Pseudotoric structures and Chekanov tori

It is a very important problem in symplectic geometry to classify all lagrangian tori in a given symplectic manifold up to Hamiltonian isotopy. Not too much is known till now even in the simplest and basic cases: only recently Yu. Chekanov constructed a family of exotic lagrangian tori in the symplectic vector spaces and certain compact symplectic manifolds – but still nobody knows does the set of types for say the projective plane is exhausted by the Clifford torus and the Chekanov torus.
The Clifford torus comes from the toric geometry setup, so if we have a toric variety then there is a standard lagrangian firbation and the smooth fiber of such a fibration gives the standard lagrangian torus. But the notion of toric structure has a generalization: the notion of pseudotoric structure. It is a structure which gives a family of lagrangian fibrations of different topological types including the standard one. F.e. for the projective plane both the Clifford and the Chekanov types can be derived from the same pseudotoric structure. On the other hand, not only toric varieties admit pseudotoric structures: we show that certain non toric varieties admit the structure so they can be endowed by lagrangian fibrations. As an illustration we construct a special lagrangian fibration a la Auroux on the flag variety F3.

^* (Format: 2 hours + tea break + 2 hours.)