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Microlocal condition for non-displaceability. (English) Zbl 1416.35019
Hitrik, Michael (ed.) et al., Algebraic and analytic microlocal analysis. AAMA, Evanston, Illinois, USA, May 14–26, 2012 and May 20–24, 2013. Contributions of the workshops. Cham: Springer. Springer Proc. Math. Stat. 269, 99-223 (2018).
Let $$M$$ be a symplectic manifold, $$A,B$$ its compact subsets. $$A$$ and $$B$$ called non-displaceable if $$A\cap X(B)\not=\emptyset$$, where $$X$$ is any Hamiltonian symplectomorophism of $$M$$ which is identity outside of a compact. Let $$\mathbb{T}^N=\{(z_0,\dots,z_N)\mid |z_0|=\cdots=|z_N|\}$$ be the Clifford torus in $$\mathbb{CP}^N$$, then non-dispaceability of $$\mathbb{T}^N$$ from itself is already proved by using Floer’s theory [C.-H. Cho, Int. Math. Res. Not. 2004, No. 35, 1803–1843 (2004; Zbl 1079.53133); M. Entov and L. Polterovich, Compos. Math. 145, No. 3, 773–826 (2009; Zbl 1230.53080)].
In this paper, based on Kashiwara-Schapira’s microlocal theory of sheaves on manifolds [M. Kashiwara and P. Schapira, Sheaves on manifolds. With a short history “Les débuts de la théorie des faisceaux” by Christian Houzel. Berlin etc.: Springer-Verlag (1990; Zbl 0709.18001)], a sufficient condition for non-displaceability of a pair of subsets in a cotangent bundle, independent of Floer’s theory, is presented (Theorem 3.1). Then constructing Lagrangian correspondence between $$T^\ast \mathrm{SU}(N)$$ and $$\mathbb{CP}^N\times(\mathbb{CP}^N)^{\mathrm{op}}$$, where symplectic form of $$(\mathbb{CP}^N)^{\mathrm{op}}$$ is opposite that of $$\mathbb{CP}^N$$, the following Theorem is derived as an application of Theorem 3.1.
Theorem 4.1.
(1)
$$\mathbb{T}^N$$ is non-displaceable from itself.
(2)
$$\mathbb{RP}^N$$ is non-displaceable from itself.
(3)
$$\mathbb{T}^N$$ and $$\mathbb{RP}^N$$ are non-displaceable from each another.
To obtain sufficient condition (Theorem 3.1), first the category $$\mathcal{D}(X)=D(X\times\mathbb{R})/C_{\leq 0}(X)$$, $$D(X\times \mathbb{R})$$ is the unbounded derived category of $$\mathbb{K}$$-vector spaces on $$X\times \mathbb{R}$$ and $$C_{\leq 0}(X)$$ is the full subcategory of objects microsupport $$\Omega_{\leq 0}$$; the closed subset of $$T^\ast(X\times\mathbb{R})$$ such that $\Omega_{\leq 0}=\{\omega|(\omega,V)\leq 0\}, \quad V=\frac{d}{dt},$ and $$\mathcal{D}_A(X)$$, the full subcategory consisting of all $$F\in \mathcal{D}(X)$$ such that $$SS(F)\cap \Omega_{>0}\subset \mathrm{Cone}(A)$$, are introduced.
(§2). In §2.2.2, from $$T_{c^*}:\mathcal{D}(X\times\mathbb{R})\to \mathcal{D}(X\times \mathbb{R})$$, $$T_c(x,t)=(x,t+c)$$, a natural transfunctor $$\tau_{c^*}:\mathrm{Id}\to T_{c^*}$$ of endfunctors on $$\mathcal{D}_A(X)$$ for any $$A$$ is constructed. Using these, the sufficient condition is stated:
Theorem 3.1. Suppose there exist objects $$F_i\in \mathcal{D}_{F_i}(X)$$, $$i=1,2$$, such that for all $$c>0$$, the natural map $\tau_c:R\mathrm{hom}(F_1,F_2)\to R\mathrm{hom}(F_1, T_cF_2),$ is not zero. Then $$F_1$$ and $$F_2$$ are mutually non-displaceable.
This is proved to use $$R\mathrm{hom}_{\mathcal{D}(x)}(F_1,F_2)=0$$, if $$F_i\in\mathcal{D}_{A_i}(X)$$, $$A_i\subset T^\ast X$$ and $$A_1\cap A_2=\emptyset$$ (Theorem 3.2), and existence of a collection of endmorphisms $$T_n:\mathcal{D}(X)\to \mathcal{D}(X)$$, $$1\leq k\leq N$$ for some $$N$$ and a collection of functors $$t_k*T_{2k}\to T_{2k+1}$$ and $$s_k:T_{2k+2}\to T_{2k+1}$$ such that $$T_N=Id$$, $$T_1(\mathcal{D}_L(X))\subset \mathcal{D}_{\Phi(L)}(X)$$ and $$\mathrm{Cone}(t_k(F)), \mathrm{Cone}(s_k(F))$$ are torsion sheaves for all $$k$$ and $$F\in\mathcal{D}(X)$$. Here $$\Phi:T^\ast X\to T^\ast X$$ is a Hamiltonian symplectomorphism which is equal to identity outside of a compact and $$L\subset T^\ast X$$ is a compact subset (Theorem 3.9).
Let $$G=\mathrm{SU}(N)$$, $$\mathfrak{g}$$ its Lie algebra and $$\mathbb{CP}^N=\mathcal{O}\subset \mathfrak{g}^*$$, and let $$I:M\to T^ast G$$ be the inclusion, $$P:M\to \mathcal{O}^{\mathrm{op}}\times \mathcal{O}$$ be the projection, and $$\Delta$$ is the diagonal of $$\mathcal{O}^{\mathrm{op}}\times\mathcal{O}$$. Then Theorem 4.1 follows from the facts that $$IP^{-1}\Delta$$ and $$IP^{-1}(\mathbb{T}\times\mathbb{T})$$, $$IP^{-1}\Delta$$ and $$IP^{-1}(\mathbb{RP}^N\times\mathbb{RP}^N)$$, and $$IP^{-1}(\mathbb{RP}^N\times\mathbb{RP}^N)$$ and $$IP^{-1}(\mathbb{T}\times\mathbb{T})$$ are non-displaceable each others (Theorem 4.3).
Theorem 4.3 is proved from Theorem 3.1 assuming existence $$u_\mathcal{O}\in\mathcal{D}_{IP^{-1}\Delta}(G)$$ which is not a torsion object, and that there exists a neighborhood $$U$$ of the unit $$G$$ such that for every $$g\in G, F\in D(G)$$ such that $$F$$ is supported on $$gU$$ and $$R\Gamma(G,F)=0$$, then $$F_{* G}u_\mathcal{O}$$ is a torsion object (Proposition 4.4).
Proof of Proposition 4.4 is the main body of this paper. In §5, assuming unique existence a special element $$\mathfrak{S}\in D(G\times\mathfrak{h})$$, $$\mathfrak{h}$$ is the Cartan algebra of $$\mathfrak{g}$$, $$u_\mathcal{O}$$ is constructed by using convolution on $$\mathfrak{h}$$; $$u_\mathcal{O}=I_0^{-1}(\mathfrak{S}\ast_\mathfrak{h}\gamma_L)$$, $$\gamma_L=\mathbb{K}_{\{(A,t)|t+\langle A,L\rangle\geq 0\}}\in D(\mathfrak{h}\times\mathbb{R})$$.
§6 constructs $$\mathfrak{S}$$ and proves its uniqueness (Theorem 6.1, 6.7). §7 computes an isomorphism type of $$\mathfrak{S}|_{z\times C^o_-}$$, $$C^o_-$$ is the interior of $$-C~+$$. The author says the computation is a version of Bott’s computation of $$H_\bullet (\Omega(G))$$ using Morse theory. $$\mathfrak{S}$$ is a strict B-sheaf. In §8, it is shown that any strict B-sheaf can be recovered from its restriction onto $$\mathbf{Z}\times C^o_-$$, $$\mathbf{Z}$$ is the center of $$G$$ (Theorem 8.4).
This paper also includes two Appendices; $$\mathrm{SU}(N)$$ and its Lie algebra: Notations and a Couple of Lemmas (§10), and Results from [Kashiwara and Schapira, loc. cit.] on fundamental properties of microsupport (§11).
For the entire collection see [Zbl 1412.32002].

##### MSC:
 35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs 53D05 Symplectic manifolds, general 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32C38 Sheaves of differential operators and their modules, $$D$$-modules
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