Let $X$ be a connected compact Kähler complex manifold of complex dimension $n$, and let $\Omega \in H^0(X,K_X)$ be a nowhere-vanishing holomorphic volume form. Then contraction with $\Omega$ defines an isomorphism of holomorphic vector bundles
where $H^{n-1,1}(X)$ denotes Dolbeault cohomology of $(n-1,1)$-forms, equivalently the corresponding Hodge summand of $H^n(X,\mathbb C)$. If $\Omega'$ is any other nowhere-vanishing holomorphic volume form, then $\Omega'=c\Omega$ for a unique scalar $c\in\mathbb C^\times$, and the induced isomorphism is multiplied by $c$.