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.

(§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].

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.

(§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].

Reviewer: Akira Asada (Takarazuka)

##### 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 |