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Cycle spaces. (English) Zbl 0811.32020
Grauert, H. (ed.) et al., Several complex variables VII. Sheaf- theoretical methods in complex analysis. Berlin: Springer-Verlag. Encycl. Math. Sci. 74, 319-349 (1994).
This is a survey article about main developments in the field during the last thirty years, starting with the work of A. Douady [Ann. Inst. Fourier 16, No. 1, 1-95 (1966; Zbl 0146.311)]. For any complex space \(X\) the Douady space \({\mathcal D} (X)\), which parametrizes all pure dimensional compact complex subspaces of \(X\), carries a natural complex structure. It generalizes the Hilbert scheme for a projective space which was earlier constructed by A. Grothendieck. In section 1 the authors list the main properties of \({\mathcal D}(X)\), in particular the fundamental theorem of A. Fujiki, that for \(X\) in class \(\mathcal C\) (i.e. \(X\) a reduced compact complex space bimeromorphic to a compact Kähler manifold) every irreducible component of \({\mathcal D}(X)\) is compact and its reduction is again in class \(\mathcal C\).
A new major step of progress was made by D. Barlet by constructing for every complex space \(X\) the cycle space \({\mathcal C}(X)\) and showing that \({\mathcal C}(X)\) has a natural complex structure [see D. Barlet, Lect. Notes Math. 482, 1-158 (1975; Zbl 0331.32008)]. The Barlet space \({\mathcal C}(X)\) is a generalization of the Chow scheme for projective \(X\). In 1978 D. Lieberman [Lect. Notes Math. 670, 140-186 (1978; Zbl 0391.32018)] proved, that for \(n \in \mathbb{N}\) the components of the subspace \({\mathcal C}_ n(X)\) of \(n\)-cycles are compact for every compact Kähler manifold \(X\).
The theory of cycle spaces is an important tool for the classification theory of compact complex spaces, in particular for spaces in class \(\mathcal C\). This is related to the fact that the irreducible components of \({\mathcal C}(X)\) are compact and even in class \(\mathcal C\) if \(X\) is so [see the first author, Math. Ann. 251, 7-18 (1980; Zbl 0445.32021)].
In section 3 the authors list several applications in this direction and in section 4 they discuss convexity properties of \({\mathcal C}(X)\) for \(q\)- complete and \(q\)-convex complex spaces \(X\).
For the entire collection see [Zbl 0793.00010].

32J27 Compact Kähler manifolds: generalizations, classification
32F10 \(q\)-convexity, \(q\)-concavity
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces