Meromorphe Äquivalenzrelationen. Anwendungen, Beispiele, Ergänzungen. (Meromorphic equivalence relations. Applications, examples, supplements) (Q1106363)

Suppose X is a normal complex space. A normal complex equivalence relation on X is a subvariety \(R\subset X\times X\) which is an equivalence relation and such that (i) each fibre \(X_ x=R_ n(X\times \{x\})\subset X\) has codimension c, (ii) the natural projection \(R\to X\) is open. In this case the quotient Q is a pure c-dimensional normal complex space. This article is concerned, however, with meromorphic equivalence relations whereby \(R\subset X\times X\) is only normal outside a nowhere dense analytic subset \(P\subset X\). In a previous paper the author has shown that, under a natural technical assumption of regularity, the quotient is still a normal complex space. Any meromorpic equivalence relation with codimension \(c=1\) is regular. However, in this article he shows that there is a 3-dimensional complex manifold X with a (non- regular) meromorphic equivalence relation of codimension 2 such that the quotient space is not complex. For codimension 2, however, he shows that a weaker notion of ``quotient'' always exists. This notion of m-Basis is due to Stein and is defined by means of a universal mapping property. For codimension 3 this fails too - although an m-Basis may exist, it can turn out to have dimension 2.