# Finite TYCZ expansions and cscK metrics

## TL;DR

This paper characterizes Kaehler manifolds with finite TYCZ expansions, showing they have polynomial TYCZ functions and no log-term in the Szeg"o kernel, and classifies such manifolds in low dimensions with specific metrics.

## Contribution

It proves that finite TYCZ expansions imply polynomial TYCZ functions and zero log-term, and provides a full classification for certain low-dimensional cases.

## Key findings

- Finite TYCZ expansion implies T_{mg} is a degree n polynomial.
- The log-term of the Szeg"o kernel vanishes for these manifolds.
- Complete classification of such manifolds in complex curves and surfaces with cscK metrics.

## Abstract

Let $(M, g)$ be a Kaehler manifold whose associated Kaehler form $\omega$ is integral and let $(L, h)\rightarrow (M, \omega)$ 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 $n$, $n=dim M$, and the log-term of the Szeg\"{o} kernel of the disc bundle $D\subset L^*$ vanishes (where $L^*$ 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.