Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topology of a complex double plane branching along a real line arrangement

Ichiro Shimada

TL;DR

This work analyzes the topology of the smooth surface $X$, the minimal desingularization of the double cover of $ mathbb{A}^2( mathbb{C})$ branched along a nodal real line arrangement. By leveraging the real structure, it constructs vanishing-cycle-like $2$-spheres $\,igl[ abla_Cigr]$ associated to bounded chambers, and explicitly computes their intersection form with the exceptional curves $[D_P]$ and with each other via two displacement techniques (C- and E-displacements). It proves that $H_2(X)$ is free and that the classes $[ abla_C]$ together with $[D_P]$ form a basis, with precise mutual and self-intersection numbers; it also relates these collections to coherence of chamber orientations and to the asymptotic lattice structure after projective completion. The paper further develops practical tools for calculating periods and lattices on these open surfaces, including explicit capping data, displacement constructions, and a suite of examples and experiments demonstrating the predicted lattice signatures and discriminants.

Abstract

We investigate the topology of a double cover of a complex affine plane branching along a nodal real line arrangement. We define certain topological 2-cycles in the double plane using the real structure of the arrangement, and calculate their intersection numbers.

Topology of a complex double plane branching along a real line arrangement

TL;DR

This work analyzes the topology of the smooth surface , the minimal desingularization of the double cover of branched along a nodal real line arrangement. By leveraging the real structure, it constructs vanishing-cycle-like -spheres associated to bounded chambers, and explicitly computes their intersection form with the exceptional curves and with each other via two displacement techniques (C- and E-displacements). It proves that is free and that the classes together with form a basis, with precise mutual and self-intersection numbers; it also relates these collections to coherence of chamber orientations and to the asymptotic lattice structure after projective completion. The paper further develops practical tools for calculating periods and lattices on these open surfaces, including explicit capping data, displacement constructions, and a suite of examples and experiments demonstrating the predicted lattice signatures and discriminants.

Abstract

We investigate the topology of a double cover of a complex affine plane branching along a nodal real line arrangement. We define certain topological 2-cycles in the double plane using the real structure of the arrangement, and calculate their intersection numbers.
Paper Structure (32 sections, 18 theorems, 228 equations, 8 figures, 2 tables)

This paper contains 32 sections, 18 theorems, 228 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Terminology
  3. Standard orientations
  4. The vector space of translations
  5. Good coordinates
  6. Proof of Theorem \ref{['thm:basis']}
  7. Capping hemispheres
  8. Good systems of local coordinates
  9. The orientation of Y(R)
  10. Capping hemispheres
  11. Displacement
  12. Notation and terminology about displacements
  13. C-displacement
  14. E-displacement
  15. Proof of Theorem \ref{['thm:intnumbs']} (1), (2), (3)
  16. ...and 17 more sections

Key Result

Theorem 1.3

The $\mathord{\mathbb Z}$-module $H_2(X; \mathord{\mathbb Z})$ is free. We fix an orientation $\gamma_C$ for each $C\in \boldsymbol{C h}_{\mathrm{b}}$. Then the homology classes $[\Sigma(C, \gamma_C)]$, where $C$ runs through $\boldsymbol{C h}_{\mathrm{b}}$, and the homology classes $[D_P]$, where $

Figures (8)

  • Figure 1.1: Vanishing cycle $\Sigma(C, \gamma_C)$
  • Figure 4.1: Orientation $\beta_{\mathord{\mathbb R}}^*\sigma_{\mathord{\mathbb A}}$
  • Figure 4.2: Location of chambers
  • Figure 4.3: $\beta^{\sharp} C$ on $Y(\mathord{\mathbb R})$
  • Figure 6.1: $E$-displacement for the proof of assertion (3)
  • ...and 3 more figures

Theorems & Definitions (54)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Proposition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 44 more