A normal projective surface with the same Betti numbers of the projective plane $\mathbb{CP}^2$ is called a rational homology projective plane or a $\mathbb{Q}$-homology $\mathbb{CP}^2$. People working in algebraic geometry and topology have long studied a $\mathbb{Q}$-homology $\mathbb{CP}^2$ with possibly quotient singularities. It is now known that it has at most five such singular points, but it is still mysterious so that there are many unsolved problems left.
In this talk, I’ll review some known results and open problems in this field which might be solved and might not be solved in near future. In particular, I’d like to review the following two topics and to report some recent progress:
1. Algebraic Montgomery-Yang problem.
2. Classification of $\mathbb{Q}$-homology $\mathbb{CP}^2$ with quotient singularities.