Basics on Douady spaces

I am interested in some basic references for Douady spaces (which are analytic analogies of Hilbert schemes).

The point is that I would be happy to work with Douady spaces of non-Kähler (even not Moishezon or Fujiki-$\mathcal{C}$) spaces.

I hope the whole intuition coming from Hilbert schemes(their existence, compactness, smooth points, dimension of tangent bundle etc.)will stay relevant, but wanted to get sure that I didn't miss any hidden rock.

Douady space plays an important role in the Griffiths result on the triviality of imbeddings see André Hirschowitz paper about it. (for me it is an amazing result)

The moduli problem for compact analytic subspaces of a given complex space was developed in the 1960s by Douady. The moduli space of compact complex submanifolds of a given complex manifold is known as the Douady space. The Kodaira theory of the complete family of compact complex submanifolds enables us to describe it, locally around a regular point, by means of global sections of the normal bundle. Let $X\subset Y$ be a compact complex submanifold of a complex manifold. If $Y$ is a holomorphic fibre bundle over $X$. According to Kodaira's theory of a complete family of compact complex submanifolds of a complex manifold, if $H^1(X,N)=0$, then the moduli space $M$ of displacements of $X$ in $Y$ (it is called the Douady space) is smooth at the reference point $t_0\in M$, and $T_{t_0}M\cong H^0(X,N)$ holds, where $N$ is the normal bundle.

When $X$ is projective, the Douady space is the complex space associated with the Hilbert scheme of $X$, while the cycle space is the complex space associated with the Chow scheme.

A typical example of symplectic manifolds is the Douady space $S^{[r]}$ of finite analytic subspaces of length $r$ on a $K3$ surface $S$. This is a canonical desingularization of the product $S^{(r)}$ of $S$.

Makoto showed the following theorem: Let $W$ be a complex manifold. $(X,\pi,S)$ is called a family of compact complex submanifolds of $W$ if

(i) $S$ is a reduced analytic space,

(ii) $X$ is a subvariety of $W×S$, and

(iii) $\pi$ is the restriction to $X$ of the projection map $W\times S\to S$ and $\pi:X\to S$ is proper and surjective.

The set of all compact complex submanifolds of $W$ forms a (not necessarily connected) analytic space $S(W)$. $S(W)$ is then identified with an open subspace of the Douady space of all compact complex subvarieties of $W$, Douady, Adrien, Le problème des modules pour les sous-espaces analytiques compacts d'un espace analytique donné, Annales de l'institut Fourier, Volume 16 (1966) no. 1 , p. 1-95.

Let me state one of the important theorems of Douady.

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 support. A. Douady has proved that $\mathcal D_X (E)$ can be naturally provided with the structure of a complex space.

Some important results which are known

-A.Fujiki introduced the relative Douady space,

-Indranil Biswas and Georg Schumacher introduced the Generalized Weil-Petersson metric on the Douady space of embedded manifolds. This WP-metric is the curvature of a (virtual) determinant bundle equipped with its Quillen metric. There is a same result of the second author with another one in another journal.

-In this paper the authors prove the standard realization of the direct image complex via the so-called Douady–Barlet (Hilbert-Chow in the algebraic case) morphism associated with a smooth complex analytic surface admits a natural decomposition in the form of an injective quasi-isomorphism of complexes. This is a more precise form of a special case of the decomposition theorems of Beilinson, Bernstein, Deligne, Gabber, and M. Saito.

-The authors study the Douady space $\mathcal D(X)$ of 0-dimensional analytic cycles of a complex surface $X$, which is the non-algebraic analog of the Hilbert scheme $H(X)$ of points of an algebraic surface. In the case when $X$ is a projective algebraic surface, the cohomology space $H(X)=H^∗(H(X),\mathbb C)$ admits a structure of an irreducible linear representation of a Heisenberg algebra which correspond to Vafa-Witten theory.

-Let $M$ be a compact complex manifold equipped with a hyper-Kähler metric, and $X$ a closed complex analytic subvariety of $M$. Misha Verbitsky showed that the Douady space of complex analytic deformations of X in M is equipped with a natural hyper-Kähler structure.

-Lieberman and Fujiki showed that if $X$ is a compact Kähler manifold, then the connected components of the Douady and Barlet spaces are compact. Barlet space is very important in the study of structure theorems in Fujiki class $\mathcal C$.

-In this paper you can see locally trivial families of Douady space.

-Let $X$ be a compact complex manifold, $Y⊂X$ a compact submanifold. Let $T_X$ be the tangent bundle of $X$. If $D$ denotes the Douady space of deformations of $Y$ in $X$, $K$ the Kuranishi space of deformations of $Y$ (as complex manifold), there is a holomorphic map $i:D\to K$. Kalka showed that $i$ is an isomorphism provided $H^0(Y,TX|_Y)=0, H^1(Y,TX|_Y)=0$

Let $D_d^{\prime}(X)$ be the reduction of a connected component of the Douady space of purely $d$-dimensional compact complex subspaces of $X$, then the natural morphism $D_d^{\prime}(X)\to C_d(X)$, where $C_d(X)$ is the Barlet space of $d$-dimensional cycles of $X$ is proper.