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

TL;DR

This paper constructs and analyzes a diverse family of monotone Lagrangian submanifolds in complex projective spaces derived from Delzant Fano polytopes, revealing their rich topological structures and providing methods to compute their quantum cohomology.

Contribution

It introduces a new method to associate monotone Lagrangians to Delzant Fano polytopes and computes their quantum cohomology, expanding the understanding of Lagrangian topology in symplectic geometry.

Findings

Constructed a wide variety of monotone Lagrangians with complex topology.

Developed an effective method for computing Lagrangian quantum cohomology groups.

Demonstrated the interplay between symplectic topology and polytope topology.

Abstract

Let $P \subset \mathbb{R}^m$ be a polytope of dimension $m$ with $n$ facets. Assume that $P$ is Delzant and Fano. We associate a monotone embedded Lagrangian $L \subset \mathbb{C}P^{n-1}$ to $P$. As an abstract manifold, the Lagrangian $L$ fibers over some torus with fiber $\mathcal{R}_P$, where $\mathcal{R}_P$ is defined by a system of quadrics in $\mathbb{R}P^{n-1}$. We find an effective method for computing the Lagrangian quantum cohomology groups of the mentioned Lagrangians. Then we construct explicitly some rich 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 $\mathcal{R}_P$.