  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.
