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Complex and Stable Complex Structures on Real Vector Bundles over Connected Sums of Quaternionic Projective Planes
Abstract
Let Mℓ,m=ℓ ℍℙ2# m ℍℙ2̅ be the connected sum of ℓ copies of ℍℙ2 with m copies of ℍℙ2 endowed with the opposite orientation. We characterise, in terms of Pontryagin classes, the real vector bundles over Mℓ,m admitting a stable complex structure, and deduce that Mℓ,m is stably almost complex if and only if ℓ−m is even. We then determine when an oriented real vector bundle of even rank over Mℓ,m admits a complex structure. Consequently, Mℓ,m admits an almost complex structure if and only if ℓ=2m+1 and m is odd. This corrects an assertion made by Sato and Suzuki in 1974, according to which Mℓ,m is never almost complex. We also identify the unjustified step in their argument.
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BibTeX
@misc{mandal2026complex,
title = {{Complex and Stable Complex Structures on Real Vector Bundles over Connected Sums of Quaternionic Projective Planes}},
author = {Souvik Mandal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27929},
eprint = {2609.27929},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27929},
}
RIS
TY - UNPB
TI - Complex and Stable Complex Structures on Real Vector Bundles over Connected Sums of Quaternionic Projective Planes
AU - Souvik Mandal
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.27929
UR - https://arxiv.org/abs/2609.27929
AB - Let M_ℓ,m=ℓ ℍℙ^2# m ℍℙ^2̅ be the connected sum of ℓ copies of ℍℙ^2 with m copies of ℍℙ^2 endowed with the opposite orientation. We characterise, in terms of Pontryagin classes, the real vector bundles over M_ℓ,m admitting a stable complex structure, and deduce that M_ℓ,m is stably almost complex if and only if ℓ−m is even. We then determine when an oriented real vector bundle of even rank over M_ℓ,m admits a complex structure. Consequently, M_ℓ,m admits an almost complex structure if and only if ℓ=2m+1 and m is odd. This corrects an assertion made by Sato and Suzuki in 1974, according to which M_ℓ,m is never almost complex. We also identify the unjustified step in their argument.
ER -