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Around Homology Planes: Old and new

Rodolfo Aguilar Aguilar

Abstract

We survey some old and new results about acyclic (affine) complex surfaces, also called homology planes. We ask several questions and leave open directions for future research.

Around Homology Planes: Old and new

Abstract

We survey some old and new results about acyclic (affine) complex surfaces, also called homology planes. We ask several questions and leave open directions for future research.
Paper Structure (21 sections, 23 theorems, 12 equations, 3 figures)

This paper contains 21 sections, 23 theorems, 12 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Cancellation property
  4. Classification
  5. Exotic structures
  6. Algebraic
  7. Analytic
  8. Symplectic
  9. Contractible four-manifolds
  10. More symplectic properties
  11. Search for more exotic symplectic structures
  12. A question for the contact structure of the boundary
  13. Invariance of log-Kodaira dimension
  14. Mirror symmetry
  15. Topology
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $X$ be a smooth affine surface which is contractible and simply connected at infinity. Then $X$ is isomorphic to $\mathbb{C}^2$ as an algebraic variety.

Figures (3)

  • Figure 1: Dual graph of the minimal embedded resolution of $C_1'\cup C_2'$
  • Figure 2: $L(1,5)$ in the projective plane
  • Figure 3: Splice of the Seifert spheres $\Sigma(a_1,\ldots,a_n, a_1\cdots a_n \pm 1)$ along its last fiber.

Theorems & Definitions (32)

  • Example 2.1: R71
  • Theorem 2.1: R71
  • Definition 2.1
  • Example 2.2
  • Theorem 2.2: GS89-1,GS89-2
  • Definition 2.2
  • Theorem 2.3: F82, GM88, tDP90
  • Conjecture 2.1
  • Theorem 2.4: P23
  • Theorem 3.1: IF77
  • ...and 22 more