Sheaf quantization and intersection of rational Lagrangian immersions
Annales de l'Institut Fourier, Volume 73 (2023) no. 4, pp. 1533-1587.

We study rational Lagrangian immersions in a cotangent bundle, based on the microlocal theory of sheaves. We construct a sheaf quantization of a rational Lagrangian immersion and investigate its properties in Tamarkin category. Using the sheaf quantization, we give an explicit bound for the displacement energy and a Betti/cup-length estimate for the number of the intersection points of the immersion and its Hamiltonian image by a purely sheaf-theoretic method.

Nous étudions les immersions lagrangiennes rationnelles dans un fibré cotangent en nous basant sur la théorie microlocale des faisceaux. Nous construisons une quantification faisceautique d’une immersion lagrangienne rationnelle et étudions ses propriétés dans la catégorie de Tamarkin. En utilisant la quantification faisceautique, nous donnons une limite explicite à l’énergie de déplacement et une estimation Betti ou cup-length pour le nombre de points d’intersection de l’immersion et de son image hamiltonienne par une méthode purement faisceautique.

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Accepted:
Published online:
DOI: 10.5802/aif.3554
Classification: 53D12, 37J10, 53D35, 35A27
Keywords: Lagrangian immersions, displacement energy, microlocal theory of sheaves.
Mot clés : Immersions lagrangiennes, énergie de déplacement, théorie microlocale des faisceaux.

Asano, Tomohiro 1; Ike, Yuichi 2

1 Graduate School of Mathematical Sciences, The University of Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo (Japan)
2 Graduate School of Information Science and Technology, The University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo (Japan)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Asano, Tomohiro; Ike, Yuichi. Sheaf quantization and intersection of rational Lagrangian immersions. Annales de l'Institut Fourier, Volume 73 (2023) no. 4, pp. 1533-1587. doi : 10.5802/aif.3554. https://aif.centre-mersenne.org/articles/10.5802/aif.3554/

[1] Akaho, Manabu Symplectic displacement energy for exact Lagrangian immersions (2015) (http://arxiv.org/abs/1505.06560)

[2] Akaho, Manabu; Joyce, Dominic Immersed Lagrangian Floer theory, J. Differential Geom., Volume 86 (2010) no. 3, pp. 381-500 | DOI | MR | Zbl

[3] Albers, Peter; Hein, Doris Cuplength estimates in Morse cohomology, J. Topol. Anal., Volume 8 (2016) no. 2, pp. 243-272 | DOI | MR | Zbl

[4] An, Byung Hee; Bae, Youngjin; Su, Tao Augmentations are sheaves for Legendrian graphs (2019) (http://arxiv.org/abs/1912.10782)

[5] Asano, Tomohiro; Ike, Yuichi 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

[6] Chekanov, Yu V. Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J., Volume 95 (1998) no. 1, pp. 213-226 | DOI | MR | Zbl

[7] Chiu, Sheng-Fu Nonsqueezing property of contact balls, Duke Math. J., Volume 166 (2017) no. 4, pp. 605-655 | DOI | MR | Zbl

[8] Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru Lagrangian intersection Floer theory. Anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009, xii+396 pages | MR | Zbl

[9] Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru Lagrangian intersection Floer theory. Anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009, p. i-xii and 397–805 | MR | Zbl

[10] Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru Displacement of polydisks and Lagrangian Floer theory, J. Symplectic Geom., Volume 11 (2013) no. 2, pp. 231-268 | DOI | MR | Zbl

[11] Guillermou, Stéphane Quantization of conic Lagrangian submanifolds of cotangent bundles (2012) (http://arxiv.org/abs/1212.5818v2)

[12] Guillermou, Stéphane Sheaves and symplectic geometry of cotangent bundles (2019) (http://arxiv.org/abs/1905.07341v2)

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

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

[15] Ike, Yuichi Compact exact Lagrangian intersections in cotangent bundles via sheaf quantization, Publ. Res. Inst. Math. Sci., Volume 55 (2019) no. 4, pp. 737-778 | DOI | MR | Zbl

[16] Jin, Xin; Treumann, David Brane structures in microlocal sheaf theory (2017) (http://arxiv.org/abs/1704.04291)

[17] Kashiwara, Masaki; Schapira, Pierre Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften, 292, Springer-Verlag, Berlin, 1990, x+512 pages | MR | Zbl

[18] Keller, Bernhard On triangulated orbit categories, Doc. Math., Volume 10 (2005), pp. 551-581 | MR | Zbl

[19] Liu, Chun-Gen Cup-length estimate for Lagrangian intersections, J. Differential Equations, Volume 209 (2005) no. 1, pp. 57-76 | DOI | MR | Zbl

[20] Nadler, David Microlocal branes are constructible sheaves, Selecta Math. (N.S.), Volume 15 (2009) no. 4, pp. 563-619 | DOI | MR | Zbl

[21] Nadler, David; Zaslow, Eric Constructible sheaves and the Fukaya category, J. Amer. Math. Soc., Volume 22 (2009) no. 1, pp. 233-286 | DOI | MR | Zbl

[22] Ng, Lenhard; Rutherford, Dan; Shende, Vivek; Sivek, Steven; Zaslow, Eric Augmentations are sheaves, Geom. Topol., Volume 24 (2020) no. 5, pp. 2149-2286 | DOI | MR | Zbl

[23] Rutherford, Dan; Sullivan, Michael G Sheaves via augmentations of Legendrian surfaces (2019) (http://arxiv.org/abs/1912.06186)

[24] Tamarkin, Dmitry Microlocal condition for non-displaceability, Algebraic and analytic microlocal analysis (Springer Proc. Math. Stat.), Volume 269, Springer, Cham, 2018, pp. 99-223 | DOI | MR | Zbl

[25] Viterbo, Claude Sheaf Quantization of Lagrangians and Floer cohomology (2019) (http://arxiv.org/abs/1901.09440)

[26] Zhang, Jun Quantitative Tamarkin theory, CRM Short Courses, Springer, Cham, 2020, x+146 pages | DOI | MR | Zbl

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