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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 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.

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