We prove infinitesimal rigidity and integrability of the moduli space for Hermitian gravitational instantons. Together with the recent proof by Biquard, Gauduchon, and LeBrun of local rigidity for Hermitian instantons, this completes the picture of the moduli space of Hermitian gravitational instantons, both for the compact and non-compact cases.
An important step in the proof is to show that provided certain boundary conditions hold, a curve of Riemannian metrics passing through a Hermitian non-Kähler Einstein metric is conformally Kähler to second perturbative order. This uses ideas of Wu and LeBrun.