Generalized asymptotic expansions of Tian-Yau-Zelditch (Announcements)
英文摘要：
This an announcement for the generalized asymptotic expansion of Tian-Yau-Zeldtich.
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The asymptotic Tian-Yau-Zelditch expansion on Riemann surfaces with Constant Curvature
Let $M$ be a regular Riemann surface with a metric which has constant scalar curvature $ρ$. We give the asymptotic expansion of the sum of the square norm of the sections of the pluricanonical bundles $K_{M}^{m}$. That is, \[\sum_{i=0}^{d_{m}-1}\|S_{i}(x_{0})\|_{h_{m}}^{2} \sim m(1+\fracρ{2 m})+O(e^{-\frac{(\log m)^{2}}{8}}),\] where $\{S_{0},...,S_{d_{m}-1}\}$ is an orthonormal basis for $H^{0}(M, K_{M}^{m})$ for sufficiently large $m$.
math.DG
Spectral measures on toric varieties and the asymptotic expansion of Tian-Yau-Zelditch
We extend a recent result of Burns, Guillemin and Uribe on the asymptotics of the spectral measure for the reduction metric on a toric variety to any toric metric on a toric variety. We show how this extended result together with the Tian-Yau-Zelditch asymptotic expansion can be used to deduce Abreu's formula for the scalar curvature of a toric metric on a toric variety in terms of polytope data.
math.DGmath.SG
The $α$-Invariant on $CP^2#2\bar{CP^2}$
In this paper, we apply the Tian-Yau-Zelditch expansion of the Bergman kernel on polarized Kähler metrics to approximate plurisubharmonic functions and compute the $α$-invariant of $CP^2#2\bar{CP^2}$, which is exactly 1/3. In addition we prove Tian's conjecture on the generalized Moser-Trudinger inequality in a special case prescribed by the $α$-invariant.
math.DG
A closed formula for the asymptotic expansion of the Bergman kernel
We prove a graph theoretic closed formula for coefficients in the Tian-Yau-Zelditch asymptotic expansion of the Bergman kernel. The formula is expressed in terms of the characteristic polynomial of the directed graphs representing Weyl invariants. The proof relies on a combinatorial interpretation of a recursive formula due to M. Englis and A. Loi.
math-phmath.DGmath.MP
TYZ expansion for the Kepler manifold
The main goal of the paper is to address the issue of the existence of Kempf's distortion function and the Tian-Yau-Zelditch (TYZ) asymptotic expansion for the Kepler manifold - an important example of non compact manfold. Motivated by the recent results for compact manifolds we construct Kempf's distortion function and derive a precise TYZ asymptotic expansion for the Kepler manifold. We get an exact formula: finite asymptotic expansion of $n-1$ terms and exponentially small error terms uniformly with respect to the discrete quantization parameter $m\to \infty $ and $ρ\to \infty$, $ρ$ being t…
math.CVmath.DG
