Universality of Fedosov's Construction for Star Products of Wick Type on Pseudo-Kähler Manilfolds

Abstract

In this paper we construct star products on a pseudo-Kähler manifold $(M,ω,I)$ using a modification of the Fedosov method based on a different fibrewise product similar to the Wick product on $\mathbb C^n$. In a first step we show that this construction is rich enough to obtain star products of every equivalence class by computing Deligne's characteristic class of these products. Among these products we uniquely characterize the ones which have the additional property to be of Wick type which means that the bidifferential operators describing the star products only differentiate with respect to holomorphic directions in the first argument and anti-holomorphic directions in the second argument. These star products are in fact strongly related to star products with separation of variables introduced and studied by Karabegov. This characterization gives rise to special conditions on the data that enter the Fedosov procedure. Moreover, we compare our results that are based on an obviously coordinate independent construction to those of Karabegov that were obtained by local considerations and give an independent proof of the fact that star products of Wick type are in bijection to formal series of closed two-forms of type $(1,1)$ on $M$. Using this result we finally succeed in showing that the given Fedosov construction is universal in the sense that it yields all star products of Wick type on a pseudo-Kähler manifold.