Algebraic Montgomery-Yang problem: the non-rational surface case (Q1951491)

A \(\mathbb{Q}\)-homology plane is a normal projective surface with the same Betti numbers as the projective plane. Kollár conjectured that a \(\mathbb{Q}\)-homology plane \(S\) with quotient singularities has at most three singular points, if \(S\backslash \mathrm{Sing}(S)\) is simply connected. In the paper under review, the authors continue their study on Kollár's conjecture (see their previous paper [Math. Ann. 350, No. 3, 721--754 (2011; Zbl 1228.14033)]) and confirm this conjecture with an extra condition that \(S\) is not rational. The main ingredients of the proof are orbifold Bogomolov-Miyaoka-Yau inequalities and lattice theory.