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

Abstract

We show that if a compact Kähler manifold M with nonnegative 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 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. Take arbitrary J ∈ J (M, ω), then we also have the decomposition with respect to J (20)T If J is sufficiently close to J then T * J M can be expressed as a graph overWe use the identification of T * J M with T J M by the Kähler metric defined by the pair (ω, J), and then µ is regarded as µ ∈ Γ(End(

Quantum moment map and obstructions to the existence of closed Fedosov star products

“…The ν 2−m -term is exactly the invariant found in [29]. See also [21] for a different derivation of this invariant using Donaldson-Fujiki type picture. This is one of the obstructions to asymptotic Chow semi-stability found by the first author in [17].…”

The formal moment map geometry of the space of symplectic connections

“…The motivation of this paper is twofold. First, it justifies the study of closed Fedosov star product, see [24,23], Futaki-Ono [17] and Futaki-L. [15,16], as a zero moment map problem, a picture that was up to now only valid at first orders in the formal parameter. Second, there is an increasing interest in the study of the quantization of Hamiltonian diffeomorphisms from the point of view of symplectic topology, see for example Ioos [19,20], Charles [6], Charles-Polterovich [9] and Charles-Le Floch [8] (in the past 3 years!).…”

“…Proof of Theorem 3.10. We have to prove the formal moment map Equation (17). From the variation formula from Lemma 3.6, we have…”

H-functional and Matsushima type decomposition theorem

“…Similar results for the Hessian formula for the norm squared functional of a moment map and the Matsushima type decomposition theorem of holomorphic vector fields were observed for various situations. See for instance, [9] (for a Calabi type functional and the perturbed extremal Kähler metric), [10] (for a Calabi type functional and the f -extremal Kähler metric), [11] (for the norm squared functional of the Cahen-Gutt moment map and the closed Fedosov star product) and [16] (for the Ricci-Calabi functional and the Mabuchi's generalized Kähler Einstein metric).…”