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 $CS$, 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 $CS$ be a finite union of projective varieties of given dimensions and codimensions in their linear spans which must be skew and generate.