The maximum number of singular points on rational homology projective planes

Abstract

A normal projective complex surface is called a rational homology projective plane if it has the same Betti numbers with the complex projective plane C P 2 \mathbb {C}\mathbb {P}^2 . It is known that a rational homology projective plane with quotient singularities has at most 5 5 singular points. So far all known examples have at most 4 4 singular points. In this paper, we prove that a rational homology projective plane S S with quotient singularities such that K S K_S is nef has at most 4 4 singular points except one case. The exceptional case comes from Enriques surfaces with a configuration of 9 smooth rational curves whose Dynkin diagram is of type 3 A 1 ⊕ 2 A 3 3A_1 \oplus 2A_3 .

We also obtain a similar result in the differentiable case and in the symplectic case under certain assumptions which all hold in the algebraic case.