Generalized complex structure on certain principal torus bundles
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866915079071465472 |
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| author | Pal, Debjit Poddar, Mainak |
| author_facet | Pal, Debjit Poddar, Mainak |
| contents | A principal torus bundle over a complex manifold with even dimensional fiber and characteristic class of type $(1,1)$ admits a family of regular generalized complex structures (GCS) with the fibers as leaves of the associated symplectic foliation. We show that such a generalized complex structure is equivalent to the product of the complex structure on the base and the symplectic structure on the fiber in a tubular neighborhood of an arbitrary fiber if and only if the bundle is flat. This has consequences for the generalized Dolbeault cohomology of the bundle that includes a Künneth formula. On a more general note, if a principal bundle over a complex manifold with a symplectic structure group admits a GCS with the fibers of the bundle as leaves of the associated symplectic foliation, and the GCS is equivalent to a product GCS in a neighborhood of every fiber, then the bundle is flat and symplectic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_07835 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generalized complex structure on certain principal torus bundles Pal, Debjit Poddar, Mainak Differential Geometry 53D18, 57R22, 57R30 A principal torus bundle over a complex manifold with even dimensional fiber and characteristic class of type $(1,1)$ admits a family of regular generalized complex structures (GCS) with the fibers as leaves of the associated symplectic foliation. We show that such a generalized complex structure is equivalent to the product of the complex structure on the base and the symplectic structure on the fiber in a tubular neighborhood of an arbitrary fiber if and only if the bundle is flat. This has consequences for the generalized Dolbeault cohomology of the bundle that includes a Künneth formula. On a more general note, if a principal bundle over a complex manifold with a symplectic structure group admits a GCS with the fibers of the bundle as leaves of the associated symplectic foliation, and the GCS is equivalent to a product GCS in a neighborhood of every fiber, then the bundle is flat and symplectic. |
| title | Generalized complex structure on certain principal torus bundles |
| topic | Differential Geometry 53D18, 57R22, 57R30 |
| url | https://arxiv.org/abs/2303.07835 |