Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, and let $(E,h)$ be a Hermitian holomorphic vector bundle on $X$. Suppose that there exists a constant $c>0$ such that, for every $x\in X$, every unitary holomorphic coframe $(dz_1,\dots,dz_n)$ for $T_x^*X$, every unitary frame $(e_\alpha)_\alpha$ for $E_x$, and every tensor

\begin{align*} u=\sum_{j,\alpha}u_{j\alpha}\,\partial_{z_j}\otimes e_\alpha\in T_xX\otimes E_x, \end{align*}

one has

\begin{align*} \sum_{j,k,\alpha,\beta} i\Theta(E,h)_{j\bar k\alpha\bar\beta}(x)\,u_{j\alpha}\,\overline{u_{k\beta}}\ge c\sum_{j,\alpha}|u_{j\alpha}|^2. \end{align*}

Then, for every integer $q\ge 1$ and every smooth $E$-valued $(n,q)$-form $\alpha\in A^{n,q}(X,E)$,

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

where $\Lambda$ is contraction by $\omega$, and the $L^2$ [inner product](/page/Inner%20Product) and norm are those induced by $\omega$ and $h$.