Let $X$ be a compact Kähler manifold and $\Delta$ be a $ℝ$-divisor with simple normal crossing support and coefficients between $1/2$ and $1$. Assuming that $K_X+\Delta$ is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on $X\setminus \mathrm{Supp}\left(\Delta \right)$ having mixed Poincaré and cone singularities according to the coefficients of $\Delta$. As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair $(X,\Delta )$.

Given a positive closed (1,1)-current $T$ defined on the regular locus of a projective variety $X$ with bounded mass near the singular part of $X$ and $Y$ an irreducible algebraic subset of $X$, we present uniform estimates for the locus inside $Y$ where the Lelong numbers of $T$ are larger than the generic Lelong number of $T$ along $Y$.

We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor of a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-Einstein metrics with cone singularities. As an application we extend to this setting classical results of Lichnerowicz and Kobayashi on the parallelism and vanishing of appropriate holomorphic tensor fields.