$L^2$-vanishing theorem and a conjecture of Kollár

In 1995, Kollár conjectured that a smooth complex projective $n$-fold $X$ with generically large fundamental group has Euler characteristic $\chi (X,K_X)\ge 0$. In this paper, we prove the conjecture assuming $X$ has linear fundamental group, i.e., there exists a representation ${\pi }_1(X)\to {\mathrm{G}\mathrm{L}}_N(\mathbb{C})$ with finite kernel. We deduce the conjecture by proving a stronger $L^2$ vanishing theorem: for the universal cover $\tilde{X}$ of such $X$, its $L^2$-Dolbeault cohomology $H_{(2)}^{n,q}(\tilde{X})=0$ for $q\mathrm{\ne }0$. The main ingredients of the proof are techniques from the linear Shafarevich conjecture along with some analytic methods.