Published online by Cambridge University Press: 21 September 2016
Let $X$ be a compact Kähler manifold, endowed with an effective reduced divisor $B=\sum Y_{k}$ having simple normal crossing support. We consider a closed form of $(1,1)$ -type $\unicode[STIX]{x1D6FC}$ on $X$ whose corresponding class $\{\unicode[STIX]{x1D6FC}\}$ is nef, such that the class $c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\}\in H^{1,1}(X,\mathbb{R})$ is pseudo-effective. A particular case of the first result we establish in this short note states the following. Let $m$ be a positive integer, and let $L$ be a line bundle on $X$ , such that there exists a generically injective morphism $L\rightarrow \bigotimes ^{m}T_{X}^{\star }\langle B\rangle$ , where we denote by $T_{X}^{\star }\langle B\rangle$ the logarithmic cotangent bundle associated to the pair $(X,B)$ . Then for any Kähler class $\{\unicode[STIX]{x1D714}\}$ on $X$ , we have the inequality