Linear Singer-Hopf conjecture

If X is a closed $2n$-dimensional aspherical manifold, i.e., the universal cover of X is contractible, then the Singer-Hopf conjecture predicts that (-1)ⁿχ(X)≥ 0. We prove this conjecture when X is a complex projective manifold whose fundamental group admits an almost faithful linear representation over any field. In fact, we prove a much stronger statement that if X is a complex projective manifold with large fundamental group and π₁(X) admits an almost faithful linear representation, then χ(X, P)≥ 0 for any perverse sheaf P on X. To prove the main result, we introduce a vanishing cycle functor of multivalued one-forms. Then using techniques from non-abelian Hodge theories in both archimedean and non-archimedean settings, we deduce the desired positivity from the geometry of pure and mixed period maps.