Bergman Kernel and Kähler Tensor Calculus
نویسنده
چکیده
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.
منابع مشابه
On the Asymptotic Expansion of Bergman Kernel
We study the asymptotic of the Bergman kernel of the spin Dirac operator on high tensor powers of a line bundle.
متن کاملSzasz Analytic Functions and Noncompact Kähler Toric Manifolds
We show that the classical Szasz analytic function SN (f)(x) is related to the Bergman kernel for the Bargmann-Fock space. Then we generalize this relation to any noncompact toric Kähler manifold, defining the generalized Szasz analytic function ShN (f)(x). Then we will prove the complete asymptotic expansion of ShN (f)(x) and its scaling limit property. As examples, we will compute the general...
متن کاملTHE FIRST COEFFICIENTS OF THE ASYMPTOTIC EXPANSION OF THE BERGMAN KERNEL OF THE spin
We establish the existence of the asymptotic expansion of the Bergman kernel associated to the spin Dirac operators acting on high tensor powers of line bundles with non-degenerate mixed curvature (negative and positive eigenvalues) by extending [15]. We compute the second coefficient b1 in the asymptotic expansion using the method of [24].
متن کاملTrees and Tensors on Kähler Manifolds
We present an organized method to convert between partial derivatives of metrics (functions) and covariant derivatives of curvature tensors (functions) on Kähler manifolds. Basically it reduces the highly recursive computation in tensor calculus to the enumeration of certain trees with external legs.
متن کاملBoundary Behaviour of the Bergman Invariant and Related Quantities
Using Fefferman’s classical result on the boundary singularity of the Bergman kernel, we give an analogous description of the boundary behaviour of various related quantities like the Bergman invariant, the coefficients of the Bergman metric, of the associated Laplace-Beltrami operator, of its curvature tensor, Ricci curvature and scalar curvature. The main point is that even though one would e...
متن کامل