Surveys in Differential Geometry

Volume 26 (2021)

Local scalar invariants of Kähler metric

Pages: 233 – 261

DOI: https://dx.doi.org/10.4310/SDG.2021.v26.n1.a7

Authors

Kefeng Liu (Mathematical Science Research Center, Chongqing University of Technology, Chongqing, China; and Department of Mathematics, University of California, Los Angeles, Calif., U.S.A.)

Hao Xu (Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China)

Abstract

On Kähler manifolds, the asymptotic coefficients of Bergman kernel, heat kernel and deformation quantization are local scalar invariants which are universal polynomials of jets of the Kähler metric. We show that they could be canonically expressed as a summation over directed graphs and the coefficients of these graphs are explicit graph invariants. The method should work for all geometrically meaningful local Kähler invariants. We survey the related works and give applications to heat coefficients of Kähler manifold. In particular, we show an explicit formula of the heat coefficients of $\mathbb{C}P^d$ as polynomials in $d$ and present a heuristic approach to Chern–Gauss–Bonnet formula of Kähler manifold.

Published 22 January 2024