#### Volume 22, issue 3 (2022)

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

### Vardan Oganesyan

Algebraic & Geometric Topology 22 (2022) 1017–1056
##### Abstract

Mironov, Panov and Kotelskiy studied Hamiltonian-minimal Lagrangians inside ${ℂ}^{n}$. They associated a closed embedded Lagrangian $L$ to each Delzant polytope $P\phantom{\rule{-0.17em}{0ex}}$. 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 $\mathsc{𝒩}\to {T}^{k}$ be some fibration over the $k$–dimensional torus with fibers equal to either ${S}^{k}×{S}^{l}$ or ${S}^{k}×{S}^{l}×{S}^{m}$ or ${#}_{5}\left({S}^{2p-1}×{S}^{n-2p-2}\right)$. We construct monotone Lagrangian embeddings $\mathsc{𝒩}\subset {ℂ}^{n}$ 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
Primary: 53D12
Secondary: 53D40