The moduli of substructures of a compact complex space
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- by Peter M. Schuster PDF
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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 $(X,\mathcal {A})$ 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
- J. Frisch, Aplatissement en géométrie analytique, Ann. Sci. École Norm. Sup. (4) 1 (1968), 305–312 (French). MR 236421, DOI 10.24033/asens.1165
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 238860
- H.-W. Schuster and A. Vogt, The moduli of quotients of a compact complex space, J. Reine Angew. Math. 364 (1986), 51–59. MR 817637
- Schuster, P. M.: Moduln singulärer Kurven mit vorgegebener Normalisierung. Diss., Univ. München 1995. Also at Verlag Mainz, Aachen 1996.
- Schuster, P. M.: Identifying variable points on a smooth curve. Manuscripta Math. 94 (1997), 195–210.
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