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

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