Algebraic Montgomery-Yang Problem: the non-rational case and the del Pezzo case

2009/06/03 by Keum, JongHae, Hwang, DongSeon
#14J17 #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.0906.0633

Abstract

Montgomery-Yang problem predicts that every pseudofree differentiable circle action on the 5-dimensional sphere 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 the second Betti number b₂(S) = 1 and with quotient singularities only has at most 3 singular points if its smooth locus S⁰ is simply-connected. In a previous paper, we have confirmed the conjecture when S has at least one non-cyclic quotient singularity. In this paper, we prove the conjecture either when S is not rational or when -K_S is ample.

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