Resumen de Zoo of monotone Lagrangians in CPn

Vardan Oganesyan

• Let P ⊂ Rm be a polytope of dimension m with n facets and a1,..., an be the normal vectors to the facets of P. Assume that P is Delzant, Fano, and a1 +···+ an = 0.

  We associate a monotone embedded Lagrangian L ⊂ CPn−1 to P. As an abstract manifold, the Lagrangian L fibers over(S1)n−m−1 with fiber RP, where RP is defined by a system of quadrics in RPn−1. The manifold RP is called the real toric space. We find an effective method for computing the Lagrangian quantum cohomology groups of the mentioned Lagrangians. Then we construct explicitly some set of wide and narrow Lagrangians. Our method yields many different monotone Lagrangians with rich topological properties, including non-trivial Massey products, complicated fundamental group and complicated singular cohomology ring. Interestingly, not only the methods of toric topology can be used to construct monotone Lagrangians, but the converse is also true: the symplectic topology of Lagrangians can be used to study the topology of RP. General formulas for the rings H∗(RP,Z), H∗(RP,Z2) are not known. Since we have a method for constructing narrow Lagrangians, the spectral sequence of Oh can be used to study the singular cohomology ring of RP. This idea will be developed in a further paper.