On non-displaceable Lagrangian submanifolds in two-step flag varieties

TL;DR

This paper demonstrates the existence of non-displaceable Lagrangian submanifolds in two-step flag varieties, including a specific fiber diffeomorphic to S^3 x T^{2n-4} and a family of torus fibers, expanding understanding of symplectic geometry.

Contribution

It establishes the presence of non-displaceable Lagrangian Gelfand--Zeitlin fibers in two-step flag varieties, including a particular non-monotone fiber and a continuum of torus fibers, which was previously unknown.

Findings

Existence of a non-displaceable, non-monotone Lagrangian Gelfand--Zeitlin fiber diffeomorphic to S^3 x T^{2n-4}.

Presence of a continuum family of non-displaceable Lagrangian Gelfand--Zeitlin torus fibers for n > 2.

Advancement in understanding the symplectic topology of two-step flag varieties.

Abstract

We prove that the two-step flag variety $\mathcal{F}\ell(1,n;n+1)$ carries a non-displaceable and non-monotone Lagrangian Gelfand--Zeitlin fiber diffeomorphic to $S^3 \times T^{2n-4}$ and a continuum family of non-displaceable Lagrangian Gelfand--Zeitlin torus fibers when $n > 2$.