A characterization of varieties whose universal cover is a bounded symmetric domain without ball factors

We give two characterizations of varieties whose universal cover is a bounded symmetric domain without ball factors in terms of the existence of a holomorphic endomorphism \s of the tensor product T\otimes T' of the tangent bundle T with the cotangent bundle T'. To such a curvature type tensor \s one associates the first Mok characteristic cone S, obtained by projecting on T the intersection of ker (\s) with the space of rank 1 tensors. The simpler characterization requires that the projective scheme associated to S be a finite union of projective varieties of given dimensions and codimensions in their linear spans which must be skew and generate.