$\mathbb{Q}$-homology projective planes with nodes or cusps

Abstract

We classify all $\mathbb{Q}$-homology projective planes with $A_1$- or $A_2$-singularities (and with no worse singularities). It turns out that such a surface is isomorphic to a global quotient $X/G$, where $X$ is a fake projective plane or the complex projective plane and $G$ a finite abelian group of bi-holomorphic automorphisms. There are only finitely many such surfaces.