# ECH capacities, Ehrhart theory, and toric varieties

## TL;DR

This paper links embedded contact homology (ECH) capacities of convex toric domains to Ehrhart theory and toric varieties, providing algebraic and combinatorial insights into symplectic embedding problems.

## Contribution

It formalizes the connection between ECH capacities and toric geometry by constructing a polarized toric variety that encodes ECH data algebraically.

## Key findings

- New algebraic interpretation of ECH capacities for convex toric domains.
- Application of Ehrhart theory yields asymptotic results for ECH capacities.
- Enhanced understanding of symplectic embedding problems via combinatorial methods.

## Abstract

ECH capacities were developed by Hutchings to study embedding problems for symplectic $4$-manifolds with boundary. They have found especial success in the case of certain toric symplectic manifolds where many of the computations resemble calculations found in cohomology of $\mathbb{Q}$-line bundles on toric varieties, or in lattice point counts for rational polytopes. We formalise this observation in the case of convex toric lattice domains $X_\Omega$ by constructing a natural polarised toric variety $(Y_{\Sigma(\Omega)},D_\Omega)$ containing the all the information of the ECH capacities of $X_\Omega$ in purely algebro-geometric terms. Applying the Ehrhart theory of the polytopes involved in this construction gives some new results in the combinatorialisation and asymptotics of ECH capacities for convex toric domains.