Weakly complex homogeneous spaces

TL;DR

This paper completes the classification of compact homogeneous spaces with weakly complex tangent bundles, extending previous results to a broader category and identifying new spaces with this property.

Contribution

It extends the classification of compact homogeneous spaces with weakly complex tangent bundles to include cases with positive Euler characteristic.

Findings

Characterization of simply connected compact equal rank homogeneous spaces with weakly complex tangent bundles.

Identification of spaces with invariant almost complex structures, stably trivial tangent bundles, or neither.

Complete classification including an explicit list of weakly complex spaces without other known structures.

Abstract

We complete our recent classification of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank homogeneous space has weakly complex tangent bundle if and only if it is a product of compact equal rank homogeneous spaces which either carry an invariant almost complex structure (and are classified by Hermann), or have stably trivial tangent bundle (and are classified by Singhof and Wemmer), or belong to an explicit list of weakly complex spaces which have neither stably trivial tangent bundle, nor carry invariant almost complex structures.