Pseudotoric Lagrangian fibrations of toric and nontoric Fano varieties (Q1956836)

The concept of a pseudotoric structure on a symplectic manifold generalizing the notion of a toric structure is introduced. It is shown that pseudotoric structure can exist on toric and nontoric symplectic manifolds. For the toric manifolds, it describes deformations of the standard toric Lagrangian fibrations. For the nontoric manifolds, it gives Lagrangian fibrations with singularities that are very close to the toric fibrations. The examples of toric manifolds with different pseudotoric structures are present.