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 Kähler 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 $1$.