A vanishing result for harmonic $(n-1,1)$-forms

We prove that every primitive harmonic $(n-1,1)$-form of a closed Kähler manifold of complex dimension $n$ vanishes provided its Kähler curvature operator is $\frac{n^2-n+2}{2}$-positive if $n\ge 4$, respectively $\frac{7}{2}$-positive if $n=3$. As a consequence, a closed Kähler manifold of complex dimension $n\ge 3$ with $3$-positive Kähler curvature operator is a real cohomology complex projective space.