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Stable Complex Structures On Real Vector Bundles Over Connected Sums Of Quaternionic Projective Spaces

  • Souvik Mandal
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Abstract

For integers n≥ 2 and integers ℓ,m≥ 0, not both zero, let M4nℓ,m=ℓ ℍℙn # m ℍℙn̅ denote the connected sum of ℓ copies of the quaternionic projective space ℍℙn and m copies of ℍℙn endowed with the opposite orientation. By analysing the image of the complexification map c : KÕ(M4nℓ,m)→K̃(M4nℓ,m), we characterise, in terms of Pontryagin classes, the real vector bundles over M4nℓ,m admitting a stable complex structure, and deduce that M4nℓ,m is stably almost complex if and only if n=2 and ℓ−m is even. Consequently no M4nℓ,m with n≥ 3 admits an almost complex structure. Sato and Suzuki asserted the non-existence of almost complex structures on M4nℓ,m in 1974 for 3≤ n≤ 10, except possibly when n=3 and ℓ=3m+1; we establish it for all n≥ 3, and in the stronger form of stable almost complex structures.

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Cite

BibTeXRIS
BibTeX
@misc{mandal2026stable,
  title = {{Stable Complex Structures On Real Vector Bundles Over Connected Sums Of Quaternionic Projective Spaces}},
  author = {Souvik Mandal},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02147},
  eprint = {2610.02147},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02147},
}
RIS
TY  - UNPB
TI  - Stable Complex Structures On Real Vector Bundles Over Connected Sums Of Quaternionic Projective Spaces
AU  - Souvik Mandal
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.02147
UR  - https://arxiv.org/abs/2610.02147
AB  - For integers n≥ 2 and integers ℓ,m≥ 0, not both zero, let M^4n_ℓ,m=ℓ ℍℙ^n # m ℍℙ^n̅ denote the connected sum of ℓ copies of the quaternionic projective space ℍℙ^n and m copies of ℍℙ^n endowed with the opposite orientation. By analysing the image of the complexification map c : KÕ(M^4n_ℓ,m)→K̃(M^4n_ℓ,m), we characterise, in terms of Pontryagin classes, the real vector bundles over M^4n_ℓ,m admitting a stable complex structure, and deduce that M^4n_ℓ,m is stably almost complex if and only if n=2 and ℓ−m is even. Consequently no M^4n_ℓ,m with n≥ 3 admits an almost complex structure. Sato and Suzuki asserted the non-existence of almost complex structures on M^4n_ℓ,m in 1974 for 3≤ n≤ 10, except possibly when n=3 and ℓ=3m+1; we establish it for all n≥ 3, and in the stronger form of stable almost complex structures.
ER  -