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

Chengwan Liu, Huijun Yang

Abstract

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 $4 \le q < a(n)$.

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

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 .
Paper Structure (3 sections, 9 theorems, 42 equations)

This paper contains 3 sections, 9 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Divisibility properties of Chern classes
  3. Non-existence of almost complex structures on sphere bundles

Key Result

Theorem 1.1

Let $\xi_{n,q}$ be a sphere bundle over $\mathbb{C} P^{n}$ with fibre $S^{2q}$. If $q \ge a(n)$, then the total space $E_{n,q}$ does not admit an almost complex structure.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1: Adams ad61
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 3.1
  • ...and 3 more