Volume preserving subgroups of $${{\mathcal A}}$$ and $${{\mathcal K}}$$ and singularities in unimodular geometry |
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Authors: | W Domitrz J H Rieger |
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Institution: | 1.Faculty of Mathematics and Information Science,Warsaw University of Technology,Warsaw,Poland;2.Institut für Mathematik,Universit?t Halle,Halle (Saale),Germany |
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Abstract: | For a germ of a smooth map f from
\mathbb Kn{{\mathbb K}^n} to
\mathbb Kp{{\mathbb K}^p} and a subgroup GWq{{{G}_{\Omega _q}}} of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form Ω
q
in
\mathbb Kq{{\mathbb K}^q} (q = n or p) we study the GWq{{{G}_{\Omega _q}}} -moduli space of f that parameterizes the GWq{{{G}_{\Omega _q}}} -orbits inside the G-orbit of f. We find, for example, that this moduli space vanishes for GWq = AWp{{{G}_{\Omega _q}} ={{\mathcal A}_{\Omega _p}}} and A{{\mathcal A}}-stable maps f and for GWq = KWn{{{G}_{\Omega _q}} ={{\mathcal K}_{\Omega _n}}} and K{{\mathcal K}}-simple maps f. On the other hand, there are A{{\mathcal A}}-stable maps f with infinite-dimensional AWn{{{\mathcal A}_{\Omega _n}}} -moduli space. |
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Keywords: | |
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