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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].

##### MSC:
 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