Real Lagrangian tori and versal deformations (Q6183442)

A real Lagrangian submanifold of a symplectic manifold \((M,\omega)\) is the fixed point set of an anti-symplectic involution of \(M\) or a connected component thereof. \textit{J. Kim} proved in [J. Symplectic Geom. 19, No. 1, 121--142 (2021; Zbl 1466.53089)] that a necessary condition for a Lagrangian submanifold \(L\) of a symplectic manifold \((M,\omega)\) to be real is that the number of \(J\)-holomorphic discs \(u: (D^2,\partial D^2) \rightarrow (M, L)\) of Maslov index 2 passing through a generic point in \(L\) is even. In the paper under review, another symplectic invariant, the displacement energy of nearby Lagrangian submanifolds, introduced by \textit{Yu. V. Chekanov} [Math. Z. 223, No. 4, 547--559 (1996; Zbl 0877.58024)] is used to give an obstruction for a closed Lagrangian submanifold to be real. The author proves that for a compact real Lagrangian submanifold \(L\) of an arbitrary symplectic manifold \((M, \omega)\), the displacement energy germ \(S_L : (H_1(L,\mathbb{R}), 0) \rightarrow \mathbb{R}\cup \{\infty\}\) is even: \(S_L(-p)=S_L(p)\), and for the class of fibers of toric symplectic manifolds the displacement energy is related to the moment polytope \(\Delta\). Then, the author studies the case of a monotone symplectic manifold, whose moment polytope is called monotone. If, moreover, \(\Delta\) has property \(FS\) (which means that every facet \(F \subset \Delta\) contains a point of the set \(S(\Delta)\) of non-zero symmetric lattice points in \(\Delta\)) and the central fibre is real, then \(\Delta\) is centrally symmetric, \(\Delta = -\Delta\). The converse of this theorem was proved in [\textit{J. Brendel} et al., Isr. J. Math. 253, No. 1, 113--156 (2023; Zbl 1527.53065)], hence on a toric monotone symplectic manifold whose moment polytope has property \(FS\), the condition for the central fibre to be real is equivalent to the central symmetry of the moment polytope. A result of \textit{J. Kim} [loc. cit.] on the Chekanov torus in \(S^2\times S^2\) is extended in the reviewed paper to show that the Chekanov torus embeds in all toric monotone symplectic manifolds. The author obtains its displacement energy germ, which reveals that the Chekanov torus is exotic and not real, since the polytope obtained as level set of the displacement energy germ is never centrally symmetric. Finally, the author presents another approach which can be used to prove the main results of the paper, namely the Eliashberg-Polterovich technique based on \(J\)-holomorphic curves, used in [\textit{J. Kim}, J. Symplectic Geom. 19, No. 1, 121--142 (2021; Zbl 1466.53089)].