Relativistic Boson Stars, Polarimetric Signatures, and Information Geometry in the Kähler-Fisher Framework

We present a relativistic formulation of fuzzy dark matter as an Einstein-Klein-Gordon-Kähler-Fisher system and apply it to model the formation, stability, and observable signatures of boson stars. Starting from the EKG action for a complex scalar field, we prove Lorentz and diffeomorphism invariance, derive the covariant Madelung equations, and show that the Madelung transform defines an isometric Kähler morphism between the Fubini-Study and Sasaki-Fisher-Rao metrics. The Bohm potential arises as the first variation of Fisher information, giving a geometric origin to relativistic quantum pressure and defining curvature corrections that regulate photon propagation and polarization transport near compact configurations. Within this framework, stationary solutions describe boson-star ground states that smoothly reduce to Schrödinger-Poisson solitons in the weak-field limit. The resulting spacetime curvature and Fisher-information tensor predict polarimetric imprints consistent with recent ALMA and GRAVITY observations, reproducing the QU-loop asymmetries and EVPA shifts reported for the ultracompact solitonic boson stars (Rosa et al., Phys. Rev. D 111(2025)). These information-geometric corrections thus provide a unified, testable bridge between quantum curvature, relativistic boson-star dynamics, and polarization-based astrophysical probes of exotic compact objects.