Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Products of locally conformal symplectic manifolds

Baptiste Chantraine, Kevin Sackel

Abstract

Given two locally conformal symplectic (LCS) structures on manifolds $M_1$ and $M_2$, we construct a natural $\R^+$-torsor of locally conformal symplectic structures on a certain covering space $M_1 \boxplus M_2$ of $M_1 \times M_2$. As the smooth construction of $M_1 \boxplus M_2$ is natural from the perspective of flat line bundles, we use this language to phrase the LCS theory. This construction shares many properties with, and in a sense generalizes, the standard symplectic product. Notably, for a Hamiltonian isotopy $φ_t$ of an LCS manifold $M$, there is an associated Lagrangian embedding $Γ(φ_1) \colon M \hookrightarrow M \boxplus M$, in which certain fixed points of $φ_1$ are in bijection with intersection points of $Γ(φ_1)$ with the diagonal $Δ= Γ(\mathrm{id})$. Using a Lagrangian intersection of result of the first author and E. Murphy, we may conclude that if $φ_t$ is a $C^0$-small Hamiltonian isotopy, then the number of fixed points of $φ_1$ is bounded below by the rank of the Novikov theory associated to the Lee class of the LCS structure on $M$. Finally, we end the paper by constructing the suspension of a Lagrangian submanifold along a Hamiltonian isotopy in the LCS theory, again generalizing the symplectic setting.

Products of locally conformal symplectic manifolds

Abstract

Given two locally conformal symplectic (LCS) structures on manifolds and , we construct a natural -torsor of locally conformal symplectic structures on a certain covering space of . As the smooth construction of is natural from the perspective of flat line bundles, we use this language to phrase the LCS theory. This construction shares many properties with, and in a sense generalizes, the standard symplectic product. Notably, for a Hamiltonian isotopy of an LCS manifold , there is an associated Lagrangian embedding , in which certain fixed points of are in bijection with intersection points of with the diagonal . Using a Lagrangian intersection of result of the first author and E. Murphy, we may conclude that if is a -small Hamiltonian isotopy, then the number of fixed points of is bounded below by the rank of the Novikov theory associated to the Lee class of the LCS structure on . Finally, we end the paper by constructing the suspension of a Lagrangian submanifold along a Hamiltonian isotopy in the LCS theory, again generalizing the symplectic setting.
Paper Structure (16 sections, 18 theorems, 157 equations)

This paper contains 16 sections, 18 theorems, 157 equations.

Table of Contents

  1. Introduction.
  2. Acknowledgements
  3. Flat line bundles
  4. Flat line bundles and the twisted de Rham complex
  5. The gauged viewpoint on flat oriented line bundles
  6. Twisted products of flat line bundles
  7. The twisted product
  8. Adding twisted differential forms
  9. An example
  10. LCS geometry
  11. Products of LCS manifolds
  12. Examples.
  13. Comparison with the contact product.
  14. Product of twisted cotangent bundles.
  15. Products of isotropics and Lagrangians
  16. ...and 1 more sections

Key Result

Theorem 1.5

Suppose $M_1$ and $M_2$ are two smooth manifolds with specified cohomology class $\mathfrak{l}_1 \in H^1(M_1;\mathbb{R})$ and $\mathfrak{l}_2 \in H^1(M_2;\mathbb{R})$. Then there exists a smooth manifold $M_1 \boxplus M_2$ (depending upon $\mathfrak{l}_1$ and $\mathfrak{l}_2$) together with a coveri canonical up to deck transformations and satisfying the following properties:

Theorems & Definitions (61)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 51 more