For general compact Kahler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finite-dimensional matrix algebras gl(N), N → ∞.

Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to\infty$

Preliminaries.- Kahler-Einstein metrics and extremal Kahler metrics.- The character f and its generalization to Kahlerian invariants.- The character f as an obstruction.- The character f as a classical invariant.- Lifting f to a group character.- The character f as a moment map.- Aubin's approach and related results.

Kähler-Einstein metrics and integral invariants

We show that if a compact Kaehler manifold $M$ of 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. 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 Kaehler manifolds. The proof uses a Hessian formula for the squared norm of Cahen-Gutt moment map.

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

Motivated by the study of coupled Kahler-Einstein metrics by Hultgren and Witt Nystrom and coupled Kahler-Ricci solitons by Hultgren, we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields and certain space of coupled basic functions related to coupled twisted Laplacians for basic functions, and obtain extensions of the well-known obstructions to the existence of Kahler-Einstein metrics to this coupled case. These results are reduced to coupled Kahler-Einstein metrics when the Sasaki structure is regular. Secondly we show the existence of toric coupled Sasaki-Einstein metrics when the basic first Chern class is positive extending the work of Hultgren.

Coupled Sasaki-Ricci solitons

Motivated by the study of coupled Kahler-Einstein metrics by Hultgren and Witt Nystrom (2018) and coupled Kahler-Ricci solitons by Hultgren (2017), we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields and certain space of coupled basic functions related to coupled twisted Laplacians for basic functions, and obtain extensions of the well-known obstructions to the existence of Kahler-Einstein metrics to this coupled case. These results are reduced to coupled Kahler-Einstein metrics when the Sasaki structure is regular. Secondly we show the existence of toric coupled Sasaki-Einstein metrics when the basic first Chern class is positive extending the work of Hultgren (2017).

https://link.springer.com/content/pdf/10.1007/s11425-018-9499-y.pdf

Futaki invariant for Fedosov star products

In the first part of this paper we outline the constructions and properties of Fedosov star product and Berezin-Toeplitz star product. In the second part we outline the basic ideas and recent developments on Yau-Tian-Donaldson conjecture on the existence of Kahler metrics of constant scalar curvature. In the third part of the paper we outline recent results of both authors, and in particular show that the constant scalar curvature Kahler metric problem and the study of deformation quantization meet at the notion of trace (density) for star product. We formulate a cohomology formula for the invariant of K-stability condition on Kahler metrics with constant Cahen-Gutt momentum.

Deformation quantization and K\"ahler geometry with moment map

We study the Cahen–Gutt moment map on the space of symplectic connections of a symplectic manifold. Given a Kahler manifold $(M,\omega ,J)$, we define a Calabi-type functional $\mathscr {F}$ on the space $\mathcal {M}_{\Theta }$ of Kahler metrics in the class $\Theta :=[\omega ]$. We study the space of zeroes of $\mathscr {F}$. When $(M,\omega ,J)$ has non-negative Ricci tensor and $\omega $ is a zero of $\mathscr {F}$, we show the space of zeroes of $\mathscr {F}$ near $\omega $ has the structure of a smooth finite dimensional submanifold. We give a new motivation, coming from deformation quantization, for the study of moment maps on infinite dimensional spaces. More precisely, we establish a strong link between trace densities for star products (obtained from Fedosov’s type methods) and moment map geometry on infinite dimensional spaces. As a byproduct, we provide, on certain Kahler manifolds, a geometric characterization of a space of Fedosov’s star products that are closed up to order 3 in $\nu $.

Infinite dimensional moment map geometry and closed Fedosov’s star products

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

We study obstructions to the existence of closed Fedosov's star products on a given K\"ahler manifold. In our previous paper, we proved that the Levi-Civita connection of a K\"ahler manifold will produce a closed (in the sense of Connes-Flato-Sternheimer) Fedosov's star product only if it is a zero of the Cahen-Gutt moment map on the space of symplectic connections. By analogy with the Futaki invariant obstructing the existence of cscK metrics, we build an obstruction for the existence of zero of the moment map and hence for the existence of closed Fedosov's star products on a K\"ahler manifold.

Laurent La Fuente-Gravy

Papers

Futaki invariant for Fedosov star products

Deformation quantization and K\"ahler geometry with moment map

Infinite dimensional moment map geometry and closed Fedosov’s star products

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

Futaki invariant for Fedosov's star products