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Let $f:X \to S$ be a morphism of complex analytic spaces. Then, just like in the case of schemes, we can define the relative Hilbert and Picard functors. For instance, if $\text{An}_{/S}$ denotes de category of complex analytic spaces over $S$, we define a functor $$ \text{Hilb}_{X/S} : \text{An}_{/S}^{\text{op}} \to \text{Set} $$ by taking $\text{Hilb}_{X/S}(S')$ as the set of closed subanalytic spaces of $X\times_SS'$ which are proper and flat over $S'$.

If $f$ is projective (that is, the composition of a closed immersion of $X$ into $\mathbb{C}\mathbb{P}^n \times S$ with the projection to $S$), then is this functor is representable?

I know I can just read carefully all the proofs for schemes and check if everything works in the case of analytic spaces, but I am curious to know if someone has already worked out the details. So this would be more like a reference request.

P.S. Similar questions to rigid analytic spaces.

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  • $\begingroup$ The analogues of Hilbert schemes in the context of proper morphisms of analytic spaces are the "Douady analytic spaces". $\endgroup$
    – Jason Starr
    Commented Oct 13, 2016 at 13:47
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    $\begingroup$ The scheme methods do not help for analytic cases without projectivity hypotheses (and with projectivity hypotheses this has all been done). Hilbert functors do not help with Picard functors in the absence of projectivity, so the latter is a separate hard problem in general. The German complex analysts in the 1970's adapted Artin's criteria to complex-analytic spaces, handling Hilb and Pic there in good generality. For non-arch. fields, formal models and Artin's results on algebraic space Hilb for infinitesimal fibers lead (with work!) to a "generic fiber" that is as good as Douday's space. $\endgroup$
    – nfdc23
    Commented Oct 13, 2016 at 21:16
  • $\begingroup$ Thank you. Do you know some references? $\endgroup$
    – user96873
    Commented Oct 14, 2016 at 10:22
  • $\begingroup$ Try Fujiki, "Closedness of the Douady spaces of compact Kähler spaces", MR0486648, or Lieberman, "Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds", MR0521918. $\endgroup$
    – pgraf
    Commented Oct 14, 2016 at 13:47
  • $\begingroup$ The paper "Relative ampleness in rigid geometry" in Annales Fourier 56 (2006) discusses Hilb, Quot, and Proj in the proper rigid-analytic case when given a line bundle that is ample on fibers. The main content there is setting up such "relative ampleness", including relative analytic Proj, after which one borrows from the scheme case. This does not help in the general proper case. For relative ampleness in the complex-analytic case, see section 1.4 of the paper "The lower semi-continuity of the plurigenera of complex varieties" by Noburu Nakayama; applications to Hilb, etc. go the same way. $\endgroup$
    – nfdc23
    Commented Oct 14, 2016 at 19:25

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For complex spaces there is this paper by Juergen Bingener. Unfortunately it is in German. For the Picard functor the result is the following:

Let $f:X \to S$ be a morphism of complex spaces which is flat, proper and cohomologically flat in dimension $0$. Then the relative Picard functor $Pic_{X/S}$ is representable.

He also gives similar results for the Douady space and some deformations.

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