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Density of fibers for the filtered Fukaya category of $T^*N$

Stéphane Guillermou, Claude Viterbo, Bingyu Zhang

Abstract

We answer a question of Biran and Cornea about the density of iterated cones of fibers in the Fukaya category of a cotangent bundle. We prove that indeed if we take a dense set of basepoints, the iterated cones of the cotangent fibres are dense in the Filtered Fukaya category.

Density of fibers for the filtered Fukaya category of $T^*N$

Abstract

We answer a question of Biran and Cornea about the density of iterated cones of fibers in the Fukaya category of a cotangent bundle. We prove that indeed if we take a dense set of basepoints, the iterated cones of the cotangent fibres are dense in the Filtered Fukaya category.
Paper Structure (17 sections, 34 theorems, 51 equations, 2 figures)

This paper contains 17 sections, 34 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Categories
  4. Quantization
  5. Continuation morphisms
  6. Quantization functor
  7. Properties of sheaf quantization functor
  8. Strategy of proof
  9. Distances on the category of sheaves
  10. Sheaf distance for a continuous norm
  11. Some properties of the distance
  12. Main example: Small wrapping of the diagonal
  13. Proof of the main result
  14. Some results on the projector
  15. Some lemmata on iterated cones
  16. ...and 2 more sections

Key Result

Theorem 1.1

For any closed exact Lagrangian $L \in {\mathscr{F}}(DT^*N)$ and $\varepsilon >0$, there are points $(x_i)_{i\in \{1, \ldots,l\}}$, real numbers $(a_i)_{i\in \{1, \ldots,l\}}$ and $C \in \langle {\mathcal{V}}_{(x_1,a_1)}, \ldots, {\mathcal{V}}_{(x_l,a_l)} \rangle$ such that $\gamma (L,C) < \varepsil

Figures (2)

  • Figure 1: Lagrangian labeling and Floer data for the bimodule $B$
  • Figure 2: The graph of $k_{C(x_0, a_0, s)}$.

Theorems & Definitions (75)

  • Theorem 1.1: Density Theorem
  • Theorem 1.2: Density Theorem
  • Remark 1.3
  • Remark 3.1
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Remark 4.3
  • Proposition 4.4: Viterbo-Sheaves
  • Corollary 4.5
  • ...and 65 more