Microlocal Condition for Non-displaceability
We formulate a sufficient condition for non-displaceability (by Hamiltonian symplectomorphisms which are identity outside of a compact) of a pair of subsets in a cotangent bundle. This condition is based on micro-local analysis of sheaves on manifolds by Kashiwara–Schapira. This condition is used to prove that the real projective space and the Clifford torus inside the complex projective space are mutually non-displaceable.
I would like to thank Boris Tsygan and Alexander Getmanenko for motivation and numerous fruitful discussions. I am grateful to Pavel Etingof, Roman Bezrukavnikov, Ivan Mirkovich, and David Kazhdan for their explanations on Peterson varieties.
- 1.Kashiwara, M., Schapira, P.: Sheaves on Manifolds, Grundlehren der mathematischen Wissenschaften , vol. 292. Springer, BerlinGoogle Scholar
- 4.Cho, C.-H.: Holomorphic disks, spin structures and Floer cohomology of Clifford torus. Int. Math. Res. Notes 35, 1803–1843 (2004)Google Scholar
- 5.Bezrukavnikov, R., Finkelberg, M., Mirkovich, I.: Equivariant \((K-)\) homology of affine Grassmanian and Toda Lattice Compos. Math. 141(3), 746–768 (2005)Google Scholar
- 6.Kostant, B.: lag manifold quantum cohomology, the Toda lattice, and the representation with highest weight \(\rho \). Selecta Math. (N.S.) 2, 43–91 (1996)Google Scholar
- 7.Nadler, D.: Zaslow, E.: Constructible sheaves and Fukaya category. J. Am. Math. Soc 22(1), 233–286 (2009)Google Scholar
- 8.Nadler, D.: Microlocal Branes and Constructible Sheaves (2006). arXiv:math/0612399v4
- 9.Nest, R., Tsygan, B.: Remarks on modules over deformation quantization algebras. Mosc. Math. J. 4(4), 911–940 (2004)Google Scholar