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.

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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)