AI 中文总结
本文通过庞特里亚金类刻画四元射影平面连通和上实向量丛的（稳定）复结构，纠正了1974年Sato-Suzuki关于殆复结构的错误断言，给出精确充要条件。
AI 中文摘要
设 $M_{\ell,m}=\ell\\,\mathbb{HP}^{2}\\# m\\,\overline{\mathbb{HP}^{2}}$ 为 $\ell$ 个 $\mathbb{HP}^{2}$ 与 $m$ 个取相反定向的 $\mathbb{HP}^{2}$ 的连通和。我们利用庞特里亚金类刻画了 $M_{\ell,m}$ 上具有稳定复结构的实向量丛，并由此推出 $M_{\ell,m}$ 是稳定殆复的当且仅当 $\ell-m$ 为偶数。随后我们确定了 $M_{\ell,m}$ 上偶数秩定向实向量丛何时具有复结构。因此，$M_{\ell,m}$ 具有殆复结构当且仅当 $\ell=2m+1$ 且 $m$ 为奇数。这纠正了 Sato 与 Suzuki 在 1974 年提出的断言，即 $M_{\ell,m}$ 永不具有殆复结构。我们还指出了他们论证中不合理的步骤。
英文摘要
Let $M_{\ell,m}=\ell\,\mathbb{HP}^{2}\# m\,\overline{\mathbb{HP}^{2}}$ be the connected sum of $\ell$ copies of $\mathbb{HP}^{2}$ with $m$ copies of $\mathbb{HP}^{2}$ endowed with the opposite orientation. We characterise, in terms of Pontryagin classes, the real vector bundles over $M_{\ell,m}$ admitting a stable complex structure, and deduce that $M_{\ell,m}$ is stably almost complex if and only if $\ell-m$ is even. We then determine when an oriented real vector bundle of even rank over $M_{\ell,m}$ admits a complex structure. Consequently, $M_{\ell,m}$ admits an almost complex structure if and only if $\ell=2m+1$ and $m$ is odd. This corrects an assertion made by Sato and Suzuki in 1974, according to which $M_{\ell,m}$ is never almost complex. We also identify the unjustified step in their argument.
Comments22 pages. Comments are welcome
