Linear Chern-Hopf-Thurston conjecture

If $X$ is a closed $2n$-dimensional aspherical manifold, i.e., the universal cover of $X$ is contractible, then the Chern-Hopf-Thurston 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 $\chi(X, \mathcal{P})\geq 0$ for any perverse sheaf $\mathcal{P}$ on $X$. To prove this, we introduce a vanishing cycle functor of multivalued one-forms and apply techniques from non-abelian Hodge theory, both in archimedean and non-archimedean settings. These techniques allow us to deduce the desired positivity from the geometric properties of pure and mixed period maps.