  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.
