Algebraic Montgomery-Yang Problem: the noncyclic case

Abstract

Montgomery-Yang problem predicts that every pseudofree differentiable circle action on the 5-dimensional sphere ${\mathbb{S}}^5$ has at most 3 non-free orbits. Using a certain one-to-one correspondence, Kollár formulated the algebraic version of the Montgomery-Yang problem: every projective surface $S$ with quotient singularities such that $b_2(S)=1$ has at most 3 singular points if its smooth locus $S^0$ is simply-connected. In this paper, we prove the conjecture under the assumption that $S$ has at least one noncyclic singularity. In the course of the proof, we classify projective surfaces $S$ with quotient singularities such that (i) $b_2(S)=1$, (ii) $H_1(S^0,\mathbb{Z})=0$, and (iii) $S$ has 4 or more singular points, not all cyclic, and prove that all such surfaces have ${\pi }_1(S^0)\cong {\mathfrak{A}}_5$, the icosahedral group.

26 pages. A statement in Theorem 1.5(4) is corrected, hence the corresponding statements in Proposition 3.5 and Example 3.6 are also corrected

References in corpus (2)

• Algebraic Montgomery-Yang Problem: the non-rational case and the del Pezzo case
• The maximum number of singular points on rational homology projective planes

Cited by in corpus (2)

• Algebraic Montgomery-Yang Problem: the non-rational case and the del Pezzo case
• Constructions of singular rational surfaces of Picard number one with ample canonical divisor