Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields

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