Algebraic Montgomery-Yang problem: the non-cyclic case

Abstract

Montgomery-Yang problem predicts that every pseudofree circle action on the 5-dimensional sphere has at most 3 non-free orbits. Using a certain one-to-one correspondence, Kollar formulated the algebraic version of the Montgomery-Yang problem: every projective surface S with quotient singularities such that the second Betti number b (2)(S) = 1 has at most 3 singular points if its smooth locus S (0) is simply connected. We prove the conjecture under the assumption that S has at least one non-cyclic singularity. In the course of the proof, we classify projective surfaces S with quotient singularities such that (i) b (2)(S) = 1, (ii) , and (iii) S has 4 or more singular points, not all cyclic, and prove that all such surfaces have , the icosahedral group.