On the non-existence of almost complex structures on sphere bundles over complex projective spaces
Chengwan Liu, Huijun Yang

TL;DR
This paper investigates conditions under which even-dimensional sphere bundles over complex projective spaces lack almost complex structures, providing explicit non-existence criteria based on Chern class divisibility and p-adic valuations.
Contribution
It establishes explicit necessary conditions for the non-existence of almost complex structures on these bundles, advancing understanding of their topological constraints.
Findings
Non-existence criteria for q ≥ a(n)
Application to bundles from the canonical line bundle
Numerical bounds for p=2 case
Abstract
We study the existence of almost complex structures on even-dimensional sphere bundles over complex projective spaces. For bundles with fibre over , we establish a necessary condition: if for an explicit function, then the total space 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 -adic valuations; for 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 .
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometry and complex manifolds · advanced mathematical theories
