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Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems. (English) Zbl 1242.53108
Let \(M\) be a smooth real manifold, \(I\) an open interval containing zero, and let \(\dot T^*M= T^* M\setminus T^*M_0\), where \(T^* M_0\) is the set of zero co-vectors. Then, adopting the microlocal theory of sheaves (M. Kashiwara and P. M. Schapira [Sheaves on manifolds. With a short history “Les débuts de la théorie des faisceaux” by Christian Houzel. Grundlehren der Mathematischen Wissenschaften, 292. Berlin etc.: Springer-Verlag. x, 512 p. (1990; Zbl 0709.18001)], hereafter referred to as [1]), existence and unicity of a quantization \(K\), i.e., a sheaf on \(M\times M\times I\), of a homogeneous Hamiltonian isotopy \(\Phi= \{\phi_t\}_{t\in I}:\dot T^* M\times I\to\dot T^* M\), \(\phi_0= \text{id}\), where \(\phi_t\) is a homogeneous symplectic isomorphism for each \(t\in I\), is proved (§3. Th.3.7). As applications, the nondisplaceability, which asserts that the image of the zero section of \(T^* M\) under a Hamiltonian isotopy always intersects the zero-section (Arnold conjecture) and its refinements, using Morse inequalities, are proved (§4, Th.4.1, Th.4.4 and Th.4.16., cf. [M. Chaperon, Sémin. Bourbaki, 35e année, 1982/83, Exp. 610, Astérisque 105–106, 231–249 (1983; Zbl 0525.53049); H. Hofer, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2, 407–462 (1985; Zbl 0591.58009); F. Laudenbach and J.-C. Sikorav, Invent. Math. 82, 349–357 (1985; Zbl 0592.58023)], hereafter, these papers are referred to as [2]).
To a Hamiltonian isotopy, we can associate a unique conic Lagrangian submanifold \(\Lambda\) of \(T^*M\times\dot T^* M\times I\) such that \(\Lambda_t= \Lambda\otimes\dot T^*_t I\), where \(\Lambda_t\) is the graph of \(\phi_t\) (Appendix, Lemma A.1 and Lemma A.2., cf. §1.6–§1.9). The quantization of \(\Phi\) on \(I\) is the sheaf \(K\) on \(M\times M\times I\) whose microsupport (singular support, the definition of microsupport is described in §1 as Def.1.1) is contained in \(\Lambda\) outside the zero-section and whose restriction at \(t=0\) is \({\mathbf k}_\Delta\), the constant sheaf of stalk \({\mathbf k}\), a field on the diagonal \(\Delta\) of \(M\times M\) extended by zero outside \(\Delta\). Existence and uniqueness of \(K\) are proved in §3 (Prop.3.2 and Th.3.7). They are main results of this paper. Examples of quantizations are also given in §3.
To apply the quantization \(K\) of \(\Phi\) to nondisplaceability, the authors use a microlocal Morse lemma ([1]). Let \(D^b({\mathbf k}_M)\) be the bounded derived category of sheaves of \({\mathbf k}\)-modules and \(F\) one of its elements. Then, by the microlocal Morse lemma, if \(\text{Supp}(F)\) is compact and \(R\Gamma(M; F)\neq 0\), and let \(\psi: M\to\mathbb{R}\) be a \(C^1\) class function, and \[ \Lambda_\psi= \{(x; d\psi(x))\}\subset T^* M, \] then \(\Lambda_\psi\in SS(F)\neq\emptyset\). Here \(SS(F)\) means the microsupport of \(F\) (§1, Cor.1.9). If \(\Lambda_\psi\cap SS(F)\) is a finite set \(\{p_1,\dots, p_N\}\), assuming \(H^j(V_i)\), \(V_i= (R\Gamma_{\{\psi(x)\geq \psi(x_i)\}}(F))_{x_i}\), \(x_i= \pi(p_i)\), are finite-dimensional for all \(1\leq i\leq N\), and \(j\in\mathbb{Z}\), the inequality \[ b^*_\ell(F)\leq \sum^N_{i=1} b^*_\ell(V_i),\quad\text{for any }\ell, \] is also hold (§1. Th.1.10). If \(F_0\in D^b({\mathbf k}_M)\) and has a compact support, then to set \(F= K\circ F_0\in D^b({\mathbf k}_{M\times I})\) and \(F_{t_0}= F_{t=t_0}\cong K_{t_0}\circ F_0\in D^b({\mathbf K}_M)\), \(t_0\in I\), we have \[ SS(F_t)\cap\dot T^* M= \phi_t(SS(F_0)\cap\dot T^* M). \] Hence we can use Cor.1.9 and conclude if \(R\Gamma(M; F_0)\neq 0\), then for any \(t\in I\), \(\phi_t(SS(F_0)\cap\dot T^* M)\cap \Lambda_\psi\neq 0\) (Th.4.1). Further nondisplaceability results such as if there exists a Lagrangian submanifold \(S_{0,\text{reg}}\) of \(S_0= SS(F_0)\cap\dot T^* M\), and \(\Lambda_\psi\cap\phi_{t=0} (S_0)\) is a finite and transversal, and contained in \(\Lambda_\psi\cap \phi_{t_0}(S_{0,\text{reg}})\), then \[ \sharp(\phi_{t_0}(S_0)\cap \Lambda_\psi)\geq \sum_j b_j(F_0) \] (Th.4.4), and if \(\Phi\) is nonnegative, \(\phi_1(\dot T^*_X M)=\dot T^*_Y M\), where \(X\), \(Y\) are compact connected submanifolds of a connected noncompact \(M\), then \(X=Y\) and \(\phi_t|_{\dot T^*_X M}= \text{id}_{T^*_X M}\) (Th.4.13), are proved in §4, the last section. This section is concluded to recover a well-known result solving a conjecture of Arnold (Th.4.16. cf.[2]).
This paper is written in the framework of the microlocal theory of sheaves. In §1, it is reviewed following [1]. §2 studies deformation of the conormal to the diagonal, which is used in §3, to prove existence of the quantization. Properties of Hamiltonian isotopy used in §3 are explained in the Appendix.

MSC:
53D35 Global theory of symplectic and contact manifolds
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
81Q99 General mathematical topics and methods in quantum theory
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[1] M. Chaperon, Quelques questions de geométrie symplectique (d’après, entre autres, Poincaré, Arnold, Conley et Zehnder) , Astérisque 105-106 , 231-249, Séminaire Bourbaki 1982/1983, Soc. Math. France, Montrouge, 1983. · Zbl 0525.53049
[2] M. Chaperon, “On generating families” in The Floer Memorial Volume , Progr. Math. 133 , Birkhäuser, Basel, 1995, 283-296. · Zbl 0837.58003
[3] Y. Chekanov, Critical points of quasifunctions and generating families of Legendrian manifolds , Funct. Anal. Appl. 30 (1996), 118-128. · Zbl 0873.58017
[4] V. Chernov and S. Nemirovski, Non-negative Legendrian isotopy in ST * M , Geom. Topol. 14 (2010), 611-626. · Zbl 1194.53066
[5] V. Colin, E. Ferrand, and P. Pushkar, Positive isotopies of Legendrian submanifolds and applications , preprint,
[6] C. Conley and E. Zehnder, The Birkhof-Lewis fixed point theorem and a conjecture of V. I. Arnol’d , Invent. Math. 73 (1983), 33-49. · Zbl 0516.58017
[7] Y. Eliashberg, S. S. Kim, and L. Polterovich, Geometry of contact transformations and domains: Orderability versus squeezing , Geom. Topol. 10 (2006), 1635-1747. · Zbl 1134.53044
[8] E. Ferrand, “On a theorem of Chekanov” in Symplectic Singularities and Geometry of Gauge Fields , Banach Center Publ. 39 , Polish Acad. Sci., Warsaw, 1997, 39-48. · Zbl 0893.53015
[9] S. Guillermou and P. Schapira, Microlocal theory of sheaves and Tamarkin’s non displaceability theorem , preprint, · Zbl 1319.32006
[10] H. Hofer, Lagrangian embeddings and critical point theory , Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 407-462. · Zbl 0591.58009
[11] M. Kashiwara and P. Schapira, Micro-support des faisceaux: Applications aux modules différentiels , C. R. Acad. Sci. Paris Série I Math. 295 (1982), 487-490. · Zbl 0501.58006
[12] M. Kashiwara and P. Schapira, Microlocal Study of Sheaves , Astérisque 128 , Soc. Math. France, Montrouge, 1985. · Zbl 0589.32019
[13] M. Kashiwara and P. Schapira, Sheaves on Manifolds , Grundlehren Math. Wiss. 292 , Springer, Berlin, 1990.
[14] M. Kashiwara and P. Schapira, Categories and Sheaves , Grundlehren Math. Wiss. 332 , Springer, Berlin, 2006.
[15] R. Kasturirangan and Y.-G. Oh, Floer homology of open subsets and a relative version of Arnold’s conjecture , Math. Z. 236 (2001), 151-189. · Zbl 0985.53039
[16] F. Laudenbach, A Morse complex on manifolds with boundary , preprint, · Zbl 1223.57020
[17] F. Laudenbach and J.-C. Sikorav, Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibré cotangent , Invent. Math. 82 (1985), 349-357. · Zbl 0592.58023
[18] D. McDuff and D. Salamon, Introduction to symplectic topology , Oxford Math. Monogr. Oxford Univ. Press, New York, 1995. · Zbl 0844.58029
[19] D. Nadler, Microlocal branes are constructible sheaves , preprint, · Zbl 1197.53116
[20] D. Nadler and E. Zaslow, Constructible sheaves and the Fukaya category , J. Amer. Math. Soc. 22 (2009), 233-286. · Zbl 1227.32019
[21] Y.-G. Oh, “Naturality of Floer homology of open subsets in Lagrangian intersection theory” in The Third Pacific Rim Geometry Conference (Seoul, 1996) , Monogr. Geom. Topol. 25 , Int. Press, Cambridge, Mass., 1998, 261-280. · Zbl 1013.53056
[22] P. Polesello and P. Schapira, Stacks of quantization-deformation modules over complex symplectic manifolds , Int. Math. Res. Notices 2004 , no. 49, 2637-2664. · Zbl 1086.53107
[23] P. Schapira and N. Tose, Morse inequalities for R-constructible sheaves , Adv. Math. 93 (1992), 1-8. · Zbl 0767.58008
[24] D. Tamarkin, Microlocal conditions for non-displaceability ,
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