Almost complex structures on quaternion-K\"ahler manifolds and inner symmetric spaces

TL;DR

The paper proves that most compact quaternionic-Kähler manifolds and irreducible inner symmetric spaces do not admit almost complex structures, with specific exceptions like complex Grassmannians, spheres, and Hermitian symmetric spaces.

Contribution

It establishes non-existence results for almost complex structures on certain classes of quaternionic-Kähler and symmetric spaces, identifying key exceptions.

Findings

Compact quaternionic-Kähler manifolds of positive scalar curvature admit no almost complex structure except for complex Grassmannians.

Irreducible inner symmetric spaces of compact type are not weakly complex, except for spheres and Hermitian symmetric spaces.

Abstract

We prove that compact quaternionic-K\"ahler manifolds of positive scalar curvature admit no almost complex structure, even in the weak sense, except for the complex Grassmannians $Gr_2(C^{n+2})$. We also prove that irreducible inner symmetric spaces $M^{4n}$ of compact type are not weakly complex, except for spheres and Hermitian symmetric spaces.