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The local Floer cohomology of indicator functions

Yoel Groman

Abstract

For a compact set $K$ with contact type boundary in a symplectic manifold $M$ we construct a spectral sequence from the local Floer homology of the Reeb orbits, as studied by \cite{Mclean2012}, to the relative symplectic cohomology of $K$ in $M$ over the Novikov ring. The spectral sequence is functorial with respect to inclusions which are not required to be exact. This functoriality is key to the closed string reconstruction problem near the singularity of an SYZ fibration. We illustrate this in the case of dimension $2n=4$ for symplectic cluster manifolds. In higher dimension, an additional ingredient, the locality spectral sequence, is required, and is the subject of a forthcoming work in progress.

The local Floer cohomology of indicator functions

Abstract

For a compact set with contact type boundary in a symplectic manifold we construct a spectral sequence from the local Floer homology of the Reeb orbits, as studied by \cite{Mclean2012}, to the relative symplectic cohomology of in over the Novikov ring. The spectral sequence is functorial with respect to inclusions which are not required to be exact. This functoriality is key to the closed string reconstruction problem near the singularity of an SYZ fibration. We illustrate this in the case of dimension for symplectic cluster manifolds. In higher dimension, an additional ingredient, the locality spectral sequence, is required, and is the subject of a forthcoming work in progress.
Paper Structure (42 sections, 51 theorems, 114 equations, 7 figures)

This paper contains 42 sections, 51 theorems, 114 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. SYZ mirror symmetry near a singularity
  4. The case $n=2$
  5. The case $n>2$
  6. Local and infinitesimal Floer homology of indicator functions
  7. Removing the exactness assumption?
  8. Acknowledgements
  9. Conventions for Floer cohomology
  10. The Floer complex for general Hamiltonians
  11. Floer cohomology of $1$-periodic orbits of autonomous Hamiltonians.
  12. Floer cohomology in small truncation windows
  13. The unweighted local Floer complex
  14. The Morse-Bott case
  15. From truncated to unweighted Floer cohomology
  16. ...and 27 more sections

Key Result

Theorem 1

For $\hbar>0$ small enough, the page $E_1$ is naturally isomorphic to

Figures (7)

  • Figure 1: An admissible inclusion
  • Figure 2: Graph of $H_{K,I}$
  • Figure 3: An admissible inclusion of compact domains.
  • Figure 4: The smoothings on the right are compatible, while those on the left are not
  • Figure 5: Non-convex boundary
  • ...and 2 more figures

Theorems & Definitions (138)

  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2
  • Theorem 3
  • Remark 1.7
  • Theorem 4
  • Remark 1.8
  • Remark 1.9
  • ...and 128 more