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Proceedings of the American Mathematical Society

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The moduli of substructures
of a compact complex space

Author: Peter M. Schuster
Journal: Proc. Amer. Math. Soc. 126 (1998), 1983-1987
MSC (1991): Primary 32G13; Secondary 32G05, 14D22
MathSciNet review: 1486750
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Abstract: We construct a space $W_X$ of fine moduli for the substructures of an arbitrary compact complex space $X$. A substructure $(X,{\cal A})$ of $X$ is given by a subalgebra ${\cal A}$ of the structure sheaf $\text{${\cal O}_{X}$}$ with the additional feature that $(X,{\cal A})$ is also a complex space; $(X,{\cal A})$ and $(X,{\cal A'})$ are called equivalent if and only if ${\cal A}$ and ${\cal A'}$ are isomorphic as subalgebras of $\text{${\cal 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 [Enhancements On Off] (What's this?)

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Additional Information

Peter M. Schuster
Affiliation: Mathematisches Institut der Universität, Theresienstr. 39, 80333 München, Germany

Keywords: Moduli spaces, substructures, subalgebras, quotients
Received by editor(s): September 3, 1996
Communicated by: Eric Bedford
Article copyright: © Copyright 1998 American Mathematical Society