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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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