I want to know the parallel properties of Douady space with respect to Hilbert scheme. For example I want to know what is the irreducible component of Douady space, what if I consider a family of fixed point rational curve in ambient complex manifold etc. or some useful reference? BTW I am not so good at other languages than English.
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3$\begingroup$ Let $X$ be a complex space and let $E$ be a coherent analytical sheaf on $X$. Let $\mathcal D_X(E)$ be the set of all coherent sheaves on $X$ that are quotients of $E$ and have compact supports. Douady has proved that $\mathcal D_X(E)$ can be naturally provided with the structure of a complex space. Later Fujiki showed: If $X$ has countable topology, then also $\mathcal D_X(E)$ $\endgroup$– user21574Dec 6, 2017 at 19:58
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3$\begingroup$ Let $X$ is projective complex space, the Douady space is the complex space associated with the Hilbert scheme of $X$. Note that the Barlet cycle space is the complex space associated with the Chow scheme, and there exists a holomorphic map between these complex spaces, $\endgroup$– user21574Dec 6, 2017 at 20:11
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4$\begingroup$ If you suppose the Douady space as a moduli space of submanifolds of a Kähler manifold , then the Kähler structures on the Douady space has been considered by Fujiki and Varouchas, and also such Douady space has canonical metric called Weil-Petersson metric which can be derived by Bismut-Soulé formula using Grothendieck-Riemann-Roch formula of determinant bundle via Quillen metric . We can extend the same result for logarithmic version of Douady space, i.e. finding logarithmic Weil-Petersson metric on "log Douady space" link.springer.com/article/10.1007/s00229-006-0020-z $\endgroup$– user21574Dec 6, 2017 at 20:21
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2$\begingroup$ Let $X$ be Moishezon, then irreducible components of Douady space are Moishezon also. $\endgroup$– user21574Dec 6, 2017 at 20:39
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I am not certain, but perhaps you are asking about the fact that Douady space of a compact complex analytic space may have non-compact connected components. This happens, for instance, if you let $X$ be the compact complex analytic space obtained from $\mathbb{C}P^3$ by identifying a line $L \subset \mathbb{C}P^3$ and a disjoint plane conic $C\subset \mathbb{C}P^3$ via an isomorphism $\phi:L\to C$.
If you begin with the irreducible component $M_1$ of the Douady space of $X$ parameterizing (among others) the images of lines in $\mathbb{C}P^3$ that are disjoint from $C\cup L$, then those lines could specialize to $L$. Similarly, the irreducible component $M_2$ parameterizing images of conics in $\mathbb{C}P^3$ can specialize to $C$. In $X$, $L$ equals $C$. Thus $M_1$ and $M_2$ intersect. However, conics in $M_2$ could also specialize to a line pair, $L\cup L'$. Now you can repeat the argument replacing $L$ by $C$.
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1$\begingroup$ Hi,Jason. Do you know Fujiki's theorem that if X is Moishezon, the irreducible component of Douady space is compact if the total space of this component is reduced. I don't know how to check the reduced property in reality. If it's right only assume it's reduced in general points? $\endgroup$ Aug 8, 2014 at 12:22
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1$\begingroup$ @user42804: I did not know Fujiki's theorem; thank you for letting me know about it. In the example I sketch above, the irreducible components are, indeed, compact. The issue is that one connected component may contain infinitely many irreducible components. $\endgroup$ Aug 8, 2014 at 13:34