$L^2$-vanishing theorem and a conjecture of Koll\'ar

TL;DR

This paper proves Kollár's conjecture on the non-negativity of the Euler characteristic for certain complex projective varieties by establishing an $L^2$-vanishing theorem for their universal covers, using analytic and algebraic techniques.

Contribution

It introduces a new $L^2$-vanishing theorem for universal covers of projective varieties with linear fundamental groups, confirming Kollár's conjecture under this condition.

Findings

Proved $L^2$-Dolbeault cohomology vanishing for universal covers.

Confirmed Kollár's conjecture for varieties with linear fundamental groups.

Applied techniques from the linear Shafarevich conjecture and analysis.

Abstract

In 1995, Koll\'ar conjectured that a smooth complex projective $n$-fold $X$ with generically large fundamental group has Euler characteristic $\chi(X, K_X)\geq 0$. In this paper, we prove the conjecture assuming $X$ has linear fundamental group, i.e., there exists a representation $\pi_1(X)\to {\rm GL}_N(\mathbb{C})$ with finite kernel. We deduce the conjecture by proving a stronger $L^2$ vanishing theorem: for the universal cover $\widetilde{X}$ of such $X$, its $L^2$-Dolbeault cohomology $H_{(2)}^{n,q}(\widetilde{X})=0$ for $q\neq 0$. The main ingredients of the proof are techniques from the linear Shafarevich conjecture along with some analytic methods.