Algebraic Montgomery-Yang problem and smooth obstructions

Abstract

Let $S$ be a rational homology complex projective plane with quotient singularities. The algebraic Montgomery-Yang problem conjectures that the number of singular points of $S$ is at most three if its smooth locus is simply-connected. In this paper, we leverage results from the study of smooth 4-manifolds, including the Donaldson diagonalization theorem and Heegaard Floer correction terms, to establish additional conditions for $S$. As a result, we eliminate the possibility of a rational homology complex projective plane of specific types with four singularities. Moreover, we identify large families encompassing infinitely many types of singularities that satisfy the orbifold BMY inequality, a key property in algebraic geometry, yet are obstructed from being a rational homology complex projective plane due to smooth conditions. Additionally, we discuss computational results related to this problem, offering new insights into the algebraic Montgomery-Yang problem.

Figures (14)

• Figure 1: The plumbing graph of $X(p,q)$.
• Figure 2: The graph of the boundary sum of four linear plumbed 4-manifolds.
• Figure 3: An essentially unique embedding of $Q_{X(2,1)}\oplus Q_{X(3,2)}\oplus Q_{X(11,2)}\oplus Q_{X(13,1)}$ into $-\mathbb{Z}^{7}$.
• Figure 4: An essentially unique embedding of $Q_{X(2,1)}\oplus Q_{X(3,2)} \oplus Q_{X(5,1)} \oplus Q_{X(9409,5519)}$ into $-\mathbb{Z}^{15}$.
• Figure 5: An essentially unique embedding of $Q_{X(2,1)}\oplus Q_{X(3,2)} \oplus Q_{X(5,1)} \oplus Q_{X(3529,1880)}$ into $-\mathbb{Z}^{17}$.

Theorems & Definitions (59)

• Conjecture 1.1: Algebraic Montgomery-Yang Problem, Kollar-2008
• Conjecture 1.2
• Remark
• Conjecture 1.4: Montgomery-Yang Problem
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Example 2.1
• Theorem 2.2
• Proposition 2.3: The perfect squareness of $D$, Hwang-Keum-2011-1