Complex and Stable Complex Structures on Real Vector Bundles over Connected Sums of Quaternionic Projective Planes

Abstract

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.