Anticanonically balanced metrics on Fano manifolds. (English) Zbl 1494.14040
Consider a Kähler-Einstein Fano manifold with no nontrivial holomorphic vector field. The main result of the paper says that there exists a sequence of anticanonically balanced metrics that converge smoothly to the Kähler-Einstein metric. It was known from the work of R. J. Berman et al. [Publ. Math., Inst. Hautes Étud. Sci. 117, 179–245 (2013; Zbl 1277.32049)] the weak convergence of the sequence of anticanonically balanced metrics. Introducing new techniques, the paper improves this result by showing the smooth convergence using Berezin-Toeplitz quantization. Moreover, the proof provides an estimate of the first positive eigenvalue of the Laplacian associated to the Kähler-Einstein metric, which confirms a computation of S. K. Donaldson [Pure Appl. Math. Q. 5, No. 2, 571–618 (2009; Zbl 1178.32018)] providing an estimate for the speed of convergence of the iterative method.
Reviewer: Julien Keller (Marseille)
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