The singular support of sheaves is $γ$-coisotropic

Abstract

Introduction 2. Notation, comments and acknowledgements 3. Reminders on the singular support of sheaves 4. Quantization of Hamiltonian isotopies 5. Spectral invariants for sheaves and Lagrangians 5.1. Definition of spectral invariants 5.2. The case of Hamiltonian maps 5.3. Defining γ-coisotropic subsets 6. The γ g -metric on sheaves 6.1. The γ g -topology 6.2. Link with spectral invariants 6.3. γ g -limits and colimits 7. Proof of Theorem 1.2 8. γ-coisotropic vs cone-coisotropic 8.1. A γ-coisotropic set is cone-coisotropic 8.2. A cone-coisotropic set that is not γ-coisotropic 9. Questions and comments Appendix A. Persistence modules and Barcodes Appendix B. Decomposition in the completion of D l c (R) B.1. Limits of constructible sheaves B.2. Decomposition in D l c (R) Appendix C. Comparing different interleaving metrics on D(N × R).

On the supports in the Humilière completion and $γ$-coisotropic sets (with an Appendix joint with Vincent Humilière)

“…This paper precedes, both logically and historically the papers [Vit22] and [GV22b] which are of course related.…”

“…It is an elementary fact that in both cases a smooth submanifold is Poisson coisitropic or cone coisotropic if and only if it is coisotropic in the usual sense. We shall prove in [GV22b] the first part of Proposition 7.10. For a subset V in (M , ω) we have the following implications…”

“…We refer to [GV22b] for extensions to the non-constructible case. It is quite clear that elements of D b c ((R, ≤)) are obtained by considering graded persistence modules V d t for d in a finite range (in Z) and with the maps r s,t we have maps δ t :…”

“…Since on G(T * N ) the metric γ coincides with the Hofer metric, we may conclude that (L(T * N ) endowed with the Hofer metric is not a Baire space either. Let F • ∈ D b (N × R) associated to L. According to [GV22b], proposition 9.9, there is an element F L in D l c (N × R) such that it extends the Lagrangian quantization map Q…”

Persistence and the Sheaf-Function Correspondence

“…Inspired by persistence theory from topological data analysis (TDA) [36,21], Kashiwara and Schapira have recently introduced the convolution distance between (derived) sheaves of k-vector spaces on a finite-dimensional real normed vector space [27]. This construction has found important applications, both in TDA -where it allows expressing stability of certain constructions with respect to noise in datasets - [6,9,7,8] and in symplectic topology [2,3,23]. A challenging research direction, of interest to these two fields, is to associate numerical invariants to a sheaf on a vector space, which satisfy a certain form of continuity with respect to the convolution distance.…”