Selection of the optimal background Kähler metric

Determine which background Kähler metric is best suited to study the moduli space of complex structures on a holomorphic vector bundle when the underlying smooth manifold, complex vector bundle, and symplectic form are fixed.

Background

The authors describe the motivation for Kähler–Yang–Mills metrics as canonical metrics associated with pairs consisting of a Kähler manifold and a holomorphic vector bundle. For a fixed complex structure on the base, the Hermitian–Einstein metric provides a canonical metric on the bundle, but the choice of background Kähler metric remains variable.

The unresolved issue concerns how this freedom should be resolved when studying the moduli space of complex structures on the bundle. Identifying the most appropriate background Kähler metric would clarify the geometric framework in which that moduli problem should be analyzed.