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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The moduli of substructures of a compact complex space
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by Peter M. Schuster PDF
Proc. Amer. Math. Soc. 126 (1998), 1983-1987 Request permission

Abstract:

We construct a space $W_X$ of fine moduli for the substructures of an arbitrary compact complex space $X$. A substructure $(X,\mathcal {A})$ of $X$ is given by a subalgebra $\mathcal {A}$ of the structure sheaf $\mathcal {O}_X$ with the additional feature that $(X,\mathcal {A})$ is also a complex space; $(X,\mathcal {A})$ and $(X,\mathcal {A’})$ are called equivalent if and only if $\mathcal {A}$ and $\mathcal {A’}$ are isomorphic as subalgebras of $\mathcal {O}_X$.

Since substructures are quotients, it is only natural to start with the fine moduli space $Q_X$ of all complex-analytic quotients of $X$. In order to obtain a representable moduli functor of substructures, we are forced to concentrate on families of quotients which satisfy some flatness condition for relative differential modules of higher order. Considering the corresponding flatification of $Q_X$, we realize that its open subset $W_X$ consisting of all substructures turns out to be a complex space which has the required universal property.

References
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Additional Information
  • Peter M. Schuster
  • Affiliation: Mathematisches Institut der Universität, Theresienstr. 39, 80333 München, Germany
  • Email: pschust@rz.mathematik.uni-muenchen.de
  • Received by editor(s): September 3, 1996
  • Communicated by: Eric Bedford
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 1983-1987
  • MSC (1991): Primary 32G13; Secondary 32G05, 14D22
  • DOI: https://doi.org/10.1090/S0002-9939-98-04815-1
  • MathSciNet review: 1486750