Real Lagrangian Tori and Versal Deformations

TL;DR

This paper investigates conditions under which Lagrangian submanifolds are fixed points of anti-symplectic involutions, revealing that real tori are rare and providing new obstructions based on displacement energy, with applications to toric and Chekanov tori.

Contribution

It introduces an obstruction criterion for a Lagrangian to be real, applies it to toric fibers, and demonstrates the non-reality of Chekanov tori in various symplectic manifolds.

Findings

Central fibers of many toric monotone manifolds are real only if the polytope is centrally symmetric.

Chekanov tori are shown to be exotic and not real in all toric monotone manifolds.

All products of Chekanov tori in products of S^2 are pairwise distinct and non-real.

Abstract

Can a given Lagrangian submanifold be realized as the fixed point set of an anti-symplectic involution? If so, it is called \emph{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 (arXiv:1909.09972) 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~$J$-holomorphic curves. Instead, we rely on symplectic reduction and the displacement energy of product tori in $\mathbb{R}^{2n}$.