Timeline for Representability of relative Hilbert and Picard functors over analytic spaces
Current License: CC BY-SA 3.0
9 events
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| Sep 23, 2021 at 9:43 | answer | added | Andrei Halanay | timeline score: 1 | |
| Oct 17, 2016 at 9:48 | comment | added | nfdc23 | The forthcoming PhD thesis of Evan Warner proves representability for Pic and related functors in broad generality in the non-archimedean case (by a version of Artin's method adapted to suitable adic spaces). | |
| Oct 14, 2016 at 19:25 | comment | added | nfdc23 | 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. | |
| Oct 14, 2016 at 13:47 | comment | added | pgraf | 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. | |
| Oct 14, 2016 at 10:22 | comment | added | user96873 | Thank you. Do you know some references? | |
| Oct 13, 2016 at 21:16 | comment | added | nfdc23 | 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. | |
| Oct 13, 2016 at 13:47 | comment | added | Jason Starr | The analogues of Hilbert schemes in the context of proper morphisms of analytic spaces are the "Douady analytic spaces". | |
| Oct 13, 2016 at 13:28 | review | First posts | |||
| Oct 13, 2016 at 13:32 | |||||
| Oct 13, 2016 at 13:19 | history | asked | user96873 | CC BY-SA 3.0 |