  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.
