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

Mironov, Panov and Kotelskiy studied Hamiltonian-minimal Lagrangians inside 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 Tk be some fibration over the k–dimensional torus with fibers equal to either Sk × Sl or Sk × Sl × Sm or #5(S2p1 × Sn2p2). We construct monotone Lagrangian embeddings 𝒩 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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monotone Lagrangians, moment-angle manifold, isotopic Lagrangians
Mathematical Subject Classification 2010
Primary: 53D12
Secondary: 53D40
Received: 26 July 2019
Revised: 8 February 2021
Accepted: 28 March 2021
Published: 25 August 2022
Vardan Oganesyan
Department of Mathematics
Stony Brook University
Stony Brook, NY
United States