Remark on the off-diagonal expansion of the Bergman kernel on compact K\"ahler manifolds

TL;DR

This paper discusses the off-diagonal expansion of the Bergman kernel on compact Kähler manifolds, highlighting connections and clarifications among previous works and recent developments in the field.

Contribution

It clarifies the relationship between earlier results and recent preprints regarding the off-diagonal expansion of the Bergman kernel, emphasizing the implicit and explicit presence of certain coefficients.

Findings

The vanishing of the p^{-1/2} coefficient is implicitly in Dai-Liu-Ma's work.

This vanishing was explicitly stated in the authors' previous book.

The paper connects recent preprints with established results in the literature.

Abstract

In this short note, we compare our previous works on the off-diagonal expansion of the Bergman kernel and the recent preprint of Lu-Shiffman (arxiv.1301.2166). In particular, we note that the vanishing of the coefficient of p^{-1/2} is implicitly contained in Dai-Liu-Ma's work (J. Differential Geom. 72 (2006), no. 1, 1-41) and was explicitly stated in our book (Holomorphic Morse inequalities and Bergman kernels, Progress in Math., vol. 254, Birkh\"auser, 2007).