  We propose singular variants of the Singer-Hopf conjecture, formulated in
terms of the Euler-Mather characteristic, intersection homology Euler
characteristic and, resp., virtual Euler characteristic of a closed irreducible
subvariety of an aspherical complex projective manifold. We prove the
conjecture under the assumption that the cotangent bundle of the ambient
variety is numerically effective (nef), or, more generally, when the ambient
manifold admits a finite morphism to a complex projective manifold with a nef
cotangent bundle.
