Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Guillermou-Kashiwara-Schapira kernels of geodesic flows

Takumi Arai

TL;DR

This work provides explicit constructions of Guillermou–Kashiwara–Schapira kernels for geodesic flows on canonical spaces, enabling concrete realizations of sheaf quantizations of homogeneous Hamiltonian isotopies. By developing iterative gluings of region-precise sheaves $(\mathbb{K}_{Z_i})$ via morphisms $(\psi_i)$ and their mapping cones, the authors produce kernels $K$ whose microsupport matches the geodesic-Lagrangian $\Lambda_\phi$ and which recover the diagonal at time $t=0$. The spheres $S^n$ and complex projective spaces $\mathbb{CP}^n$ are treated in parallel, with $S^n$ using distance-based $Z_i$ and $\mathbb{CP}^n$ using $D_x$, $D_{x,y}$ to accommodate their respective geometry; in each case $K$ is shown to satisfy the GKS criteria. The results provide concrete tools for applications in displaceability, wrapping, and Legendrian isotopies within microlocal sheaf theory, extending the reach of GKS quantization to important geometric settings.

Abstract

Guillermou-Kashiwara-Schapira proved that there exists a unique sheaf quantization of any homogeneous Hamiltonian isotopy on a cotangent bundle. In this paper, we explicitly construct a sheaf quantization of geodesic flows on spheres and complex projective spaces.

Guillermou-Kashiwara-Schapira kernels of geodesic flows

TL;DR

This work provides explicit constructions of Guillermou–Kashiwara–Schapira kernels for geodesic flows on canonical spaces, enabling concrete realizations of sheaf quantizations of homogeneous Hamiltonian isotopies. By developing iterative gluings of region-precise sheaves via morphisms and their mapping cones, the authors produce kernels whose microsupport matches the geodesic-Lagrangian and which recover the diagonal at time . The spheres and complex projective spaces are treated in parallel, with using distance-based and using , to accommodate their respective geometry; in each case is shown to satisfy the GKS criteria. The results provide concrete tools for applications in displaceability, wrapping, and Legendrian isotopies within microlocal sheaf theory, extending the reach of GKS quantization to important geometric settings.

Abstract

Guillermou-Kashiwara-Schapira proved that there exists a unique sheaf quantization of any homogeneous Hamiltonian isotopy on a cotangent bundle. In this paper, we explicitly construct a sheaf quantization of geodesic flows on spheres and complex projective spaces.

Paper Structure

This paper contains 6 sections, 30 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symplectic Geometry
  4. Sheaf Theory
  5. GKS kernel of the geodesic flow on $S^n$
  6. GKS kernel of the geodesic flow on $\mathbb{CP}^n$

Key Result

Theorem 1.1

Let $\phi : T^*M \setminus 0_M \times I \to T^*M \setminus 0_M$ be a homogeneous Hamiltonian isotopy. Then there exists a category equivalence $\Phi_t : \mathrm{Sh}(M) \to \mathrm{Sh}(M)$ satisfying $\phi_t({ \mathrm{SS} (F) }\setminus 0_M ) = \mathrm{SS} (\Phi_t(F)) \setminus 0_M$ for any $F \in \m

Theorems & Definitions (65)

  • Theorem 1.1: GKS
  • Theorem 1.2: GKS
  • Theorem 1.3
  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5: KS
  • Proposition 2.6
  • proof
  • ...and 55 more