The $\gamma $-support as a micro-support

Tomohiro Asano; Stéphane Guillermou; Vincent Humilière; Yuichi Ike; Claude Viterbo

doi:10.5802/crmath.499

Géométrie et Topologie

The

γ

-support as a micro-support

Tomohiro Asano ¹ ; Stéphane Guillermou ² ; Vincent Humilière ³ ; Yuichi Ike ⁴ ; Claude Viterbo ⁵

¹ Faculty of Electrical, Information and Communication Engineering, Institute of Science and Engineering, Kanazawa University, Kakumamachi, Kanazawa, 920-1192, Japan
² UMR CNRS 6629 du CNRS Laboratoire de Mathématiques Jean LERAY 2 Chemin de la Houssinière, BP 92208, F-44322 NANTES Cedex 3 France
³ Sorbonne Université and Université de Paris, CNRS, IMJ-PRG, F-75005 Paris, France and Institut Universitaire de France
⁴ Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
⁵ Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1333-1340.

Résumé

We prove that for any element $L$ in the completion of the space of smooth compact exact Lagrangian submanifolds of a cotangent bundle equipped with the spectral distance, the $γ$ -support of $L$ coincides with the reduced micro-support of its sheaf quantization. As an application, we give a characterization of the Vichery subdifferential in terms of $γ$ -support.

Reçu le : 2022-12-14
Révisé le : 2023-02-17
Accepté le : 2023-03-27
Publié le : 2023-10-31

DOI : 10.5802/crmath.499

Affiliations des auteurs :

Tomohiro Asano ¹ ; Stéphane Guillermou ² ; Vincent Humilière ³ ; Yuichi Ike ⁴ ; Claude Viterbo ⁵

¹ Faculty of Electrical, Information and Communication Engineering, Institute of Science and Engineering, Kanazawa University, Kakumamachi, Kanazawa, 920-1192, Japan
² UMR CNRS 6629 du CNRS Laboratoire de Mathématiques Jean LERAY 2 Chemin de la Houssinière, BP 92208, F-44322 NANTES Cedex 3 France
³ Sorbonne Université and Université de Paris, CNRS, IMJ-PRG, F-75005 Paris, France and Institut Universitaire de France
⁴ Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
⁵ Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2023__361_G8_1333_0,
     author = {Tomohiro Asano and St\'ephane Guillermou and Vincent Humili\`ere and Yuichi Ike and Claude Viterbo},
     title = {The $\gamma $-support as a micro-support},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1333--1340},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.499},
     language = {en},
}

TY  - JOUR
AU  - Tomohiro Asano
AU  - Stéphane Guillermou
AU  - Vincent Humilière
AU  - Yuichi Ike
AU  - Claude Viterbo
TI  - The $\gamma $-support as a micro-support
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1333
EP  - 1340
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.499
LA  - en
ID  - CRMATH_2023__361_G8_1333_0
ER  -

%0 Journal Article
%A Tomohiro Asano
%A Stéphane Guillermou
%A Vincent Humilière
%A Yuichi Ike
%A Claude Viterbo
%T The $\gamma $-support as a micro-support
%J Comptes Rendus. Mathématique
%D 2023
%P 1333-1340
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.499
%G en
%F CRMATH_2023__361_G8_1333_0

Tomohiro Asano; Stéphane Guillermou; Vincent Humilière; Yuichi Ike; Claude Viterbo. The $\gamma $-support as a micro-support. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1333-1340. doi : 10.5802/crmath.499. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.499/

Bibliographie
Cité par

[1] Marie-Claude Arnaud; Vincent Humilière; Claude Viterbo Higher-dimensional Birkhoff attractors (2023) (in preparation)

[2] Tomohiro Asano; Yuichi Ike Persistence-like distance on Tamarkin’s category and symplectic displacement energy, J. Symplectic Geom., Volume 18 (2020) no. 3, pp. 613-649 | DOI | MR | Zbl

[3] Tomohiro Asano; Yuichi Ike Completeness of derived interleaving distances and sheaf quantization of non-smooth objects (2022) | arXiv

[4] Kenji Fukaya; Paul Seidel; Ivan Smith Exact Lagrangian submanifolds in simply-connected cotangent bundles, Invent. Math., Volume 172 (2008) no. 1, pp. 1-27 | DOI | MR | Zbl

[5] Stéphane Guillermou Quantization of conic Lagrangian submanifolds of cotangent bundles (2012) | arXiv

[6] Stéphane Guillermou Sheaves and symplectic geometry of cotangent bundles (2019) | arXiv

[7] Stéphane Guillermou; Masaki Kashiwara; Pierre Schapira Sheaf quantization of Hamiltonian isotopies and applications to non-displaceability problems, Duke Math. J., Volume 161 (2012) no. 2, pp. 201-245 | DOI | Zbl

[8] Stéphane Guillermou; Pierre Schapira Microlocal theory of sheaves and Tamarkin’s non displaceability theorem, Homological mirror symmetry and tropical geometry (Lecture Notes of the Unione Matematica Italiana), Volume 15, Springer, 2014, pp. 43-85 | DOI | MR | Zbl

[9] Stéphane Guillermou; Claude Viterbo The singular support of sheaves is $γ$ -coisotropic (2022) | arXiv

[10] Vincent Humilière On some completions of the space of Hamiltonian maps., Bull. Soc. Math. Fr., Volume 136 (2008) no. 3, pp. 373-404 | DOI | Numdam | MR | Zbl

[11] Vincent Humilière; Rémi Leclercq; Sobhan Seyfaddini Reduction of symplectic homeomorphisms, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016) no. 3, pp. 633-668 | DOI | MR | Zbl

[12] Masaki Kashiwara; Pierre Schapira Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften, 292, Springer, 1990 | DOI

[13] Alexandra Monzner; Nicolas Vichery; Frol Zapolsky Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization, J. Mod. Dyn., Volume 6 (2012) no. 2, pp. 205-249 | DOI | MR | Zbl

[14] Yong-Geun Oh Symplectic topology as the geometry of action functional. I. Relative Floer theory on the cotangent bundle, J. Differ. Geom., Volume 46 (1997) no. 3, pp. 499-577 | MR | Zbl

[15] Marco Robalo; Pierre Schapira A lemma for microlocal sheaf theory in the $\infty$ -categorical setting, Publ. Res. Inst. Math. Sci., Volume 54 (2018) no. 2, pp. 379-391 | DOI | MR | Zbl

[16] Paul Seidel Graded Lagrangian submanifolds, Bull. Soc. Math. Fr., Volume 128 (2000) no. 1, pp. 103-149 | DOI | Numdam | MR | Zbl

[17] Dmitry Tamarkin Microlocal Condition for Non-displaceability, Algebraic and Analytic Microlocal Analysis (Michael Hitrik; Dmitry Tamarkin; Boris Tsygan; Steve Zelditch, eds.), Springer (2018), pp. 99-223 | DOI | Zbl

[18] Michael Usher Observations on the Hofer distance between closed subsets, Math. Res. Lett., Volume 22 (2015) no. 6, pp. 1805-1820 | DOI | MR | Zbl

[19] Nicolas Vichery Homogénéisation symplectique et Applications de la théorie des faisceaux à la topologie symplectique, Ph. D. Thesis, École polytechnique (2012) https://tel.archives-ouvertes.fr/pastel-00780016/

[20] Nicolas Vichery Homological differential calculus (2013) | arXiv

[21] Claude Viterbo Symplectic topology as the geometry of generating functions, Math. Ann., Volume 292 (1992) no. 4, pp. 685-710 | DOI | MR | Zbl

[22] Claude Viterbo Symplectic Homogenization (2008) | arXiv

[23] Claude Viterbo Sheaf Quantization of Lagrangians and Floer cohomology (2019) | arXiv

[24] Claude Viterbo On the supports in the Humilière completion and $γ$ -coisotropic sets (with an Appendix joint with Vincent Humilière) (2022) | arXiv

Cité par Sources :

Commentaires - Politique