Real Lagrangian tori and versal deformations

Can a given Lagrangian submanifold be realized as the fixed point set of an anti-symplectic involution? If so, it is called real. We give an obstruction for a closed Lagrangian submanifold to be real in terms of the displacement energy of nearby Lagrangians. Applying this obstruction to toric fibres, we obtain that the central fibre of many (and probably all) toric monotone symplectic manifolds is real only if the corresponding moment polytope is centrally symmetric. Furthermore, we embed the Chekanov torus in all toric monotone symplectic manifolds and show that it is exotic and not real, extending Kim’s result $\href{https://dx.doi.org/10.4310/JSG.2021.v19.n1.a3}{[24]}$ for $S^2 \times S^2$. Inside products of $S^2$, we show that all products of Chekanov tori are pairwise distinct and not real either. These results indicate that real tori are rare. Our methods are elementary in the sense that we do not use Jholomorphic curves. Instead, we rely on symplectic reduction and the displacement energy of product tori in $\mathbb{R}^{2n}$.

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The author was partially supported by SNF grant 200020-144432/1.

Received 25 May 2020

Accepted 15 August 2022

Published 22 December 2023