Journal of Symplectic Geometry

Volume 18 (2020)

Number 4

Optimal convergence speed of Bergman metrics on symplectic manifolds

Pages: 1091 – 1126



Wen Lu (School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China)

Xiaonan Ma (UFR de Mathématiques, Université de Paris and Université Paris Diderot, Paris, France)

George Marinescu (Mathematisches institut, Universität zu Köln, Germany; and Institute of Mathematics, Romanian Academy, Bucharest, Romania)


It is known that a compact symplectic manifold endowed with a prequantum line bundle can be embedded in the projective space generated by the eigensections of low energy of the Bochner Laplacian acting on high $p$-tensor powers of the prequantum line bundle. We show that the Fubini-Study forms induced by these embeddings converge at speed rate $1 / p^2$ to the symplectic form. This result implies the generalization to the almost-Kähler case of the lower bounds on the Calabi functional given by Donaldson for Kähler manifolds, as shown by Lejmi and Keller.

W.L. was supported by National Natural Science Foundation of China (Grant Nos. 11401232, 11871233).

X.M. was partially supported by NNSFC No. 11829102 and funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative.

G.M. was partially supported by DFG funded project SFB TRR 191.

Received 26 July 2017

Accepted 6 September 2019

Published 28 October 2020