ECH capacities, Ehrhart theory, and toric varieties

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 rational convex toric domains $X_\Omega$ by constructing a natural polarised toric variety $(Y_{\Sigma(\Omega)} , D_\Omega)$ containing 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.

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Received 4 October 2019

Accepted 8 September 2020

Published 27 May 2021