Non-Compact Symplectic Toric Manifolds
A key result in equivariant symplectic geometry is Delzant's classification of compact connected symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a unimodular ('&#...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2015 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2015
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/147125 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Non-Compact Symplectic Toric Manifolds / Y. Karshon, E. Lerman // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 32 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862712124099264512 |
|---|---|
| author | Karshon, Y. Lerman, E. |
| author_facet | Karshon, Y. Lerman, E. |
| citation_txt | Non-Compact Symplectic Toric Manifolds / Y. Karshon, E. Lerman // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 32 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | A key result in equivariant symplectic geometry is Delzant's classification of compact connected symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a unimodular (''Delzant'') polytope; this gives a bijection between unimodular polytopes and isomorphism classes of compact connected symplectic toric manifolds. In this paper we extend Delzant's classification to non-compact symplectic toric manifolds. For a non-compact symplectic toric manifold the image of the moment map need not be convex and the induced map on the quotient need not be an embedding. Moreover, even when the map on the quotient is an embedding, its image no longer determines the symplectic toric manifold; a degree two characteristic class on the quotient makes an appearance. Nevertheless, the quotient is a manifold with corners, and the induced map from the quotient to the dual of the Lie algebra is what we call a unimodular local embedding. We classify non-compact symplectic toric manifolds in terms of manifolds with corners equipped with degree two cohomology classes and unimodular local embeddings into the dual of the Lie algebra of the corresponding torus. The main new ingredient is the construction of a symplectic toric manifold from such data. The proof passes through an equivalence of categories between symplectic toric manifolds and symplectic toric bundles over a fixed unimodular local embedding. This equivalence also gives a geometric interpretation of the degree two cohomology class.
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| first_indexed | 2025-12-07T17:34:31Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-147125 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:34:31Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Karshon, Y. Lerman, E. 2019-02-13T17:06:54Z 2019-02-13T17:06:54Z 2015 Non-Compact Symplectic Toric Manifolds / Y. Karshon, E. Lerman // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D20; 53035; 14M25; 37J35 DOI:10.3842/SIGMA.2015.055 https://nasplib.isofts.kiev.ua/handle/123456789/147125 A key result in equivariant symplectic geometry is Delzant's classification of compact connected symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a unimodular (''Delzant'') polytope; this gives a bijection between unimodular polytopes and isomorphism classes of compact connected symplectic toric manifolds. In this paper we extend Delzant's classification to non-compact symplectic toric manifolds. For a non-compact symplectic toric manifold the image of the moment map need not be convex and the induced map on the quotient need not be an embedding. Moreover, even when the map on the quotient is an embedding, its image no longer determines the symplectic toric manifold; a degree two characteristic class on the quotient makes an appearance. Nevertheless, the quotient is a manifold with corners, and the induced map from the quotient to the dual of the Lie algebra is what we call a unimodular local embedding. We classify non-compact symplectic toric manifolds in terms of manifolds with corners equipped with degree two cohomology classes and unimodular local embeddings into the dual of the Lie algebra of the corresponding torus. The main new ingredient is the construction of a symplectic toric manifold from such data. The proof passes through an equivalence of categories between symplectic toric manifolds and symplectic toric bundles over a fixed unimodular local embedding. This equivalence also gives a geometric interpretation of the degree two cohomology class. This paper is a contribution to the Special Issue on Poisson Geometry in Mathematics and Physics. The full
 collection is available at http://www.emis.de/journals/SIGMA/Poisson2014.html.
 This research is partially supported by the Natural Sciences and Engineering Research Council
 of Canada and by the National Science Foundation.
 We are grateful to Chris Woodward and to Daniele Sepe for valuable discussions. We are
 also grateful to our anonymous referees for their valuable comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Non-Compact Symplectic Toric Manifolds Article published earlier |
| spellingShingle | Non-Compact Symplectic Toric Manifolds Karshon, Y. Lerman, E. |
| title | Non-Compact Symplectic Toric Manifolds |
| title_full | Non-Compact Symplectic Toric Manifolds |
| title_fullStr | Non-Compact Symplectic Toric Manifolds |
| title_full_unstemmed | Non-Compact Symplectic Toric Manifolds |
| title_short | Non-Compact Symplectic Toric Manifolds |
| title_sort | non-compact symplectic toric manifolds |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147125 |
| work_keys_str_mv | AT karshony noncompactsymplectictoricmanifolds AT lermane noncompactsymplectictoricmanifolds |