Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor

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

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