Almost complex structures on quaternion-Kähler manifolds and inner symmetric spaces

We prove that compact quaternionic-Kähler 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.

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