Non-Abelian Lefschetz Hyperplane Theorems

Let X be a smooth projective variety over the complex numbers, and let D be an ample divisor in X. For which spaces Y is the restriction map r: Hom(X, Y) -> Hom(D, Y) an isomorphism? Using positive characteristic methods, we give a fairly exhaustive answer to this question. An example application of our techniques is: if dim(X) > 2, Y is smooth, the cotangent bundle of Y is nef, and dim(Y) < dim(D), the restriction map r is an isomorphism. Taking Y to be the classifying space of a finite group BG, the moduli space of pointed curves M_{g,n}, the moduli space of principally polarized Abelian varieties A_g, certain period domains, and various other moduli spaces, one obtains many new and classical Lefschetz hyperplane theorems.