Zoo of monotone Lagrangians in ${\mathbb {C}}P^n$

Abstract

Let $P \subset {\mathbb {R}}^m$ be a polytope of dimension m with n facets and $a_1,\ldots , a_{n}$ be the normal vectors to the facets of P. Assume that P is Delzant, Fano, and $a_1 + \cdots + a_{n} = 0$. We associate a monotone embedded Lagrangian $L \subset {\mathbb {C}}P^{n-1}$ to P. As an abstract manifold, the Lagrangian L fibers over $(S^1)^{n-m-1}$ with fiber ${\mathcal {R}}_P$, where ${\mathcal {R}}_P$ is defined by a system of quadrics in ${\mathbb {R}}P^{n-1}$. The manifold ${\mathcal {R}}_P$ 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 ${\mathcal {R}}_P$. General formulas for the rings $H^{*}({\mathcal {R}}_P, {\mathbb {Z}})$, $H^{*}({\mathcal {R}}_P, {\mathbb {Z}}_2)$ 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 ${\mathcal {R}}_P$. This idea will be developed in a further paper.