Intersections of quadrics and Hamiltonian-minimal Lagrangian submanifolds

Hamiltonian minimality (H-minimality) for Lagrangian submanifolds is a symplectic analogue of minimality in Riemannian geometry. A Lagrangian immersion is called H-minimal if the variations of its volume along all Hamiltonian vector fields are zero.
We study the topology of H-minimal Lagrangian submanifolds $N$ in $\mathbb C^m$ constructed from intersections of real quadrics in the work of Mironov. This construction is linked via an embedding criterion to the well-known Delzant construction of Hamiltonian toric manifolds.
By applying the methods of toric topology we produce new examples of H-minimal Lagrangian submanifolds with quite complicated topology. The interpretation of our construction in terms of symplectic reduction leads to its generalisation providing new examples of H-minimal submanifolds in toric varieties.
The talk is based on a joint work with Andrey E. Mironov.