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 σ   of the tensor product T⊗T∨T⊗T∨ of the tangent bundle T   with the cotangent bundle T∨T∨. To such a curvature type tensor σ   one associates the first Mok characteristic cone CSCS, obtained by projecting on T   the intersection of ker(σ)ker(σ) with the space of rank 1 tensors. The simpler characterization requires that the projective scheme associated to CSCS be a finite union of projective varieties of given dimensions and codimensions in their linear spans which must be skew and generate.