| Résume | This talk is based on joint works with Botong Wang. A conjecture by Chern-Hopf-Thurston states that an aspherical closed real $2n$-manifold $X$ satisfies $(-1)^n\chi(X) \geq 0$, where $\chi(X)$ denotes the Euler characteristic of $X$. I will focus on the case where $X$ has the structure of a complex algebraic variety, which implies that $X$ has large fundamental group. Inspired by this, in 1995, Kollár proposed the following conjecture: a complex projective manifold $X$ satisfies $\chi(K_X) \geq 0$ if it has generically large fundamental group. In this talk, I will outline the proofs of both conjectures under the assumption that $\pi_1(X)$ is linear. |
