Global moduli spaces and simple holomorphic bundles
The authors prove the following theorem: Let S be a complex space and F a groupoid defined on the category (An/S) of complex spaces over S with values in the category of sets. Let \(| F|\) denote the associated isomorphism class functor and let \(k(s)\cong {\mathbb{C}}\) be te residue field of \(s\in S\). Assume that: (1) Each object of \(| F| (Spec(k(s)))\), \(s\in S\), possesses a versal deformation and versality is an open property. (2) For all objects of \(a,b\in F(T)\), \(T\in (An/S)\), the isomorphism functor \(Isom^ F_ T(a,b)\) is representable by a separated morphism \(Z(a,b)\to T\) such that \(Z(a,a)\) is smooth over T and the operation of \(Z(a,a)\) on \(Z(a,b)\) is locally proper (resp. proper). Then \(| F|\) has a coarse moduli-space (resp. one with a separated structure map). The proof consists of a local construction and a gluing procedure and shows actually a bit more. Namely that the sheafified functor associated to \(| F|\) is representable. The theorem is then applied to \(F=Coh_ S(X/S)\) (where \(X\to S\) is a fixed separated map of complex spaces) which associates to \(T\in (An/S)\) the set of coherent \({\mathcal O}_{X_ T}\)-modules F, flat and proper over S, which are simple (i.e. \(T\times {\mathbb{C}}\to End_ X(F)\), \((t,\lambda)\mapsto \lambda id_{F(t)}\), is an isomorphism of linear fibre spaces over T). The result is that the functor \(| Coh_ S(X/S)|\) of isomorphism classes of simple coherent sheaves on X/S has a coarse moduli space. In particular, simple coherent sheaves and simple holomorphic bundles on a fixed compact complex space have a coarse moduli space. Note that these moduli spaces are in general not separated. It should be mentioned that if \(X\to S\) is a locally projective, finitely presented morphism of schemes, then \(| Coh_ S(X/S)|\) has a coarse moduli space which is a quasi-separated algebraic space in the sense of M. Artin. This was proved before by \textit{A. B. Altman} and \textit{S. L. Kleiman} [Adv. Math. 35, 50-112 (1980; Zbl 0427.14015), Theorem 7.4]. Moreover, the results of the authors have been generalized by Flenner and Sundararaman to complex superspaces (Preprint, Göttingen 1989).
moduli space of simple holomorphic bundles
coarse moduli space of analytic functors
scientific article; zbMATH DE number 4113002
scientific article