. If $\chi(X,\OO_X)\neq0$, then $\kappa(X,K_X)\geq0$.  If there exist a resolution $\pi\colon Y\to X$, a positive integer $m$ such that $mK_X$ is Cartier, and a singular metric $h$ on $\pi^*\OO_X(mK_X)$ with semipositive curvature current, such that the multiplier ideal sheaf $\mathcal I(h)$ is different from $\OO_Y$, then the result follows from Corollary \[cor:multiplier\].  Otherwise, pick a resolution $\pi\colon Y\to X$, a positive integer $m$ such that $mK_X$ is Cartier, and a singular metric $h$ on $\pi^*\OO_X(mK_X)$ with semipositive curvature current. Then the result follows from 