Nontoric foliations by Lagrangian tori of toric Fano varieties

Special Lagrangian fibrations on the flag variety F 3

“…Let us extend the class of symplectic toric manifolds. Here and in other papers, [3], [4], we study compact smooth symplectic manifolds which admit lagrangian fibrations of almost the same type as in the toric case unless the allowance of the existence of singular tori in the fibrations. We require that generic fiber must be smooth, and the types of the allowed singularties are exhausted by the following models.…”

“…This fibration is not unique given by our construction, but we distinguish it and call it minimal since it has minimal set of singular tori. The construction is based on the fact that F 3 admits a pseudo toric structure, a generalization of the toric structure, proposed in [3] and [4]. Any pseudo toric manifold admits a variety of lagrangian fibrations of the desired type.…”

“…We construct a big family of lagrangian fibrations on F 3 using the notion of pseudotoric structures introduced in [3], [4]. We have the following Recall that pseudotoric structure on a symplectic manifold (M, ω M ) of real dimension 2n is a set of data (f 1 , ..., f k , B, ψ, (Y, ω Y )) where f i are smooth Morse functions in involution, ψ defines a family of symplectic toric submanifolds with the base set B ⊂ M parameterized by a symplectic toric manifold…”

Exotic Chekanov tori in toric symplectic varieties

“…The main reason for the introduction of this new structure is the possibility to construct lagrangian fibrations on whole X starting with lagrangian fibrations on the base toric manifolds and using the toric nature of the fibers. If we choose a system (h 1 , ..., h n−k ) of commuting moment maps on Y (since Y by the definition is toric) we get a lagrangian fibration on the base Y but at the same time we have the following Theorem ( [6]): Choice of moment maps (h 1 , ..., h n−k ) on the base Y of a regular pseudotoric structure (f 1 , ..., f k , ψ, Y ) on a given X defines a lagrangian foliation on X whose generic fiber is a smooth lagrangian torus.…”

Chekanov tori and pseudotoric structures

“…This tree of examples grew up over the first one -the torus Θ ∈ R 4 , constructured in [2] and [3]. On the other hand, in [4] and [5] we proposed a new structure which generalizes the notion of toric structure. One calls this generalization pseudo toric structure, and several examples were given which show that certain toric symplectic manifolds can carry the structre and that certain non toric symplectic manifolds do the same.…”