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Integration of the elliptic tangent bundle and elliptic Poisson structures

Bas Wensink

TL;DR

This work provides an explicit Lie groupoid integration for the elliptic tangent bundle $\mathcal{A}_{|D|}$ associated to (possibly non-coorientable) elliptic divisors $D$ on a manifold $M$, and establishes precise topological criteria for the existence of Hausdorff, $s$-connected integrations. It develops the integration in progressively general cases, from a smooth coorientable divisor via a complex blow-up, to coorientation coverings, non-coorientable cases, and finally normal-crossing divisors, using a mix of local blow-ups, double covers, and pushouts. The paper also constructs a canonical elliptic ideal on the resulting groupoids and provides local models for the symplectic integration of elliptic Poisson structures, including nonzero and zero elliptic residue cases, connecting with known holomorphic integrations. These results bridge the infinitesimal elliptic tangent geometry with explicit global groupoid models, enabling potential pseudodifferential calculus and symplectic integration in singular settings along codimension-two loci. The constructions yield concrete criteria and models with explicit isotropy and orbit structures, informing both the geometry of elliptic divisors and applications to elliptic Poisson/symplectic geometry.

Abstract

We explicitly construct a Lie groupoid integrating the elliptic tangent bundle associated to a (possibly normal crossing) elliptic divisor, providing a necessary and sufficient topological condition for the existence of a Hausdorff integration. We also produce an explicit local model for the symplectic integration of an elliptic Poisson structure.

Integration of the elliptic tangent bundle and elliptic Poisson structures

TL;DR

This work provides an explicit Lie groupoid integration for the elliptic tangent bundle associated to (possibly non-coorientable) elliptic divisors on a manifold , and establishes precise topological criteria for the existence of Hausdorff, -connected integrations. It develops the integration in progressively general cases, from a smooth coorientable divisor via a complex blow-up, to coorientation coverings, non-coorientable cases, and finally normal-crossing divisors, using a mix of local blow-ups, double covers, and pushouts. The paper also constructs a canonical elliptic ideal on the resulting groupoids and provides local models for the symplectic integration of elliptic Poisson structures, including nonzero and zero elliptic residue cases, connecting with known holomorphic integrations. These results bridge the infinitesimal elliptic tangent geometry with explicit global groupoid models, enabling potential pseudodifferential calculus and symplectic integration in singular settings along codimension-two loci. The constructions yield concrete criteria and models with explicit isotropy and orbit structures, informing both the geometry of elliptic divisors and applications to elliptic Poisson/symplectic geometry.

Abstract

We explicitly construct a Lie groupoid integrating the elliptic tangent bundle associated to a (possibly normal crossing) elliptic divisor, providing a necessary and sufficient topological condition for the existence of a Hausdorff integration. We also produce an explicit local model for the symplectic integration of an elliptic Poisson structure.

Paper Structure

This paper contains 12 sections, 31 theorems, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on ideal Lie algebroids
  3. Integration case I: $D$ is smooth and coorientable
  4. Integration case II: $(M,D)$ has a coorientation normal covering
  5. $(M,D)$ has a coorientation double cover
  6. $(M,D)$ has a coorientation normal covering
  7. Integration case III: $D$ is smooth but not coorientable
  8. Integration case IV: $D$ has normal crossings
  9. Elliptic divisors on elliptic groupoids
  10. Symplectic integrations of elliptic symplectic structures
  11. Nonzero elliptic residue
  12. Zero elliptic residue

Key Result

Lemma 2.10

Let $\mathcal{A}\to M$ be a holomorphic Lie algebroid. Consider the underlying vector bundle $\mathcal{A}^\mathbb{R}\to M^\mathbb{R}$ obtained by forgetting the complex structures on $\mathcal{A}$ and on $M\,$. Then under the isomorphism $TM\xrightarrow{\varphi}T^{1,0}M\,,$$\mathcal{A}^\mathbb{R}$ b

Figures (1)

  • Figure 1: Four-fold cover of the embedding of the Klein bottle in the four-torus. Each black square represents a four-torus, where the horizontal direction is the $S^1$-factor representing the cycle $\xi\,,$ whereas the vertical direction is a three-torus. The red double helix is the image of $\gamma\,,$ which represents the cycle $2\xi\,.$

Theorems & Definitions (80)

  • Definition 2.1: Ideal Lie algebroid
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5: Elliptic tangent bundle
  • Remark 2.6
  • Example 2.7
  • Example 2.8
  • Definition 2.9: Complex primitive
  • Lemma 2.10: LSX07
  • ...and 70 more