The Chern-Hopf-Thurston conjecture predicts that if a closed $2d$-dimensional manifold $X$ is aspherical--meaning its universal cover is contractible, then its Euler characteristic satisfies $(-1)^d\chi(X)\geq 0$. The Singer conjecture further asserts that all the $L^2$-Betti numbers of $X$ vanish except in degree $d$. In this talk, I will discuss various generalizations of these conjectures for complex projective varieties, focusing on perverse sheaves on projective varieties with large fundamental groups. In particular, I will explore connections between certain cases of these conjectures and the Lagrangian positivity of the cotangent bundle. Under the assumption that the fundamental group is linear, we establish Lagrangian positivity using techniques from non-abelian Hodge theory. This talk is based on joint work with Donu Arapura and Ya Deng.