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Lagrangian cobordism and shadow distance in Tamarkin category

Tomohiro Asano, Yuichi Ike, Wenyuan Li

TL;DR

The paper addresses quantitative and rigidity questions for exact Lagrangian cobordisms in cotangent bundles by developing a microlocal, sheaf-theoretic framework. It constructs a simple sheaf quantization for cobordisms within the Tamarkin category $\mathcal{T}(T^*M)$, proves an iterated cone decomposition that organizes the end data, and establishes a fundamental bound $d'_{\mathcal{T}(T^*M)}(\tilde{F}_{-1},\tilde{F}_{+1}) \le \mathcal{S}(V)$ linking interleaving distance to shadow distance $\mathcal{S}(V)$. This bound implies an isomorphism in the torsion localization $\mathcal{T}_\infty(T^*M)$ and yields a rigidity statement: small shadow distance costs less energy than to separate or connect Lagrangians via cobordisms, quantifying Lagrangian intersection constraints. The work integrates Kashiwara–Schapira stacks and Guillermou–Kashiwara–Schapira's doubling with Tamarkin category techniques to extend cobordism methods beyond smooth, embedded settings, broadening the algebraic-geometry perspective on symplectic topological phenomena and offering new tools for understanding Lagrangian intersections via sheaf theory.

Abstract

We study exact Lagrangian cobordisms between exact Lagrangians in a cotangent bundle in the sense of Arnol'd, using microlocal theory of sheaves. We construct a sheaf quantization for an exact Lagrangian cobordism between Lagrangians with conical ends, prove an iterated cone decomposition of the sheaf quantization for cobordisms with multiple ends, and show that the interleaving distance of sheaves is bounded by the shadow distance of the cobordism. Using the result, we prove a rigidity result on Lagrangian intersection by estimating the energy cost of splitting and connecting Lagrangians through cobordisms.

Lagrangian cobordism and shadow distance in Tamarkin category

TL;DR

The paper addresses quantitative and rigidity questions for exact Lagrangian cobordisms in cotangent bundles by developing a microlocal, sheaf-theoretic framework. It constructs a simple sheaf quantization for cobordisms within the Tamarkin category , proves an iterated cone decomposition that organizes the end data, and establishes a fundamental bound linking interleaving distance to shadow distance . This bound implies an isomorphism in the torsion localization and yields a rigidity statement: small shadow distance costs less energy than to separate or connect Lagrangians via cobordisms, quantifying Lagrangian intersection constraints. The work integrates Kashiwara–Schapira stacks and Guillermou–Kashiwara–Schapira's doubling with Tamarkin category techniques to extend cobordism methods beyond smooth, embedded settings, broadening the algebraic-geometry perspective on symplectic topological phenomena and offering new tools for understanding Lagrangian intersections via sheaf theory.

Abstract

We study exact Lagrangian cobordisms between exact Lagrangians in a cotangent bundle in the sense of Arnol'd, using microlocal theory of sheaves. We construct a sheaf quantization for an exact Lagrangian cobordism between Lagrangians with conical ends, prove an iterated cone decomposition of the sheaf quantization for cobordisms with multiple ends, and show that the interleaving distance of sheaves is bounded by the shadow distance of the cobordism. Using the result, we prove a rigidity result on Lagrangian intersection by estimating the energy cost of splitting and connecting Lagrangians through cobordisms.
Paper Structure (16 sections, 28 theorems, 147 equations, 4 figures)

This paper contains 16 sections, 28 theorems, 147 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Related work
  4. Preliminaries
  5. Microsupports of sheaves
  6. Tamarkin category and interleaving distance
  7. Sheaf quantization of Hamiltonian isotopy
  8. Sheaf quantization of Lagrangian cobordism
  9. Kashiwara-Schapira stack
  10. Guillermou doubling functor
  11. Sheaf quantization of Lagrangian cobordisms
  12. Lagrangian shadows and interleaving distance
  13. Twisted complex and cone decomposition
  14. Shadow distance and interleaving distance
  15. Application to Lagrangian intersection
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $V \subset T^*(M \times \mathbb{R})$ be an exact Lagrangian cobordism with conical ends between $(L_0,\dots,L_{m-1})$ and $(L'_0,\dots,L'_{n-1})$ with vanishing Maslov class and relative second Stiefel--Whitney class. Then there is a simple sheaf quantization $F \in \mathcal{T}(T^*(M \times \mat

Figures (4)

  • Figure 1: The cone $\gamma_i$ and dual cone $\gamma_i^*$ for $i = 0, 1, \dots, m$ in the proof of \ref{['theorem:iterated_cone']}.
  • Figure 2: The schematic picture of the Hamiltonian isotopy in \ref{['lem:deform-shadow']} such that the Lagrangian shadow is bounded in between the sections $0$ and $\sigma_+ \colon [-1, 1] \to T^*[-1, 1]$.
  • Figure 3: The shift $c$ as the signed area of the red region in \ref{['proposition:inequality_one_end']}.
  • Figure 4: The shifts $c_i$ and $c_j'$ ($i = 1, \dots, m-1$, $j = 0, 1, \cdots, n-1$) in \ref{['theorem:inequality_multiple_ends']}.

Theorems & Definitions (61)

  • Theorem 1.1: see \ref{['thm:sheaf-quan-cob-filling']}
  • Remark 1.2: see \ref{['rem:quan-jintreumann']}
  • Theorem 1.3: see \ref{['theorem:iterated_cone']}
  • Theorem 1.4: see \ref{['theorem:inequality_multiple_ends']}
  • Theorem 1.5: see \ref{['theorem:intersection']} for a more precise statement
  • Remark 1.6
  • Definition 2.1
  • Proposition 2.2: cf. GS14
  • Definition 2.3
  • Remark 2.4
  • ...and 51 more