Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, and let $(E,h)$ be a Hermitian holomorphic vector bundle on $X$. Let $A^{n,q}(X,E)$ denote the space of smooth $E$-valued $(n,q)$-forms, let $\bar\partial_E:A^{n,q}(X,E)\to A^{n,q+1}(X,E)$ be the Dolbeault operator induced by the holomorphic structure on $E$, and let $\Delta_{\bar\partial,E}=\bar\partial_E\bar\partial_E^*+\bar\partial_E^*\bar\partial_E$ be the Dolbeault Laplacian with respect to the $L^2$ [inner product](/page/Inner%20Product) induced by $\omega$ and $h$. Fix an integer $q\ge 1$. Suppose there exists a constant $c>0$ such that every $\alpha\in A^{n,q}(X,E)$ satisfies

\begin{align*} \bigl([i\Theta(E,h),\Lambda]\alpha,\alpha\bigr)_{L^2}\ge c\|\alpha\|_{L^2}^2, \end{align*}

where $\Theta(E,h)$ is the Chern curvature of $(E,h)$ and $\Lambda$ is contraction with the Kähler form $\omega$. Then every $\bar\partial_E$-harmonic $E$-valued $(n,q)$-form, meaning every element of $\ker\Delta_{\bar\partial,E}\subset A^{n,q}(X,E)$, is zero. Consequently,

\begin{align*} H^q(X,K_X\otimes E)=0. \end{align*}