Classifying affine line bundles on a compact complex space

The classification of affine line bundles on a compact complex space is a difficult problem. We study the affine analogue of the Picard functor and the representability problem for this functor. Let be a compact complex space with . We introduce the affine Picard functor which assigns to a complex space the set of families of linearly -framed affine line bundles on parameterized by . Our main result states that the functor is representable if and only if the map is constant. If this is the case, the space which represents this functor is a linear space over whose underlying set is , where is a Poincaré line bundle normalized at . The main idea idea of the proof is to compare the representability of to the representability of a functor considered by Bingener related to the deformation theory of -cohomology classes. Our arguments show in particular that, for = 1, the converse of Bingener’s representability criterion holds

Normal Families of meromorphic mappings of several complex variables into PN(C)

The first aim in this article is to give some sufficient conditions for a family of meromorphic mappings of a domain D in Cn into PN(C) omitting hypersurfaces to be meromorphically normal. Our result is a generalization of the results of Fujimoto and Tu. The second aim is to investigate extending holomorphic mappings into the compact complex space from the viewpoint of the theory of meromorphically normal families of meromorphic mappings.

A definability result for compact complex spaces

A compact complex space X is viewed as a 1-st order structure by taking predicates for analytic subsets of X, X x X, … as basic relations. Let f: X → Y be a proper surjective holomorphic map between complex spaces and set Xy ≔ f−1(y). We show that the setis analytically constructible, i.e.. is a definable set when X and Y are compact complex spaces and f: X → Y is a holomorphic map. The analogous result in the context of algebraic geometry gives rise to the definability of Morley degree.

Discreteness For the Set of Complex Structures On a Real Variety

Let X, Y be reduced and irreducible compact complex spaces and S the set of all isomorphism classes of reduced and irreducible compact complex spaces W such that X × Y ≅ X × W. Here we prove that S is at most countable. We apply this result to show that for every reduced and irreducible compact complex space X the set S(X) of all complex reduced compact complex spaces W with X × Xσ ≅ W × Wσ (where Aσ denotes the complex conjugate of any variety A) is at most countable.

On the douady space of a compact complex space in the category

Let X be a complex space. Let Dx be the Douady space of compact complex subspaces of X [6] and px: Zx→ Dx the corresponding universal family of subspaces of X.