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 Funktsional. Anal. i Prilozhen., 2013, Volume 47, Issue 1, Pages 47–61 (Mi faa3094)

Intersections of Quadrics, Moment-Angle Manifolds, and Hamiltonian-Minimal Lagrangian Embeddings

A. E. Mironovab, T. E. Panovcde

a N. N. Bogoljubov Laboratory of Geometric Methods in Mathematical Physics
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
c M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
d A. A. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow
e Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center), Moscow

Abstract: We study the topology of Hamiltonian-minimal Lagrangian submanifolds $N$ in $\mathbb{C}^m$ constructed from intersections of real quadrics in a work of the first author. This construction is linked via an embedding criterion to the well-known Delzant construction of Hamiltonian toric manifolds. We establish the following topological properties of $N$: every $N$ embeds as a submanifold in the corresponding moment-angle manifold $\mathcal Z$, and every $N$ is the total space of two different fibrations, one over the torus $T^{m-n}$ with fiber a real moment-angle manifold $\mathcal{R}$ and the other over a quotient of $\mathcal{R}$ by a finite group with fiber a torus. These properties are used to produce new examples of Hamiltonian-minimal Lagrangian submanifolds with quite complicated topology.

Keywords: moment-angle manifold, simplicial fan, simple polytope

DOI: https://doi.org/10.4213/faa3094

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English version:
Functional Analysis and Its Applications, 2013, 47:1, 38–49

Bibliographic databases:

UDC: 514.76+515.16

Citation: A. E. Mironov, T. E. Panov, “Intersections of Quadrics, Moment-Angle Manifolds, and Hamiltonian-Minimal Lagrangian Embeddings”, Funktsional. Anal. i Prilozhen., 47:1 (2013), 47–61; Funct. Anal. Appl., 47:1 (2013), 38–49

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa3094
• https://doi.org/10.4213/faa3094
• http://mi.mathnet.ru/eng/faa/v47/i1/p47

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. E. Mironov, T. E. Panov, “Hamiltonian-minimal Lagrangian submanifolds in toric varieties”, Russian Math. Surveys, 68:2 (2013), 392–394
2. T. E. Panov, “Geometric structures on moment-angle manifolds”, Russian Math. Surveys, 68:3 (2013), 503–568
3. S. López de Medrano, “Singularities of homogeneous quadratic mappings”, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 108:1 (2014), 95–112
4. B. T. Saparbayeva, “On the Schrödinger operator connected with a family of Hamiltonian-minimal Lagrangian surfaces in $\mathbb CP^2$”, Siberian Math. J., 57:6 (2016), 1077–1081
5. Kotelskiy A., “Minimal and H-minimal submanifolds in toric geometry”, J. Symplectic Geom., 14:2 (2016), 431–448
6. V. M. Buchstaber, N. Yu. Erokhovets, M. Masuda, T. E. Panov, S. Park, “Cohomological rigidity of manifolds defined by 3-dimensional polytopes”, Russian Math. Surveys, 72:2 (2017), 199–256
7. N. A. Tyurin, “Pseudotoric structures: Lagrangian submanifolds and Lagrangian fibrations”, Russian Math. Surveys, 72:3 (2017), 513–546
8. M. A. Ovcharenko, “On Hamiltonian-minimal isotropic homogeneous tori in $\mathbb C^n$ and $\mathbb C\mathrm P^n$”, Siberian Math. J., 59:5 (2018), 931–937
9. M. A. Ovcharenko, “On Hamiltonian Minimality of Isotropic Nonhomogeneous Tori in $\mathbb{H}^n$ and $\mathbb C\mathrm P^{2n+1}$”, Math. Notes, 108:1 (2020), 108–116
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