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On rational homology projective planes with quotient singularities of small indices

Woohyeok Jo, Jongil Park, Kyungbae Park

Abstract

In this article, we study the effects of topological and smooth obstructions on the existence of rational homology complex projective planes that admit quotient singularities of small indices. In particular, we provide a classification of the types of quotient singularities that can be realized on rational homology complex projective planes with indices up to three, whose smooth loci have trivial first integral homology group.

On rational homology projective planes with quotient singularities of small indices

Abstract

In this article, we study the effects of topological and smooth obstructions on the existence of rational homology complex projective planes that admit quotient singularities of small indices. In particular, we provide a classification of the types of quotient singularities that can be realized on rational homology complex projective planes with indices up to three, whose smooth loci have trivial first integral homology group.

Paper Structure

This paper contains 12 sections, 35 theorems, 57 equations, 18 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results from Algebraic Geometry
  4. Topological and Smooth Obstructions
  5. Topological Constraints
  6. Donaldson's Diagonalization Theorem
  7. Heegaard Floer $d$-invariants
  8. The Index One Case (Proof of Theorem \ref{['thm:Gorenstein_H1=0']})
  9. The Index Two Case (Proof of Theorem \ref{['thm:index2_list_H1=0']})
  10. The Index Three Case (Proof of Theorem \ref{['thm:index3_list']})
  11. Obstructions
  12. Realizations

Key Result

Theorem 1.5

The singularity type of a Gorenstein $\mathbb{Q}$-homology $\mathbb{CP}^2$ is one of the following $58$ types:

Figures (18)

  • Figure 1: The weighted dual graph of a singularity of type $K_n$.
  • Figure 2: The plumbing graph for the negative definite 4-manifold $X(p,q)$ bounded by the lens space $L(p,q)$.
  • Figure 3: The plumbing graph of $X(n+1,1)$.
  • Figure 4: The plumbing graph of $X(4n,2n+1)$.
  • Figure 5: An embedding of $Q_{X(9,1)}$ into $-\mathbb{Z}^2$.
  • ...and 13 more figures

Theorems & Definitions (60)

  • Conjecture 1.2: Algebraic Montgomery-Yang Problem, Kollar-2008
  • Theorem 1.5: Hwang-Keum-Ohashi-2015
  • Theorem 1.6: Miyanishi-Zhang-1988Gurjar-Pradeep-Zhang-2002
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9: Alexeev-Nikulin-2006
  • Theorem 1.10: Kojima-2003
  • Theorem 1.11
  • Corollary 1.12
  • Remark 1.13
  • ...and 50 more