A New Class of Domains of Holomorphy (III) (Reinhardt domains of holomorphy on a 3-dimensional analytic set with a $(ℂ^*)^2$-action)

The present paper is the third part of the study on domains of holomorphy under the same title ([1] and [2]). In this paper we shall give a supplement of the second paper and complete the discussions there. We remark on the notations in this paper. If we say an analytic set M9 we mean that it is a 3-dimensional analytic set which is defined by polynomials in C and with an isolated singularity pQ. p0 is assumed to be the origin of C,. In the first paper, a new class of domains of holomorphy is introduced and they are called L-mam'folds ([!]). In the second paper, we have treated domains of holomorphy on 3-dimensional Stein spaces and we have given examples of L-manifolds ([2]). There we have shown that certain domains of holomorphy are L-manifolds under certain conditions. The condition is stated as the condition A and domains are called simple domains (see Introduction in [2]). Unfortunately function-theoretic meanings of the condition A and simple domains have not been given there. In this paper we shall remove these additional restrictions and generalize the examples to a certain general situation. For this purpose we consider a 3-dimensional analytic set with an isolated singularity which admits a (C*) -action (§ 1). Then we see that the condition A can be satisfied on such an analytic set. Moreover, if we define the concept