Toric Hyperkahler Varieties
Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and spe...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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2002
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| _version_ | 1797079249164697600 |
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| author | Hausel, T Sturmfels, B |
| author_facet | Hausel, T Sturmfels, B |
| author_sort | Hausel, T |
| collection | OXFORD |
| description | Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions. A toric hyperkahler variety is a complete intersection in a Lawrence toric variety. Both varieties are non-compact, and they share the same cohomology ring, namely, the Stanley-Reisner ring of a matroid modulo a linear system of parameters. Familiar applications of toric geometry to combinatorics, including the Hard Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov, are extended to the hyperkahler setting. When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of Nakajima. |
| first_indexed | 2024-03-07T00:43:05Z |
| format | Journal article |
| id | oxford-uuid:83b9f409-dcf1-4c7e-a54b-f36c54a6801a |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T00:43:05Z |
| publishDate | 2002 |
| record_format | dspace |
| spelling | oxford-uuid:83b9f409-dcf1-4c7e-a54b-f36c54a6801a2022-03-26T21:46:07ZToric Hyperkahler VarietiesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:83b9f409-dcf1-4c7e-a54b-f36c54a6801aEnglishSymplectic Elements at Oxford2002Hausel, TSturmfels, BExtending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions. A toric hyperkahler variety is a complete intersection in a Lawrence toric variety. Both varieties are non-compact, and they share the same cohomology ring, namely, the Stanley-Reisner ring of a matroid modulo a linear system of parameters. Familiar applications of toric geometry to combinatorics, including the Hard Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov, are extended to the hyperkahler setting. When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of Nakajima. |
| spellingShingle | Hausel, T Sturmfels, B Toric Hyperkahler Varieties |
| title | Toric Hyperkahler Varieties |
| title_full | Toric Hyperkahler Varieties |
| title_fullStr | Toric Hyperkahler Varieties |
| title_full_unstemmed | Toric Hyperkahler Varieties |
| title_short | Toric Hyperkahler Varieties |
| title_sort | toric hyperkahler varieties |
| work_keys_str_mv | AT hauselt torichyperkahlervarieties AT sturmfelsb torichyperkahlervarieties |