Intersections of quadrics, moment-angle manifolds, and Hamiltonian-minimal Lagrangian embeddings

TL;DR

This paper explores the topology of Hamiltonian-minimal Lagrangian submanifolds constructed from intersections of quadrics, revealing their embedding properties and fibrations, and providing new examples with complex topology.

Contribution

It establishes the embedding of these submanifolds into moment-angle manifolds and describes their fibrations, linking to Delzant's construction and expanding the class of known Hamiltonian-minimal Lagrangian examples.

Findings

N embeds as a submanifold in the moment-angle manifold Z

N admits two distinct fibrations over tori and quotients of real moment-angle manifolds

New examples of Hamiltonian-minimal Lagrangian submanifolds with complex topology

Abstract

We study the topology of Hamiltonian-minimal Lagrangian submanifolds N in 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 Z, and every N is the total space of two different fibrations, one over the torus T^{m-n} with fibre a real moment-angle manifold R, and another over a quotient of R by a finite group with fibre a torus. These properties are used to produce new examples of Hamiltonian-minimal Lagrangian submanifolds with quite complicated topology.