On the non-existence of almost complex structures on sphere bundles over complex projective spaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910023925366784 |
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| author | Liu, Chengwan Yang, Huijun |
| author_facet | Liu, Chengwan Yang, Huijun |
| contents | We study the existence of almost complex structures on even-dimensional sphere bundles over complex projective spaces. For bundles $ξ_{n,q}$ with fibre $S^{2q}$ over $\mathbb{C} P^n$, we establish a necessary condition: if $q \ge a(n)$ for an explicit function, then the total space $E_{n,q}$ does not admit an almost complex structure. As an application, we analyse a concrete family associated with the canonical line bundle and obtain non-existence criteria in terms of $p$-adic valuations; for $p=2$ this yields a simple numerical bound. The proofs rely on Chern class computations and divisibility properties of characteristic classes. The results leave open the question of existence in the range $4 \le q < a(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14945 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the non-existence of almost complex structures on sphere bundles over complex projective spaces Liu, Chengwan Yang, Huijun Algebraic Topology Algebraic Geometry Differential Geometry 57R15, 55R40, 55R25 We study the existence of almost complex structures on even-dimensional sphere bundles over complex projective spaces. For bundles $ξ_{n,q}$ with fibre $S^{2q}$ over $\mathbb{C} P^n$, we establish a necessary condition: if $q \ge a(n)$ for an explicit function, then the total space $E_{n,q}$ does not admit an almost complex structure. As an application, we analyse a concrete family associated with the canonical line bundle and obtain non-existence criteria in terms of $p$-adic valuations; for $p=2$ this yields a simple numerical bound. The proofs rely on Chern class computations and divisibility properties of characteristic classes. The results leave open the question of existence in the range $4 \le q < a(n)$. |
| title | On the non-existence of almost complex structures on sphere bundles over complex projective spaces |
| topic | Algebraic Topology Algebraic Geometry Differential Geometry 57R15, 55R40, 55R25 |
| url | https://arxiv.org/abs/2602.14945 |