On the non-existence of almost complex structures on sphere bundles over complex projective spaces

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 $\xi_{n,q}$ 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 $4 \le q < a(n)$.