Toric Varieties Associated with Moduli Spaces
Any genus $g$ surface, $\Sigma_{g,n},$ with $n$ boundary components may be given a trinion decomposition: a realization of the surface as a union of $2g-2+n$ trinions glued together along $3g-3+n$ of their boundary circles. Together with the flows of Goldman, Jeffrey and Weitsman use the trinion bo...
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ndltd-LACETR-oai-collectionscanada.gc.ca-OTU.1807-319602013-04-17T04:19:17ZToric Varieties Associated with Moduli SpacesUren, JamesSymplectic GeometryToric Geometry0405Any genus $g$ surface, $\Sigma_{g,n},$ with $n$ boundary components may be given a trinion decomposition: a realization of the surface as a union of $2g-2+n$ trinions glued together along $3g-3+n$ of their boundary circles. Together with the flows of Goldman, Jeffrey and Weitsman use the trinion boundary circles in a decomposition of $\Sigma_{g,n}$ to obtain a Hamiltonian action of a compact torus $(S^1)^{3g-3+n'} $ on an open dense subset of the moduli space of certain gauge equivalence classes of flat $SU(2)-$connections on $\Sigma_{g,n}.$ Jeffrey and Weitsman also provide a complete description of the moment polytopes for these torus actions, and we make use of this description to study the cohomology of associated toric varieties. While we are able to make use of the work of Danilov to obtain the integral (rational) cohomology ring in the smooth (orbifold) case, we show that the aforementioned toric varieties almost always possess singularities worse than those of an orbifold. In these cases we use an algorithm of Bressler and Lunts to recover the intersection cohomology Betti numbers using the combinatorial information provided by the corresponding moment polytopes. The main contribution of this thesis is a computation of the intersection cohomology Betti numbers for the toric varieties associated to trinion decomposed surfaces $\Sigma_{2,0},\Sigma_{2,1},\Sigma_{3,0}, \Sigma_{3,1}, \Sigma_{4,0},$ and $\Sigma_{4,1}.$Jeffrey, LisaSelick, Paul2011-112012-01-11T21:41:48ZNO_RESTRICTION2012-01-11T21:41:48Z2012-01-11Thesishttp://hdl.handle.net/1807/31960en_ca |
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en_ca |
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Symplectic Geometry Toric Geometry 0405 |
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Symplectic Geometry Toric Geometry 0405 Uren, James Toric Varieties Associated with Moduli Spaces |
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Any genus $g$ surface, $\Sigma_{g,n},$ with $n$ boundary components may be given a trinion decomposition: a realization of the surface as a union of $2g-2+n$ trinions glued together along $3g-3+n$ of their boundary circles. Together with the flows of Goldman, Jeffrey and Weitsman use the trinion boundary circles in a decomposition of $\Sigma_{g,n}$ to obtain a Hamiltonian action of a compact torus $(S^1)^{3g-3+n'} $ on an open dense subset of the moduli space of certain gauge equivalence classes of flat $SU(2)-$connections on $\Sigma_{g,n}.$ Jeffrey and Weitsman also provide a complete description of the moment polytopes for these torus actions, and we make use of this description to study the cohomology of associated toric varieties.
While we are able to make use of the work of Danilov to obtain the integral (rational) cohomology ring in the smooth (orbifold) case, we show that the aforementioned toric varieties almost always possess singularities worse than those of an orbifold. In these cases we use an algorithm of Bressler and Lunts to recover the intersection cohomology Betti numbers using the combinatorial information provided by the corresponding moment polytopes. The main contribution of this thesis is a computation of the intersection cohomology Betti numbers for the toric varieties associated to trinion decomposed surfaces $\Sigma_{2,0},\Sigma_{2,1},\Sigma_{3,0}, \Sigma_{3,1}, \Sigma_{4,0},$ and $\Sigma_{4,1}.$ |
author2 |
Jeffrey, Lisa |
author_facet |
Jeffrey, Lisa Uren, James |
author |
Uren, James |
author_sort |
Uren, James |
title |
Toric Varieties Associated with Moduli Spaces |
title_short |
Toric Varieties Associated with Moduli Spaces |
title_full |
Toric Varieties Associated with Moduli Spaces |
title_fullStr |
Toric Varieties Associated with Moduli Spaces |
title_full_unstemmed |
Toric Varieties Associated with Moduli Spaces |
title_sort |
toric varieties associated with moduli spaces |
publishDate |
2011 |
url |
http://hdl.handle.net/1807/31960 |
work_keys_str_mv |
AT urenjames toricvarietiesassociatedwithmodulispaces |
_version_ |
1716580813396508672 |