On the coefficients of TYZ expansion of locally Hermitian symmetric spaces

TL;DR

This paper investigates the TYZ expansion coefficients of locally Hermitian symmetric spaces, proving that only flat spaces have the first two coefficients vanish and identifying nonflat examples with all odd coefficients zero.

Contribution

It establishes a characterization of flatness in locally Hermitian symmetric spaces based on the vanishing of initial TYZ coefficients and reveals the existence of nonflat spaces with all odd coefficients zero.

Findings

First two TYZ coefficients vanish only in flat spaces

Existence of nonflat spaces with all odd coefficients zero

Characterization of flatness via TYZ coefficient vanishing

Abstract

In this paper we address the problem of studying those K\"ahler manifolds whose first two coefficients of the associated TYZ expansion vanish and we prove that for a locally Hermitian symmetric space this happens only in the flat case. We also prove that there exist nonflat locally Hermitian symmetric spaces where all the odd coefficients vanish.