Towards non-reductive geometric invariant theory
We study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of "quotients" under such actions, by generalizing (from reductive to non-reductive group actions) existing methods involving Mumford's geometric invaria...
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| Format: | Journal article |
| Language: | English |
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2007
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| _version_ | 1826305213371252736 |
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| author | Doran, B Kirwan, F |
| author_facet | Doran, B Kirwan, F |
| author_sort | Doran, B |
| collection | OXFORD |
| description | We study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of "quotients" under such actions, by generalizing (from reductive to non-reductive group actions) existing methods involving Mumford's geometric invariant theory (GIT). We concentrate on actions of unipotent groups H, and define sets of stable points X s and semistable points X ss, often explicitly computable via the methods of reductive GIT, which reduce to the standard definitions due to Mumford in the case of reductive actions. We compare these with definitions in the literature. Results include (1) a geometric criterion determining whether or not a ring of invariants is finitely generated, (2) the existence of a geometric quotient of X s, and (3) the existence of a canonical "enveloping quotient" variety of X ss, denoted X//H, which (4) has a projective completion given by a reductive GIT quotient and (5) is itself projective and isomorphic to Proj(k[X] H) when k[X] H is finitely generated. |
| first_indexed | 2024-03-07T06:29:29Z |
| format | Journal article |
| id | oxford-uuid:f57baf54-5b8e-4b6b-a11c-358b18bfa3aa |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:29:29Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | oxford-uuid:f57baf54-5b8e-4b6b-a11c-358b18bfa3aa2022-03-27T12:27:35ZTowards non-reductive geometric invariant theoryJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:f57baf54-5b8e-4b6b-a11c-358b18bfa3aaEnglishSymplectic Elements at Oxford2007Doran, BKirwan, FWe study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of "quotients" under such actions, by generalizing (from reductive to non-reductive group actions) existing methods involving Mumford's geometric invariant theory (GIT). We concentrate on actions of unipotent groups H, and define sets of stable points X s and semistable points X ss, often explicitly computable via the methods of reductive GIT, which reduce to the standard definitions due to Mumford in the case of reductive actions. We compare these with definitions in the literature. Results include (1) a geometric criterion determining whether or not a ring of invariants is finitely generated, (2) the existence of a geometric quotient of X s, and (3) the existence of a canonical "enveloping quotient" variety of X ss, denoted X//H, which (4) has a projective completion given by a reductive GIT quotient and (5) is itself projective and isomorphic to Proj(k[X] H) when k[X] H is finitely generated. |
| spellingShingle | Doran, B Kirwan, F Towards non-reductive geometric invariant theory |
| title | Towards non-reductive geometric invariant theory |
| title_full | Towards non-reductive geometric invariant theory |
| title_fullStr | Towards non-reductive geometric invariant theory |
| title_full_unstemmed | Towards non-reductive geometric invariant theory |
| title_short | Towards non-reductive geometric invariant theory |
| title_sort | towards non reductive geometric invariant theory |
| work_keys_str_mv | AT doranb towardsnonreductivegeometricinvarianttheory AT kirwanf towardsnonreductivegeometricinvarianttheory |