Finite TYCZ expansions and cscK metrics

Let (M, g) be a Kahler manifold whose associated Kahler form omega is integral and let (L, h) -> (M, omega) be a quantization hermitian line bundle. In this paper we study those Kahle': manifolds (M, g) admitting a finite TYCZ expansion, namely those for which the associated Kempf distortion function T-mg is of the form: T-mg (p) = f(s) (p)m(s) + f(s-1) (p)m(s-1)+ . . . + fr(p)m(T), f(j) is an element of C-infinity(M), s, r is an element of Z. We show that if the TYCZ expansion is finite then T-mg is indeed a polynomial in m of degree n, n = dim(C) M, and the log-term of the Szego kernel of the disc bundle D C L* vanishes (where L* is the dual bundle of L). Moreover, we provide a complete classification of the Kdhler manifolds admitting finite TYCZ expansion either when M is a complex curve or when M is a complex surface with a cscK metric which M admits a radial Kahler potential.