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Completeness of derived interleaving distances and sheaf quantization of non-smooth objects

Tomohiro Asano, Yuichi Ike

Abstract

We develop sheaf-theoretic methods to deal with non-smooth objects in symplectic geometry. We show the completeness of a derived category of sheaves with respect to the interleaving distance and construct a sheaf quantization of a Hamiltonian homeomorphism. We also develop Lusternik--Schnirelmann theory in the microlocal theory of sheaves. With these new sheaf-theoretic methods, we prove an Arnold-type theorem for the image of a compact exact Lagrangian submanifold under a Hamiltonian homeomorphism.

Completeness of derived interleaving distances and sheaf quantization of non-smooth objects

Abstract

We develop sheaf-theoretic methods to deal with non-smooth objects in symplectic geometry. We show the completeness of a derived category of sheaves with respect to the interleaving distance and construct a sheaf quantization of a Hamiltonian homeomorphism. We also develop Lusternik--Schnirelmann theory in the microlocal theory of sheaves. With these new sheaf-theoretic methods, we prove an Arnold-type theorem for the image of a compact exact Lagrangian submanifold under a Hamiltonian homeomorphism.
Paper Structure (24 sections, 46 theorems, 147 equations, 1 figure)

This paper contains 24 sections, 46 theorems, 147 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our results
  3. Related work
  4. Organization
  5. Microlocal theory of sheaves
  6. Geometric notions
  7. Microsupports of sheaves
  8. Composition and convolution
  9. Homotopy colimits
  10. Interleaving distance for sheaves and sheaf quantization
  11. Interleaving distance for sheaves
  12. Tamarkin category
  13. Sheaf quantization of Hamiltonian isotopies
  14. Completeness of derived category of sheaves
  15. Sheaf quantization of Hamiltonian diffeomorphisms and homeomorphisms
  16. ...and 9 more sections

Key Result

Theorem 1.1

The Tamarkin category $\mathcal{D}(X)$ is complete with respect to the pseudo-distance $d_{\mathcal{D}(X)}$.

Figures (1)

  • Figure 5.1: $D(a,b)$

Theorems & Definitions (103)

  • Theorem 1.1: see \ref{['corollary:limit_Tamarkin']}
  • Theorem 1.2: see \ref{['theorem:stability_hofer']}
  • Theorem 1.3: see \ref{['theorem:sheaf_spectral_invariant']} for a more precise statement
  • Theorem 1.4: see \ref{['theorem:ineq_hameo']}
  • Definition 2.1: KS90
  • Proposition 2.2: KS90 and robalo2018lemma
  • Proposition 2.3: KS90
  • Proposition 2.4: KS90
  • Lemma 2.5: cf. KS90
  • Proposition 2.6
  • ...and 93 more