Anticanonically balanced metrics on Fano manifolds

We show that if a Fano manifold has discrete automorphism group and admits a polarized Kähler–Einstein metric, then there exists a sequence of anticanonically balanced metrics converging smoothly to the Kähler–Einstein metric. Our proof is based on a simplification of Donaldson’s proof of the analogous result for balanced metrics, replacing a delicate geometric argument by the use of Berezin–Toeplitz quantization. We then apply this result to compute the asymptotics of the optimal rate of convergence to the fixed point of Donaldson’s iterations in the anticanonical setting.