Futaki invariant for Fedosov star products

Abstract

We study obstructions to the existence of closed Fedosov star products on a given Kähler manifold (M,ω, 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 μ 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 μ and hence for the existence of closed Fedosov star product on a Kähler manifold.