Orbifolds, special varieties and classification theory: appendix. (Q1890165): Difference between revisions

This article is a merely technical appendix to the author's foregoing treatise in the same volume [Ann. Inst. Fourier 54, No. 3, 499--630 (2004; Zbl 1062.14014)]. Its purpose is to give, in full detail, some results used there concerning geometric quotients, Zariski regularity, and \(S\)-stability. These results are important in their own right, as they have much wider applications, and they are discussed here for the sake of generality, completeness, coherence, and reference. The first part of this appendix gives a simplified and detailed proof of the existence of geometric quotients of compact connected normal complex spaces modulo certain analytic equivalence relations. This existence theorem was proved by the author himself more than 20 years ago, and that in a special case [Invent. Math. 63, 187--223 (1981; Zbl 0436.32024)]. Now he offers a general proof of the existence of such geometric quotients, which actually can be reduced to the earlier proof in the special case. This requires some subtle technical work of remarkable length, all of which is carried out in full detail and rigor. Basically, the author follows closely his original approach, but essentially simplifying it in one crucial step. Part 2 introduces the notion of Zariski regularity for subsets of arbitrary analytic spaces. This is a weak notion of countable constructibility which is useful in the study of certain fibres of holomorphic maps between complex spaces. Also, the notion of Zariski regularity is shown to have important applications to the construction of meromorphic quotients. Part 3 deals with a stability condition for Zariski-regular subsets of the Chow scheme of a compact connected normal complex space \(X\), again with applications to the study of fibrations. This stability condition was already used by the author in an earlier paper on connectedness properties of compact Kähler manifolds [Contemp. Math. 241, 85--96 (1999; Zbl 0965.32021)].