Infinitesimal rigidity of Hermitian gravitational instantons

TL;DR

This paper establishes the infinitesimal rigidity and integrability of the moduli space of Hermitian gravitational instantons, completing the understanding of their deformation theory in both compact and non-compact cases.

Contribution

It proves the infinitesimal rigidity and integrability of the moduli space for Hermitian gravitational instantons, building on recent results and using boundary condition analysis.

Findings

Proves infinitesimal rigidity of Hermitian gravitational instantons.

Shows the moduli space is integrable, completing the deformation picture.

Demonstrates that certain metric curves are conformally K"ahler to second order.

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-K\"ahler Einstein metric is conformally K\"ahler to second perturbative order. This uses ideas of Wu and LeBrun.