Computing the $U$ maps on ECH of prequantization bundles

Abstract

The purpose of this paper is to study the $U$-module structure of the embedded contact homology of a class of prequantization circle bundles over closed symplectic surfaces. We assume that the Euler number of each circle bundle is strictly less than the Euler characteristic of the corresponding surface. We give an explicit description of the $U$-map with respect to a basis of filtered ECH defined via cobordism maps. We use the formula for the $U$-map to compute the ECH spectrum and, as an application, the ECH capacities of certain complements of symplectic divisors. Furthermore, we show that the ECH capacities of closed integral symplectic manifolds are infinite.