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Deformation quantization and K\"ahler geometry with moment map
TLDR
In this article, the authors formulate a cohomology formula for the invariant of K-stability condition on Kahler metrics with constant Cahen-Gutt momentum, and 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.Abstract:
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.read more
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Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to\infty$
TL;DR: For general compact Kahler manifolds, it was shown in this paper that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit.
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Kähler-Einstein metrics and integral invariants
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Anticanonically balanced metrics on Fano manifolds
TL;DR: In this paper, it was shown that if a Fano manifold has discrete automorphism group and admits a polarized Kahler-Einstein metric, then there exists a sequence of anticanonically balanced metrics converging smoothly to the KE metric.
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Quantum moment map and obstructions to the existence of closed Fedosov star products
TL;DR: In this paper, it was shown that the normalized trace of Fedosov star product for quantum moment map depends only on the path component in the cohomology class of the symplectic form and the closed formal 2-form.
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A new construction of strict deformation quantization for Lagrangian fiber bundles
TL;DR: In this paper, a new construction of strict deformation quantization of symplectic manifolds equipped with a proper Lagrangian fiber bundle structure, whose representation spaces are the quantum Hilbert spaces obtained by geometric quantization, is presented.
References
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On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*
TL;DR: In this paper, the Ricci form of some Kahler metric is shown to be closed and its cohomology class must represent the first Chern class of M. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.
Book
Heat Kernels and Dirac Operators
TL;DR: In this article, the authors present a formal solution for the trace of the heat kernel on Euclidean space, and show that the trace can be used to construct a heat kernel of an equivariant vector bundle.
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Kähler-Einstein metrics with positive scalar curvature
TL;DR: In this article, it was shown that the existence of Kahler-Einstein metrics implies the stability of the underlying Kahler manifold in a suitable sense, which disproves a long-standing conjecture that a compact KG admits KG metrics if it has positive first Chern class and no nontrivial holomorphic vector fields.
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A simple geometrical construction of deformation quantization
TL;DR: In this paper, a canonical star product associated with any symplectic connection on a regular Poisson manifold is considered. And a trace construction of the star product is given for coefficients in the bundle Hom(£, E) of the manifold.
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Scalar Curvature and Stability of Toric Varieties
TL;DR: In this paper, a stability condition for a polarised algebraic variety is defined and a conjecture relating this to the existence of a Kahler metric of constant scalar curvature.