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Poisson geometry of truncated polynomials and hypersurface algebroids

Francis Bischoff, Aldo Witte

Abstract

We study symplectic forms on hypersurface algebroids. These are a broad generalization of the $b^{k}$-Poisson structures studied extensively by Miranda, Scott, and collaborators, and their geometry is intimately related to the group of truncated polynomials under composition. They induce Poisson structures that are generically symplectic and drop rank along a codimension $1$ submanifold $W$. However, unlike in the case of $b^{k}$-Poisson structures, the symplectic foliation along $W$ can have non-zero symplectic variation, reflecting the obstruction to extending the order of vanishing of a hypersurface algebroid. In addition to studying the symplectic geometry of these algebroids, in this paper we carry out a detailed study of the Lie algebroid de Rham complex, and develop a method for deforming symplectic forms along paths in a $k$-jet character variety. As a result, we are able to produce a large class of new examples of Poisson structures. Finally, we construct universal hypersurface algebroids and show that in even dimensions they admit canonical Poisson structures.

Poisson geometry of truncated polynomials and hypersurface algebroids

Abstract

We study symplectic forms on hypersurface algebroids. These are a broad generalization of the -Poisson structures studied extensively by Miranda, Scott, and collaborators, and their geometry is intimately related to the group of truncated polynomials under composition. They induce Poisson structures that are generically symplectic and drop rank along a codimension submanifold . However, unlike in the case of -Poisson structures, the symplectic foliation along can have non-zero symplectic variation, reflecting the obstruction to extending the order of vanishing of a hypersurface algebroid. In addition to studying the symplectic geometry of these algebroids, in this paper we carry out a detailed study of the Lie algebroid de Rham complex, and develop a method for deforming symplectic forms along paths in a -jet character variety. As a result, we are able to produce a large class of new examples of Poisson structures. Finally, we construct universal hypersurface algebroids and show that in even dimensions they admit canonical Poisson structures.
Paper Structure (42 sections, 61 theorems, 294 equations)

This paper contains 42 sections, 61 theorems, 294 equations.

Table of Contents

  1. Introduction
  2. Review of hypersurface algebroids
  3. $k$-th order foliations
  4. $k$-th order Atiyah algebroid
  5. The canonical representation
  6. Hypersurface algebroids and the Maurer-Cartan equation
  7. The structure group
  8. Cohomology
  9. Classification
  10. Extension class
  11. Examples
  12. Surfaces
  13. The Heisenberg group
  14. Arnold's cat map
  15. The universal hypersurface algebroids
  16. ...and 27 more sections

Key Result

proposition 2.2

foliationpaper Let $A \Rightarrow M$ be a HS algebroid of order $k$ for $(W, M)$. Then there are local coordinates $(z, x_2, ..., x_n)$ for $M$ such that $W = \{ z = 0 \}$, and such that $A$ is spanned by the following sections

Theorems & Definitions (137)

  • remark 1
  • remark 2
  • definition 2.1
  • remark 3
  • proposition 2.2
  • proposition 2.3
  • definition 2.4
  • theorem 2.5
  • definition 2.6
  • lemma 2.7
  • ...and 127 more