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

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).