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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2603.02755 |
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| _version_ | 1866910039161176064 |
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| author | Hasegawa, Kazuyuki |
| author_facet | Hasegawa, Kazuyuki |
| contents | In this paper, we study fixed-point sets of $S^{1}$-actions and compatible complex structures on quaternionic manifolds. We obtain an equation involving the first Chern classes of the fixed-point set and of a quaternionically flat manifold with compatible complex structure of closed type. In addition, if the first Chern class of the fixed-point set is not trivial, then the quaternionic manifold does not admit hypercomplex structures containing given compatible complex structure on any open set containing the fixed-point set. Moreover, we determine the connected components of the fixed-point set arising from quaternionic $S^{1}$-actions on the quaternionic projective space. We apply these results to Pontecorvo's example $\mathrm{SO}^{\ast}(2n+2)/\mathrm{SO}^{\ast}(2n) \times \mathrm{SO}^{\ast}(2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_02755 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quaternionic complex manifolds and fixed-point sets of $S^{1}$-actions Hasegawa, Kazuyuki Differential Geometry 53C10, 53C26 In this paper, we study fixed-point sets of $S^{1}$-actions and compatible complex structures on quaternionic manifolds. We obtain an equation involving the first Chern classes of the fixed-point set and of a quaternionically flat manifold with compatible complex structure of closed type. In addition, if the first Chern class of the fixed-point set is not trivial, then the quaternionic manifold does not admit hypercomplex structures containing given compatible complex structure on any open set containing the fixed-point set. Moreover, we determine the connected components of the fixed-point set arising from quaternionic $S^{1}$-actions on the quaternionic projective space. We apply these results to Pontecorvo's example $\mathrm{SO}^{\ast}(2n+2)/\mathrm{SO}^{\ast}(2n) \times \mathrm{SO}^{\ast}(2)$. |
| title | Quaternionic complex manifolds and fixed-point sets of $S^{1}$-actions |
| topic | Differential Geometry 53C10, 53C26 |
| url | https://arxiv.org/abs/2603.02755 |