Addendum to ?The Universal Central Extension of the Holomorphic Current Algebra?

Abstract

We supply an argument missing in the proof of Theorem 3.3 in [2]. If X is a complex manifold, then the algebra O(X) of holomorphic complex-valued functions on X is a commutative Frechet algebra. The main result of [2] (Theorem 3.3) is that the de Rham differential d : O(X) → Ω(X) into the Frechet O(X)-module Ω(X) of holomorphic 1-forms on X is universal whenever X is a Riemannian domain over a Stein manifold. However, in the proof of this theorem we have assumed implicitly that the canonical map iX : X → X of X into its enevelope of holomorphy X := Hom(O(X),C ) is injective, i.e., that the holomorphic functions on X separate points. This is not always the case, as the following example ([1], p.101) shows. Example 0.1 In C 2 we consider the two Reinhardt domains X1 := {(z1, z2) ∈ C 2 : |z1| < 2, |z2| < 2} \ {(z1, z2) ∈ C 2 : |z1| = 1, |z2| ≤ 1} and X2 := {(z1, z2) ∈ C 2 : |z1| < 2, |z2| < 1}. We define the set X := (X1∪X2)/ ∼ where the equivalence relation on the disjoint union of X1 and X2 is defined by identifying all point (z1, z2) ∈ X1 with 1 < |z1| < 2 and |z2| < 1 with the corresponding points in X2. We write [(z1, z2)] for the equivalence class of (z1, z2) ∈ X1∪X2 in X . Then the obvious