Journal of Symplectic Geometry

Volume 17 (2019)

Number 5

Futaki invariant for Fedosov star products

Pages: 1317 – 1330



Laurent La Fuente-Gravy (Mathematics Research Unit, Université du Luxembourg, Esch-sur-Alzette, Luxembourg)


We study obstructions to the existence of closed Fedosov star products on a given Kähler manifold $(M, \omega, J)$. In our previous paper [14], we proved that the Levi–Civita connection of a Kähler manifold will produce a closed Fedosov star product (closed in the sense of Connes–Flato–Sternheimer [4]) only if it is a zero of a moment map $\mu$ on the space of symplectic connections. By analogy with the Futaki invariant obstructing the existence of constant scalar curvature Kähler metric, we build an obstruction for the existence of zero of $\mu$ and hence for the existence of closed Fedosov star product on a Kähler manifold.

Part of this work benefitted from the Belgian Interuniversity Attraction Pole (IAP) DYGEST.

Received 25 October 2017

Accepted 5 September 2018

Published 20 November 2019