On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch

Lu, Zhiqin

doi:10.1353/ajm.2000.0013

Cited by 217 publications

(206 citation statements)

References 11 publications

Supporting

Mentioning

201

Contrasting

Order By: Relevance

“…In [12], Lu computed S 0 , S 1 , S 2 , S 3 and S 4 and with his method we can also compute the other coefficients. Since 1 * is the unit of (C ∞ (M ) [[ ]], * ), we can also compute it from the formulas for the star product * .…”

Section: Introductionmentioning

confidence: 99%

Berezin-Toeplitz Operators, a Semi-Classical Approach

Charles

2003

Communications in Mathematical Physics

101

175

View full text Add to dashboard Cite

This article is devoted to the Toeplitz Operators [4] in the context of the geometric quantization [11], [15]. We propose an ansatz for their Schwartz kernel. From this, we deduce the main known properties of the principal symbol of these operators and obtain new results : we define their covariant and contravariant symbols, which are full symbol, and compute the product of these symbols in terms of the Kähler metric. This gives canonical star products on the Kählerian manifolds. This ansatz is also useful to introduce the notion of microsupport.

show abstract

Section: Introductionmentioning

confidence: 99%

Berezin-Toeplitz Operators, a Semi-Classical Approach

Charles

2003

Communications in Mathematical Physics

101

175

View full text Add to dashboard Cite

show abstract

“…The computation of a 3 , independently done by Lu [45] and Engliš [17], is a marvelous feat; this requires technical and hard calculations occupying more than ten pages in both papers.…”

Section: Local and Global Bergman Kernels: An Explicit Computationmentioning

confidence: 99%

Bergman kernel and Kähler tensor calculus

2013

Pure and Applied Mathematics Quarterly

View full text Add to dashboard Cite

Fefferman [22] initiated a program of expressing the asymptotic expansion of the boundary singularity of the Bergman kernel for strictly pseudoconvex domains explicitly in terms of boundary invariants. Hirachi, Komatsu and Nakazawa [30] carried out computations of the first few terms of Fefferman's asymptotic expansion building partly on Graham's work on CR invariants and Nakazawa's work on the asymptotic expansion of the Bergman kernel for strictly pseudoconvex complete Reinhardt domains. In this paper, we prove a formula for coefficients in Nakazawa's asymptotic expansion as explicit summations over strongly connected graphs, and a formula expressing partial derivatives of Kähler metrics (resp. functions) as summations over rooted trees encoding covariant derivatives of curvature tensors (resp. functions). These formulae shall be useful in studying general patterns of Fefferman's asymptotic expansion.

show abstract

“…For simplicity, we study only the vacuum expectation value of the identity operator using the naïve vacuum state |Ω x,κ . Z. Lu computed the lower order terms in powers of κ −1 of the squared norm of the naïve vacuum state, [18],…”

Section: Vacuum Energy In Geometric Quantizationmentioning

confidence: 99%

Balanced Metrics and Noncommutative Kähler Geometry

Lukić¹

2010

SIGMA

View full text Add to dashboard Cite

Abstract. In this paper we show how Einstein metrics are naturally described using the quantization of the algebra of functions C ∞ (M ) on a Kähler manifold M . In this setup one interprets M as the phase space itself, equipped with the Poisson brackets inherited from the Kähler 2-form. We compare the geometric quantization framework with several deformation quantization approaches. We find that the balanced metrics appear naturally as a result of requiring the vacuum energy to be the constant function on the moduli space of semiclassical vacua. In the classical limit these metrics become Kähler-Einstein (when M admits such metrics). Finally, we sketch several applications of this formalism, such as explicit constructions of special Lagrangian submanifolds in compact Calabi-Yau manifolds.

show abstract

On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch

Abstract: We compute the first three coefficients of the asymptotic expansion of Zelditch. We also prove that in general, the k th coefficient is a polynomial of the curvature and its derivative of weight k .

Cited by 217 publications

References 11 publications

Berezin-Toeplitz Operators, a Semi-Classical Approach

Berezin-Toeplitz Operators, a Semi-Classical Approach

Bergman kernel and Kähler tensor calculus

Balanced Metrics and Noncommutative Kähler Geometry

Contact Info

Product

Resources

About