Infinitesimal rigidity of Hermitian gravitational instantons
Lars Andersson, Bernardo Araneda
TL;DR
The paper proves infinitesimal rigidity and integrability for the moduli space of Hermitian gravitational instantons in the ALF setting. It introduces a divergence identity valid for generic 4-manifolds with non‑self-dual Weyl tensor and uses a conformal transformation framework together with ALF perturbation theory to show that infinitesimal Einstein deformations remain tangent to the Hermitian moduli, yielding conformal‑Kähler behavior to second order. By combining these analytical tools with known classifications and smoothness of the moduli space, the authors conclude that ALF Ricci-flat perturbations are integrable and streamline the moduli-space picture for both compact and non-compact Hermitian instantons. This result strengthens the understanding of the deformation theory of Hermitian instantons and supports conjectures about the global structure of their moduli spaces.
Abstract
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.