Monotone Lagrangian submanifolds of ℂn and toric topology

Oganesyan, Vardan

doi:10.2140/agt.2022.22.1017

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Monotone Lagrangian submanifolds of\/ $\mathbb{C}^n$ and toric topology

Vardan Oganesyan

Algebraic & Geometric Topology 22 (2022) 1017–1056

DOI: 10.2140/agt.2022.22.1017

Abstract

Mironov, Panov and Kotelskiy studied Hamiltonian-minimal Lagrangians inside $ℂ^{n}$ . They associated a closed embedded Lagrangian $L$ to each Delzant polytope $P$ . We develop their ideas and prove that $L$ is monotone if and only if the polytope $P$ is Fano.

In some examples, we further compute the minimal Maslov numbers. Namely, let $𝒩 \to T^{k}$ be some fibration over the $k$ –dimensional torus with fibers equal to either $S^{k} \times S^{l}$ or $S^{k} \times S^{l} \times S^{m}$ or $#_{5} (S^{2 p - 1} \times S^{n - 2 p - 2})$ . We construct monotone Lagrangian embeddings $𝒩 \subset ℂ^{n}$ with different minimal Maslov number, which are therefore distinct up to Lagrangian isotopy. Moreover, we show that some of our embeddings are smoothly isotopic but not Lagrangian isotopic.

Keywords

monotone Lagrangians, moment-angle manifold, isotopic Lagrangians

Mathematical Subject Classification 2010

Primary: 53D12

Secondary: 53D40

References

Publication

Received: 26 July 2019

Revised: 8 February 2021

Accepted: 28 March 2021

Published: 25 August 2022

Authors

Vardan Oganesyan
Department of Mathematics Stony Brook University Stony Brook, NY United States

Volume 22, issue 3 (2022)