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

We prove the existence of non-positively curved K"ahler-Einstein metrics with cone singularities along a given simple normal crossing divisor on a compact K"ahler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved K"ahler-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.