Symplectic embeddings of toric domains with boundary a lens space

Abstract

We give a combinatorial description of the embedded contact complex (ECC) of a certain family of contact toric lens spaces that we call concave lens spaces. We also define a notion of a concave toric domain that generalizes the usual concave toric domain in a way that possesses a singularity point and has a boundary a lens space. After desingularization these toric domains include the unitary cotangent bundle of $\mathbb{S}^2$ and the unitary cotangent bundle of $\mathbb{R}P^2$. We use the combinatorial expression of the ECC to compute the ECH capacities of these toric domains. Furthermore, for certain concave toric domains we describe a packing of symplectic manifolds that recovers their ECH capacities.