharpen the Theorem by showing that the condition (a) alone is sufficient to derive the conclusion of the theorem. This sharpened theorem enables us to solve a problem posed by K. Kodaira in [2]. The authors are grateful to Professor Mudumbai S. Narasimhan for calling our attention to the gap, to him, to Takahiro Kawai and to other friends for discussion about this. § 1. Vanishing of Cohomology In the proof of theorem 1 in Q4], we made too easy use of Serre's duality theorem. Let V be a connected paracompact complex analytic manifold of dimension n and E a holomorphic vector bundle over V. Let us denote by Cp>q=-Cp'q(V, E) the space of E -valued differential forms of type (p, q) and of class C°°, with the topology as given in Q5]. (Cp>q stands for Ap*9 in [5].) The space K%~pt*~q of £*-valued currents of type (n—p, n — q) with compact supports is isomorphic to the topological dual of Cp>9, and the transpose of the sequence