Let $X$ be a compact complex manifold which admits a holomorphic $(p,0)$ form $\eta$ which is everywhere non-degenerate. If dim $H^0(X,\Omega^p)=1$, then is the contraction map $$\lrcorner \eta:H^1(X,T_X)\to H^1(X,\Omega^{p-1})$$ an isomorphism?
From Voisin p.26 example 7, maybe we can conclude it is true for $p=2$? But for a general $p$, does it still holds?