[proofplan] We prove the vanishing by replacing the coherent sheaf $K_X \otimes L \otimes \mathcal I(h)$ with its $L^2$ Dolbeault resolution. The curvature hypothesis \begin{align*} i\Theta_h(L) \geq \varepsilon\omega \end{align*} puts the singular metric in the strictly positive range where the singular Bochner-Kodaira-Hörmander estimate solves every $\bar\partial$-closed $L$-valued $(n,q)$-form in degree $q \geq 1$. Exactness of the global $L^2$ Dolbeault complex in positive degree then identifies the relevant sheaf cohomology groups with zero. [/proofplan] [step:Define the multiplier ideal and the $L^2$ Dolbeault sheaves] Let $dV_\omega := \omega^n/n!$ denote the volume measure determined by the Kähler form $\omega$. For an [open set](/page/Open%20Set) $U \subset X$ and a holomorphic frame $e: L|_U \to U \times \mathbb C$, write $|e|_h^2 = e^{-\varphi}$, where $\varphi: U \to [-\infty,\infty)$ is the local weight of the singular Hermitian metric $h$. The multiplier ideal sheaf $\mathcal I(h) \subset \mathcal O_X$ is the sheaf whose sections over $U$ are the holomorphic functions $f: U \to \mathbb C$ such that \begin{align*} \int_K |f|^2 e^{-\varphi}\,dV_\omega < \infty \end{align*} for every compact set $K \subset U$. For each integer $q$ with $0 \leq q \leq \dim_{\mathbb C}X$, let $\mathcal A_h^{n,q}(L)$ be the sheaf assigning to each open set $U \subset X$ the [vector space](/page/Vector%20Space) of measurable $L$-valued $(n,q)$-forms $u$ on $U$ such that $u$ and its distributional Dolbeault derivative $\bar\partial u$ are locally square-integrable with respect to $h$ and $dV_\omega$. Thus, in the local frame $e$, the condition is that the coefficient functions of $u$ and $\bar\partial u$ have locally finite integral against $e^{-\varphi}\,dV_\omega$. [/step] [step:Resolve $K_X \otimes L \otimes \mathcal I(h)$ by the $L^2$ Dolbeault complex] The local $L^2$ Dolbeault resolution theorem for multiplier ideals applies because $X$ is a complex manifold, $L$ is holomorphic, and the local weights of $h$ are plurisubharmonic up to a smooth term under the curvature-current lower bound. It gives an exact sequence of sheaves \begin{align*} 0 \longrightarrow K_X \otimes L \otimes \mathcal I(h) \longrightarrow \mathcal A_h^{n,0}(L) \xrightarrow{\bar\partial} \mathcal A_h^{n,1}(L) \xrightarrow{\bar\partial} \cdots \xrightarrow{\bar\partial} \mathcal A_h^{n,m}(L) \longrightarrow 0, \end{align*} where $m := \dim_{\mathbb C}X$. The sheaves $\mathcal A_h^{n,q}(L)$ are fine because multiplication by smooth partitions of unity preserves local $L^2$ integrability and commutes with the distributional Leibniz rule up to locally bounded smooth coefficients. Therefore \begin{align*} H^q\left(X,K_X\otimes L\otimes\mathcal I(h)\right) \cong \frac{\ker\left(\bar\partial: \Gamma(X,\mathcal A_h^{n,q}(L)) \to \Gamma(X,\mathcal A_h^{n,q+1}(L))\right)}{\operatorname{im}\left(\bar\partial: \Gamma(X,\mathcal A_h^{n,q-1}(L)) \to \Gamma(X,\mathcal A_h^{n,q}(L))\right)} \end{align*} for every $q \geq 1$. [/step] [step:Apply the singular $L^2$ estimate to solve $\bar\partial u=f$ in positive degree] Let $q \geq 1$, and let \begin{align*} f \in \Gamma(X,\mathcal A_h^{n,q}(L)) \end{align*} be a global $L$-valued $(n,q)$-form satisfying $\bar\partial f = 0$ in the distributional sense. Since $X$ is compact Kähler, the [metric space](/page/Metric%20Space) associated to $\omega$ is complete and the total volume measure $dV_\omega$ is finite. The curvature-current hypothesis gives the strict Nakano-positive lower bound required for the singular Bochner-Kodaira-Hörmander estimate in bidegree $(n,q)$: \begin{align*} i\Theta_h(L) \geq \varepsilon\omega. \end{align*} Demailly regularization of the quasi-plurisubharmonic local weights of $h$ supplies smooth approximating metrics with curvature bounded below by $\varepsilon_\nu\omega$, where $\varepsilon_\nu > 0$ and $\varepsilon_\nu \to \varepsilon$; passing to the limit in the regularized estimates gives the singular estimate. Hence there exists \begin{align*} u \in \Gamma(X,\mathcal A_h^{n,q-1}(L)) \end{align*} such that $\bar\partial u = f$ and \begin{align*} \int_X |u|_{\omega,h}^2\,dV_\omega \leq \frac{1}{q\varepsilon}\int_X |f|_{\omega,h}^2\,dV_\omega. \end{align*} Thus every global $\bar\partial$-closed section of $\mathcal A_h^{n,q}(L)$ with $q \geq 1$ is globally $\bar\partial$-exact. [/step] [step:Identify the sheaf cohomology class with zero] The cohomology group in degree $q \geq 1$ of the global complex from the fine resolution is represented by a $\bar\partial$-closed section $f \in \Gamma(X,\mathcal A_h^{n,q}(L))$. The previous step produces $u \in \Gamma(X,\mathcal A_h^{n,q-1}(L))$ with $\bar\partial u=f$, so the class of $f$ in the quotient complex is zero. Therefore \begin{align*} H^q\left(X,K_X\otimes L\otimes\mathcal I(h)\right)=0 \end{align*} for every $q \geq 1$, which is the asserted Nadel vanishing theorem. [/step]