Chekanov exotic tori and pseudotoric structures

One problem we have inherited after Andery Tyurin is the interest in lagrangian geometry of algebraic varieties. The lagrangian geometry comes when we consider an additional data - a fixed Kähler form of the Hodge type. E.g. for Fano varieties one has an embedding by certain power of the anticanonical system to the projective space and so the standard Kähler form on the latter induces an (anti) canonical symplectic structure of the variety. Then the following questions arise: which types of submanifolds are realized by lagrangian submanifolds; what is the classification of lagrangian submanifolds up to deformation or, more restrictively, up to Hamiltonian isotopy.

Not much is known till now. Even for the projective plane we have two types of lagrangian tori - the Clifford and the Chekanov ones - and nobody knows whether there are other types. In the talk we present exotic Chekanov tori in terms of pseudotoric structures and show that any toric variety admits non standard lagrangian tori of the Chekanov type.