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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2412.15424 |
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| _version_ | 1866910755636379648 |
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| author | Biliotti, Leonardo Minuzzo, Alessandro |
| author_facet | Biliotti, Leonardo Minuzzo, Alessandro |
| contents | Let $M_i$, for $i=1,2$, be a Kähler manifold, and let $G$ be a Lie group acting on $M_i$ by Kähler isometries. Suppose that the action admits a momentum map $μ_i$ and let $N_i:=μ_i^{-1}(0)$ be a regular level set. When the action of $G$ on $N_i$ is proper and free, the Meyer--Marsden--Weinstein quotient $P_i:=N_i/G$ is a Kähler manifold and $π_i:N_i\to P_i$ is a principal fiber bundle with base $P_i$ and characteristic fiber $G$. In this paper, we define an almost complex structure for the manifold $N_1\times N_2$ and give necessary and sufficient conditions for its integrability. In the integrable case, we find explicit holomorphic charts for $N_1\times N_2$. As applications, we consider a non integrable almost-complex structure on the product of two complex Stiefel manifolds and the infinite Calabi-Eckmann manifolds $\mathbb S^{2n+1}\times S(\mathcal{H})$, for $n\geq 1$, where $S(\mathcal{H})$ denotes the unit sphere of an infinite dimensional Hilbert space $\mathcal{H}$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15424 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complex Structures on Product Manifolds Biliotti, Leonardo Minuzzo, Alessandro Differential Geometry 53D20 Let $M_i$, for $i=1,2$, be a Kähler manifold, and let $G$ be a Lie group acting on $M_i$ by Kähler isometries. Suppose that the action admits a momentum map $μ_i$ and let $N_i:=μ_i^{-1}(0)$ be a regular level set. When the action of $G$ on $N_i$ is proper and free, the Meyer--Marsden--Weinstein quotient $P_i:=N_i/G$ is a Kähler manifold and $π_i:N_i\to P_i$ is a principal fiber bundle with base $P_i$ and characteristic fiber $G$. In this paper, we define an almost complex structure for the manifold $N_1\times N_2$ and give necessary and sufficient conditions for its integrability. In the integrable case, we find explicit holomorphic charts for $N_1\times N_2$. As applications, we consider a non integrable almost-complex structure on the product of two complex Stiefel manifolds and the infinite Calabi-Eckmann manifolds $\mathbb S^{2n+1}\times S(\mathcal{H})$, for $n\geq 1$, where $S(\mathcal{H})$ denotes the unit sphere of an infinite dimensional Hilbert space $\mathcal{H}$ |
| title | Complex Structures on Product Manifolds |
| topic | Differential Geometry 53D20 |
| url | https://arxiv.org/abs/2412.15424 |