  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
$(-1)^n\chi(X)\geq 0$. 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
$\pi_1(X)$ admits an almost faithful linear representation, then $\chi(X,
{P})\geq 0$ 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.
