Finite TYCZ expansions and cscK metrics

Abstract

Let $(M,g)$ be a Kaehler manifold whose associated Kaehler form $\omega$ is integral and let $(L,h)\to (M,\omega )$ be a quantization hermitian line bundle. In this paper we study those Kaehler manifolds $(M,g)$ admitting a finite TYCZ expansion. We show that if the TYCZ expansion is finite then $T_{mg}$ is indeed a polynomial in $m$ of degree $n$, $n=dimM$, and the log-term of the Szegö kernel of the disc bundle $D\subset L^{\ast }$ vanishes (where $L^{\ast }$ is the dual bundle of $L$). Moreover, we provide a complete classification of the Kaehler manifolds admitting finite TYCZ expansion either when $M$ is a complex curve or when $M$ is a complex surface with a cscK metric which admits a radial Kaehler potential.