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A novel approach to the cohomology of symplectic quotients
We develop a novel approach to the topology of singular symplectic quotients by extending Sjamaar’s complex of differential forms to the complex of resolution differential forms. The motivation for this is to extend Sjamaar’s complex in a way which makes the definition of a Kirwan map possible. I...
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| Format: | Doctoral Thesis |
| Language: | English |
| Published: |
Philipps-Universität Marburg
2023
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| Online Access: | PDF Full Text |
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| Summary: | We develop a novel approach to the topology of singular symplectic quotients by extending
Sjamaar’s complex of differential forms to the complex of resolution differential forms. The
motivation for this is to extend Sjamaar’s complex in a way which makes the definition of a
Kirwan map possible. In his theory this is not possible due to the singularities of any
connection form at the fixed points of the action. Thus, the idea is to resolve the group action
by blow-ups. Doing this using real blow-ups results in a locally free action on the whole
manifold but also in difficult exceptional bundles. Carrying out the construction using
symplectic blow-ups the exceptional bundles turn out to be more controllable. This approach
then allows to define a Kirwan map, whose surjectivity we study in case that the fixed point
set has vanishing cohomology in odd degrees. It turns out that this map is surjective in even
degrees while it is not surjective in odd degrees. |
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| DOI: | 10.17192/z2023.0523 |