Exotic Chekanov tori in toric symplectic varieties

Abstract

We present a generalization of toric structures on compact symplectic manifolds called pseudotoric structure. In the present talk we show that every toric manifold admits pseudotoric structures and then we show that the construction of exotic Chekanov tori can be peformed in terms of pseudotoric structures.

On exotic Lagrangian tori in ℂℙ2

“…Therefore we get that our family of Lagrangian tori is given by the following. Definition 2.2 Given r > 0, and a real number 2 R, set Remark All the pairs consisting of a symplectic fibration together with a map from the symplectic manifold to R (real data) used to define the Lagrangian fibrations considered in this paper form pseudotoric structures as defined by Tyurin in [19].…”

“…The families of discs in the classes 2H 5ˇC k˛, given by (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17), have been shown to converge uniformly to the corresponding ones in CP .1; 1; 4/ given by (5-20). Smoothness of the moduli space M.X ; L/ near the families of discs given by (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) in X 0 guarantees that each family fu  I j  2 OE0; 2 g has a unique family in X t converging to it, for all t sufficiently small. Hence the counts of Maslov index 2 holomorphic discs in the classes 2H 5ˇC k˛for X t are the same as the ones computed in X 0 .…”

“…Theorem 5. 19 The holomorphic discs in X 0 computed in Theorem 5.10 are regular. By Lemma 5.15, for small t , the corresponding holomorphic discs in the classes 2H 5ˇC m˛in X t are also regular.…”

“…Now we basically need to show that D O.t 1=2 /, since the asymptotic behavior of the Á j described in Lemma 5.11 follows from(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21) and D O.t 1=2 /, after we separate the terms that are O.t 1=2 /, more precisely, ".w/ D w.r C cw 5 / C w 7 O.t 1=2 / C O.t 1=2 /.…”