Journal of Symplectic Geometry

Volume 18 (2020)

Number 1

Cahen–Gutt moment map, closed Fedosov star product and structure of the automorphism group

Pages: 123 – 145



Akito Futaki (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Hajime Ono (Department of Mathematics, Saitama University, Saitama, Japan)


We show that if a compact Kähler manifold $M$ with non-negative Ricci curvature admits closed Fedosov star product then the reduced Lie algebra of holomorphic vector fields on $M$ is reductive. This comes in pair with the obstruction previously found by La Fuente–Gravy [20]. More generally we consider the squared norm of Cahen–Gutt moment map as in the same spirit of Calabi functional for the scalar curvature in $\operatorname{cscK}$ problem, and prove a Cahen–Gutt version of Calabi’s theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kähler manifolds.

Received 8 March 2018

Accepted 13 February 2019

Published 25 March 2020