  We investigate shrinkage priors on power spectral densities for
complex-valued circular-symmetric autoregressive processes. We construct
shrinkage predictive power spectral densities, which asymptotically dominate
(i) the Bayesian predictive power spectral density based on the Jeffreys prior
and (ii) the estimative power spectral density with the maximal likelihood
estimator, where the Kullback-Leibler divergence from the true power spectral
density to a predictive power spectral density is adopted as a risk.
Furthermore, we propose general constructions of objective priors for K\"ahler
parameter spaces, utilizing a positive continuous eigenfunction of the
Laplace-Beltrami operator with a negative eigenvalue. We present numerical
experiments on a complex-valued stationary autoregressive model of order $1$.
